### 3.1109 $$\int (b d+2 c d x)^4 (a+b x+c x^2) \, dx$$

Optimal. Leaf size=45 $\frac{d^4 (b+2 c x)^7}{56 c^2}-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^5}{40 c^2}$

[Out]

-((b^2 - 4*a*c)*d^4*(b + 2*c*x)^5)/(40*c^2) + (d^4*(b + 2*c*x)^7)/(56*c^2)

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Rubi [A]  time = 0.0751143, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {683} $\frac{d^4 (b+2 c x)^7}{56 c^2}-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^5}{40 c^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2),x]

[Out]

-((b^2 - 4*a*c)*d^4*(b + 2*c*x)^5)/(40*c^2) + (d^4*(b + 2*c*x)^7)/(56*c^2)

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin{align*} \int (b d+2 c d x)^4 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^4}{4 c}+\frac{(b d+2 c d x)^6}{4 c d^2}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5}{40 c^2}+\frac{d^4 (b+2 c x)^7}{56 c^2}\\ \end{align*}

Mathematica [B]  time = 0.0171741, size = 102, normalized size = 2.27 $d^4 \left (\frac{8}{5} c^3 x^5 \left (2 a c+7 b^2\right )+8 b c^2 x^4 \left (a c+b^2\right )+b^2 c x^3 \left (8 a c+3 b^2\right )+\frac{1}{2} b^3 x^2 \left (8 a c+b^2\right )+a b^4 x+8 b c^4 x^6+\frac{16 c^5 x^7}{7}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2),x]

[Out]

d^4*(a*b^4*x + (b^3*(b^2 + 8*a*c)*x^2)/2 + b^2*c*(3*b^2 + 8*a*c)*x^3 + 8*b*c^2*(b^2 + a*c)*x^4 + (8*c^3*(7*b^2
+ 2*a*c)*x^5)/5 + 8*b*c^4*x^6 + (16*c^5*x^7)/7)

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Maple [B]  time = 0.04, size = 137, normalized size = 3. \begin{align*}{\frac{16\,{c}^{5}{d}^{4}{x}^{7}}{7}}+8\,b{d}^{4}{c}^{4}{x}^{6}+{\frac{ \left ( 16\,{c}^{4}{d}^{4}a+56\,{b}^{2}{d}^{4}{c}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 32\,b{d}^{4}{c}^{3}a+32\,{b}^{3}{d}^{4}{c}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 24\,{b}^{2}{d}^{4}{c}^{2}a+9\,{b}^{4}{d}^{4}c \right ){x}^{3}}{3}}+{\frac{ \left ( 8\,{b}^{3}{d}^{4}ca+{b}^{5}{d}^{4} \right ){x}^{2}}{2}}+{b}^{4}{d}^{4}ax \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^4*(c*x^2+b*x+a),x)

[Out]

16/7*c^5*d^4*x^7+8*b*d^4*c^4*x^6+1/5*(16*a*c^4*d^4+56*b^2*c^3*d^4)*x^5+1/4*(32*a*b*c^3*d^4+32*b^3*c^2*d^4)*x^4
+1/3*(24*a*b^2*c^2*d^4+9*b^4*c*d^4)*x^3+1/2*(8*a*b^3*c*d^4+b^5*d^4)*x^2+b^4*d^4*a*x

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Maxima [B]  time = 1.22842, size = 162, normalized size = 3.6 \begin{align*} \frac{16}{7} \, c^{5} d^{4} x^{7} + 8 \, b c^{4} d^{4} x^{6} + a b^{4} d^{4} x + \frac{8}{5} \,{\left (7 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d^{4} x^{5} + 8 \,{\left (b^{3} c^{2} + a b c^{3}\right )} d^{4} x^{4} +{\left (3 \, b^{4} c + 8 \, a b^{2} c^{2}\right )} d^{4} x^{3} + \frac{1}{2} \,{\left (b^{5} + 8 \, a b^{3} c\right )} d^{4} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

16/7*c^5*d^4*x^7 + 8*b*c^4*d^4*x^6 + a*b^4*d^4*x + 8/5*(7*b^2*c^3 + 2*a*c^4)*d^4*x^5 + 8*(b^3*c^2 + a*b*c^3)*d
^4*x^4 + (3*b^4*c + 8*a*b^2*c^2)*d^4*x^3 + 1/2*(b^5 + 8*a*b^3*c)*d^4*x^2

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Fricas [B]  time = 2.02305, size = 286, normalized size = 6.36 \begin{align*} \frac{16}{7} x^{7} d^{4} c^{5} + 8 x^{6} d^{4} c^{4} b + \frac{56}{5} x^{5} d^{4} c^{3} b^{2} + \frac{16}{5} x^{5} d^{4} c^{4} a + 8 x^{4} d^{4} c^{2} b^{3} + 8 x^{4} d^{4} c^{3} b a + 3 x^{3} d^{4} c b^{4} + 8 x^{3} d^{4} c^{2} b^{2} a + \frac{1}{2} x^{2} d^{4} b^{5} + 4 x^{2} d^{4} c b^{3} a + x d^{4} b^{4} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

16/7*x^7*d^4*c^5 + 8*x^6*d^4*c^4*b + 56/5*x^5*d^4*c^3*b^2 + 16/5*x^5*d^4*c^4*a + 8*x^4*d^4*c^2*b^3 + 8*x^4*d^4
*c^3*b*a + 3*x^3*d^4*c*b^4 + 8*x^3*d^4*c^2*b^2*a + 1/2*x^2*d^4*b^5 + 4*x^2*d^4*c*b^3*a + x*d^4*b^4*a

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Sympy [B]  time = 0.123086, size = 143, normalized size = 3.18 \begin{align*} a b^{4} d^{4} x + 8 b c^{4} d^{4} x^{6} + \frac{16 c^{5} d^{4} x^{7}}{7} + x^{5} \left (\frac{16 a c^{4} d^{4}}{5} + \frac{56 b^{2} c^{3} d^{4}}{5}\right ) + x^{4} \left (8 a b c^{3} d^{4} + 8 b^{3} c^{2} d^{4}\right ) + x^{3} \left (8 a b^{2} c^{2} d^{4} + 3 b^{4} c d^{4}\right ) + x^{2} \left (4 a b^{3} c d^{4} + \frac{b^{5} d^{4}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**4*(c*x**2+b*x+a),x)

[Out]

a*b**4*d**4*x + 8*b*c**4*d**4*x**6 + 16*c**5*d**4*x**7/7 + x**5*(16*a*c**4*d**4/5 + 56*b**2*c**3*d**4/5) + x**
4*(8*a*b*c**3*d**4 + 8*b**3*c**2*d**4) + x**3*(8*a*b**2*c**2*d**4 + 3*b**4*c*d**4) + x**2*(4*a*b**3*c*d**4 + b
**5*d**4/2)

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Giac [B]  time = 1.21602, size = 185, normalized size = 4.11 \begin{align*} \frac{16}{7} \, c^{5} d^{4} x^{7} + 8 \, b c^{4} d^{4} x^{6} + \frac{56}{5} \, b^{2} c^{3} d^{4} x^{5} + \frac{16}{5} \, a c^{4} d^{4} x^{5} + 8 \, b^{3} c^{2} d^{4} x^{4} + 8 \, a b c^{3} d^{4} x^{4} + 3 \, b^{4} c d^{4} x^{3} + 8 \, a b^{2} c^{2} d^{4} x^{3} + \frac{1}{2} \, b^{5} d^{4} x^{2} + 4 \, a b^{3} c d^{4} x^{2} + a b^{4} d^{4} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

16/7*c^5*d^4*x^7 + 8*b*c^4*d^4*x^6 + 56/5*b^2*c^3*d^4*x^5 + 16/5*a*c^4*d^4*x^5 + 8*b^3*c^2*d^4*x^4 + 8*a*b*c^3
*d^4*x^4 + 3*b^4*c*d^4*x^3 + 8*a*b^2*c^2*d^4*x^3 + 1/2*b^5*d^4*x^2 + 4*a*b^3*c*d^4*x^2 + a*b^4*d^4*x
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