Ling Jihua pleaded guilty and "decided not to appeal", state media reported.
Ling was in effect chief-of-staff to Mr Hu, but was demoted in 2012 amid a high profile anti-corruption campaign.
His downfall came after reports of an alleged cover-up over his son, who died alongside two semi-clad women while driving his Ferrari in Beijing in 2012.
There was also speculation that Ling's demotion may have been the result of political manoeuvring ahead of a leadership change that year.
The court in the northern city of Tianjin held a closed-door hearing on 7 June, Xinhua reported.
Ling was convicted of taking $11.57m (Â£8.71m) in bribes and to have used his influence to secure property deals and promotions for his associates, among other favours, the state news agency said.
Until the scandal involving his son, Ling had been tipped for promotion to the party's Politburo.
Instead, he was moved from head of the Central Committee's General Office - in effect chief of staff to China's leader - and appointed to the less influential post of head of the United Front Work Department.
The identity of those inside the Ferrari was never revealed and censors blocked online comments mentioning the crash. Ling was accused of trying to cover up the scandal.
The authorities announced an investigation into him in late 2014, and he was expelled from the Communist Party and arrested in July 2015.
President Xi Jinping has overseen a campaign against corruption since coming to power in late 2012 and many high-profile political figures have been jailed.
Critics say President Xi's campaign is as much about eliminating political rivals as it is about tackling corruption.
An ex-aide to retired Chinese president Hu Jintao has been jailed for life for corruption, illegally obtaining state secrets and abuse of power.