\(\int \frac {x^2}{(a+b e^{c+d x})^3} \, dx\) [18]

Optimal result

Integrand size = 17, antiderivative size = 243 \[ \int \frac {x^2}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {x}{a^3 d^2}-\frac {x}{a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac {3 x^2}{2 a^3 d}+\frac {x^2}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^2}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^3}{3 a^3}-\frac {\log \left (a+b e^{c+d x}\right )}{a^3 d^3}+\frac {3 x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}-\frac {x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d}+\frac {3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}-\frac {2 x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}+\frac {2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3} \] Output:

x/a^3/d^2-x/a^2/d^2/(a+b*exp(d*x+c))-3/2*x^2/a^3/d+1/2*x^2/a/d/(a+b*exp(d* 
x+c))^2+x^2/a^2/d/(a+b*exp(d*x+c))+1/3*x^3/a^3-ln(a+b*exp(d*x+c))/a^3/d^3+ 
3*x*ln(1+b*exp(d*x+c)/a)/a^3/d^2-x^2*ln(1+b*exp(d*x+c)/a)/a^3/d+3*polylog( 
2,-b*exp(d*x+c)/a)/a^3/d^3-2*x*polylog(2,-b*exp(d*x+c)/a)/a^3/d^2+2*polylo 
g(3,-b*exp(d*x+c)/a)/a^3/d^3
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {\frac {6 x}{d^2}-\frac {6 a x}{d^2 \left (a+b e^{c+d x}\right )}-\frac {9 x^2}{d}+\frac {3 a^2 x^2}{d \left (a+b e^{c+d x}\right )^2}+\frac {6 a x^2}{a d+b d e^{c+d x}}+2 x^3-\frac {6 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{d^3}+\frac {18 x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{d^2}-\frac {6 x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{d}-\frac {6 (-3+2 d x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d^3}+\frac {12 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{d^3}}{6 a^3} \] Input:

Integrate[x^2/(a + b*E^(c + d*x))^3,x]
 

Output:

((6*x)/d^2 - (6*a*x)/(d^2*(a + b*E^(c + d*x))) - (9*x^2)/d + (3*a^2*x^2)/( 
d*(a + b*E^(c + d*x))^2) + (6*a*x^2)/(a*d + b*d*E^(c + d*x)) + 2*x^3 - (6* 
Log[1 + (b*E^(c + d*x))/a])/d^3 + (18*x*Log[1 + (b*E^(c + d*x))/a])/d^2 - 
(6*x^2*Log[1 + (b*E^(c + d*x))/a])/d - (6*(-3 + 2*d*x)*PolyLog[2, -((b*E^( 
c + d*x))/a)])/d^3 + (12*PolyLog[3, -((b*E^(c + d*x))/a)])/d^3)/(6*a^3)
 

Rubi [A] (verified)

Time = 4.38 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.56, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.118, Rules used = {2616, 2616, 2615, 2620, 2621, 2615, 2616, 2615, 2620, 2621, 2715, 2720, 47, 14, 16, 2838, 3011, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2}{\left (a+b e^{c+d x}\right )^3} \, dx\)

\(\Big \downarrow \) 2616

\(\displaystyle \frac {\int \frac {x^2}{\left (a+b e^{c+d x}\right )^2}dx}{a}-\frac {b \int \frac {e^{c+d x} x^2}{\left (a+b e^{c+d x}\right )^3}dx}{a}\)

\(\Big \downarrow \) 2616

\(\displaystyle \frac {\frac {\int \frac {x^2}{a+b e^{c+d x}}dx}{a}-\frac {b \int \frac {e^{c+d x} x^2}{\left (a+b e^{c+d x}\right )^2}dx}{a}}{a}-\frac {b \int \frac {e^{c+d x} x^2}{\left (a+b e^{c+d x}\right )^3}dx}{a}\)

\(\Big \downarrow \) 2615

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \int \frac {e^{c+d x} x^2}{a+b e^{c+d x}}dx}{a}}{a}-\frac {b \int \frac {e^{c+d x} x^2}{\left (a+b e^{c+d x}\right )^2}dx}{a}}{a}-\frac {b \int \frac {e^{c+d x} x^2}{\left (a+b e^{c+d x}\right )^3}dx}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \int \frac {e^{c+d x} x^2}{\left (a+b e^{c+d x}\right )^2}dx}{a}}{a}-\frac {b \int \frac {e^{c+d x} x^2}{\left (a+b e^{c+d x}\right )^3}dx}{a}\)

\(\Big \downarrow \) 2621

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \int \frac {x}{a+b e^{c+d x}}dx}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\int \frac {x}{\left (a+b e^{c+d x}\right )^2}dx}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 2615

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \int \frac {e^{c+d x} x}{a+b e^{c+d x}}dx}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\int \frac {x}{\left (a+b e^{c+d x}\right )^2}dx}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 2616

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \int \frac {e^{c+d x} x}{a+b e^{c+d x}}dx}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\int \frac {x}{a+b e^{c+d x}}dx}{a}-\frac {b \int \frac {e^{c+d x} x}{\left (a+b e^{c+d x}\right )^2}dx}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 2615

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \int \frac {e^{c+d x} x}{a+b e^{c+d x}}dx}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\frac {x^2}{2 a}-\frac {b \int \frac {e^{c+d x} x}{a+b e^{c+d x}}dx}{a}}{a}-\frac {b \int \frac {e^{c+d x} x}{\left (a+b e^{c+d x}\right )^2}dx}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \int \frac {e^{c+d x} x}{\left (a+b e^{c+d x}\right )^2}dx}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 2621

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {\int \frac {1}{a+b e^{c+d x}}dx}{b d}-\frac {x}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a}+1\right )de^{c+d x}}{b d^2}\right )}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a}+1\right )de^{c+d x}}{b d^2}\right )}{a}}{a}-\frac {b \left (\frac {\int \frac {1}{a+b e^{c+d x}}dx}{b d}-\frac {x}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a}+1\right )de^{c+d x}}{b d^2}\right )}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a}+1\right )de^{c+d x}}{b d^2}\right )}{a}}{a}-\frac {b \left (\frac {\int \frac {e^{-c-d x}}{a+b e^{c+d x}}de^{c+d x}}{b d^2}-\frac {x}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 47

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a}+1\right )de^{c+d x}}{b d^2}\right )}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a}+1\right )de^{c+d x}}{b d^2}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\int e^{-c-d x}de^{c+d x}}{a}-\frac {b \int \frac {1}{a+b e^{c+d x}}de^{c+d x}}{a}}{b d^2}-\frac {x}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a}+1\right )de^{c+d x}}{b d^2}\right )}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a}+1\right )de^{c+d x}}{b d^2}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\log \left (e^{c+d x}\right )}{a}-\frac {b \int \frac {1}{a+b e^{c+d x}}de^{c+d x}}{a}}{b d^2}-\frac {x}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a}+1\right )de^{c+d x}}{b d^2}\right )}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a}+1\right )de^{c+d x}}{b d^2}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\log \left (e^{c+d x}\right )}{a}-\frac {\log \left (a+b e^{c+d x}\right )}{a}}{b d^2}-\frac {x}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{b d^2}+\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}\right )}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{b d^2}+\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\log \left (e^{c+d x}\right )}{a}-\frac {\log \left (a+b e^{c+d x}\right )}{a}}{b d^2}-\frac {x}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \left (\frac {\int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )dx}{d}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{b d^2}+\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}\right )}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{b d^2}+\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\log \left (e^{c+d x}\right )}{a}-\frac {\log \left (a+b e^{c+d x}\right )}{a}}{b d^2}-\frac {x}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \left (\frac {\int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )de^{c+d x}}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{b d^2}+\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}\right )}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{b d^2}+\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\log \left (e^{c+d x}\right )}{a}-\frac {\log \left (a+b e^{c+d x}\right )}{a}}{b d^2}-\frac {x}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\frac {\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \left (\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {2 \left (\frac {x^2}{2 a}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{b d^2}+\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}\right )}{a}\right )}{b d}-\frac {x^2}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{b d^2}+\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\log \left (e^{c+d x}\right )}{a}-\frac {\log \left (a+b e^{c+d x}\right )}{a}}{b d^2}-\frac {x}{b d \left (a+b e^{c+d x}\right )}\right )}{a}}{b d}-\frac {x^2}{2 b d \left (a+b e^{c+d x}\right )^2}\right )}{a}\)

Input:

Int[x^2/(a + b*E^(c + d*x))^3,x]
 

Output:

-((b*(-1/2*x^2/(b*d*(a + b*E^(c + d*x))^2) + (-((b*(-(x/(b*d*(a + b*E^(c + 
 d*x)))) + (Log[E^(c + d*x)]/a - Log[a + b*E^(c + d*x)]/a)/(b*d^2)))/a) + 
(x^2/(2*a) - (b*((x*Log[1 + (b*E^(c + d*x))/a])/(b*d) + PolyLog[2, -((b*E^ 
(c + d*x))/a)]/(b*d^2)))/a)/a)/(b*d)))/a) + (-((b*(-(x^2/(b*d*(a + b*E^(c 
+ d*x)))) + (2*(x^2/(2*a) - (b*((x*Log[1 + (b*E^(c + d*x))/a])/(b*d) + Pol 
yLog[2, -((b*E^(c + d*x))/a)]/(b*d^2)))/a))/(b*d)))/a) + (x^3/(3*a) - (b*( 
(x^2*Log[1 + (b*E^(c + d*x))/a])/(b*d) - (2*(-((x*PolyLog[2, -((b*E^(c + d 
*x))/a)])/d) + PolyLog[3, -((b*E^(c + d*x))/a)]/d^2))/(b*d)))/a)/a)/a
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 2615

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x 
_))))^(n_.)), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Simp[ 
b/a   Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n)), x] 
, x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2616

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + 
(d_.)*(x_))^(m_.), x_Symbol] :> Simp[1/a   Int[(c + d*x)^m*(a + b*(F^(g*(e 
+ f*x)))^n)^(p + 1), x], x] - Simp[b/a   Int[(c + d*x)^m*(F^(g*(e + f*x)))^ 
n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n 
}, x] && ILtQ[p, 0] && IGtQ[m, 0]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2621

Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*( 
(e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> 
 Simp[(c + d*x)^m*((a + b*(F^(g*(e + f*x)))^n)^(p + 1)/(b*f*g*n*(p + 1)*Log 
[F])), x] - Simp[d*(m/(b*f*g*n*(p + 1)*Log[F]))   Int[(c + d*x)^(m - 1)*(a 
+ b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, 
m, n, p}, x] && NeQ[p, -1]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.58

method	result	size
risch	\(\frac {x \left (2 x b d \,{\mathrm e}^{d x +c}+3 x d a -2 b \,{\mathrm e}^{d x +c}-2 a \right )}{2 d^{2} a^{2} \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}-\frac {c^{2} x}{a^{3} d^{2}}-\frac {x^{2} \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d}-\frac {2 x \operatorname {polylog}\left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{2}}-\frac {3 c x}{a^{3} d^{2}}+\frac {3 x \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{2}}-\frac {2 c^{3}}{3 a^{3} d^{3}}+\frac {2 \operatorname {polylog}\left (3, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{3}}-\frac {c^{2} \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a^{3} d^{3}}+\frac {3 c \ln \left ({\mathrm e}^{d x +c}\right )}{a^{3} d^{3}}-\frac {3 c \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a^{3} d^{3}}+\frac {\ln \left ({\mathrm e}^{d x +c}\right )}{a^{3} d^{3}}-\frac {\ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a^{3} d^{3}}+\frac {x^{3}}{3 a^{3}}+\frac {\ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right ) c^{2}}{a^{3} d^{3}}-\frac {3 x^{2}}{2 a^{3} d}+\frac {3 \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right ) c}{a^{3} d^{3}}+\frac {3 \operatorname {polylog}\left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{3}}+\frac {c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a^{3} d^{3}}-\frac {3 c^{2}}{2 a^{3} d^{3}}\)	\(385\)

Input:

int(x^2/(a+b*exp(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

1/2*x*(2*x*b*d*exp(d*x+c)+3*x*d*a-2*b*exp(d*x+c)-2*a)/d^2/a^2/(a+b*exp(d*x 
+c))^2-1/a^3/d^2*c^2*x-x^2*ln(1+b*exp(d*x+c)/a)/a^3/d-2*x*polylog(2,-b*exp 
(d*x+c)/a)/a^3/d^2-3/a^3/d^2*c*x+3*x*ln(1+b*exp(d*x+c)/a)/a^3/d^2-2/3/a^3/ 
d^3*c^3+2*polylog(3,-b*exp(d*x+c)/a)/a^3/d^3-1/a^3/d^3*c^2*ln(a+b*exp(d*x+ 
c))+3/a^3/d^3*c*ln(exp(d*x+c))-3/a^3/d^3*c*ln(a+b*exp(d*x+c))+1/a^3/d^3*ln 
(exp(d*x+c))-ln(a+b*exp(d*x+c))/a^3/d^3+1/3*x^3/a^3+1/a^3/d^3*ln(1+b*exp(d 
*x+c)/a)*c^2-3/2*x^2/a^3/d+3/a^3/d^3*ln(1+b*exp(d*x+c)/a)*c+3*polylog(2,-b 
*exp(d*x+c)/a)/a^3/d^3+1/a^3/d^3*c^2*ln(exp(d*x+c))-3/2/a^3/d^3*c^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (226) = 452\).

Time = 0.08 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.14 \[ \int \frac {x^2}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {2 \, a^{2} d^{3} x^{3} + 2 \, a^{2} c^{3} + 9 \, a^{2} c^{2} + 6 \, a^{2} c - 6 \, {\left (2 \, a^{2} d x - 3 \, a^{2} + {\left (2 \, b^{2} d x - 3 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (2 \, a b d x - 3 \, a b\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )} + a}{a} + 1\right ) + {\left (2 \, b^{2} d^{3} x^{3} - 9 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{3} + 9 \, b^{2} c^{2} + 6 \, b^{2} d x + 6 \, b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (2 \, a b d^{3} x^{3} - 6 \, a b d^{2} x^{2} + 2 \, a b c^{3} + 9 \, a b c^{2} + 3 \, a b d x + 6 \, a b c\right )} e^{\left (d x + c\right )} - 6 \, {\left (a^{2} c^{2} + 3 \, a^{2} c + a^{2} + {\left (b^{2} c^{2} + 3 \, b^{2} c + b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (a b c^{2} + 3 \, a b c + a b\right )} e^{\left (d x + c\right )}\right )} \log \left (b e^{\left (d x + c\right )} + a\right ) - 6 \, {\left (a^{2} d^{2} x^{2} - a^{2} c^{2} - 3 \, a^{2} d x - 3 \, a^{2} c + {\left (b^{2} d^{2} x^{2} - b^{2} c^{2} - 3 \, b^{2} d x - 3 \, b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (a b d^{2} x^{2} - a b c^{2} - 3 \, a b d x - 3 \, a b c\right )} e^{\left (d x + c\right )}\right )} \log \left (\frac {b e^{\left (d x + c\right )} + a}{a}\right ) + 12 \, {\left (b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (d x + c\right )} + a^{2}\right )} {\rm polylog}\left (3, -\frac {b e^{\left (d x + c\right )}}{a}\right )}{6 \, {\left (a^{3} b^{2} d^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} b d^{3} e^{\left (d x + c\right )} + a^{5} d^{3}\right )}} \] Input:

integrate(x^2/(a+b*exp(d*x+c))^3,x, algorithm="fricas")
 

Output:

1/6*(2*a^2*d^3*x^3 + 2*a^2*c^3 + 9*a^2*c^2 + 6*a^2*c - 6*(2*a^2*d*x - 3*a^ 
2 + (2*b^2*d*x - 3*b^2)*e^(2*d*x + 2*c) + 2*(2*a*b*d*x - 3*a*b)*e^(d*x + c 
))*dilog(-(b*e^(d*x + c) + a)/a + 1) + (2*b^2*d^3*x^3 - 9*b^2*d^2*x^2 + 2* 
b^2*c^3 + 9*b^2*c^2 + 6*b^2*d*x + 6*b^2*c)*e^(2*d*x + 2*c) + 2*(2*a*b*d^3* 
x^3 - 6*a*b*d^2*x^2 + 2*a*b*c^3 + 9*a*b*c^2 + 3*a*b*d*x + 6*a*b*c)*e^(d*x 
+ c) - 6*(a^2*c^2 + 3*a^2*c + a^2 + (b^2*c^2 + 3*b^2*c + b^2)*e^(2*d*x + 2 
*c) + 2*(a*b*c^2 + 3*a*b*c + a*b)*e^(d*x + c))*log(b*e^(d*x + c) + a) - 6* 
(a^2*d^2*x^2 - a^2*c^2 - 3*a^2*d*x - 3*a^2*c + (b^2*d^2*x^2 - b^2*c^2 - 3* 
b^2*d*x - 3*b^2*c)*e^(2*d*x + 2*c) + 2*(a*b*d^2*x^2 - a*b*c^2 - 3*a*b*d*x 
- 3*a*b*c)*e^(d*x + c))*log((b*e^(d*x + c) + a)/a) + 12*(b^2*e^(2*d*x + 2* 
c) + 2*a*b*e^(d*x + c) + a^2)*polylog(3, -b*e^(d*x + c)/a))/(a^3*b^2*d^3*e 
^(2*d*x + 2*c) + 2*a^4*b*d^3*e^(d*x + c) + a^5*d^3)
 

Sympy [F]

\[ \int \frac {x^2}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {3 a d x^{2} - 2 a x + \left (2 b d x^{2} - 2 b x\right ) e^{c + d x}}{2 a^{4} d^{2} + 4 a^{3} b d^{2} e^{c + d x} + 2 a^{2} b^{2} d^{2} e^{2 c + 2 d x}} + \frac {\int \left (- \frac {3 d x}{a + b e^{c} e^{d x}}\right )\, dx + \int \frac {d^{2} x^{2}}{a + b e^{c} e^{d x}}\, dx + \int \frac {1}{a + b e^{c} e^{d x}}\, dx}{a^{2} d^{2}} \] Input:

integrate(x**2/(a+b*exp(d*x+c))**3,x)
 

Output:

(3*a*d*x**2 - 2*a*x + (2*b*d*x**2 - 2*b*x)*exp(c + d*x))/(2*a**4*d**2 + 4* 
a**3*b*d**2*exp(c + d*x) + 2*a**2*b**2*d**2*exp(2*c + 2*d*x)) + (Integral( 
-3*d*x/(a + b*exp(c)*exp(d*x)), x) + Integral(d**2*x**2/(a + b*exp(c)*exp( 
d*x)), x) + Integral(1/(a + b*exp(c)*exp(d*x)), x))/(a**2*d**2)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.96 \[ \int \frac {x^2}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {3 \, a d x^{2} - 2 \, a x + 2 \, {\left (b d x^{2} e^{c} - b x e^{c}\right )} e^{\left (d x\right )}}{2 \, {\left (a^{2} b^{2} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b d^{2} e^{\left (d x + c\right )} + a^{4} d^{2}\right )}} + \frac {x}{a^{3} d^{2}} + \frac {2 \, d^{3} x^{3} - 9 \, d^{2} x^{2}}{6 \, a^{3} d^{3}} - \frac {d^{2} x^{2} \log \left (\frac {b e^{\left (d x + c\right )}}{a} + 1\right ) + 2 \, d x {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )}}{a}\right ) - 2 \, {\rm Li}_{3}(-\frac {b e^{\left (d x + c\right )}}{a})}{a^{3} d^{3}} + \frac {3 \, {\left (d x \log \left (\frac {b e^{\left (d x + c\right )}}{a} + 1\right ) + {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )}}{a}\right )\right )}}{a^{3} d^{3}} - \frac {\log \left (b e^{\left (d x + c\right )} + a\right )}{a^{3} d^{3}} \] Input:

integrate(x^2/(a+b*exp(d*x+c))^3,x, algorithm="maxima")
 

Output:

1/2*(3*a*d*x^2 - 2*a*x + 2*(b*d*x^2*e^c - b*x*e^c)*e^(d*x))/(a^2*b^2*d^2*e 
^(2*d*x + 2*c) + 2*a^3*b*d^2*e^(d*x + c) + a^4*d^2) + x/(a^3*d^2) + 1/6*(2 
*d^3*x^3 - 9*d^2*x^2)/(a^3*d^3) - (d^2*x^2*log(b*e^(d*x + c)/a + 1) + 2*d* 
x*dilog(-b*e^(d*x + c)/a) - 2*polylog(3, -b*e^(d*x + c)/a))/(a^3*d^3) + 3* 
(d*x*log(b*e^(d*x + c)/a + 1) + dilog(-b*e^(d*x + c)/a))/(a^3*d^3) - log(b 
*e^(d*x + c) + a)/(a^3*d^3)
 

Giac [F]

\[ \int \frac {x^2}{\left (a+b e^{c+d x}\right )^3} \, dx=\int { \frac {x^{2}}{{\left (b e^{\left (d x + c\right )} + a\right )}^{3}} \,d x } \] Input:

integrate(x^2/(a+b*exp(d*x+c))^3,x, algorithm="giac")
 

Output:

integrate(x^2/(b*e^(d*x + c) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (a+b e^{c+d x}\right )^3} \, dx=\int \frac {x^2}{{\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}^3} \,d x \] Input:

int(x^2/(a + b*exp(c + d*x))^3,x)
 

Output:

int(x^2/(a + b*exp(c + d*x))^3, x)
 

Reduce [F]

\[ \int \frac {x^2}{\left (a+b e^{c+d x}\right )^3} \, dx=\int \frac {x^{2}}{e^{3 d x +3 c} b^{3}+3 e^{2 d x +2 c} a \,b^{2}+3 e^{d x +c} a^{2} b +a^{3}}d x \] Input:

int(x^2/(a+b*exp(d*x+c))^3,x)
                                                                                    
                                                                                    
 

Output:

int(x**2/(e**(3*c + 3*d*x)*b**3 + 3*e**(2*c + 2*d*x)*a*b**2 + 3*e**(c + d* 
x)*a**2*b + a**3),x)