3.1.60 \(\int \frac {c+d x}{(a+b (F^{g (e+f x)})^n)^3} \, dx\) [60]

3.1.60.1 Optimal result

Integrand size = 23, antiderivative size = 276 \[ \int \frac {c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\frac {(c+d x)^2}{2 a^3 d}-\frac {d}{2 a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac {3 d x}{2 a^3 f g n \log (F)}+\frac {c+d x}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {c+d x}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {3 d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}-\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)} \]

output

1/2*(d*x+c)^2/a^3/d-1/2*d/a^2/f^2/(a+b*(F^(g*(f*x+e)))^n)/g^2/n^2/ln(F)^2- 
3/2*d*x/a^3/f/g/n/ln(F)+1/2*(d*x+c)/a/f/(a+b*(F^(g*(f*x+e)))^n)^2/g/n/ln(F 
)+(d*x+c)/a^2/f/(a+b*(F^(g*(f*x+e)))^n)/g/n/ln(F)+3/2*d*ln(a+b*(F^(g*(f*x+ 
e)))^n)/a^3/f^2/g^2/n^2/ln(F)^2-(d*x+c)*ln(1+b*(F^(g*(f*x+e)))^n/a)/a^3/f/ 
g/n/ln(F)-d*polylog(2,-b*(F^(g*(f*x+e)))^n/a)/a^3/f^2/g^2/n^2/ln(F)^2
 

3.1.60.2 Mathematica [F]

\[ \int \frac {c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx \]

input

Integrate[(c + d*x)/(a + b*(F^(g*(e + f*x)))^n)^3,x]
 

output

Integrate[(c + d*x)/(a + b*(F^(g*(e + f*x)))^n)^3, x]
 

3.1.60.3 Rubi [A] (verified)

Time = 1.58 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.25, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2616, 2616, 2615, 2620, 2621, 2715, 2720, 798, 47, 14, 16, 54, 2009, 2838}

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.

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

\(\Big \downarrow \) 2616

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

\(\Big \downarrow \) 2616

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

\(\Big \downarrow \) 2615

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2621

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {F^{-g (e+f x)}}{b \left (F^{g (e+f x)}\right )^n+a}dF^{g (e+f x)}}{b f^2 g^2 n \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {F^{-g (e+f x)}}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dF^{g (e+f x)}}{2 b f^2 g^2 n \log ^2(F)}-\frac {c+d x}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {F^{-g (e+f x)}}{b \left (F^{g (e+f x)}\right )^n+a}d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {F^{-g (e+f x)}}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}d\left (F^{g (e+f x)}\right )^n}{2 b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\)

\(\Big \downarrow \) 47

\(\displaystyle \frac {\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\int F^{-g (e+f x)}d\left (F^{g (e+f x)}\right )^n}{a}-\frac {b \int \frac {1}{b \left (F^{g (e+f x)}\right )^n+a}d\left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {F^{-g (e+f x)}}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}d\left (F^{g (e+f x)}\right )^n}{2 b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a}-\frac {b \int \frac {1}{b \left (F^{g (e+f x)}\right )^n+a}d\left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {F^{-g (e+f x)}}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}d\left (F^{g (e+f x)}\right )^n}{2 b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {F^{-g (e+f x)}}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}d\left (F^{g (e+f x)}\right )^n}{2 b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \int \left (\frac {F^{-g (e+f x)}}{a^2}-\frac {b}{a^2 \left (b \left (F^{g (e+f x)}\right )^n+a\right )}-\frac {b}{a \left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}\right )d\left (F^{g (e+f x)}\right )^n}{2 b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2}+\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a^2}+\frac {1}{a \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{2 b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\)

\(\Big \downarrow \) 2838

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

input

Int[(c + d*x)/(a + b*(F^(g*(e + f*x)))^n)^3,x]
 

output

-((b*(-1/2*(c + d*x)/(b*f*(a + b*(F^(g*(e + f*x)))^n)^2*g*n*Log[F]) + (d*( 
1/(a*(a + b*(F^(g*(e + f*x)))^n)) + Log[(F^(g*(e + f*x)))^n]/a^2 - Log[a + 
 b*(F^(g*(e + f*x)))^n]/a^2))/(2*b*f^2*g^2*n^2*Log[F]^2)))/a) + (-((b*(-(( 
c + d*x)/(b*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F])) + (d*(Log[(F^(g*(e 
+ f*x)))^n]/a - Log[a + b*(F^(g*(e + f*x)))^n]/a))/(b*f^2*g^2*n^2*Log[F]^2 
)))/a) + ((c + d*x)^2/(2*a*d) - (b*(((c + d*x)*Log[1 + (b*(F^(g*(e + f*x)) 
)^n)/a])/(b*f*g*n*Log[F]) + (d*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/( 
b*f^2*g^2*n^2*Log[F]^2)))/a)/a)/a
 

3.1.60.3.1 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 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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]
 

3.1.60.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(667\) vs. \(2(266)=532\).

Time = 0.28 (sec) , antiderivative size = 668, normalized size of antiderivative = 2.42

method	result	size
risch	\(\frac {2 \left (F^{g \left (f x +e \right )}\right )^{n} \ln \left (F \right ) b d f g n x +3 \ln \left (F \right ) a d f g n x +2 \left (F^{g \left (f x +e \right )}\right )^{n} \ln \left (F \right ) b c f g n +3 c \ln \left (F \right ) a f g n -\left (F^{g \left (f x +e \right )}\right )^{n} b d -a d}{2 n^{2} g^{2} f^{2} \ln \left (F \right )^{2} a^{2} {\left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}^{2}}+\frac {3 d \ln \left (\left (F^{g \left (f x +e \right )}\right )^{n} F^{-n g f x} F^{n g f x} b +a \right )}{2 \ln \left (F \right )^{2} f^{2} g^{2} n^{2} a^{3}}-\frac {3 d \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{2 \ln \left (F \right )^{2} f^{2} g^{2} n^{2} a^{3}}+\frac {d \ln \left (F^{g \left (f x +e \right )}\right )^{2}}{2 \ln \left (F \right )^{2} f^{2} g^{2} a^{3}}-\frac {d \ln \left (F^{g \left (f x +e \right )}\right ) \ln \left (1+\frac {b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n \,a^{3}}-\frac {d \,\operatorname {Li}_{2}\left (-\frac {b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n^{2} a^{3}}-\frac {d \ln \left (\left (F^{g \left (f x +e \right )}\right )^{n} F^{-n g f x} F^{n g f x} b +a \right ) x}{\ln \left (F \right ) f g n \,a^{3}}+\frac {d \ln \left (\left (F^{g \left (f x +e \right )}\right )^{n} F^{-n g f x} F^{n g f x} b +a \right ) \ln \left (F^{g \left (f x +e \right )}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n \,a^{3}}+\frac {d \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right ) x}{\ln \left (F \right ) f g n \,a^{3}}-\frac {d \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right ) \ln \left (F^{g \left (f x +e \right )}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n \,a^{3}}-\frac {c \ln \left (\left (F^{g \left (f x +e \right )}\right )^{n} F^{-n g f x} F^{n g f x} b +a \right )}{\ln \left (F \right ) f g n \,a^{3}}+\frac {c \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right ) f g n \,a^{3}}\)	\(668\)

input

int((d*x+c)/(a+b*(F^(g*(f*x+e)))^n)^3,x,method=_RETURNVERBOSE)
 

output

1/2*(2*(F^(g*(f*x+e)))^n*ln(F)*b*d*f*g*n*x+3*ln(F)*a*d*f*g*n*x+2*(F^(g*(f* 
x+e)))^n*ln(F)*b*c*f*g*n+3*c*ln(F)*a*f*g*n-(F^(g*(f*x+e)))^n*b*d-a*d)/n^2/ 
g^2/f^2/ln(F)^2/a^2/(a+b*(F^(g*(f*x+e)))^n)^2+3/2/ln(F)^2/f^2/g^2/n^2/a^3* 
d*ln((F^(g*(f*x+e)))^n*F^(-n*g*f*x)*F^(n*g*f*x)*b+a)-3/2/ln(F)^2/f^2/g^2/n 
^2/a^3*d*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)+1/2/ln(F)^2/f^2/g^ 
2/a^3*d*ln(F^(g*(f*x+e)))^2-1/ln(F)^2/f^2/g^2/n/a^3*d*ln(F^(g*(f*x+e)))*ln 
(1+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)-1/ln(F)^2/f^2/g^2/n^2/a 
^3*d*polylog(2,-b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)-1/ln(F)/f/ 
g/n/a^3*d*ln((F^(g*(f*x+e)))^n*F^(-n*g*f*x)*F^(n*g*f*x)*b+a)*x+1/ln(F)^2/f 
^2/g^2/n/a^3*d*ln((F^(g*(f*x+e)))^n*F^(-n*g*f*x)*F^(n*g*f*x)*b+a)*ln(F^(g* 
(f*x+e)))+1/ln(F)/f/g/n/a^3*d*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^ 
n)*x-1/ln(F)^2/f^2/g^2/n/a^3*d*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e))) 
^n)*ln(F^(g*(f*x+e)))-1/ln(F)/f/g/n/a^3*c*ln((F^(g*(f*x+e)))^n*F^(-n*g*f*x 
)*F^(n*g*f*x)*b+a)+1/ln(F)/f/g/n/a^3*c*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*( 
f*x+e)))^n)
 

3.1.60.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 696 vs. \(2 (265) = 530\).

Time = 0.26 (sec) , antiderivative size = 696, normalized size of antiderivative = 2.52 \[ \int \frac {c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=-\frac {3 \, {\left (a^{2} d e - a^{2} c f\right )} g n \log \left (F\right ) + a^{2} d - {\left (a^{2} d f^{2} g^{2} n^{2} x^{2} + 2 \, a^{2} c f^{2} g^{2} n^{2} x - {\left (a^{2} d e^{2} - 2 \, a^{2} c e f\right )} g^{2} n^{2}\right )} \log \left (F\right )^{2} - {\left ({\left (b^{2} d f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c f^{2} g^{2} n^{2} x - {\left (b^{2} d e^{2} - 2 \, b^{2} c e f\right )} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 3 \, {\left (b^{2} d f g n x + b^{2} d e g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} + {\left (a b d - 2 \, {\left (a b d f^{2} g^{2} n^{2} x^{2} + 2 \, a b c f^{2} g^{2} n^{2} x - {\left (a b d e^{2} - 2 \, a b c e f\right )} g^{2} n^{2}\right )} \log \left (F\right )^{2} + 2 \, {\left (2 \, a b d f g n x + {\left (3 \, a b d e - a b c f\right )} g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n} + 2 \, {\left (2 \, F^{f g n x + e g n} a b d + F^{2 \, f g n x + 2 \, e g n} b^{2} d + a^{2} d\right )} {\rm Li}_2\left (-\frac {F^{f g n x + e g n} b + a}{a} + 1\right ) - {\left (2 \, {\left (a^{2} d e - a^{2} c f\right )} g n \log \left (F\right ) + 3 \, a^{2} d + {\left (2 \, {\left (b^{2} d e - b^{2} c f\right )} g n \log \left (F\right ) + 3 \, b^{2} d\right )} F^{2 \, f g n x + 2 \, e g n} + 2 \, {\left (2 \, {\left (a b d e - a b c f\right )} g n \log \left (F\right ) + 3 \, a b d\right )} F^{f g n x + e g n}\right )} \log \left (F^{f g n x + e g n} b + a\right ) + 2 \, {\left ({\left (b^{2} d f g n x + b^{2} d e g n\right )} F^{2 \, f g n x + 2 \, e g n} \log \left (F\right ) + 2 \, {\left (a b d f g n x + a b d e g n\right )} F^{f g n x + e g n} \log \left (F\right ) + {\left (a^{2} d f g n x + a^{2} d e g n\right )} \log \left (F\right )\right )} \log \left (\frac {F^{f g n x + e g n} b + a}{a}\right )}{2 \, {\left (2 \, F^{f g n x + e g n} a^{4} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + F^{2 \, f g n x + 2 \, e g n} a^{3} b^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{5} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\right )}} \]

input

integrate((d*x+c)/(a+b*(F^(g*(f*x+e)))^n)^3,x, algorithm="fricas")
 

output

-1/2*(3*(a^2*d*e - a^2*c*f)*g*n*log(F) + a^2*d - (a^2*d*f^2*g^2*n^2*x^2 + 
2*a^2*c*f^2*g^2*n^2*x - (a^2*d*e^2 - 2*a^2*c*e*f)*g^2*n^2)*log(F)^2 - ((b^ 
2*d*f^2*g^2*n^2*x^2 + 2*b^2*c*f^2*g^2*n^2*x - (b^2*d*e^2 - 2*b^2*c*e*f)*g^ 
2*n^2)*log(F)^2 - 3*(b^2*d*f*g*n*x + b^2*d*e*g*n)*log(F))*F^(2*f*g*n*x + 2 
*e*g*n) + (a*b*d - 2*(a*b*d*f^2*g^2*n^2*x^2 + 2*a*b*c*f^2*g^2*n^2*x - (a*b 
*d*e^2 - 2*a*b*c*e*f)*g^2*n^2)*log(F)^2 + 2*(2*a*b*d*f*g*n*x + (3*a*b*d*e 
- a*b*c*f)*g*n)*log(F))*F^(f*g*n*x + e*g*n) + 2*(2*F^(f*g*n*x + e*g*n)*a*b 
*d + F^(2*f*g*n*x + 2*e*g*n)*b^2*d + a^2*d)*dilog(-(F^(f*g*n*x + e*g*n)*b 
+ a)/a + 1) - (2*(a^2*d*e - a^2*c*f)*g*n*log(F) + 3*a^2*d + (2*(b^2*d*e - 
b^2*c*f)*g*n*log(F) + 3*b^2*d)*F^(2*f*g*n*x + 2*e*g*n) + 2*(2*(a*b*d*e - a 
*b*c*f)*g*n*log(F) + 3*a*b*d)*F^(f*g*n*x + e*g*n))*log(F^(f*g*n*x + e*g*n) 
*b + a) + 2*((b^2*d*f*g*n*x + b^2*d*e*g*n)*F^(2*f*g*n*x + 2*e*g*n)*log(F) 
+ 2*(a*b*d*f*g*n*x + a*b*d*e*g*n)*F^(f*g*n*x + e*g*n)*log(F) + (a^2*d*f*g* 
n*x + a^2*d*e*g*n)*log(F))*log((F^(f*g*n*x + e*g*n)*b + a)/a))/(2*F^(f*g*n 
*x + e*g*n)*a^4*b*f^2*g^2*n^2*log(F)^2 + F^(2*f*g*n*x + 2*e*g*n)*a^3*b^2*f 
^2*g^2*n^2*log(F)^2 + a^5*f^2*g^2*n^2*log(F)^2)
 

3.1.60.6 Sympy [F]

\[ \int \frac {c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {c + d x}{\left (a + b \left (F^{e g + f g x}\right )^{n}\right )^{3}}\, dx \]

input

integrate((d*x+c)/(a+b*(F**(g*(f*x+e)))**n)**3,x)
 

output

Integral((c + d*x)/(a + b*(F**(e*g + f*g*x))**n)**3, x)
 

3.1.60.7 Maxima [F]

\[ \int \frac {c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int { \frac {d x + c}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{3}} \,d x } \]

input

integrate((d*x+c)/(a+b*(F^(g*(f*x+e)))^n)^3,x, algorithm="maxima")
 

output

1/2*d*((3*a*f*g*n*x*log(F) + (2*F^(e*g*n)*b*f*g*n*x*log(F) - F^(e*g*n)*b)* 
F^(f*g*n*x) - a)/(2*F^(f*g*n*x)*F^(e*g*n)*a^3*b*f^2*g^2*n^2*log(F)^2 + F^( 
2*f*g*n*x)*F^(2*e*g*n)*a^2*b^2*f^2*g^2*n^2*log(F)^2 + a^4*f^2*g^2*n^2*log( 
F)^2) + 2*integrate(1/2*(2*f*g*n*x*log(F) - 3)/(F^(f*g*n*x)*F^(e*g*n)*a^2* 
b*f*g*n*log(F) + a^3*f*g*n*log(F)), x)) + 1/2*c*((2*F^(f*g*n*x + e*g*n)*b 
+ 3*a)/((2*F^(f*g*n*x + e*g*n)*a^3*b + F^(2*f*g*n*x + 2*e*g*n)*a^2*b^2 + a 
^4)*f*g*n*log(F)) + 2*(f*g*n*x + e*g*n)/(a^3*f*g*n) - 2*log(F^(f*g*n*x + e 
*g*n)*b + a)/(a^3*f*g*n*log(F)))
 

3.1.60.8 Giac [F]

\[ \int \frac {c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int { \frac {d x + c}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{3}} \,d x } \]

input

integrate((d*x+c)/(a+b*(F^(g*(f*x+e)))^n)^3,x, algorithm="giac")
 

output

integrate((d*x + c)/((F^((f*x + e)*g))^n*b + a)^3, x)
 

3.1.60.9 Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {c+d\,x}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^3} \,d x \]

input

int((c + d*x)/(a + b*(F^(g*(e + f*x)))^n)^3,x)
 

output

int((c + d*x)/(a + b*(F^(g*(e + f*x)))^n)^3, x)