Optimal. Leaf size=496 \[ -\frac{m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (2,\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac{2 m n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (3,\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac{m \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c-a d}-\frac{2 m n \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c-a d}-\frac{2 m n^2 \text{PolyLog}\left (4,\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac{2 m n^2 \text{PolyLog}\left (4,\frac{d (a+b x)}{b (c+d x)}\right )}{b c-a d}+\frac{\log \left (h (f+g x)^m\right ) \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 n (b c-a d)}-\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{3 n (b c-a d)}+\frac{m \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 n (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.818883, antiderivative size = 517, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 7, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {2507, 2489, 2488, 2506, 2508, 6610, 2503} \[ -\frac{m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (2,1-\frac{(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac{2 m n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (3,1-\frac{(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac{m \text{PolyLog}\left (2,1-\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c-a d}-\frac{2 m n \text{PolyLog}\left (3,1-\frac{b c-a d}{b (c+d x)}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c-a d}-\frac{2 m n^2 \text{PolyLog}\left (4,1-\frac{(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac{2 m n^2 \text{PolyLog}\left (4,1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac{\log \left (h (f+g x)^m\right ) \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 n (b c-a d)}-\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{3 n (b c-a d)}+\frac{m \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 n (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2507
Rule 2489
Rule 2488
Rule 2506
Rule 2508
Rule 6610
Rule 2503
Rubi steps
\begin{align*} \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx &=\frac{\log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-\frac{(g m) \int \frac{\log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{f+g x} \, dx}{3 (b c-a d) n}\\ &=\frac{\log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-\frac{(d m) \int \frac{\log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3 (b c-a d) n}+\frac{((d f-c g) m) \int \frac{\log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x) (f+g x)} \, dx}{3 (b c-a d) n}\\ &=\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{b c-a d}{b (c+d x)}\right )}{3 (b c-a d) n}-\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{3 (b c-a d) n}+\frac{\log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-m \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx+m \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (-\frac{(-b c+a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{b c-a d}{b (c+d x)}\right )}{3 (b c-a d) n}-\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{3 (b c-a d) n}+\frac{\log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}+\frac{m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac{m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-(2 m n) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx+(2 m n) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{(-b c+a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{b c-a d}{b (c+d x)}\right )}{3 (b c-a d) n}-\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{3 (b c-a d) n}+\frac{\log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}+\frac{m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac{m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac{2 m n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac{2 m n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_3\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\left (2 m n^2\right ) \int \frac{\text{Li}_3\left (1+\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx-\left (2 m n^2\right ) \int \frac{\text{Li}_3\left (1+\frac{(-b c+a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{b c-a d}{b (c+d x)}\right )}{3 (b c-a d) n}-\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{3 (b c-a d) n}+\frac{\log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}+\frac{m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac{m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac{2 m n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac{2 m n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_3\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac{2 m n^2 \text{Li}_4\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac{2 m n^2 \text{Li}_4\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}\\ \end{align*}
Mathematica [B] time = 10.071, size = 9211, normalized size = 18.57 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 5.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( h \left ( gx+f \right ) ^{m} \right ) }{ \left ( bx+a \right ) \left ( dx+c \right ) } \left ( \ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2}}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]