Optimal. Leaf size=414 \[ -\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right )}{2 e^2 g m^2}-\frac{(g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{2 g m}-\frac{f^2 k x^{2 m} \log (x) (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right )}{2 e^2 g}+\frac{f^2 k x^{2 m} (g x)^{-2 m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 g m}-\frac{f k x^m (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right )}{2 e g m}-\frac{b n (g x)^{-2 m} \log \left (d \left (e+f x^m\right )^k\right )}{4 g m^2}+\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log \left (e+f x^m\right )}{4 e^2 g m^2}-\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{2 e^2 g m^2}+\frac{b f^2 k n x^{2 m} \log ^2(x) (g x)^{-2 m}}{4 e^2 g}-\frac{b f^2 k n x^{2 m} \log (x) (g x)^{-2 m}}{4 e^2 g m}-\frac{3 b f k n x^m (g x)^{-2 m}}{4 e g m^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.521206, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2455, 20, 266, 44, 2376, 30, 19, 2301, 2454, 2394, 2315, 16} \[ -\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right )}{2 e^2 g m^2}-\frac{(g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{2 g m}-\frac{f^2 k x^{2 m} \log (x) (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right )}{2 e^2 g}+\frac{f^2 k x^{2 m} (g x)^{-2 m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 g m}-\frac{f k x^m (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right )}{2 e g m}-\frac{b n (g x)^{-2 m} \log \left (d \left (e+f x^m\right )^k\right )}{4 g m^2}+\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log \left (e+f x^m\right )}{4 e^2 g m^2}-\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{2 e^2 g m^2}+\frac{b f^2 k n x^{2 m} \log ^2(x) (g x)^{-2 m}}{4 e^2 g}-\frac{b f^2 k n x^{2 m} \log (x) (g x)^{-2 m}}{4 e^2 g m}-\frac{3 b f k n x^m (g x)^{-2 m}}{4 e g m^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2455
Rule 20
Rule 266
Rule 44
Rule 2376
Rule 30
Rule 19
Rule 2301
Rule 2454
Rule 2394
Rule 2315
Rule 16
Rubi steps
\begin{align*} \int (g x)^{-1-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx &=-\frac{f k x^m (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right )}{2 e g m}-\frac{f^2 k x^{2 m} (g x)^{-2 m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 g}+\frac{f^2 k x^{2 m} (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{2 e^2 g m}-\frac{(g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{2 g m}-(b n) \int \left (-\frac{f k x^{-1+m} (g x)^{-2 m}}{2 e g m}-\frac{f^2 k x^{-1+2 m} (g x)^{-2 m} \log (x)}{2 e^2 g}+\frac{f^2 k x^{-1+2 m} (g x)^{-2 m} \log \left (e+f x^m\right )}{2 e^2 g m}-\frac{(g x)^{-2 m} \log \left (d \left (e+f x^m\right )^k\right )}{2 g m x}\right ) \, dx\\ &=-\frac{f k x^m (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right )}{2 e g m}-\frac{f^2 k x^{2 m} (g x)^{-2 m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 g}+\frac{f^2 k x^{2 m} (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{2 e^2 g m}-\frac{(g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{2 g m}+\frac{\left (b f^2 k n\right ) \int x^{-1+2 m} (g x)^{-2 m} \log (x) \, dx}{2 e^2 g}+\frac{(b n) \int \frac{(g x)^{-2 m} \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx}{2 g m}+\frac{(b f k n) \int x^{-1+m} (g x)^{-2 m} \, dx}{2 e g m}-\frac{\left (b f^2 k n\right ) \int x^{-1+2 m} (g x)^{-2 m} \log \left (e+f x^m\right ) \, dx}{2 e^2 g m}\\ &=-\frac{f k x^m (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right )}{2 e g m}-\frac{f^2 k x^{2 m} (g x)^{-2 m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 g}+\frac{f^2 k x^{2 m} (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{2 e^2 g m}-\frac{(g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{2 g m}+\frac{(b n) \int (g x)^{-1-2 m} \log \left (d \left (e+f x^m\right )^k\right ) \, dx}{2 m}+\frac{\left (b f^2 k n x^{2 m} (g x)^{-2 m}\right ) \int \frac{\log (x)}{x} \, dx}{2 e^2 g}+\frac{\left (b f k n x^{2 m} (g x)^{-2 m}\right ) \int x^{-1-m} \, dx}{2 e g m}-\frac{\left (b f^2 k n x^{2 m} (g x)^{-2 m}\right ) \int \frac{\log \left (e+f x^m\right )}{x} \, dx}{2 e^2 g m}\\ &=-\frac{b f k n x^m (g x)^{-2 m}}{2 e g m^2}+\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log ^2(x)}{4 e^2 g}-\frac{f k x^m (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right )}{2 e g m}-\frac{f^2 k x^{2 m} (g x)^{-2 m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 g}+\frac{f^2 k x^{2 m} (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{2 e^2 g m}-\frac{b n (g x)^{-2 m} \log \left (d \left (e+f x^m\right )^k\right )}{4 g m^2}-\frac{(g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{2 g m}+\frac{(b f k n) \int \frac{x^{-1+m} (g x)^{-2 m}}{e+f x^m} \, dx}{4 g m}-\frac{\left (b f^2 k n x^{2 m} (g x)^{-2 m}\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,x^m\right )}{2 e^2 g m^2}\\ &=-\frac{b f k n x^m (g x)^{-2 m}}{2 e g m^2}+\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log ^2(x)}{4 e^2 g}-\frac{f k x^m (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right )}{2 e g m}-\frac{f^2 k x^{2 m} (g x)^{-2 m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 g}-\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{2 e^2 g m^2}+\frac{f^2 k x^{2 m} (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{2 e^2 g m}-\frac{b n (g x)^{-2 m} \log \left (d \left (e+f x^m\right )^k\right )}{4 g m^2}-\frac{(g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{2 g m}+\frac{\left (b f^3 k n x^{2 m} (g x)^{-2 m}\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,x^m\right )}{2 e^2 g m^2}+\frac{\left (b f k n x^{2 m} (g x)^{-2 m}\right ) \int \frac{x^{-1-m}}{e+f x^m} \, dx}{4 g m}\\ &=-\frac{b f k n x^m (g x)^{-2 m}}{2 e g m^2}+\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log ^2(x)}{4 e^2 g}-\frac{f k x^m (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right )}{2 e g m}-\frac{f^2 k x^{2 m} (g x)^{-2 m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 g}-\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{2 e^2 g m^2}+\frac{f^2 k x^{2 m} (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{2 e^2 g m}-\frac{b n (g x)^{-2 m} \log \left (d \left (e+f x^m\right )^k\right )}{4 g m^2}-\frac{(g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{2 g m}-\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \text{Li}_2\left (1+\frac{f x^m}{e}\right )}{2 e^2 g m^2}+\frac{\left (b f k n x^{2 m} (g x)^{-2 m}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (e+f x)} \, dx,x,x^m\right )}{4 g m^2}\\ &=-\frac{b f k n x^m (g x)^{-2 m}}{2 e g m^2}+\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log ^2(x)}{4 e^2 g}-\frac{f k x^m (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right )}{2 e g m}-\frac{f^2 k x^{2 m} (g x)^{-2 m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 g}-\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{2 e^2 g m^2}+\frac{f^2 k x^{2 m} (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{2 e^2 g m}-\frac{b n (g x)^{-2 m} \log \left (d \left (e+f x^m\right )^k\right )}{4 g m^2}-\frac{(g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{2 g m}-\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \text{Li}_2\left (1+\frac{f x^m}{e}\right )}{2 e^2 g m^2}+\frac{\left (b f k n x^{2 m} (g x)^{-2 m}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{e x^2}-\frac{f}{e^2 x}+\frac{f^2}{e^2 (e+f x)}\right ) \, dx,x,x^m\right )}{4 g m^2}\\ &=-\frac{3 b f k n x^m (g x)^{-2 m}}{4 e g m^2}-\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log (x)}{4 e^2 g m}+\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log ^2(x)}{4 e^2 g}-\frac{f k x^m (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right )}{2 e g m}-\frac{f^2 k x^{2 m} (g x)^{-2 m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 g}+\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log \left (e+f x^m\right )}{4 e^2 g m^2}-\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{2 e^2 g m^2}+\frac{f^2 k x^{2 m} (g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{2 e^2 g m}-\frac{b n (g x)^{-2 m} \log \left (d \left (e+f x^m\right )^k\right )}{4 g m^2}-\frac{(g x)^{-2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{2 g m}-\frac{b f^2 k n x^{2 m} (g x)^{-2 m} \text{Li}_2\left (1+\frac{f x^m}{e}\right )}{2 e^2 g m^2}\\ \end{align*}
Mathematica [A] time = 0.355511, size = 302, normalized size = 0.73 \[ \frac{(g x)^{-2 m} \left (2 b f^2 k n x^{2 m} \text{PolyLog}\left (2,-\frac{f x^m}{e}\right )-f^2 k m x^{2 m} \log (x) \left (2 a m+2 b m \log \left (c x^n\right )-2 b n \log \left (\frac{f x^m}{e}+1\right )+2 b n \log \left (f-f x^{-m}\right )+b n\right )-2 a e^2 m \log \left (d \left (e+f x^m\right )^k\right )-2 a e f k m x^m+2 a f^2 k m x^{2 m} \log \left (f-f x^{-m}\right )-2 b e^2 m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-2 b e f k m x^m \log \left (c x^n\right )+2 b f^2 k m x^{2 m} \log \left (c x^n\right ) \log \left (f-f x^{-m}\right )-b e^2 n \log \left (d \left (e+f x^m\right )^k\right )-3 b e f k n x^m+b f^2 k m^2 n x^{2 m} \log ^2(x)+b f^2 k n x^{2 m} \log \left (f-f x^{-m}\right )\right )}{4 e^2 g m^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.237, size = 0, normalized size = 0. \begin{align*} \int \left ( gx \right ) ^{-1-2\,m} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ( e+f{x}^{m} \right ) ^{k} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.900203, size = 828, normalized size = 2. \begin{align*} \frac{2 \, b f^{2} g^{-2 \, m - 1} k m n x^{2 \, m} \log \left (x\right ) \log \left (\frac{f x^{m} + e}{e}\right ) + 2 \, b f^{2} g^{-2 \, m - 1} k n x^{2 \, m}{\rm Li}_2\left (-\frac{f x^{m} + e}{e} + 1\right ) -{\left (b f^{2} k m^{2} n \log \left (x\right )^{2} +{\left (2 \, b f^{2} k m^{2} \log \left (c\right ) + 2 \, a f^{2} k m^{2} + b f^{2} k m n\right )} \log \left (x\right )\right )} g^{-2 \, m - 1} x^{2 \, m} -{\left (2 \, b e f k m n \log \left (x\right ) + 2 \, b e f k m \log \left (c\right ) + 2 \, a e f k m + 3 \, b e f k n\right )} g^{-2 \, m - 1} x^{m} -{\left (2 \, b e^{2} m n \log \left (d\right ) \log \left (x\right ) +{\left (2 \, b e^{2} m \log \left (c\right ) + 2 \, a e^{2} m + b e^{2} n\right )} \log \left (d\right )\right )} g^{-2 \, m - 1} +{\left ({\left (2 \, b f^{2} k m \log \left (c\right ) + 2 \, a f^{2} k m + b f^{2} k n\right )} g^{-2 \, m - 1} x^{2 \, m} -{\left (2 \, b e^{2} k m n \log \left (x\right ) + 2 \, b e^{2} k m \log \left (c\right ) + 2 \, a e^{2} k m + b e^{2} k n\right )} g^{-2 \, m - 1}\right )} \log \left (f x^{m} + e\right )}{4 \, e^{2} m^{2} x^{2 \, m}} \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{\left (b \log \left (c x^{n}\right ) + a\right )} \left (g x\right )^{-2 \, m - 1} \log \left ({\left (f x^{m} + e\right )}^{k} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]