Optimal. Leaf size=363 \[ \frac{b e^2 k n x^{-2 m} (g x)^{2 m} \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right )}{2 f^2 g m^2}+\frac{(g x)^{2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{2 g m}-\frac{e^2 k x^{-2 m} (g x)^{2 m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^2 g m}+\frac{e k x^{-m} (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 f g m}-\frac{k (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{4 g m}-\frac{b n (g x)^{2 m} \log \left (d \left (e+f x^m\right )^k\right )}{4 g m^2}+\frac{b e^2 k n x^{-2 m} (g x)^{2 m} \log \left (e+f x^m\right )}{4 f^2 g m^2}+\frac{b e^2 k n x^{-2 m} (g x)^{2 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{2 f^2 g m^2}-\frac{3 b e k n x^{-m} (g x)^{2 m}}{4 f g m^2}+\frac{b k n (g x)^{2 m}}{4 g m^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.41775, antiderivative size = 363, 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, 43, 2376, 16, 32, 30, 19, 2454, 2394, 2315} \[ \frac{b e^2 k n x^{-2 m} (g x)^{2 m} \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right )}{2 f^2 g m^2}+\frac{(g x)^{2 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{2 g m}-\frac{e^2 k x^{-2 m} (g x)^{2 m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^2 g m}+\frac{e k x^{-m} (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 f g m}-\frac{k (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{4 g m}-\frac{b n (g x)^{2 m} \log \left (d \left (e+f x^m\right )^k\right )}{4 g m^2}+\frac{b e^2 k n x^{-2 m} (g x)^{2 m} \log \left (e+f x^m\right )}{4 f^2 g m^2}+\frac{b e^2 k n x^{-2 m} (g x)^{2 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{2 f^2 g m^2}-\frac{3 b e k n x^{-m} (g x)^{2 m}}{4 f g m^2}+\frac{b k n (g x)^{2 m}}{4 g m^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2455
Rule 20
Rule 266
Rule 43
Rule 2376
Rule 16
Rule 32
Rule 30
Rule 19
Rule 2454
Rule 2394
Rule 2315
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{k (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{4 g m}+\frac{e k x^{-m} (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 f g m}-\frac{e^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 f^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{k (g x)^{2 m}}{4 g m x}+\frac{e k x^{-1-m} (g x)^{2 m}}{2 f g m}-\frac{e^2 k x^{-1-2 m} (g x)^{2 m} \log \left (e+f x^m\right )}{2 f^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{k (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{4 g m}+\frac{e k x^{-m} (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 f g m}-\frac{e^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 f^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 \frac{(g x)^{2 m} \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx}{2 g m}+\frac{(b k n) \int \frac{(g x)^{2 m}}{x} \, dx}{4 g m}+\frac{\left (b e^2 k n\right ) \int x^{-1-2 m} (g x)^{2 m} \log \left (e+f x^m\right ) \, dx}{2 f^2 g m}-\frac{(b e k n) \int x^{-1-m} (g x)^{2 m} \, dx}{2 f g m}\\ &=-\frac{k (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{4 g m}+\frac{e k x^{-m} (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 f g m}-\frac{e^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 f^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{(b k n) \int (g x)^{-1+2 m} \, dx}{4 m}+\frac{\left (b e^2 k n x^{-2 m} (g x)^{2 m}\right ) \int \frac{\log \left (e+f x^m\right )}{x} \, dx}{2 f^2 g m}-\frac{\left (b e k n x^{-2 m} (g x)^{2 m}\right ) \int x^{-1+m} \, dx}{2 f g m}\\ &=\frac{b k n (g x)^{2 m}}{8 g m^2}-\frac{b e k n x^{-m} (g x)^{2 m}}{2 f g m^2}-\frac{k (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{4 g m}+\frac{e k x^{-m} (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 f g m}-\frac{e^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 f^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 e^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 f^2 g m^2}\\ &=\frac{b k n (g x)^{2 m}}{8 g m^2}-\frac{b e k n x^{-m} (g x)^{2 m}}{2 f g m^2}-\frac{k (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{4 g m}+\frac{e k x^{-m} (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 f g m}+\frac{b e^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 f^2 g m^2}-\frac{e^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 f^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 e^2 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 f g m^2}+\frac{\left (b f k n x^{-2 m} (g x)^{2 m}\right ) \int \frac{x^{-1+3 m}}{e+f x^m} \, dx}{4 g m}\\ &=\frac{b k n (g x)^{2 m}}{8 g m^2}-\frac{b e k n x^{-m} (g x)^{2 m}}{2 f g m^2}-\frac{k (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{4 g m}+\frac{e k x^{-m} (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 f g m}+\frac{b e^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 f^2 g m^2}-\frac{e^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 f^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 e^2 k n x^{-2 m} (g x)^{2 m} \text{Li}_2\left (1+\frac{f x^m}{e}\right )}{2 f^2 g m^2}+\frac{\left (b f k n x^{-2 m} (g x)^{2 m}\right ) \operatorname{Subst}\left (\int \frac{x^2}{e+f x} \, dx,x,x^m\right )}{4 g m^2}\\ &=\frac{b k n (g x)^{2 m}}{8 g m^2}-\frac{b e k n x^{-m} (g x)^{2 m}}{2 f g m^2}-\frac{k (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{4 g m}+\frac{e k x^{-m} (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 f g m}+\frac{b e^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 f^2 g m^2}-\frac{e^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 f^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 e^2 k n x^{-2 m} (g x)^{2 m} \text{Li}_2\left (1+\frac{f x^m}{e}\right )}{2 f^2 g m^2}+\frac{\left (b f k n x^{-2 m} (g x)^{2 m}\right ) \operatorname{Subst}\left (\int \left (-\frac{e}{f^2}+\frac{x}{f}+\frac{e^2}{f^2 (e+f x)}\right ) \, dx,x,x^m\right )}{4 g m^2}\\ &=\frac{b k n (g x)^{2 m}}{4 g m^2}-\frac{3 b e k n x^{-m} (g x)^{2 m}}{4 f g m^2}-\frac{k (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{4 g m}+\frac{e k x^{-m} (g x)^{2 m} \left (a+b \log \left (c x^n\right )\right )}{2 f g m}+\frac{b e^2 k n x^{-2 m} (g x)^{2 m} \log \left (e+f x^m\right )}{4 f^2 g m^2}+\frac{b e^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 f^2 g m^2}-\frac{e^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 f^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 e^2 k n x^{-2 m} (g x)^{2 m} \text{Li}_2\left (1+\frac{f x^m}{e}\right )}{2 f^2 g m^2}\\ \end{align*}
Mathematica [A] time = 0.379153, size = 352, normalized size = 0.97 \[ \frac{x^{-2 m} (g x)^{2 m} \left (2 b e^2 k n \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right )+e^2 k m \log (x) \left (-2 a m-2 b m \log \left (c x^n\right )-2 b n \log \left (e+f x^m\right )+2 b n \log \left (e-e x^m\right )+b n\right )+2 a f^2 m x^{2 m} \log \left (d \left (e+f x^m\right )^k\right )-2 a e^2 k m \log \left (e-e x^m\right )+2 a e f k m x^m-a f^2 k m x^{2 m}+2 b f^2 m x^{2 m} \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-2 b e^2 k m \log \left (c x^n\right ) \log \left (e-e x^m\right )+2 b e f k m x^m \log \left (c x^n\right )-b f^2 k m x^{2 m} \log \left (c x^n\right )-b f^2 n x^{2 m} \log \left (d \left (e+f x^m\right )^k\right )+2 b e^2 k n \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )+2 b e^2 k m^2 n \log ^2(x)+b e^2 k n \log \left (e-e x^m\right )-3 b e f k n x^m+b f^2 k n x^{2 m}\right )}{4 f^2 g m^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.24, 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 = 0.987284, size = 730, normalized size = 2.01 \begin{align*} -\frac{2 \, b e^{2} g^{2 \, m - 1} k m n \log \left (x\right ) \log \left (\frac{f x^{m} + e}{e}\right ) + 2 \, b e^{2} g^{2 \, m - 1} k n{\rm Li}_2\left (-\frac{f x^{m} + e}{e} + 1\right ) +{\left (b f^{2} k m \log \left (c\right ) + a f^{2} k m - b f^{2} k n -{\left (2 \, b f^{2} m \log \left (c\right ) + 2 \, a f^{2} m - b f^{2} n\right )} \log \left (d\right ) +{\left (b f^{2} k m n - 2 \, b f^{2} m n \log \left (d\right )\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 ({\left (2 \, b f^{2} k m n \log \left (x\right ) + 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 \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 \, f^{2} m^{2}} \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]