Optimal. Leaf size=433 \[ -\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right )}{3 f^3 g m^2}+\frac{(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}+\frac{e^3 k x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3 g m}-\frac{e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac{e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac{k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac{b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac{4 b e^2 k n x^{-2 m} (g x)^{3 m}}{9 f^2 g m^2}-\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right )}{9 f^3 g m^2}-\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}-\frac{5 b e k n x^{-m} (g x)^{3 m}}{36 f g m^2}+\frac{2 b k n (g x)^{3 m}}{27 g m^2} \]
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Rubi [A] time = 0.602527, antiderivative size = 433, normalized size of antiderivative = 1., number of steps used = 18, 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} \[ -\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right )}{3 f^3 g m^2}+\frac{(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}+\frac{e^3 k x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3 g m}-\frac{e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac{e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac{k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac{b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac{4 b e^2 k n x^{-2 m} (g x)^{3 m}}{9 f^2 g m^2}-\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right )}{9 f^3 g m^2}-\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}-\frac{5 b e k n x^{-m} (g x)^{3 m}}{36 f g m^2}+\frac{2 b k n (g x)^{3 m}}{27 g m^2} \]
Antiderivative was successfully verified.
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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+3 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)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac{e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac{e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}+\frac{e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}+\frac{(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-(b n) \int \left (-\frac{k (g x)^{3 m}}{9 g m x}-\frac{e^2 k x^{-1-2 m} (g x)^{3 m}}{3 f^2 g m}+\frac{e k x^{-1-m} (g x)^{3 m}}{6 f g m}+\frac{e^3 k x^{-1-3 m} (g x)^{3 m} \log \left (e+f x^m\right )}{3 f^3 g m}+\frac{(g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{3 g m x}\right ) \, dx\\ &=-\frac{k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac{e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac{e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}+\frac{e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}+\frac{(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-\frac{(b n) \int \frac{(g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx}{3 g m}+\frac{(b k n) \int \frac{(g x)^{3 m}}{x} \, dx}{9 g m}-\frac{\left (b e^3 k n\right ) \int x^{-1-3 m} (g x)^{3 m} \log \left (e+f x^m\right ) \, dx}{3 f^3 g m}+\frac{\left (b e^2 k n\right ) \int x^{-1-2 m} (g x)^{3 m} \, dx}{3 f^2 g m}-\frac{(b e k n) \int x^{-1-m} (g x)^{3 m} \, dx}{6 f g m}\\ &=-\frac{k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac{e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac{e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}+\frac{e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}+\frac{(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-\frac{(b n) \int (g x)^{-1+3 m} \log \left (d \left (e+f x^m\right )^k\right ) \, dx}{3 m}+\frac{(b k n) \int (g x)^{-1+3 m} \, dx}{9 m}-\frac{\left (b e^3 k n x^{-3 m} (g x)^{3 m}\right ) \int \frac{\log \left (e+f x^m\right )}{x} \, dx}{3 f^3 g m}+\frac{\left (b e^2 k n x^{-3 m} (g x)^{3 m}\right ) \int x^{-1+m} \, dx}{3 f^2 g m}-\frac{\left (b e k n x^{-3 m} (g x)^{3 m}\right ) \int x^{-1+2 m} \, dx}{6 f g m}\\ &=\frac{b k n (g x)^{3 m}}{27 g m^2}+\frac{b e^2 k n x^{-2 m} (g x)^{3 m}}{3 f^2 g m^2}-\frac{b e k n x^{-m} (g x)^{3 m}}{12 f g m^2}-\frac{k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac{e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac{e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}+\frac{e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}-\frac{b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac{(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}+\frac{(b f k n) \int \frac{x^{-1+m} (g x)^{3 m}}{e+f x^m} \, dx}{9 g m}-\frac{\left (b e^3 k n x^{-3 m} (g x)^{3 m}\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,x^m\right )}{3 f^3 g m^2}\\ &=\frac{b k n (g x)^{3 m}}{27 g m^2}+\frac{b e^2 k n x^{-2 m} (g x)^{3 m}}{3 f^2 g m^2}-\frac{b e k n x^{-m} (g x)^{3 m}}{12 f g m^2}-\frac{k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac{e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac{e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}+\frac{e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}-\frac{b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac{(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}+\frac{\left (b e^3 k n x^{-3 m} (g x)^{3 m}\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,x^m\right )}{3 f^2 g m^2}+\frac{\left (b f k n x^{-3 m} (g x)^{3 m}\right ) \int \frac{x^{-1+4 m}}{e+f x^m} \, dx}{9 g m}\\ &=\frac{b k n (g x)^{3 m}}{27 g m^2}+\frac{b e^2 k n x^{-2 m} (g x)^{3 m}}{3 f^2 g m^2}-\frac{b e k n x^{-m} (g x)^{3 m}}{12 f g m^2}-\frac{k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac{e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac{e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}+\frac{e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}-\frac{b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac{(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \text{Li}_2\left (1+\frac{f x^m}{e}\right )}{3 f^3 g m^2}+\frac{\left (b f k n x^{-3 m} (g x)^{3 m}\right ) \operatorname{Subst}\left (\int \frac{x^3}{e+f x} \, dx,x,x^m\right )}{9 g m^2}\\ &=\frac{b k n (g x)^{3 m}}{27 g m^2}+\frac{b e^2 k n x^{-2 m} (g x)^{3 m}}{3 f^2 g m^2}-\frac{b e k n x^{-m} (g x)^{3 m}}{12 f g m^2}-\frac{k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac{e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac{e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}+\frac{e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}-\frac{b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac{(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \text{Li}_2\left (1+\frac{f x^m}{e}\right )}{3 f^3 g m^2}+\frac{\left (b f k n x^{-3 m} (g x)^{3 m}\right ) \operatorname{Subst}\left (\int \left (\frac{e^2}{f^3}-\frac{e x}{f^2}+\frac{x^2}{f}-\frac{e^3}{f^3 (e+f x)}\right ) \, dx,x,x^m\right )}{9 g m^2}\\ &=\frac{2 b k n (g x)^{3 m}}{27 g m^2}+\frac{4 b e^2 k n x^{-2 m} (g x)^{3 m}}{9 f^2 g m^2}-\frac{5 b e k n x^{-m} (g x)^{3 m}}{36 f g m^2}-\frac{k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac{e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac{e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right )}{9 f^3 g m^2}-\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}+\frac{e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}-\frac{b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac{(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-\frac{b e^3 k n x^{-3 m} (g x)^{3 m} \text{Li}_2\left (1+\frac{f x^m}{e}\right )}{3 f^3 g m^2}\\ \end{align*}
Mathematica [A] time = 0.426978, size = 410, normalized size = 0.95 \[ \frac{x^{-3 m} (g x)^{3 m} \left (-36 b e^3 k n \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right )+12 e^3 k m \log (x) \left (3 a m+3 b m \log \left (c x^n\right )+3 b n \log \left (e+f x^m\right )-3 b n \log \left (e-e x^m\right )-b n\right )+36 a f^3 m x^{3 m} \log \left (d \left (e+f x^m\right )^k\right )-36 a e^2 f k m x^m+36 a e^3 k m \log \left (e-e x^m\right )+18 a e f^2 k m x^{2 m}-12 a f^3 k m x^{3 m}+36 b f^3 m x^{3 m} \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-36 b e^2 f k m x^m \log \left (c x^n\right )+36 b e^3 k m \log \left (c x^n\right ) \log \left (e-e x^m\right )+18 b e f^2 k m x^{2 m} \log \left (c x^n\right )-12 b f^3 k m x^{3 m} \log \left (c x^n\right )-12 b f^3 n x^{3 m} \log \left (d \left (e+f x^m\right )^k\right )+48 b e^2 f k n x^m-36 b e^3 k n \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )-36 b e^3 k m^2 n \log ^2(x)-12 b e^3 k n \log \left (e-e x^m\right )-15 b e f^2 k n x^{2 m}+8 b f^3 k n x^{3 m}\right )}{108 f^3 g m^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.238, size = 0, normalized size = 0. \begin{align*} \int \left ( gx \right ) ^{-1+3\,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.
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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.
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Fricas [A] time = 0.990596, size = 910, normalized size = 2.1 \begin{align*} \frac{36 \, b e^{3} g^{3 \, m - 1} k m n \log \left (x\right ) \log \left (\frac{f x^{m} + e}{e}\right ) + 36 \, b e^{3} g^{3 \, m - 1} k n{\rm Li}_2\left (-\frac{f x^{m} + e}{e} + 1\right ) - 4 \,{\left (3 \, b f^{3} k m \log \left (c\right ) + 3 \, a f^{3} k m - 2 \, b f^{3} k n - 3 \,{\left (3 \, b f^{3} m \log \left (c\right ) + 3 \, a f^{3} m - b f^{3} n\right )} \log \left (d\right ) + 3 \,{\left (b f^{3} k m n - 3 \, b f^{3} m n \log \left (d\right )\right )} \log \left (x\right )\right )} g^{3 \, m - 1} x^{3 \, m} + 3 \,{\left (6 \, b e f^{2} k m n \log \left (x\right ) + 6 \, b e f^{2} k m \log \left (c\right ) + 6 \, a e f^{2} k m - 5 \, b e f^{2} k n\right )} g^{3 \, m - 1} x^{2 \, m} - 12 \,{\left (3 \, b e^{2} f k m n \log \left (x\right ) + 3 \, b e^{2} f k m \log \left (c\right ) + 3 \, a e^{2} f k m - 4 \, b e^{2} f k n\right )} g^{3 \, m - 1} x^{m} + 12 \,{\left ({\left (3 \, b f^{3} k m n \log \left (x\right ) + 3 \, b f^{3} k m \log \left (c\right ) + 3 \, a f^{3} k m - b f^{3} k n\right )} g^{3 \, m - 1} x^{3 \, m} +{\left (3 \, b e^{3} k m \log \left (c\right ) + 3 \, a e^{3} k m - b e^{3} k n\right )} g^{3 \, m - 1}\right )} \log \left (f x^{m} + e\right )}{108 \, f^{3} m^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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 )^{3 \, 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.
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