Optimal. Leaf size=451 \[ \frac{3 b^2 f m n^2 \text{PolyLog}\left (2,-\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac{3 b^2 f m n^2 \text{PolyLog}\left (3,-\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac{3 b f m n \text{PolyLog}\left (2,-\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac{3 b^3 f m n^3 \text{PolyLog}\left (2,-\frac{e}{f x^2}\right )}{8 e}+\frac{3 b^3 f m n^3 \text{PolyLog}\left (3,-\frac{e}{f x^2}\right )}{8 e}+\frac{3 b^3 f m n^3 \text{PolyLog}\left (4,-\frac{e}{f x^2}\right )}{8 e}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{3 b^2 f m n^2 \log \left (\frac{e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac{3 b f m n \log \left (\frac{e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac{f m \log \left (\frac{e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac{3 b^3 f m n^3 \log \left (e+f x^2\right )}{8 e}+\frac{3 b^3 f m n^3 \log (x)}{4 e} \]
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Rubi [A] time = 0.569481, antiderivative size = 451, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {2305, 2304, 2378, 266, 36, 29, 31, 2345, 2391, 2374, 6589, 2383} \[ \frac{3 b^2 f m n^2 \text{PolyLog}\left (2,-\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac{3 b^2 f m n^2 \text{PolyLog}\left (3,-\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac{3 b f m n \text{PolyLog}\left (2,-\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac{3 b^3 f m n^3 \text{PolyLog}\left (2,-\frac{e}{f x^2}\right )}{8 e}+\frac{3 b^3 f m n^3 \text{PolyLog}\left (3,-\frac{e}{f x^2}\right )}{8 e}+\frac{3 b^3 f m n^3 \text{PolyLog}\left (4,-\frac{e}{f x^2}\right )}{8 e}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{3 b^2 f m n^2 \log \left (\frac{e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac{3 b f m n \log \left (\frac{e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac{f m \log \left (\frac{e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac{3 b^3 f m n^3 \log \left (e+f x^2\right )}{8 e}+\frac{3 b^3 f m n^3 \log (x)}{4 e} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rule 2378
Rule 266
Rule 36
Rule 29
Rule 31
Rule 2345
Rule 2391
Rule 2374
Rule 6589
Rule 2383
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx &=-\frac{3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-(2 f m) \int \left (-\frac{3 b^3 n^3}{8 x \left (e+f x^2\right )}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x \left (e+f x^2\right )}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x \left (e+f x^2\right )}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 x \left (e+f x^2\right )}\right ) \, dx\\ &=-\frac{3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+(f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x \left (e+f x^2\right )} \, dx+\frac{1}{2} (3 b f m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (e+f x^2\right )} \, dx+\frac{1}{2} \left (3 b^2 f m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x \left (e+f x^2\right )} \, dx+\frac{1}{4} \left (3 b^3 f m n^3\right ) \int \frac{1}{x \left (e+f x^2\right )} \, dx\\ &=-\frac{3 b^2 f m n^2 \log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{3 b f m n \log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac{f m \log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac{(3 b f m n) \int \frac{\log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e}+\frac{\left (3 b^2 f m n^2\right ) \int \frac{\log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{2 e}+\frac{1}{8} \left (3 b^3 f m n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x (e+f x)} \, dx,x,x^2\right )+\frac{\left (3 b^3 f m n^3\right ) \int \frac{\log \left (1+\frac{e}{f x^2}\right )}{x} \, dx}{4 e}\\ &=-\frac{3 b^2 f m n^2 \log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{3 b f m n \log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac{f m \log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac{3 b^3 f m n^3 \text{Li}_2\left (-\frac{e}{f x^2}\right )}{8 e}+\frac{3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e}{f x^2}\right )}{4 e}+\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{e}{f x^2}\right )}{4 e}-\frac{\left (3 b^2 f m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e}{f x^2}\right )}{x} \, dx}{2 e}+\frac{\left (3 b^3 f m n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{8 e}-\frac{\left (3 b^3 f m n^3\right ) \int \frac{\text{Li}_2\left (-\frac{e}{f x^2}\right )}{x} \, dx}{4 e}-\frac{\left (3 b^3 f^2 m n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+f x} \, dx,x,x^2\right )}{8 e}\\ &=\frac{3 b^3 f m n^3 \log (x)}{4 e}-\frac{3 b^2 f m n^2 \log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{3 b f m n \log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac{f m \log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{3 b^3 f m n^3 \log \left (e+f x^2\right )}{8 e}-\frac{3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac{3 b^3 f m n^3 \text{Li}_2\left (-\frac{e}{f x^2}\right )}{8 e}+\frac{3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e}{f x^2}\right )}{4 e}+\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{e}{f x^2}\right )}{4 e}+\frac{3 b^3 f m n^3 \text{Li}_3\left (-\frac{e}{f x^2}\right )}{8 e}+\frac{3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{e}{f x^2}\right )}{4 e}-\frac{\left (3 b^3 f m n^3\right ) \int \frac{\text{Li}_3\left (-\frac{e}{f x^2}\right )}{x} \, dx}{4 e}\\ &=\frac{3 b^3 f m n^3 \log (x)}{4 e}-\frac{3 b^2 f m n^2 \log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{3 b f m n \log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac{f m \log \left (1+\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{3 b^3 f m n^3 \log \left (e+f x^2\right )}{8 e}-\frac{3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac{3 b^3 f m n^3 \text{Li}_2\left (-\frac{e}{f x^2}\right )}{8 e}+\frac{3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e}{f x^2}\right )}{4 e}+\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{e}{f x^2}\right )}{4 e}+\frac{3 b^3 f m n^3 \text{Li}_3\left (-\frac{e}{f x^2}\right )}{8 e}+\frac{3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{e}{f x^2}\right )}{4 e}+\frac{3 b^3 f m n^3 \text{Li}_4\left (-\frac{e}{f x^2}\right )}{8 e}\\ \end{align*}
Mathematica [C] time = 0.863149, size = 2248, normalized size = 4.98 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 5.322, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \right ) }{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{3}}, x\right ) \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 \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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