Optimal. Leaf size=221 \[ \frac{b e^2 m n \text{PolyLog}\left (2,\frac{f x^2}{e}+1\right )}{8 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{e^2 m \log \left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac{1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac{b e^2 m n \log \left (e+f x^2\right )}{16 f^2}+\frac{b e^2 m n \log \left (-\frac{f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac{3 b e m n x^2}{16 f}+\frac{1}{16} b m n x^4 \]
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Rubi [A] time = 0.223533, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2454, 2395, 43, 2376, 2394, 2315} \[ \frac{b e^2 m n \text{PolyLog}\left (2,\frac{f x^2}{e}+1\right )}{8 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{e^2 m \log \left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac{1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac{b e^2 m n \log \left (e+f x^2\right )}{16 f^2}+\frac{b e^2 m n \log \left (-\frac{f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac{3 b e m n x^2}{16 f}+\frac{1}{16} b m n x^4 \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2395
Rule 43
Rule 2376
Rule 2394
Rule 2315
Rubi steps
\begin{align*} \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac{1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-(b n) \int \left (\frac{e m x}{4 f}-\frac{m x^3}{8}-\frac{e^2 m \log \left (e+f x^2\right )}{4 f^2 x}+\frac{1}{4} x^3 \log \left (d \left (e+f x^2\right )^m\right )\right ) \, dx\\ &=-\frac{b e m n x^2}{8 f}+\frac{1}{32} b m n x^4+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac{1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{4} (b n) \int x^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx+\frac{\left (b e^2 m n\right ) \int \frac{\log \left (e+f x^2\right )}{x} \, dx}{4 f^2}\\ &=-\frac{b e m n x^2}{8 f}+\frac{1}{32} b m n x^4+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac{1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{8} (b n) \operatorname{Subst}\left (\int x \log \left (d (e+f x)^m\right ) \, dx,x,x^2\right )+\frac{\left (b e^2 m n\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,x^2\right )}{8 f^2}\\ &=-\frac{b e m n x^2}{8 f}+\frac{1}{32} b m n x^4+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac{1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^2 m n \log \left (-\frac{f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}-\frac{1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{\left (b e^2 m n\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,x^2\right )}{8 f}+\frac{1}{16} (b f m n) \operatorname{Subst}\left (\int \frac{x^2}{e+f x} \, dx,x,x^2\right )\\ &=-\frac{b e m n x^2}{8 f}+\frac{1}{32} b m n x^4+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac{1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^2 m n \log \left (-\frac{f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}-\frac{1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{b e^2 m n \text{Li}_2\left (1+\frac{f x^2}{e}\right )}{8 f^2}+\frac{1}{16} (b f m n) \operatorname{Subst}\left (\int \left (-\frac{e}{f^2}+\frac{x}{f}+\frac{e^2}{f^2 (e+f x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 b e m n x^2}{16 f}+\frac{1}{16} b m n x^4+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac{1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^2 m n \log \left (e+f x^2\right )}{16 f^2}+\frac{b e^2 m n \log \left (-\frac{f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}-\frac{1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{b e^2 m n \text{Li}_2\left (1+\frac{f x^2}{e}\right )}{8 f^2}\\ \end{align*}
Mathematica [C] time = 0.158681, size = 324, normalized size = 1.47 \[ -\frac{4 b e^2 m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+4 b e^2 m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )-4 a f^2 x^4 \log \left (d \left (e+f x^2\right )^m\right )+4 a e^2 m \log \left (e+f x^2\right )-4 a e f m x^2+2 a f^2 m x^4-4 b f^2 x^4 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+4 b e^2 m \log \left (c x^n\right ) \log \left (e+f x^2\right )-4 b e f m x^2 \log \left (c x^n\right )+2 b f^2 m x^4 \log \left (c x^n\right )+b f^2 n x^4 \log \left (d \left (e+f x^2\right )^m\right )-b e^2 m n \log \left (e+f x^2\right )-4 b e^2 m n \log (x) \log \left (e+f x^2\right )+4 b e^2 m n \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+4 b e^2 m n \log (x) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )+3 b e f m n x^2-b f^2 m n x^4}{16 f^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.354, size = 2259, normalized size = 10.2 \begin{align*} \text{result too large to display} \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*} \frac{1}{16} \,{\left (4 \, b x^{4} \log \left (x^{n}\right ) -{\left (b{\left (n - 4 \, \log \left (c\right )\right )} - 4 \, a\right )} x^{4}\right )} \log \left ({\left (f x^{2} + e\right )}^{m}\right ) + \int -\frac{{\left (4 \,{\left (f m - 2 \, f \log \left (d\right )\right )} a -{\left (f m n - 4 \,{\left (f m - 2 \, f \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x^{5} - 8 \,{\left (b e \log \left (c\right ) \log \left (d\right ) + a e \log \left (d\right )\right )} x^{3} + 4 \,{\left ({\left (f m - 2 \, f \log \left (d\right )\right )} b x^{5} - 2 \, b e x^{3} \log \left (d\right )\right )} \log \left (x^{n}\right )}{8 \,{\left (f x^{2} + e\right )}}\,{d x} \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 ({\left (b x^{3} \log \left (c x^{n}\right ) + a x^{3}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ), 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{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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