Optimal. Leaf size=356 \[ -\frac{b f^2 m n \text{PolyLog}\left (2,-\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}-\frac{b^2 f^2 m n^2 \text{PolyLog}\left (2,-\frac{e}{f x^2}\right )}{16 e^2}-\frac{b^2 f^2 m n^2 \text{PolyLog}\left (3,-\frac{e}{f x^2}\right )}{8 e^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}+\frac{f^2 m \log \left (\frac{e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}+\frac{b f^2 m n \log \left (\frac{e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac{b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}+\frac{b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac{b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac{7 b^2 f m n^2}{32 e x^2} \]
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Rubi [A] time = 0.671996, antiderivative size = 408, normalized size of antiderivative = 1.15, number of steps used = 20, number of rules used = 14, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2305, 2304, 2378, 266, 44, 2351, 2301, 2337, 2391, 2353, 2302, 30, 2374, 6589} \[ \frac{b f^2 m n \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}+\frac{b^2 f^2 m n^2 \text{PolyLog}\left (2,-\frac{f x^2}{e}\right )}{16 e^2}-\frac{b^2 f^2 m n^2 \text{PolyLog}\left (3,-\frac{f x^2}{e}\right )}{8 e^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac{f^2 m \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}+\frac{b f^2 m n \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac{b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}+\frac{b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac{b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac{7 b^2 f m n^2}{32 e x^2} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rule 2378
Rule 266
Rule 44
Rule 2351
Rule 2301
Rule 2337
Rule 2391
Rule 2353
Rule 2302
Rule 30
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx &=-\frac{b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-(2 f m) \int \left (-\frac{b^2 n^2}{32 x^3 \left (e+f x^2\right )}-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{8 x^3 \left (e+f x^2\right )}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{4 x^3 \left (e+f x^2\right )}\right ) \, dx\\ &=-\frac{b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac{1}{2} (f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^3 \left (e+f x^2\right )} \, dx+\frac{1}{4} (b f m n) \int \frac{a+b \log \left (c x^n\right )}{x^3 \left (e+f x^2\right )} \, dx+\frac{1}{16} \left (b^2 f m n^2\right ) \int \frac{1}{x^3 \left (e+f x^2\right )} \, dx\\ &=-\frac{b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac{1}{2} (f m) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{e x^3}-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e^2 x}+\frac{f^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 \left (e+f x^2\right )}\right ) \, dx+\frac{1}{4} (b f m n) \int \left (\frac{a+b \log \left (c x^n\right )}{e x^3}-\frac{f \left (a+b \log \left (c x^n\right )\right )}{e^2 x}+\frac{f^2 x \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (e+f x^2\right )}\right ) \, dx+\frac{1}{32} \left (b^2 f m n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (e+f x)} \, dx,x,x^2\right )\\ &=-\frac{b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac{(f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx}{2 e}-\frac{\left (f^2 m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e^2}+\frac{\left (f^3 m\right ) \int \frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx}{2 e^2}+\frac{(b f m n) \int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx}{4 e}-\frac{\left (b f^2 m n\right ) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{4 e^2}+\frac{\left (b f^3 m n\right ) \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{e+f x^2} \, dx}{4 e^2}+\frac{1}{32} \left (b^2 f m n^2\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^2\right )\\ &=-\frac{3 b^2 f m n^2}{32 e x^2}-\frac{b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac{b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}+\frac{b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac{b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac{b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x^2}{e}\right )}{8 e^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x^2}{e}\right )}{4 e^2}-\frac{\left (f^2 m\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b e^2 n}+\frac{(b f m n) \int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx}{2 e}-\frac{\left (b f^2 m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x^2}{e}\right )}{x} \, dx}{2 e^2}-\frac{\left (b^2 f^2 m n^2\right ) \int \frac{\log \left (1+\frac{f x^2}{e}\right )}{x} \, dx}{8 e^2}\\ &=-\frac{7 b^2 f m n^2}{32 e x^2}-\frac{b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac{b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac{b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac{b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x^2}{e}\right )}{8 e^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x^2}{e}\right )}{4 e^2}+\frac{b^2 f^2 m n^2 \text{Li}_2\left (-\frac{f x^2}{e}\right )}{16 e^2}+\frac{b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x^2}{e}\right )}{4 e^2}-\frac{\left (b^2 f^2 m n^2\right ) \int \frac{\text{Li}_2\left (-\frac{f x^2}{e}\right )}{x} \, dx}{4 e^2}\\ &=-\frac{7 b^2 f m n^2}{32 e x^2}-\frac{b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac{b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac{b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac{b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x^2}{e}\right )}{8 e^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x^2}{e}\right )}{4 e^2}+\frac{b^2 f^2 m n^2 \text{Li}_2\left (-\frac{f x^2}{e}\right )}{16 e^2}+\frac{b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x^2}{e}\right )}{4 e^2}-\frac{b^2 f^2 m n^2 \text{Li}_3\left (-\frac{f x^2}{e}\right )}{8 e^2}\\ \end{align*}
Mathematica [C] time = 0.46181, size = 1111, normalized size = 3.12 \[ -\frac{16 b^2 f^2 m n^2 \log ^3(x) x^4-12 b^2 f^2 m n^2 \log ^2(x) x^4-48 a b f^2 m n \log ^2(x) x^4+48 b^2 f^2 m \log (x) \log ^2\left (c x^n\right ) x^4+6 b^2 f^2 m n^2 \log (x) x^4+48 a^2 f^2 m \log (x) x^4+24 a b f^2 m n \log (x) x^4-48 b^2 f^2 m n \log ^2(x) \log \left (c x^n\right ) x^4+96 a b f^2 m \log (x) \log \left (c x^n\right ) x^4+24 b^2 f^2 m n \log (x) \log \left (c x^n\right ) x^4+24 b^2 f^2 m n^2 \log ^2(x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^4-12 b^2 f^2 m n^2 \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^4-48 a b f^2 m n \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^4-48 b^2 f^2 m n \log (x) \log \left (c x^n\right ) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^4+24 b^2 f^2 m n^2 \log ^2(x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) x^4-12 b^2 f^2 m n^2 \log (x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) x^4-48 a b f^2 m n \log (x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) x^4-48 b^2 f^2 m n \log (x) \log \left (c x^n\right ) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) x^4-3 b^2 f^2 m n^2 \log \left (f x^2+e\right ) x^4-24 b^2 f^2 m n^2 \log ^2(x) \log \left (f x^2+e\right ) x^4-24 b^2 f^2 m \log ^2\left (c x^n\right ) \log \left (f x^2+e\right ) x^4-24 a^2 f^2 m \log \left (f x^2+e\right ) x^4-12 a b f^2 m n \log \left (f x^2+e\right ) x^4+12 b^2 f^2 m n^2 \log (x) \log \left (f x^2+e\right ) x^4+48 a b f^2 m n \log (x) \log \left (f x^2+e\right ) x^4-48 a b f^2 m \log \left (c x^n\right ) \log \left (f x^2+e\right ) x^4-12 b^2 f^2 m n \log \left (c x^n\right ) \log \left (f x^2+e\right ) x^4+48 b^2 f^2 m n \log (x) \log \left (c x^n\right ) \log \left (f x^2+e\right ) x^4-12 b f^2 m n \left (4 a+b n+4 b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^4-12 b f^2 m n \left (4 a+b n+4 b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^4+48 b^2 f^2 m n^2 \text{PolyLog}\left (3,-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^4+48 b^2 f^2 m n^2 \text{PolyLog}\left (3,\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^4+21 b^2 e f m n^2 x^2+24 b^2 e f m \log ^2\left (c x^n\right ) x^2+24 a^2 e f m x^2+36 a b e f m n x^2+48 a b e f m \log \left (c x^n\right ) x^2+36 b^2 e f m n \log \left (c x^n\right ) x^2+24 a^2 e^2 \log \left (d \left (f x^2+e\right )^m\right )+3 b^2 e^2 n^2 \log \left (d \left (f x^2+e\right )^m\right )+24 b^2 e^2 \log ^2\left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )+12 a b e^2 n \log \left (d \left (f x^2+e\right )^m\right )+48 a b e^2 \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )+12 b^2 e^2 n \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )}{96 e^2 x^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.876, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \right ) }{{x}^{5}}}\, 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*} -\frac{{\left (8 \, b^{2} \log \left (x^{n}\right )^{2} +{\left (n^{2} + 4 \, n \log \left (c\right ) + 8 \, \log \left (c\right )^{2}\right )} b^{2} + 4 \, a b{\left (n + 4 \, \log \left (c\right )\right )} + 8 \, a^{2} + 4 \,{\left (b^{2}{\left (n + 4 \, \log \left (c\right )\right )} + 4 \, a b\right )} \log \left (x^{n}\right )\right )} \log \left ({\left (f x^{2} + e\right )}^{m}\right )}{32 \, x^{4}} + \int \frac{16 \, b^{2} e \log \left (c\right )^{2} \log \left (d\right ) + 32 \, a b e \log \left (c\right ) \log \left (d\right ) + 16 \, a^{2} e \log \left (d\right ) +{\left (8 \,{\left (f m + 2 \, f \log \left (d\right )\right )} a^{2} + 4 \,{\left (f m n + 4 \,{\left (f m + 2 \, f \log \left (d\right )\right )} \log \left (c\right )\right )} a b +{\left (f m n^{2} + 4 \, f m n \log \left (c\right ) + 8 \,{\left (f m + 2 \, f \log \left (d\right )\right )} \log \left (c\right )^{2}\right )} b^{2}\right )} x^{2} + 8 \,{\left ({\left (f m + 2 \, f \log \left (d\right )\right )} b^{2} x^{2} + 2 \, b^{2} e \log \left (d\right )\right )} \log \left (x^{n}\right )^{2} + 4 \,{\left (8 \, b^{2} e \log \left (c\right ) \log \left (d\right ) + 8 \, a b e \log \left (d\right ) +{\left (4 \,{\left (f m + 2 \, f \log \left (d\right )\right )} a b +{\left (f m n + 4 \,{\left (f m + 2 \, f \log \left (d\right )\right )} \log \left (c\right )\right )} b^{2}\right )} x^{2}\right )} \log \left (x^{n}\right )}{16 \,{\left (f x^{7} + e x^{5}\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 (\frac{{\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{5}}, 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 )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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