Optimal. Leaf size=356 \[ -\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {b f^2 m n \text {Li}_2\left (-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}+\frac {f^2 m \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}+\frac {b f^2 m n \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {e}{f x^2}\right )}{16 e^2}-\frac {b^2 f^2 m n^2 \text {Li}_3\left (-\frac {e}{f x^2}\right )}{8 e^2}+\frac {b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac {b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac {7 b^2 f m n^2}{32 e x^2} \]
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Rubi [A] time = 0.67, antiderivative size = 408, normalized size of antiderivative = 1.15, number of steps used = 20, number of rules used = 14, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 30
Rule 44
Rule 266
Rule 2301
Rule 2302
Rule 2304
Rule 2305
Rule 2337
Rule 2351
Rule 2353
Rule 2374
Rule 2378
Rule 2391
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*}
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Mathematica [C] time = 0.48, 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 {Li}_2\left (-\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 {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right ) x^4+48 b^2 f^2 m n^2 \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) x^4+48 b^2 f^2 m n^2 \text {Li}_3\left (\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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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.54, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (8 \, b^{2} m \log \left (x^{n}\right )^{2} + 4 \, {\left (m n + 4 \, m \log \relax (c)\right )} a b + {\left (m n^{2} + 4 \, m n \log \relax (c) + 8 \, m \log \relax (c)^{2}\right )} b^{2} + 8 \, a^{2} m + 4 \, {\left ({\left (m n + 4 \, m \log \relax (c)\right )} b^{2} + 4 \, a b m\right )} \log \left (x^{n}\right )\right )} \log \left (f x^{2} + e\right )}{32 \, x^{4}} + \int \frac {16 \, b^{2} e \log \relax (c)^{2} \log \relax (d) + 32 \, a b e \log \relax (c) \log \relax (d) + 16 \, a^{2} e \log \relax (d) + {\left (8 \, {\left (f m + 2 \, f \log \relax (d)\right )} a^{2} + 4 \, {\left (f m n + 4 \, {\left (f m + 2 \, f \log \relax (d)\right )} \log \relax (c)\right )} a b + {\left (f m n^{2} + 4 \, f m n \log \relax (c) + 8 \, {\left (f m + 2 \, f \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{2}\right )} x^{2} + 8 \, {\left ({\left (f m + 2 \, f \log \relax (d)\right )} b^{2} x^{2} + 2 \, b^{2} e \log \relax (d)\right )} \log \left (x^{n}\right )^{2} + 4 \, {\left (8 \, b^{2} e \log \relax (c) \log \relax (d) + 8 \, a b e \log \relax (d) + {\left (4 \, {\left (f m + 2 \, f \log \relax (d)\right )} a b + {\left (f m n + 4 \, {\left (f m + 2 \, f \log \relax (d)\right )} \log \relax (c)\right )} b^{2}\right )} x^{2}\right )} \log \left (x^{n}\right )}{16 \, {\left (f x^{7} + e x^{5}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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