Optimal. Leaf size=248 \[ -\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {f^2 m \log \left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{4 e x^2}-\frac {b n \log \left (d \left (e+f x^2\right )^m\right )}{16 x^4}-\frac {b f^2 m n \text {Li}_2\left (\frac {f x^2}{e}+1\right )}{8 e^2}+\frac {b f^2 m n \log \left (e+f x^2\right )}{16 e^2}-\frac {b f^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 e^2}+\frac {b f^2 m n \log ^2(x)}{4 e^2}-\frac {b f^2 m n \log (x)}{8 e^2}-\frac {3 b f m n}{16 e x^2} \]
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Rubi [A] time = 0.23, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2454, 2395, 44, 2376, 2301, 2394, 2315} \[ -\frac {b f^2 m n \text {PolyLog}\left (2,\frac {f x^2}{e}+1\right )}{8 e^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {f^2 m \log \left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{4 e x^2}-\frac {b n \log \left (d \left (e+f x^2\right )^m\right )}{16 x^4}+\frac {b f^2 m n \log \left (e+f x^2\right )}{16 e^2}-\frac {b f^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 e^2}+\frac {b f^2 m n \log ^2(x)}{4 e^2}-\frac {b f^2 m n \log (x)}{8 e^2}-\frac {3 b f m n}{16 e x^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2301
Rule 2315
Rule 2376
Rule 2394
Rule 2395
Rule 2454
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx &=-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{4 e x^2}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 e^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-(b n) \int \left (-\frac {f m}{4 e x^3}-\frac {f^2 m \log (x)}{2 e^2 x}+\frac {f^2 m \log \left (e+f x^2\right )}{4 e^2 x}-\frac {\log \left (d \left (e+f x^2\right )^m\right )}{4 x^5}\right ) \, dx\\ &=-\frac {b f m n}{8 e x^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{4 e x^2}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 e^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {1}{4} (b n) \int \frac {\log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx-\frac {\left (b f^2 m n\right ) \int \frac {\log \left (e+f x^2\right )}{x} \, dx}{4 e^2}+\frac {\left (b f^2 m n\right ) \int \frac {\log (x)}{x} \, dx}{2 e^2}\\ &=-\frac {b f m n}{8 e x^2}+\frac {b f^2 m n \log ^2(x)}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{4 e x^2}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 e^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {1}{8} (b n) \operatorname {Subst}\left (\int \frac {\log \left (d (e+f x)^m\right )}{x^3} \, dx,x,x^2\right )-\frac {\left (b f^2 m n\right ) \operatorname {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,x^2\right )}{8 e^2}\\ &=-\frac {b f m n}{8 e x^2}+\frac {b f^2 m n \log ^2(x)}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{4 e x^2}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {b f^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 e^2}-\frac {b n \log \left (d \left (e+f x^2\right )^m\right )}{16 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {1}{16} (b f m n) \operatorname {Subst}\left (\int \frac {1}{x^2 (e+f x)} \, dx,x,x^2\right )+\frac {\left (b f^3 m n\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,x^2\right )}{8 e^2}\\ &=-\frac {b f m n}{8 e x^2}+\frac {b f^2 m n \log ^2(x)}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{4 e x^2}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {b f^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 e^2}-\frac {b n \log \left (d \left (e+f x^2\right )^m\right )}{16 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac {b f^2 m n \text {Li}_2\left (1+\frac {f x^2}{e}\right )}{8 e^2}+\frac {1}{16} (b f m n) \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 f m n}{16 e x^2}-\frac {b f^2 m n \log (x)}{8 e^2}+\frac {b f^2 m n \log ^2(x)}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{4 e x^2}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {b f^2 m n \log \left (e+f x^2\right )}{16 e^2}-\frac {b f^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 e^2}-\frac {b n \log \left (d \left (e+f x^2\right )^m\right )}{16 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac {b f^2 m n \text {Li}_2\left (1+\frac {f x^2}{e}\right )}{8 e^2}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 363, normalized size = 1.46 \[ -\frac {4 a e^2 \log \left (d \left (e+f x^2\right )^m\right )-4 a f^2 m x^4 \log \left (e+f x^2\right )+4 a e f m x^2+8 a f^2 m x^4 \log (x)+4 b e^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-4 b f^2 m x^4 \log \left (c x^n\right ) \log \left (e+f x^2\right )+4 b e f m x^2 \log \left (c x^n\right )+8 b f^2 m x^4 \log (x) \log \left (c x^n\right )+b e^2 n \log \left (d \left (e+f x^2\right )^m\right )-4 b f^2 m n x^4 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-4 b f^2 m n x^4 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )-4 b f^2 m n x^4 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-4 b f^2 m n x^4 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-b f^2 m n x^4 \log \left (e+f x^2\right )+4 b f^2 m n x^4 \log (x) \log \left (e+f x^2\right )+3 b e f m n x^2-4 b f^2 m n x^4 \log ^2(x)+2 b f^2 m n x^4 \log (x)}{16 e^2 x^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{5}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.73, size = 2313, normalized size = 9.33 \[ \text {Expression too large to display} \]
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 \[ -\frac {{\left (4 \, b m \log \left (x^{n}\right ) + {\left (m n + 4 \, m \log \relax (c)\right )} b + 4 \, a m\right )} \log \left (f x^{2} + e\right )}{16 \, x^{4}} + \int \frac {8 \, b e \log \relax (c) \log \relax (d) + {\left (4 \, {\left (f m + 2 \, f \log \relax (d)\right )} a + {\left (f m n + 4 \, {\left (f m + 2 \, f \log \relax (d)\right )} \log \relax (c)\right )} b\right )} x^{2} + 8 \, a e \log \relax (d) + 4 \, {\left ({\left (f m + 2 \, f \log \relax (d)\right )} b x^{2} + 2 \, b e \log \relax (d)\right )} \log \left (x^{n}\right )}{8 \, {\left (f x^{7} + e x^{5}\right )}}\,{d x} \]
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 \[ \int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^5} \,d x \]
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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