Optimal. Leaf size=121 \[ -\frac {e^2 \log \left (\frac {d}{e x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac {a+b \log \left (c x^n\right )}{4 d x^4}+\frac {b e^2 n \text {Li}_2\left (-\frac {d}{e x^2}\right )}{4 d^3}+\frac {b e n}{4 d^2 x^2}-\frac {b n}{16 d x^4} \]
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Rubi [A] time = 0.21, antiderivative size = 149, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {266, 44, 2351, 2304, 2301, 2337, 2391} \[ -\frac {b e^2 n \text {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 d^3}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac {e^2 \log \left (\frac {e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac {a+b \log \left (c x^n\right )}{4 d x^4}+\frac {b e n}{4 d^2 x^2}-\frac {b n}{16 d x^4} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 2301
Rule 2304
Rule 2337
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^5 \left (d+e x^2\right )} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{d x^5}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x^3}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^3 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x^5} \, dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{d^2}+\frac {e^2 \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^3}-\frac {e^3 \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{d^3}\\ &=-\frac {b n}{16 d x^4}+\frac {b e n}{4 d^2 x^2}-\frac {a+b \log \left (c x^n\right )}{4 d x^4}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 d^3}+\frac {\left (b e^2 n\right ) \int \frac {\log \left (1+\frac {e x^2}{d}\right )}{x} \, dx}{2 d^3}\\ &=-\frac {b n}{16 d x^4}+\frac {b e n}{4 d^2 x^2}-\frac {a+b \log \left (c x^n\right )}{4 d x^4}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 d^3}-\frac {b e^2 n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{4 d^3}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 196, normalized size = 1.62 \[ -\frac {\frac {4 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^4}+8 e^2 \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )+8 e^2 \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {8 d e \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+\frac {b d^2 n}{x^4}+8 b e^2 n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )+8 b e^2 n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )-\frac {4 b d e n}{x^2}}{16 d^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e x^{7} + d x^{5}}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 805, normalized size = 6.65 \[ \text {result too large to display} \]
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 \[ -\frac {1}{4} \, a {\left (\frac {2 \, e^{2} \log \left (e x^{2} + d\right )}{d^{3}} - \frac {4 \, e^{2} \log \relax (x)}{d^{3}} - \frac {2 \, e x^{2} - d}{d^{2} x^{4}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e x^{7} + d x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^5\,\left (e\,x^2+d\right )} \,d x \]
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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