Optimal. Leaf size=126 \[ \frac {e \log \left (\frac {d}{e x^2}+1\right ) \left (4 a+4 b \log \left (c x^n\right )-b n\right )}{4 d^3}-\frac {4 a+4 b \log \left (c x^n\right )-b n}{4 d^2 x^2}+\frac {a+b \log \left (c x^n\right )}{2 d x^2 \left (d+e x^2\right )}-\frac {b e n \text {Li}_2\left (-\frac {d}{e x^2}\right )}{2 d^3}-\frac {b n}{2 d^2 x^2} \]
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Rubi [A] time = 0.29, antiderivative size = 159, normalized size of antiderivative = 1.26, number of steps used = 7, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2340, 266, 44, 2351, 2304, 2301, 2337, 2391} \[ \frac {b e n \text {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{2 d^3}-\frac {e \left (4 a+4 b \log \left (c x^n\right )-b n\right )^2}{16 b d^3 n}+\frac {e \log \left (\frac {e x^2}{d}+1\right ) \left (4 a+4 b \log \left (c x^n\right )-b n\right )}{4 d^3}-\frac {4 a+4 b \log \left (c x^n\right )-b n}{4 d^2 x^2}+\frac {a+b \log \left (c x^n\right )}{2 d x^2 \left (d+e x^2\right )}-\frac {b n}{2 d^2 x^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 2301
Rule 2304
Rule 2337
Rule 2340
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^2} \, dx &=\frac {a+b \log \left (c x^n\right )}{2 d x^2 \left (d+e x^2\right )}-\frac {\int \frac {-4 a+b n-4 b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )} \, dx}{2 d}\\ &=\frac {a+b \log \left (c x^n\right )}{2 d x^2 \left (d+e x^2\right )}-\frac {\int \left (\frac {-4 a+b n-4 b \log \left (c x^n\right )}{d x^3}-\frac {e \left (-4 a+b n-4 b \log \left (c x^n\right )\right )}{d^2 x}+\frac {e^2 x \left (-4 a+b n-4 b \log \left (c x^n\right )\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx}{2 d}\\ &=\frac {a+b \log \left (c x^n\right )}{2 d x^2 \left (d+e x^2\right )}-\frac {\int \frac {-4 a+b n-4 b \log \left (c x^n\right )}{x^3} \, dx}{2 d^2}+\frac {e \int \frac {-4 a+b n-4 b \log \left (c x^n\right )}{x} \, dx}{2 d^3}-\frac {e^2 \int \frac {x \left (-4 a+b n-4 b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{2 d^3}\\ &=-\frac {b n}{2 d^2 x^2}+\frac {a+b \log \left (c x^n\right )}{2 d x^2 \left (d+e x^2\right )}-\frac {4 a-b n+4 b \log \left (c x^n\right )}{4 d^2 x^2}-\frac {e \left (4 a-b n+4 b \log \left (c x^n\right )\right )^2}{16 b d^3 n}+\frac {e \left (4 a-b n+4 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{4 d^3}-\frac {(b e n) \int \frac {\log \left (1+\frac {e x^2}{d}\right )}{x} \, dx}{d^3}\\ &=-\frac {b n}{2 d^2 x^2}+\frac {a+b \log \left (c x^n\right )}{2 d x^2 \left (d+e x^2\right )}-\frac {4 a-b n+4 b \log \left (c x^n\right )}{4 d^2 x^2}-\frac {e \left (4 a-b n+4 b \log \left (c x^n\right )\right )^2}{16 b d^3 n}+\frac {e \left (4 a-b n+4 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{4 d^3}+\frac {b e n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{2 d^3}\\ \end {align*}
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Mathematica [C] time = 0.57, size = 334, normalized size = 2.65 \[ \frac {4 e \log \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-\frac {2 d e \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{d+e x^2}-\frac {2 d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{x^2}-8 e \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+b n \left (\frac {e^{3/2} x \log (x)}{\sqrt {e} x-i \sqrt {d}}+\frac {e \left (-\sqrt {d}+i \sqrt {e} x\right ) \log \left (\sqrt {e} x+i \sqrt {d}\right )-i e^{3/2} x \log (x)}{\sqrt {d}-i \sqrt {e} x}+4 e \left (\text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )+4 e \left (\text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )-e \log \left (-\sqrt {e} x+i \sqrt {d}\right )-\frac {2 d \log (x)+d}{x^2}-4 e \log ^2(x)\right )}{4 d^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e^{2} x^{7} + 2 \, d e x^{5} + d^{2} x^{3}}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 817, normalized size = 6.48 \[ \text {result too large to display} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {2 \, e x^{2} + d}{d^{2} e x^{4} + d^{3} x^{2}} - \frac {2 \, e \log \left (e x^{2} + d\right )}{d^{3}} + \frac {4 \, e \log \relax (x)}{d^{3}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e^{2} x^{7} + 2 \, d e x^{5} + d^{2} x^{3}}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,{\left (e\,x^2+d\right )}^2} \,d x \]
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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