Optimal. Leaf size=83 \[ \frac {e \log \left (\frac {d}{e x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}-\frac {b e n \text {Li}_2\left (-\frac {d}{e x^2}\right )}{4 d^2}-\frac {b n}{4 d x^2} \]
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Rubi [A] time = 0.18, antiderivative size = 109, normalized size of antiderivative = 1.31, number of steps used = 6, 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} \[ \frac {b e n \text {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 d^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac {e \log \left (\frac {e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}-\frac {b n}{4 d x^2} \]
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^3 \left (d+e x^2\right )} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{d x^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^2}+\frac {e^2 \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{d^2}\\ &=-\frac {b n}{4 d x^2}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac {e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 d^2}-\frac {(b e n) \int \frac {\log \left (1+\frac {e x^2}{d}\right )}{x} \, dx}{2 d^2}\\ &=-\frac {b n}{4 d x^2}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac {e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 d^2}+\frac {b e n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{4 d^2}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 157, normalized size = 1.89 \[ \frac {2 e \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )+2 e \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{b n}+2 b e n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )+2 b e n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )-\frac {b d n}{x^2}}{4 d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e x^{5} + d x^{3}}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 611, normalized size = 7.36 \[ \frac {b e n \dilog \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}+\frac {b e n \dilog \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}+\frac {b e \ln \left (x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{2 d^{2}}-\frac {b e \ln \relax (x ) \ln \left (x^{n}\right )}{d^{2}}+\frac {b e \ln \relax (c ) \ln \left (e \,x^{2}+d \right )}{2 d^{2}}+\frac {b e n \ln \relax (x )^{2}}{2 d^{2}}-\frac {b e \ln \relax (c ) \ln \relax (x )}{d^{2}}+\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4 d \,x^{2}}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4 d \,x^{2}}+\frac {i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e \,x^{2}+d \right )}{4 d^{2}}-\frac {i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )}{2 d^{2}}-\frac {i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )}{2 d^{2}}+\frac {i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e \,x^{2}+d \right )}{4 d^{2}}+\frac {i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )}{2 d^{2}}-\frac {b \ln \left (x^{n}\right )}{2 d \,x^{2}}-\frac {b e n \ln \relax (x ) \ln \left (e \,x^{2}+d \right )}{2 d^{2}}+\frac {b e n \ln \relax (x ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}-\frac {b \ln \relax (c )}{2 d \,x^{2}}+\frac {a e \ln \left (e \,x^{2}+d \right )}{2 d^{2}}-\frac {a}{2 d \,x^{2}}+\frac {b e n \ln \relax (x ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}-\frac {a e \ln \relax (x )}{d^{2}}-\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 d \,x^{2}}+\frac {i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (x )}{2 d^{2}}-\frac {i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{4 d^{2}}-\frac {b n}{4 d \,x^{2}}-\frac {i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e \,x^{2}+d \right )}{4 d^{2}}-\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 d \,x^{2}} \]
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 \[ \frac {1}{2} \, a {\left (\frac {e \log \left (e x^{2} + d\right )}{d^{2}} - \frac {2 \, e \log \relax (x)}{d^{2}} - \frac {1}{d x^{2}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e x^{5} + d x^{3}}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,\left (e\,x^2+d\right )} \,d x \]
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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