Optimal. Leaf size=49 \[ \frac {\log \left (\frac {e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {b n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{4 e} \]
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Rubi [A] time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2337, 2391} \[ \frac {b n \text {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e}+\frac {\log \left (\frac {e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e} \]
Antiderivative was successfully verified.
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Rule 2337
Rule 2391
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e}-\frac {(b n) \int \frac {\log \left (1+\frac {e x^2}{d}\right )}{x} \, dx}{2 e}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e}+\frac {b n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{4 e}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 94, normalized size = 1.92 \[ \frac {\left (\log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )+\log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right )\right ) \left (a+b \log \left (c x^n\right )\right )+b n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )+b n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{2 e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x \log \left (c x^{n}\right ) + a x}{e x^{2} + d}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{e x^{2} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 299, normalized size = 6.10 \[ -\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{4 e}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e \,x^{2}+d \right )}{4 e}+\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e \,x^{2}+d \right )}{4 e}-\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e \,x^{2}+d \right )}{4 e}+\frac {b n \ln \relax (x ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e}+\frac {b n \ln \relax (x ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e}-\frac {b n \ln \relax (x ) \ln \left (e \,x^{2}+d \right )}{2 e}+\frac {b n \dilog \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e}+\frac {b n \dilog \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e}+\frac {b \ln \relax (c ) \ln \left (e \,x^{2}+d \right )}{2 e}+\frac {b \ln \left (x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{2 e}+\frac {a \ln \left (e \,x^{2}+d \right )}{2 e} \]
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 \[ b \int \frac {x \log \relax (c) + x \log \left (x^{n}\right )}{e x^{2} + d}\,{d x} + \frac {a \log \left (e x^{2} + d\right )}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{e\,x^2+d} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.83, size = 119, normalized size = 2.43 \[ \frac {a \log {\left (d + e x^{2} \right )}}{2 e} - \frac {b n \left (\begin {cases} \log {\relax (d )} \log {\relax (x )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (d )} \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (d )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (d )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {otherwise} \end {cases}\right )}{2 e} + \frac {b \log {\left (c x^{n} \right )} \log {\left (d + e x^{2} \right )}}{2 e} \]
Verification of antiderivative is not currently implemented for this CAS.
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