Optimal. Leaf size=95 \[ \frac {\log \left (\frac {e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {b n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{4 e^2}+\frac {b n \log \left (d+e x^2\right )}{4 e^2} \]
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Rubi [A] time = 0.19, antiderivative size = 95, normalized size of antiderivative = 1.00, 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, 43, 2351, 2335, 260, 2337, 2391} \[ \frac {b n \text {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e^2}+\frac {\log \left (\frac {e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {b n \log \left (d+e x^2\right )}{4 e^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 260
Rule 266
Rule 2335
Rule 2337
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac {d x \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{e}-\frac {d \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx}{e}\\ &=-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^2}-\frac {(b n) \int \frac {\log \left (1+\frac {e x^2}{d}\right )}{x} \, dx}{2 e^2}+\frac {(b n) \int \frac {x}{d+e x^2} \, dx}{2 e}\\ &=-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {b n \log \left (d+e x^2\right )}{4 e^2}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^2}+\frac {b n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{4 e^2}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 321, normalized size = 3.38 \[ \frac {2 \log \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+\frac {2 d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{d+e x^2}+\frac {b n \left (2 \left (d+e x^2\right ) \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 \left (d+e x^2\right ) \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )+e x^2 \log \left (-\sqrt {e} x+i \sqrt {d}\right )+e x^2 \log \left (\sqrt {e} x+i \sqrt {d}\right )+2 e x^2 \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 e x^2 \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+d \log \left (-\sqrt {e} x+i \sqrt {d}\right )+d \log \left (\sqrt {e} x+i \sqrt {d}\right )+2 d \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 d \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )-2 e x^2 \log (x)\right )}{d+e x^2}}{4 e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{3} \log \left (c x^{n}\right ) + a x^{3}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 511, normalized size = 5.38 \[ -\frac {i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4 \left (e \,x^{2}+d \right ) e^{2}}+\frac {i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 \left (e \,x^{2}+d \right ) e^{2}}+\frac {i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 \left (e \,x^{2}+d \right ) e^{2}}-\frac {i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4 \left (e \,x^{2}+d \right ) e^{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 ) \ln \left (e \,x^{2}+d \right )}{4 e^{2}}+\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^{2}}+\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^{2}}-\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e \,x^{2}+d \right )}{4 e^{2}}+\frac {b n \ln \relax (x ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}+\frac {b n \ln \relax (x ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}-\frac {b n \ln \relax (x ) \ln \left (e \,x^{2}+d \right )}{2 e^{2}}+\frac {b d \ln \relax (c )}{2 \left (e \,x^{2}+d \right ) e^{2}}+\frac {b d \ln \left (x^{n}\right )}{2 \left (e \,x^{2}+d \right ) e^{2}}+\frac {b n \dilog \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}+\frac {b n \dilog \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}-\frac {b n \ln \relax (x )}{2 e^{2}}+\frac {b n \ln \left (e \,x^{2}+d \right )}{4 e^{2}}+\frac {b \ln \relax (c ) \ln \left (e \,x^{2}+d \right )}{2 e^{2}}+\frac {b \ln \left (x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{2 e^{2}}+\frac {a d}{2 \left (e \,x^{2}+d \right ) e^{2}}+\frac {a \ln \left (e \,x^{2}+d \right )}{2 e^{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 \[ \frac {1}{2} \, a {\left (\frac {d}{e^{3} x^{2} + d e^{2}} + \frac {\log \left (e x^{2} + d\right )}{e^{2}}\right )} + b \int \frac {x^{3} \log \relax (c) + x^{3} \log \left (x^{n}\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \]
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 \[ \int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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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