Optimal. Leaf size=68 \[ \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n}{8 e^2 \left (d+e x^2\right )}-\frac {b n \log \left (d+e x^2\right )}{8 d e^2} \]
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Rubi [A] time = 0.08, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2335, 266, 43} \[ \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n}{8 e^2 \left (d+e x^2\right )}-\frac {b n \log \left (d+e x^2\right )}{8 d e^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2335
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx &=\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b n) \int \frac {x^3}{\left (d+e x^2\right )^2} \, dx}{4 d}\\ &=\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b n) \operatorname {Subst}\left (\int \frac {x}{(d+e x)^2} \, dx,x,x^2\right )}{8 d}\\ &=\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b n) \operatorname {Subst}\left (\int \left (-\frac {d}{e (d+e x)^2}+\frac {1}{e (d+e x)}\right ) \, dx,x,x^2\right )}{8 d}\\ &=-\frac {b n}{8 e^2 \left (d+e x^2\right )}+\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n \log \left (d+e x^2\right )}{8 d e^2}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 129, normalized size = 1.90 \[ -\frac {2 a d^2+4 a d e x^2+2 b d \left (d+2 e x^2\right ) \log \left (c x^n\right )+b d^2 n \log \left (d+e x^2\right )+b d^2 n+b e^2 n x^4 \log \left (d+e x^2\right )+b d e n x^2+2 b d e n x^2 \log \left (d+e x^2\right )-2 b n \log (x) \left (d+e x^2\right )^2}{8 d e^2 \left (d+e x^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 126, normalized size = 1.85 \[ \frac {2 \, b e^{2} n x^{4} \log \relax (x) - b d^{2} n - 2 \, a d^{2} - {\left (b d e n + 4 \, a d e\right )} x^{2} - {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \log \left (e x^{2} + d\right ) - 2 \, {\left (2 \, b d e x^{2} + b d^{2}\right )} \log \relax (c)}{8 \, {\left (d e^{4} x^{4} + 2 \, d^{2} e^{3} x^{2} + d^{3} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 140, normalized size = 2.06 \[ -\frac {b n x^{4} e^{2} \log \left (x^{2} e + d\right ) - 2 \, b n x^{4} e^{2} \log \relax (x) + 2 \, b d n x^{2} e \log \left (x^{2} e + d\right ) + b d n x^{2} e + 4 \, b d x^{2} e \log \relax (c) + 4 \, a d x^{2} e + b d^{2} n \log \left (x^{2} e + d\right ) + b d^{2} n + 2 \, b d^{2} \log \relax (c) + 2 \, a d^{2}}{8 \, {\left (d x^{4} e^{4} + 2 \, d^{2} x^{2} e^{3} + d^{3} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.24, size = 369, normalized size = 5.43 \[ -\frac {\left (2 e \,x^{2}+d \right ) b \ln \left (x^{n}\right )}{4 \left (e \,x^{2}+d \right )^{2} e^{2}}-\frac {-2 b \,e^{2} n \,x^{4} \ln \relax (x )+b \,e^{2} n \,x^{4} \ln \left (e \,x^{2}+d \right )-2 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+2 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi b d e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 b d e n \,x^{2} \ln \relax (x )+2 b d e n \,x^{2} \ln \left (e \,x^{2}+d \right )-i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+b d e n \,x^{2}+4 b d e \,x^{2} \ln \relax (c )+4 a d e \,x^{2}-2 b \,d^{2} n \ln \relax (x )+b \,d^{2} n \ln \left (e \,x^{2}+d \right )+b \,d^{2} n +2 b \,d^{2} \ln \relax (c )+2 a \,d^{2}}{8 \left (e \,x^{2}+d \right )^{2} d \,e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 128, normalized size = 1.88 \[ -\frac {1}{8} \, b n {\left (\frac {1}{e^{3} x^{2} + d e^{2}} + \frac {\log \left (e x^{2} + d\right )}{d e^{2}} - \frac {\log \left (x^{2}\right )}{d e^{2}}\right )} - \frac {{\left (2 \, e x^{2} + d\right )} b \log \left (c x^{n}\right )}{4 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )}} - \frac {{\left (2 \, e x^{2} + d\right )} a}{4 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.74, size = 129, normalized size = 1.90 \[ \frac {b\,n\,\ln \relax (x)}{4\,d\,e^2}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,x^2}{2\,e}+\frac {b\,d}{4\,e^2}\right )}{d^2+2\,d\,e\,x^2+e^2\,x^4}-\frac {b\,n\,\ln \left (e\,x^2+d\right )}{8\,d\,e^2}-\frac {\left (2\,a\,e+\frac {b\,e\,n}{2}\right )\,x^2+a\,d+\frac {b\,d\,n}{2}}{4\,d^2\,e^2+8\,d\,e^3\,x^2+4\,e^4\,x^4} \]
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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