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 = 0.079546, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, 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 2335
Rule 266
Rule 43
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*}
Mathematica [A] time = 0.144152, size = 129, normalized size = 1.9 \[ -\frac{2 a d^2+4 a d e x^2+2 b d \left (d+2 e x^2\right ) \log \left (c x^n\right )+b d^2 n \log \left (d+e x^2\right )+b d^2 n+b e^2 n x^4 \log \left (d+e x^2\right )+b d e n x^2+2 b d e n x^2 \log \left (d+e x^2\right )-2 b n \log (x) \left (d+e x^2\right )^2}{8 d e^2 \left (d+e x^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.117, size = 369, normalized size = 5.4 \begin{align*} -{\frac{b \left ( 2\,e{x}^{2}+d \right ) \ln \left ({x}^{n} \right ) }{4\, \left ( e{x}^{2}+d \right ) ^{2}{e}^{2}}}-{\frac{-2\,i\pi \,bde{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,\ln \left ( x \right ) b{e}^{2}n{x}^{4}+\ln \left ( e{x}^{2}+d \right ) b{e}^{2}n{x}^{4}+2\,i\pi \,bde{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,i\pi \,bde{x}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +2\,i\pi \,bde{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-4\,\ln \left ( x \right ) bden{x}^{2}+2\,\ln \left ( e{x}^{2}+d \right ) bden{x}^{2}+4\,\ln \left ( c \right ) bde{x}^{2}+bden{x}^{2}-2\,\ln \left ( x \right ) b{d}^{2}n+\ln \left ( e{x}^{2}+d \right ) b{d}^{2}n+4\,ade{x}^{2}+2\,\ln \left ( c \right ) b{d}^{2}+b{d}^{2}n+2\,a{d}^{2}}{8\,d{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19274, size = 173, normalized size = 2.54 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.34009, size = 273, normalized size = 4.01 \begin{align*} \frac{2 \, b e^{2} n x^{4} \log \left (x\right ) - 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 \left (c\right )}{8 \,{\left (d e^{4} x^{4} + 2 \, d^{2} e^{3} x^{2} + d^{3} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.33218, size = 189, normalized size = 2.78 \begin{align*} -\frac{b n x^{4} e^{2} \log \left (x^{2} e + d\right ) - 2 \, b n x^{4} e^{2} \log \left (x\right ) + 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 \left (c\right ) + 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 \left (c\right ) + 2 \, a d^{2}}{8 \,{\left (d x^{4} e^{4} + 2 \, d^{2} x^{2} e^{3} + d^{3} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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