Optimal. Leaf size=50 \[ \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac{b n \log \left (d+e x^2\right )}{4 d e} \]
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Rubi [A] time = 0.041939, antiderivative size = 50, normalized size of antiderivative = 1., 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 = {2335, 260} \[ \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac{b n \log \left (d+e x^2\right )}{4 d e} \]
Antiderivative was successfully verified.
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Rule 2335
Rule 260
Rubi steps
\begin{align*} \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx &=\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac{(b n) \int \frac{x}{d+e x^2} \, dx}{2 d}\\ &=\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac{b n \log \left (d+e x^2\right )}{4 d e}\\ \end{align*}
Mathematica [A] time = 0.0648603, size = 74, normalized size = 1.48 \[ -\frac{2 a d+2 b d \log \left (c x^n\right )+b e n x^2 \log \left (d+e x^2\right )-2 b n \log (x) \left (d+e x^2\right )+b d n \log \left (d+e x^2\right )}{4 d e \left (d+e x^2\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.096, size = 179, normalized size = 3.6 \begin{align*} -{\frac{b\ln \left ({x}^{n} \right ) }{2\, \left ( e{x}^{2}+d \right ) e}}-{\frac{i\pi \,bd{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,bd{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -i\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,\ln \left ( x \right ) ben{x}^{2}+\ln \left ( e{x}^{2}+d \right ) ben{x}^{2}-2\,\ln \left ( x \right ) bdn+\ln \left ( e{x}^{2}+d \right ) bdn+2\,\ln \left ( c \right ) bd+2\,ad}{ \left ( 4\,e{x}^{2}+4\,d \right ) ed}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19873, size = 96, normalized size = 1.92 \begin{align*} -\frac{1}{4} \, b n{\left (\frac{\log \left (e x^{2} + d\right )}{d e} - \frac{\log \left (x^{2}\right )}{d e}\right )} - \frac{b \log \left (c x^{n}\right )}{2 \,{\left (e^{2} x^{2} + d e\right )}} - \frac{a}{2 \,{\left (e^{2} x^{2} + d e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34862, size = 143, normalized size = 2.86 \begin{align*} \frac{2 \, b e n x^{2} \log \left (x\right ) - 2 \, b d \log \left (c\right ) - 2 \, a d -{\left (b e n x^{2} + b d n\right )} \log \left (e x^{2} + d\right )}{4 \,{\left (d e^{2} x^{2} + d^{2} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 78.7155, size = 366, normalized size = 7.32 \begin{align*} \begin{cases} \tilde{\infty } \left (- \frac{a}{2 x^{2}} - \frac{b n \log{\left (x \right )}}{2 x^{2}} - \frac{b n}{4 x^{2}} - \frac{b \log{\left (c \right )}}{2 x^{2}}\right ) & \text{for}\: d = 0 \wedge e = 0 \\\frac{\frac{a x^{2}}{2} + \frac{b n x^{2} \log{\left (x \right )}}{2} - \frac{b n x^{2}}{4} + \frac{b x^{2} \log{\left (c \right )}}{2}}{d^{2}} & \text{for}\: e = 0 \\\frac{- \frac{a}{2 x^{2}} - \frac{b n \log{\left (x \right )}}{2 x^{2}} - \frac{b n}{4 x^{2}} - \frac{b \log{\left (c \right )}}{2 x^{2}}}{e^{2}} & \text{for}\: d = 0 \\- \frac{2 a d}{4 d^{2} e + 4 d e^{2} x^{2}} - \frac{b d n \log{\left (- i \sqrt{d} \sqrt{\frac{1}{e}} + x \right )}}{4 d^{2} e + 4 d e^{2} x^{2}} - \frac{b d n \log{\left (i \sqrt{d} \sqrt{\frac{1}{e}} + x \right )}}{4 d^{2} e + 4 d e^{2} x^{2}} + \frac{2 b e n x^{2} \log{\left (x \right )}}{4 d^{2} e + 4 d e^{2} x^{2}} - \frac{b e n x^{2} \log{\left (- i \sqrt{d} \sqrt{\frac{1}{e}} + x \right )}}{4 d^{2} e + 4 d e^{2} x^{2}} - \frac{b e n x^{2} \log{\left (i \sqrt{d} \sqrt{\frac{1}{e}} + x \right )}}{4 d^{2} e + 4 d e^{2} x^{2}} + \frac{2 b e x^{2} \log{\left (c \right )}}{4 d^{2} e + 4 d e^{2} x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33931, size = 95, normalized size = 1.9 \begin{align*} -\frac{b n x^{2} e \log \left (x^{2} e + d\right ) - 2 \, b n x^{2} e \log \left (x\right ) + b d n \log \left (x^{2} e + d\right ) + 2 \, b d \log \left (c\right ) + 2 \, a d}{4 \,{\left (d x^{2} e^{2} + d^{2} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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