Optimal. Leaf size=83 \[ -\frac{\log \left (c \left (d+e x^2\right )^p\right )}{2 g \left (f+g x^2\right )}+\frac{e p \log \left (d+e x^2\right )}{2 g (e f-d g)}-\frac{e p \log \left (f+g x^2\right )}{2 g (e f-d g)} \]
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Rubi [A] time = 0.074348, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2475, 2395, 36, 31} \[ -\frac{\log \left (c \left (d+e x^2\right )^p\right )}{2 g \left (f+g x^2\right )}+\frac{e p \log \left (d+e x^2\right )}{2 g (e f-d g)}-\frac{e p \log \left (f+g x^2\right )}{2 g (e f-d g)} \]
Antiderivative was successfully verified.
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Rule 2475
Rule 2395
Rule 36
Rule 31
Rubi steps
\begin{align*} \int \frac{x \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{(f+g x)^2} \, dx,x,x^2\right )\\ &=-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{2 g \left (f+g x^2\right )}+\frac{(e p) \operatorname{Subst}\left (\int \frac{1}{(d+e x) (f+g x)} \, dx,x,x^2\right )}{2 g}\\ &=-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{2 g \left (f+g x^2\right )}-\frac{(e p) \operatorname{Subst}\left (\int \frac{1}{f+g x} \, dx,x,x^2\right )}{2 (e f-d g)}+\frac{\left (e^2 p\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x} \, dx,x,x^2\right )}{2 g (e f-d g)}\\ &=\frac{e p \log \left (d+e x^2\right )}{2 g (e f-d g)}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{2 g \left (f+g x^2\right )}-\frac{e p \log \left (f+g x^2\right )}{2 g (e f-d g)}\\ \end{align*}
Mathematica [A] time = 0.051894, size = 63, normalized size = 0.76 \[ \frac{\frac{e p \left (\log \left (d+e x^2\right )-\log \left (f+g x^2\right )\right )}{e f-d g}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2}}{2 g} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.382, size = 371, normalized size = 4.5 \begin{align*} -{\frac{\ln \left ( \left ( e{x}^{2}+d \right ) ^{p} \right ) }{2\,g \left ( g{x}^{2}+f \right ) }}-{\frac{i\pi \,dg{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}-i\pi \,dg{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \right ) -i\pi \,dg \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}+i\pi \,dg \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \,ef{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}+i\pi \,ef{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \right ) +i\pi \,ef \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}-i\pi \,ef \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,\ln \left ( -e{x}^{2}-d \right ) egp{x}^{2}-2\,\ln \left ( g{x}^{2}+f \right ) egp{x}^{2}+2\,\ln \left ( -e{x}^{2}-d \right ) efp-2\,\ln \left ( g{x}^{2}+f \right ) efp+2\,\ln \left ( c \right ) dg-2\,\ln \left ( c \right ) ef}{4\,g \left ( g{x}^{2}+f \right ) \left ( dg-fe \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0145, size = 100, normalized size = 1.2 \begin{align*} \frac{e p{\left (\frac{\log \left (e x^{2} + d\right )}{e f - d g} - \frac{\log \left (g x^{2} + f\right )}{e f - d g}\right )}}{2 \, g} - \frac{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{2 \,{\left (g x^{2} + f\right )} g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06877, size = 194, normalized size = 2.34 \begin{align*} \frac{{\left (e g p x^{2} + d g p\right )} \log \left (e x^{2} + d\right ) -{\left (e g p x^{2} + e f p\right )} \log \left (g x^{2} + f\right ) -{\left (e f - d g\right )} \log \left (c\right )}{2 \,{\left (e f^{2} g - d f g^{2} +{\left (e f g^{2} - d g^{3}\right )} x^{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 = 1.16787, size = 246, normalized size = 2.96 \begin{align*} -\frac{{\left (x^{2} e + d\right )} g p e \log \left (x^{2} e + d\right ) -{\left (x^{2} e + d\right )} g p e \log \left ({\left (x^{2} e + d\right )} g - d g + f e\right ) + d g p e \log \left ({\left (x^{2} e + d\right )} g - d g + f e\right ) - f p e^{2} \log \left ({\left (x^{2} e + d\right )} g - d g + f e\right ) + d g e \log \left (c\right ) - f e^{2} \log \left (c\right )}{2 \,{\left ({\left (x^{2} e + d\right )} d g^{3} - d^{2} g^{3} -{\left (x^{2} e + d\right )} f g^{2} e + 2 \, d f g^{2} e - f^{2} g e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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