Optimal. Leaf size=176 \[ \frac{g p \text{PolyLog}\left (2,-\frac{g \left (d+e x^2\right )}{e f-d g}\right )}{2 f^2}-\frac{g p \text{PolyLog}\left (2,\frac{e x^2}{d}+1\right )}{2 f^2}-\frac{g \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac{g \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{2 f x^2}-\frac{e p \log \left (d+e x^2\right )}{2 d f}+\frac{e p \log (x)}{d f} \]
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Rubi [A] time = 0.26662, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {2475, 44, 2416, 2395, 36, 29, 31, 2394, 2315, 2393, 2391} \[ \frac{g p \text{PolyLog}\left (2,-\frac{g \left (d+e x^2\right )}{e f-d g}\right )}{2 f^2}-\frac{g p \text{PolyLog}\left (2,\frac{e x^2}{d}+1\right )}{2 f^2}-\frac{g \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac{g \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{2 f x^2}-\frac{e p \log \left (d+e x^2\right )}{2 d f}+\frac{e p \log (x)}{d f} \]
Antiderivative was successfully verified.
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Rule 2475
Rule 44
Rule 2416
Rule 2395
Rule 36
Rule 29
Rule 31
Rule 2394
Rule 2315
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{x^3 \left (f+g x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{x^2 (f+g x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{\log \left (c (d+e x)^p\right )}{f x^2}-\frac{g \log \left (c (d+e x)^p\right )}{f^2 x}+\frac{g^2 \log \left (c (d+e x)^p\right )}{f^2 (f+g x)}\right ) \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{x^2} \, dx,x,x^2\right )}{2 f}-\frac{g \operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )}{2 f^2}+\frac{g^2 \operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )}{2 f^2}\\ &=-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{2 f x^2}-\frac{g \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac{g \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}+\frac{(e p) \operatorname{Subst}\left (\int \frac{1}{x (d+e x)} \, dx,x,x^2\right )}{2 f}+\frac{(e g p) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx,x,x^2\right )}{2 f^2}-\frac{(e g p) \operatorname{Subst}\left (\int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx,x,x^2\right )}{2 f^2}\\ &=-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{2 f x^2}-\frac{g \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac{g \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}-\frac{g p \text{Li}_2\left (1+\frac{e x^2}{d}\right )}{2 f^2}+\frac{(e p) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 d f}-\frac{\left (e^2 p\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x} \, dx,x,x^2\right )}{2 d f}-\frac{(g p) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x^2\right )}{2 f^2}\\ &=\frac{e p \log (x)}{d f}-\frac{e p \log \left (d+e x^2\right )}{2 d f}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{2 f x^2}-\frac{g \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac{g \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}+\frac{g p \text{Li}_2\left (-\frac{g \left (d+e x^2\right )}{e f-d g}\right )}{2 f^2}-\frac{g p \text{Li}_2\left (1+\frac{e x^2}{d}\right )}{2 f^2}\\ \end{align*}
Mathematica [A] time = 0.0727305, size = 147, normalized size = 0.84 \[ \frac{-g \left (p \text{PolyLog}\left (2,\frac{e x^2}{d}+1\right )+\log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )\right )+g p \text{PolyLog}\left (2,\frac{g \left (d+e x^2\right )}{d g-e f}\right )+g \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+\frac{e f p \left (2 \log (x)-\log \left (d+e x^2\right )\right )}{d}}{2 f^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.686, size = 942, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.92295, size = 240, normalized size = 1.36 \begin{align*} \frac{1}{2} \, e p{\left (\frac{{\left (2 \, \log \left (\frac{e x^{2}}{d} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{e x^{2}}{d}\right )\right )} g}{e f^{2}} - \frac{{\left (\log \left (g x^{2} + f\right ) \log \left (-\frac{e g x^{2} + e f}{e f - d g} + 1\right ) +{\rm Li}_2\left (\frac{e g x^{2} + e f}{e f - d g}\right )\right )} g}{e f^{2}} - \frac{\log \left (e x^{2} + d\right )}{d f} + \frac{2 \, \log \left (x\right )}{d f}\right )} + \frac{1}{2} \,{\left (\frac{g \log \left (g x^{2} + f\right )}{f^{2}} - \frac{g \log \left (x^{2}\right )}{f^{2}} - \frac{1}{f x^{2}}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{5} + f x^{3}}, x\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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