Optimal. Leaf size=93 \[ \frac{1}{2} g p \text{PolyLog}\left (2,\frac{e x^2}{d}+1\right )-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac{1}{2} g \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )-\frac{e f p \log \left (d+e x^2\right )}{2 d}+\frac{e f p \log (x)}{d} \]
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Rubi [A] time = 0.126835, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {2475, 43, 2416, 2395, 36, 29, 31, 2394, 2315} \[ \frac{1}{2} g p \text{PolyLog}\left (2,\frac{e x^2}{d}+1\right )-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac{1}{2} g \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )-\frac{e f p \log \left (d+e x^2\right )}{2 d}+\frac{e f p \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 2475
Rule 43
Rule 2416
Rule 2395
Rule 36
Rule 29
Rule 31
Rule 2394
Rule 2315
Rubi steps
\begin{align*} \int \frac{\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(f+g x) \log \left (c (d+e x)^p\right )}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{f \log \left (c (d+e x)^p\right )}{x^2}+\frac{g \log \left (c (d+e x)^p\right )}{x}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2} f \operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{x^2} \, dx,x,x^2\right )+\frac{1}{2} g \operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )\\ &=-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac{1}{2} g \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} (e f p) \operatorname{Subst}\left (\int \frac{1}{x (d+e x)} \, dx,x,x^2\right )-\frac{1}{2} (e g p) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx,x,x^2\right )\\ &=-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac{1}{2} g \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} g p \text{Li}_2\left (1+\frac{e x^2}{d}\right )+\frac{(e f p) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 d}-\frac{\left (e^2 f p\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x} \, dx,x,x^2\right )}{2 d}\\ &=\frac{e f p \log (x)}{d}-\frac{e f p \log \left (d+e x^2\right )}{2 d}-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac{1}{2} g \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} g p \text{Li}_2\left (1+\frac{e x^2}{d}\right )\\ \end{align*}
Mathematica [A] time = 0.0276093, size = 92, normalized size = 0.99 \[ \frac{1}{2} g \left (p \text{PolyLog}\left (2,\frac{d+e x^2}{d}\right )+\log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )\right )-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}-\frac{e f p \log \left (d+e x^2\right )}{2 d}+\frac{e f p \log (x)}{d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.499, size = 421, normalized size = 4.5 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{3}}\,{d x} \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{{\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (f + g x^{2}\right ) \log{\left (c \left (d + e x^{2}\right )^{p} \right )}}{x^{3}}\, dx \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{{\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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