Optimal. Leaf size=153 \[ \frac{1}{2} f^2 p \text{PolyLog}\left (2,\frac{e x^2}{d}+1\right )+\frac{1}{2} f^2 \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac{f g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac{1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )-\frac{d^2 g^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac{d g^2 p x^2}{4 e}-f g p x^2-\frac{1}{8} g^2 p x^4 \]
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Rubi [A] time = 0.202965, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {2475, 43, 2416, 2389, 2295, 2394, 2315, 2395} \[ \frac{1}{2} f^2 p \text{PolyLog}\left (2,\frac{e x^2}{d}+1\right )+\frac{1}{2} f^2 \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac{f g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac{1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )-\frac{d^2 g^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac{d g^2 p x^2}{4 e}-f g p x^2-\frac{1}{8} g^2 p x^4 \]
Antiderivative was successfully verified.
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Rule 2475
Rule 43
Rule 2416
Rule 2389
Rule 2295
Rule 2394
Rule 2315
Rule 2395
Rubi steps
\begin{align*} \int \frac{\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(f+g x)^2 \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (2 f g \log \left (c (d+e x)^p\right )+\frac{f^2 \log \left (c (d+e x)^p\right )}{x}+g^2 x \log \left (c (d+e x)^p\right )\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2} f^2 \operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )+(f g) \operatorname{Subst}\left (\int \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )+\frac{1}{2} g^2 \operatorname{Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )\\ &=\frac{1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f^2 \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac{(f g) \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{e}-\frac{1}{2} \left (e f^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx,x,x^2\right )-\frac{1}{4} \left (e g^2 p\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+e x} \, dx,x,x^2\right )\\ &=-f g p x^2+\frac{1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{f g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac{1}{2} f^2 \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f^2 p \text{Li}_2\left (1+\frac{e x^2}{d}\right )-\frac{1}{4} \left (e g^2 p\right ) \operatorname{Subst}\left (\int \left (-\frac{d}{e^2}+\frac{x}{e}+\frac{d^2}{e^2 (d+e x)}\right ) \, dx,x,x^2\right )\\ &=-f g p x^2+\frac{d g^2 p x^2}{4 e}-\frac{1}{8} g^2 p x^4-\frac{d^2 g^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac{1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{f g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac{1}{2} f^2 \log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f^2 p \text{Li}_2\left (1+\frac{e x^2}{d}\right )\\ \end{align*}
Mathematica [A] time = 0.0815272, size = 121, normalized size = 0.79 \[ \frac{4 e^2 f^2 p \text{PolyLog}\left (2,\frac{e x^2}{d}+1\right )+2 e \log \left (c \left (d+e x^2\right )^p\right ) \left (2 e f^2 \log \left (-\frac{e x^2}{d}\right )+g \left (4 d f+4 e f x^2+e g x^4\right )\right )-2 d^2 g^2 p \log \left (d+e x^2\right )-e g p x^2 \left (-2 d g+8 e f+e g x^2\right )}{8 e^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.569, size = 652, normalized size = 4.3 \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 )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x}\,{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^{2} x^{4} + 2 \, f g x^{2} + f^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x}, 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 )^{2} \log{\left (c \left (d + e x^{2}\right )^{p} \right )}}{x}\, 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 )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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