Optimal. Leaf size=61 \[ \frac{\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b}-\frac{p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}+p^2 x^2 \]
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Rubi [A] time = 0.0503193, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2454, 2389, 2296, 2295} \[ \frac{\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b}-\frac{p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}+p^2 x^2 \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2389
Rule 2296
Rule 2295
Rubi steps
\begin{align*} \int x \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \log ^2\left (c (a+b x)^p\right ) \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b}\\ &=\frac{\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b}-\frac{p \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{b}\\ &=p^2 x^2-\frac{p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}+\frac{\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.008556, size = 63, normalized size = 1.03 \[ \frac{1}{2} \left (\frac{\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-2 p \left (\frac{\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}-p x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.497, size = 1034, normalized size = 17. \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 = 1.03056, size = 131, normalized size = 2.15 \begin{align*} -b p{\left (\frac{x^{2}}{b} - \frac{a \log \left (b x^{2} + a\right )}{b^{2}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) + \frac{1}{2} \, x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} + \frac{{\left (2 \, b x^{2} - a \log \left (b x^{2} + a\right )^{2} - 2 \, a \log \left (b x^{2} + a\right )\right )} p^{2}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97339, size = 216, normalized size = 3.54 \begin{align*} \frac{2 \, b p^{2} x^{2} - 2 \, b p x^{2} \log \left (c\right ) + b x^{2} \log \left (c\right )^{2} +{\left (b p^{2} x^{2} + a p^{2}\right )} \log \left (b x^{2} + a\right )^{2} - 2 \,{\left (b p^{2} x^{2} + a p^{2} -{\left (b p x^{2} + a p\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.58321, size = 139, normalized size = 2.28 \begin{align*} \begin{cases} \frac{a p^{2} \log{\left (a + b x^{2} \right )}^{2}}{2 b} - \frac{a p^{2} \log{\left (a + b x^{2} \right )}}{b} + \frac{a p \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{b} + \frac{p^{2} x^{2} \log{\left (a + b x^{2} \right )}^{2}}{2} - p^{2} x^{2} \log{\left (a + b x^{2} \right )} + p^{2} x^{2} + p x^{2} \log{\left (c \right )} \log{\left (a + b x^{2} \right )} - p x^{2} \log{\left (c \right )} + \frac{x^{2} \log{\left (c \right )}^{2}}{2} & \text{for}\: b \neq 0 \\\frac{x^{2} \log{\left (a^{p} c \right )}^{2}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28606, size = 130, normalized size = 2.13 \begin{align*} \frac{{\left (2 \, b x^{2} +{\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right )^{2} - 2 \,{\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + 2 \, a\right )} p^{2} - 2 \,{\left (b x^{2} -{\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + a\right )} p \log \left (c\right ) +{\left (b x^{2} + a\right )} \log \left (c\right )^{2}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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