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 = 0.05, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2295
Rule 2296
Rule 2389
Rule 2454
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.48, size = 96, normalized size = 1.57 \[ \frac {2 \, b p^{2} x^{2} - 2 \, b p x^{2} \log \relax (c) + b x^{2} \log \relax (c)^{2} + {\left (b p^{2} x^{2} + a p^{2}\right )} \log \left (b x^{2} + a\right )^{2} - 2 \, {\left (b p^{2} x^{2} + a p^{2} - {\left (b p x^{2} + a p\right )} \log \relax (c)\right )} \log \left (b x^{2} + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 96, normalized size = 1.57 \[ \frac {{\left (2 \, b x^{2} + {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right )^{2} - 2 \, {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + 2 \, a\right )} p^{2} - 2 \, {\left (b x^{2} - {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + a\right )} p \log \relax (c) + {\left (b x^{2} + a\right )} \log \relax (c)^{2}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.53, size = 1034, normalized size = 16.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 97, normalized size = 1.59 \[ -b p {\left (\frac {x^{2}}{b} - \frac {a \log \left (b x^{2} + a\right )}{b^{2}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) + \frac {1}{2} \, x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} + \frac {{\left (2 \, b x^{2} - a \log \left (b x^{2} + a\right )^{2} - 2 \, a \log \left (b x^{2} + a\right )\right )} p^{2}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 70, normalized size = 1.15 \[ p^2\,x^2+{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2\,\left (\frac {a}{2\,b}+\frac {x^2}{2}\right )-p\,x^2\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )-\frac {a\,p^2\,\ln \left (b\,x^2+a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.13, size = 139, normalized size = 2.28 \[ \begin {cases} \frac {a p^{2} \log {\left (a + b x^{2} \right )}^{2}}{2 b} - \frac {a p^{2} \log {\left (a + b x^{2} \right )}}{b} + \frac {a p \log {\relax (c )} \log {\left (a + b x^{2} \right )}}{b} + \frac {p^{2} x^{2} \log {\left (a + b x^{2} \right )}^{2}}{2} - p^{2} x^{2} \log {\left (a + b x^{2} \right )} + p^{2} x^{2} + p x^{2} \log {\relax (c )} \log {\left (a + b x^{2} \right )} - p x^{2} \log {\relax (c )} + \frac {x^{2} \log {\relax (c )}^{2}}{2} & \text {for}\: b \neq 0 \\\frac {x^{2} \log {\left (a^{p} c \right )}^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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