Optimal. Leaf size=35 \[ \frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b}-\frac {p x^2}{2} \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2454, 2389, 2295} \[ \frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b}-\frac {p x^2}{2} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rule 2454
Rubi steps
\begin {align*} \int x \log \left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b}\\ &=-\frac {p x^2}{2}+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 0.97 \[ \frac {1}{2} \left (\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}-p x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 40, normalized size = 1.14 \[ -\frac {b p x^{2} - b x^{2} \log \relax (c) - {\left (b p x^{2} + a p\right )} \log \left (b x^{2} + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 43, normalized size = 1.23 \[ -\frac {{\left (b x^{2} - {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + a\right )} p - {\left (b x^{2} + a\right )} \log \relax (c)}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 50, normalized size = 1.43 \[ -\frac {p \,x^{2}}{2}+\frac {x^{2} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{2}-\frac {a p}{2 b}+\frac {a \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 44, normalized size = 1.26 \[ -\frac {1}{2} \, b p {\left (\frac {x^{2}}{b} - \frac {a \log \left (b x^{2} + a\right )}{b^{2}}\right )} + \frac {1}{2} \, x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 39, normalized size = 1.11 \[ \frac {x^2\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{2}-\frac {p\,x^2}{2}+\frac {a\,p\,\ln \left (b\,x^2+a\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.10, size = 56, normalized size = 1.60 \[ \begin {cases} \frac {a p \log {\left (a + b x^{2} \right )}}{2 b} + \frac {p x^{2} \log {\left (a + b x^{2} \right )}}{2} - \frac {p x^{2}}{2} + \frac {x^{2} \log {\relax (c )}}{2} & \text {for}\: b \neq 0 \\\frac {x^{2} \log {\left (a^{p} c \right )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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