Optimal. Leaf size=37 \[ \frac {1}{2} x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {b p \log \left (a x^2+b\right )}{2 a} \]
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Rubi [A] time = 0.01, antiderivative size = 37, 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 = {2455, 263, 260} \[ \frac {1}{2} x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {b p \log \left (a x^2+b\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 260
Rule 263
Rule 2455
Rubi steps
\begin {align*} \int x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx &=\frac {1}{2} x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+(b p) \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x} \, dx\\ &=\frac {1}{2} x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+(b p) \int \frac {x}{b+a x^2} \, dx\\ &=\frac {1}{2} x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {b p \log \left (b+a x^2\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 45, normalized size = 1.22 \[ \frac {1}{2} x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {b p \log \left (a+\frac {b}{x^2}\right )}{2 a}+\frac {b p \log (x)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 42, normalized size = 1.14 \[ \frac {a p x^{2} \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + a x^{2} \log \relax (c) + b p \log \left (a x^{2} + b\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 47, normalized size = 1.27 \[ \frac {1}{2} \, p x^{2} \log \left (a x^{2} + b\right ) - \frac {1}{2} \, p x^{2} \log \left (x^{2}\right ) + \frac {1}{2} \, x^{2} \log \relax (c) + \frac {b p \log \left (a x^{2} + b\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int x \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 33, normalized size = 0.89 \[ \frac {1}{2} \, x^{2} \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) + \frac {b p \log \left (a x^{2} + b\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 33, normalized size = 0.89 \[ \frac {x^2\,\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{2}+\frac {b\,p\,\ln \left (a\,x^2+b\right )}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.35, size = 71, normalized size = 1.92 \[ \begin {cases} \frac {p x^{2} \log {\left (a + \frac {b}{x^{2}} \right )}}{2} + \frac {x^{2} \log {\relax (c )}}{2} + \frac {b p \log {\left (a x^{2} + b \right )}}{2 a} & \text {for}\: a \neq 0 \\\frac {p x^{2} \log {\relax (b )}}{2} - p x^{2} \log {\relax (x )} + \frac {p x^{2}}{2} + \frac {x^{2} \log {\relax (c )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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