Optimal. Leaf size=35 \[ -\frac {p x^2}{2}+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b} \]
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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 = {2504, 2436,
2332} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2436
Rule 2504
Rubi steps
\begin {align*} \int x \log \left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,x^2\right )\\ &=\frac {\text {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 \begin {gather*} \frac {1}{2} \left (-p x^2+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 37, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) \left (b \,x^{2}+a \right )-\left (b \,x^{2}+a \right ) p}{2 b}\) | \(37\) |
default | \(\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) \left (b \,x^{2}+a \right )-\left (b \,x^{2}+a \right ) p}{2 b}\) | \(37\) |
norman | \(-\frac {p \,x^{2}}{2}+\frac {x^{2} \ln \left (c \,{\mathrm e}^{p \ln \left (b \,x^{2}+a \right )}\right )}{2}+\frac {p a \ln \left (b \,x^{2}+a \right )}{2 b}\) | \(42\) |
risch | \(\frac {x^{2} \ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{2}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{4}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{4}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{4}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{4}+\frac {\ln \left (c \right ) x^{2}}{2}-\frac {p \,x^{2}}{2}+\frac {p a \ln \left (b \,x^{2}+a \right )}{2 b}\) | \(171\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 44, normalized size = 1.26 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 40, normalized size = 1.14 \begin {gather*} -\frac {b p x^{2} - b x^{2} \log \left (c\right ) - {\left (b p x^{2} + a p\right )} \log \left (b x^{2} + a\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 51, normalized size = 1.46 \begin {gather*} \begin {cases} \frac {a \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{2 b} - \frac {p x^{2}}{2} + \frac {x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{2} & \text {for}\: b \neq 0 \\\frac {x^{2} \log {\left (a^{p} c \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.78, size = 43, normalized size = 1.23 \begin {gather*} -\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 \left (c\right )}{2 \, b} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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