Optimal. Leaf size=35 \[ \frac {\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b}-\frac {p x^3}{3} \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2454, 2389, 2295} \[ \frac {\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b}-\frac {p x^3}{3} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rule 2454
Rubi steps
\begin {align*} \int x^2 \log \left (c \left (a+b x^3\right )^p\right ) \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,x^3\right )\\ &=\frac {\operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^3\right )}{3 b}\\ &=-\frac {p x^3}{3}+\frac {\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 0.97 \[ \frac {1}{3} \left (\frac {\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{b}-p x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 40, normalized size = 1.14 \[ -\frac {b p x^{3} - b x^{3} \log \relax (c) - {\left (b p x^{3} + a p\right )} \log \left (b x^{3} + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 43, normalized size = 1.23 \[ -\frac {{\left (b x^{3} - {\left (b x^{3} + a\right )} \log \left (b x^{3} + a\right ) + a\right )} p - {\left (b x^{3} + a\right )} \log \relax (c)}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 50, normalized size = 1.43 \[ -\frac {p \,x^{3}}{3}+\frac {x^{3} \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{3}-\frac {a p}{3 b}+\frac {a \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 44, normalized size = 1.26 \[ \frac {1}{3} \, x^{3} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) - \frac {1}{3} \, {\left (\frac {x^{3}}{b} - \frac {a \log \left (b x^{3} + a\right )}{b^{2}}\right )} b p \]
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 \[ \frac {x^3\,\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{3}-\frac {p\,x^3}{3}+\frac {a\,p\,\ln \left (b\,x^3+a\right )}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.70, size = 56, normalized size = 1.60 \[ \begin {cases} \frac {a p \log {\left (a + b x^{3} \right )}}{3 b} + \frac {p x^{3} \log {\left (a + b x^{3} \right )}}{3} - \frac {p x^{3}}{3} + \frac {x^{3} \log {\relax (c )}}{3} & \text {for}\: b \neq 0 \\\frac {x^{3} \log {\left (a^{p} c \right )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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