Optimal. Leaf size=35 \[ -\frac {p x^3}{3}+\frac {\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b} \]
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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 = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2436,
2332} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2436
Rule 2504
Rubi steps
\begin {align*} \int x^2 \log \left (c \left (a+b x^3\right )^p\right ) \, dx &=\frac {1}{3} \text {Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,x^3\right )\\ &=\frac {\text {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 \begin {gather*} \frac {1}{3} \left (-p x^3+\frac {\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 37, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {\ln \left (c \left (x^{3} b +a \right )^{p}\right ) \left (x^{3} b +a \right )-\left (x^{3} b +a \right ) p}{3 b}\) | \(37\) |
default | \(\frac {\ln \left (c \left (x^{3} b +a \right )^{p}\right ) \left (x^{3} b +a \right )-\left (x^{3} b +a \right ) p}{3 b}\) | \(37\) |
risch | \(\frac {x^{3} \ln \left (\left (x^{3} b +a \right )^{p}\right )}{3}-\frac {i \pi \,x^{3} \mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{6}+\frac {i \pi \,x^{3} \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{6}+\frac {i \pi \,x^{3} \mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2}}{6}-\frac {i \pi \,x^{3} \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{3}}{6}+\frac {\ln \left (c \right ) x^{3}}{3}-\frac {p \,x^{3}}{3}+\frac {a p \ln \left (x^{3} b +a \right )}{3 b}\) | \(171\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 44, normalized size = 1.26 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 40, normalized size = 1.14 \begin {gather*} -\frac {b p x^{3} - b x^{3} \log \left (c\right ) - {\left (b p x^{3} + a p\right )} \log \left (b x^{3} + a\right )}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.83, size = 51, normalized size = 1.46 \begin {gather*} \begin {cases} \frac {a \log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{3 b} - \frac {p x^{3}}{3} + \frac {x^{3} \log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{3} & \text {for}\: b \neq 0 \\\frac {x^{3} \log {\left (a^{p} c \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.81, size = 43, normalized size = 1.23 \begin {gather*} -\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 \left (c\right )}{3 \, 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^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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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