Optimal. Leaf size=59 \[ -\frac{a^2 p \log \left (a+b x^3\right )}{6 b^2}+\frac{1}{6} x^6 \log \left (c \left (a+b x^3\right )^p\right )+\frac{a p x^3}{6 b}-\frac{p x^6}{12} \]
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Rubi [A] time = 0.0494319, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2454, 2395, 43} \[ -\frac{a^2 p \log \left (a+b x^3\right )}{6 b^2}+\frac{1}{6} x^6 \log \left (c \left (a+b x^3\right )^p\right )+\frac{a p x^3}{6 b}-\frac{p x^6}{12} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2395
Rule 43
Rubi steps
\begin{align*} \int x^5 \log \left (c \left (a+b x^3\right )^p\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x \log \left (c (a+b x)^p\right ) \, dx,x,x^3\right )\\ &=\frac{1}{6} x^6 \log \left (c \left (a+b x^3\right )^p\right )-\frac{1}{6} (b p) \operatorname{Subst}\left (\int \frac{x^2}{a+b x} \, dx,x,x^3\right )\\ &=\frac{1}{6} x^6 \log \left (c \left (a+b x^3\right )^p\right )-\frac{1}{6} (b p) \operatorname{Subst}\left (\int \left (-\frac{a}{b^2}+\frac{x}{b}+\frac{a^2}{b^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{a p x^3}{6 b}-\frac{p x^6}{12}-\frac{a^2 p \log \left (a+b x^3\right )}{6 b^2}+\frac{1}{6} x^6 \log \left (c \left (a+b x^3\right )^p\right )\\ \end{align*}
Mathematica [A] time = 0.0126688, size = 59, normalized size = 1. \[ -\frac{a^2 p \log \left (a+b x^3\right )}{6 b^2}+\frac{1}{6} x^6 \log \left (c \left (a+b x^3\right )^p\right )+\frac{a p x^3}{6 b}-\frac{p x^6}{12} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.556, size = 183, normalized size = 3.1 \begin{align*}{\frac{{x}^{6}\ln \left ( \left ( b{x}^{3}+a \right ) ^{p} \right ) }{6}}+{\frac{i}{12}}\pi \,{x}^{6}{\it csgn} \left ( i \left ( b{x}^{3}+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}-{\frac{i}{12}}\pi \,{x}^{6}{\it csgn} \left ( i \left ( b{x}^{3}+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{12}}\pi \,{x}^{6} \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{3}+{\frac{i}{12}}\pi \,{x}^{6} \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ){x}^{6}}{6}}-{\frac{p{x}^{6}}{12}}+{\frac{ap{x}^{3}}{6\,b}}-{\frac{{a}^{2}p\ln \left ( b{x}^{3}+a \right ) }{6\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17267, size = 74, normalized size = 1.25 \begin{align*} \frac{1}{6} \, x^{6} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) - \frac{1}{12} \, b p{\left (\frac{2 \, a^{2} \log \left (b x^{3} + a\right )}{b^{3}} + \frac{b x^{6} - 2 \, a x^{3}}{b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8724, size = 128, normalized size = 2.17 \begin{align*} -\frac{b^{2} p x^{6} - 2 \, b^{2} x^{6} \log \left (c\right ) - 2 \, a b p x^{3} - 2 \,{\left (b^{2} p x^{6} - a^{2} p\right )} \log \left (b x^{3} + a\right )}{12 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 34.7721, size = 70, normalized size = 1.19 \begin{align*} \begin{cases} - \frac{a^{2} p \log{\left (a + b x^{3} \right )}}{6 b^{2}} + \frac{a p x^{3}}{6 b} + \frac{p x^{6} \log{\left (a + b x^{3} \right )}}{6} - \frac{p x^{6}}{12} + \frac{x^{6} \log{\left (c \right )}}{6} & \text{for}\: b \neq 0 \\\frac{x^{6} \log{\left (a^{p} c \right )}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1641, size = 131, normalized size = 2.22 \begin{align*} \frac{\frac{{\left (2 \,{\left (b x^{3} + a\right )}^{2} \log \left (b x^{3} + a\right ) - 4 \,{\left (b x^{3} + a\right )} a \log \left (b x^{3} + a\right ) -{\left (b x^{3} + a\right )}^{2} + 4 \,{\left (b x^{3} + a\right )} a\right )} p}{b} + \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{2} - 2 \,{\left (b x^{3} + a\right )} a\right )} \log \left (c\right )}{b}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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