Optimal. Leaf size=43 \[ \frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{b \tan ^{-1}\left (c x^3\right )}{6 c^2}-\frac{b x^3}{6 c} \]
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Rubi [A] time = 0.029365, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5033, 275, 321, 203} \[ \frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{b \tan ^{-1}\left (c x^3\right )}{6 c^2}-\frac{b x^3}{6 c} \]
Antiderivative was successfully verified.
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Rule 5033
Rule 275
Rule 321
Rule 203
Rubi steps
\begin{align*} \int x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{1}{2} (b c) \int \frac{x^8}{1+c^2 x^6} \, dx\\ &=\frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{x^2}{1+c^2 x^2} \, dx,x,x^3\right )\\ &=-\frac{b x^3}{6 c}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{b \operatorname{Subst}\left (\int \frac{1}{1+c^2 x^2} \, dx,x,x^3\right )}{6 c}\\ &=-\frac{b x^3}{6 c}+\frac{b \tan ^{-1}\left (c x^3\right )}{6 c^2}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0052992, size = 48, normalized size = 1.12 \[ \frac{a x^6}{6}+\frac{b \tan ^{-1}\left (c x^3\right )}{6 c^2}-\frac{b x^3}{6 c}+\frac{1}{6} b x^6 \tan ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 41, normalized size = 1. \begin{align*}{\frac{{x}^{6}a}{6}}+{\frac{b{x}^{6}\arctan \left ( c{x}^{3} \right ) }{6}}-{\frac{b{x}^{3}}{6\,c}}+{\frac{b\arctan \left ( c{x}^{3} \right ) }{6\,{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48283, size = 58, normalized size = 1.35 \begin{align*} \frac{1}{6} \, a x^{6} + \frac{1}{6} \,{\left (x^{6} \arctan \left (c x^{3}\right ) - c{\left (\frac{x^{3}}{c^{2}} - \frac{\arctan \left (c x^{3}\right )}{c^{3}}\right )}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.58688, size = 85, normalized size = 1.98 \begin{align*} \frac{a c^{2} x^{6} - b c x^{3} +{\left (b c^{2} x^{6} + b\right )} \arctan \left (c x^{3}\right )}{6 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 166.108, size = 48, normalized size = 1.12 \begin{align*} \begin{cases} \frac{a x^{6}}{6} + \frac{b x^{6} \operatorname{atan}{\left (c x^{3} \right )}}{6} - \frac{b x^{3}}{6 c} + \frac{b \operatorname{atan}{\left (c x^{3} \right )}}{6 c^{2}} & \text{for}\: c \neq 0 \\\frac{a x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10493, size = 58, normalized size = 1.35 \begin{align*} \frac{a c x^{6} + \frac{{\left (c^{2} x^{6} \arctan \left (c x^{3}\right ) - c x^{3} + \arctan \left (c x^{3}\right )\right )} b}{c}}{6 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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