Optimal. Leaf size=47 \[ \frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^2\right )\right )+\frac{b \log \left (c^2 x^4+1\right )}{12 c^3}-\frac{b x^4}{12 c} \]
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Rubi [A] time = 0.0308981, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5033, 266, 43} \[ \frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^2\right )\right )+\frac{b \log \left (c^2 x^4+1\right )}{12 c^3}-\frac{b x^4}{12 c} \]
Antiderivative was successfully verified.
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Rule 5033
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac{1}{3} (b c) \int \frac{x^7}{1+c^2 x^4} \, dx\\ &=\frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^4\right )\\ &=\frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^4\right )\\ &=-\frac{b x^4}{12 c}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^2\right )\right )+\frac{b \log \left (1+c^2 x^4\right )}{12 c^3}\\ \end{align*}
Mathematica [A] time = 0.0132921, size = 52, normalized size = 1.11 \[ \frac{a x^6}{6}+\frac{b \log \left (c^2 x^4+1\right )}{12 c^3}-\frac{b x^4}{12 c}+\frac{1}{6} b x^6 \tan ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 45, normalized size = 1. \begin{align*}{\frac{{x}^{6}a}{6}}+{\frac{b{x}^{6}\arctan \left ( c{x}^{2} \right ) }{6}}-{\frac{b{x}^{4}}{12\,c}}+{\frac{b\ln \left ({c}^{2}{x}^{4}+1 \right ) }{12\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983944, size = 65, normalized size = 1.38 \begin{align*} \frac{1}{6} \, a x^{6} + \frac{1}{12} \,{\left (2 \, x^{6} \arctan \left (c x^{2}\right ) -{\left (\frac{x^{4}}{c^{2}} - \frac{\log \left (c^{2} x^{4} + 1\right )}{c^{4}}\right )} c\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.58199, size = 115, normalized size = 2.45 \begin{align*} \frac{2 \, b c^{3} x^{6} \arctan \left (c x^{2}\right ) + 2 \, a c^{3} x^{6} - b c^{2} x^{4} + b \log \left (c^{2} x^{4} + 1\right )}{12 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 76.1698, size = 80, normalized size = 1.7 \begin{align*} \begin{cases} \frac{a x^{6}}{6} + \frac{b x^{6} \operatorname{atan}{\left (c x^{2} \right )}}{6} - \frac{b x^{4}}{12 c} + \frac{b \log{\left (x^{2} + i \sqrt{\frac{1}{c^{2}}} \right )}}{6 c^{3}} + \frac{i b \operatorname{atan}{\left (c x^{2} \right )}}{6 c^{8} \left (\frac{1}{c^{2}}\right )^{\frac{5}{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 = 1.12515, size = 63, normalized size = 1.34 \begin{align*} \frac{2 \, a c x^{6} +{\left (2 \, c x^{6} \arctan \left (c x^{2}\right ) - x^{4} + \frac{\log \left (c^{2} x^{4} + 1\right )}{c^{2}}\right )} b}{12 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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