Optimal. Leaf size=62 \[ \frac{1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{b \left (1-c^2 x^4\right )^{3/2}}{18 c^3}+\frac{b \sqrt{1-c^2 x^4}}{6 c^3} \]
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Rubi [A] time = 0.0479316, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4842, 12, 266, 43} \[ \frac{1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{b \left (1-c^2 x^4\right )^{3/2}}{18 c^3}+\frac{b \sqrt{1-c^2 x^4}}{6 c^3} \]
Antiderivative was successfully verified.
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Rule 4842
Rule 12
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{1}{6} b \int \frac{2 c x^7}{\sqrt{1-c^2 x^4}} \, dx\\ &=\frac{1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{1}{3} (b c) \int \frac{x^7}{\sqrt{1-c^2 x^4}} \, dx\\ &=\frac{1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-c^2 x}} \, dx,x,x^4\right )\\ &=\frac{1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{c^2 \sqrt{1-c^2 x}}-\frac{\sqrt{1-c^2 x}}{c^2}\right ) \, dx,x,x^4\right )\\ &=\frac{b \sqrt{1-c^2 x^4}}{6 c^3}-\frac{b \left (1-c^2 x^4\right )^{3/2}}{18 c^3}+\frac{1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )\\ \end{align*}
Mathematica [A] time = 0.039535, size = 70, normalized size = 1.13 \[ \frac{a x^6}{6}+\frac{b x^4 \sqrt{1-c^2 x^4}}{18 c}+\frac{b \sqrt{1-c^2 x^4}}{9 c^3}+\frac{1}{6} b x^6 \sin ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 62, normalized size = 1. \begin{align*}{\frac{{x}^{6}a}{6}}+b \left ({\frac{{x}^{6}\arcsin \left ( c{x}^{2} \right ) }{6}}-{\frac{ \left ( c{x}^{2}-1 \right ) \left ( c{x}^{2}+1 \right ) \left ({c}^{2}{x}^{4}+2 \right ) }{18\,{c}^{3}}{\frac{1}{\sqrt{-{c}^{2}{x}^{4}+1}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4572, size = 80, normalized size = 1.29 \begin{align*} \frac{1}{6} \, a x^{6} + \frac{1}{18} \,{\left (3 \, x^{6} \arcsin \left (c x^{2}\right ) - c{\left (\frac{{\left (-c^{2} x^{4} + 1\right )}^{\frac{3}{2}}}{c^{4}} - \frac{3 \, \sqrt{-c^{2} x^{4} + 1}}{c^{4}}\right )}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.39229, size = 123, normalized size = 1.98 \begin{align*} \frac{3 \, b c^{3} x^{6} \arcsin \left (c x^{2}\right ) + 3 \, a c^{3} x^{6} +{\left (b c^{2} x^{4} + 2 \, b\right )} \sqrt{-c^{2} x^{4} + 1}}{18 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.02234, size = 65, normalized size = 1.05 \begin{align*} \begin{cases} \frac{a x^{6}}{6} + \frac{b x^{6} \operatorname{asin}{\left (c x^{2} \right )}}{6} + \frac{b x^{4} \sqrt{- c^{2} x^{4} + 1}}{18 c} + \frac{b \sqrt{- c^{2} x^{4} + 1}}{9 c^{3}} & \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 = 1.15378, size = 117, normalized size = 1.89 \begin{align*} \frac{3 \, a c x^{6} +{\left (\frac{3 \,{\left (c^{2} x^{4} - 1\right )} x^{2} \arcsin \left (c x^{2}\right )}{c} + \frac{3 \, x^{2} \arcsin \left (c x^{2}\right )}{c} - \frac{{\left (-c^{2} x^{4} + 1\right )}^{\frac{3}{2}}}{c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{4} + 1}}{c^{2}}\right )} b}{18 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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