Optimal. Leaf size=84 \[ \frac{1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )+\frac{b \left (1-c^2 x^4\right )^{5/2}}{50 c^5}-\frac{b \left (1-c^2 x^4\right )^{3/2}}{15 c^5}+\frac{b \sqrt{1-c^2 x^4}}{10 c^5} \]
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Rubi [A] time = 0.064852, antiderivative size = 84, 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} \[ \frac{1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )+\frac{b \left (1-c^2 x^4\right )^{5/2}}{50 c^5}-\frac{b \left (1-c^2 x^4\right )^{3/2}}{15 c^5}+\frac{b \sqrt{1-c^2 x^4}}{10 c^5} \]
Antiderivative was successfully verified.
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Rule 4842
Rule 12
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^9 \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{1}{10} b \int \frac{2 c x^{11}}{\sqrt{1-c^2 x^4}} \, dx\\ &=\frac{1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{1}{5} (b c) \int \frac{x^{11}}{\sqrt{1-c^2 x^4}} \, dx\\ &=\frac{1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{1}{20} (b c) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-c^2 x}} \, dx,x,x^4\right )\\ &=\frac{1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{1}{20} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{c^4 \sqrt{1-c^2 x}}-\frac{2 \sqrt{1-c^2 x}}{c^4}+\frac{\left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^4\right )\\ &=\frac{b \sqrt{1-c^2 x^4}}{10 c^5}-\frac{b \left (1-c^2 x^4\right )^{3/2}}{15 c^5}+\frac{b \left (1-c^2 x^4\right )^{5/2}}{50 c^5}+\frac{1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0550731, size = 60, normalized size = 0.71 \[ \frac{1}{150} \left (15 a x^{10}+\frac{b \sqrt{1-c^2 x^4} \left (3 c^4 x^8+4 c^2 x^4+8\right )}{c^5}+15 b x^{10} \sin ^{-1}\left (c x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 71, normalized size = 0.9 \begin{align*}{\frac{{x}^{10}a}{10}}+b \left ({\frac{{x}^{10}\arcsin \left ( c{x}^{2} \right ) }{10}}-{\frac{ \left ( c{x}^{2}-1 \right ) \left ( c{x}^{2}+1 \right ) \left ( 3\,{c}^{4}{x}^{8}+4\,{c}^{2}{x}^{4}+8 \right ) }{150\,{c}^{5}}{\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 = 1.60451, size = 103, normalized size = 1.23 \begin{align*} \frac{1}{10} \, a x^{10} + \frac{1}{150} \,{\left (15 \, x^{10} \arcsin \left (c x^{2}\right ) + c{\left (\frac{3 \,{\left (-c^{2} x^{4} + 1\right )}^{\frac{5}{2}}}{c^{6}} - \frac{10 \,{\left (-c^{2} x^{4} + 1\right )}^{\frac{3}{2}}}{c^{6}} + \frac{15 \, \sqrt{-c^{2} x^{4} + 1}}{c^{6}}\right )}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33201, size = 151, normalized size = 1.8 \begin{align*} \frac{15 \, b c^{5} x^{10} \arcsin \left (c x^{2}\right ) + 15 \, a c^{5} x^{10} +{\left (3 \, b c^{4} x^{8} + 4 \, b c^{2} x^{4} + 8 \, b\right )} \sqrt{-c^{2} x^{4} + 1}}{150 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 31.1882, size = 90, normalized size = 1.07 \begin{align*} \begin{cases} \frac{a x^{10}}{10} + \frac{b x^{10} \operatorname{asin}{\left (c x^{2} \right )}}{10} + \frac{b x^{8} \sqrt{- c^{2} x^{4} + 1}}{50 c} + \frac{2 b x^{4} \sqrt{- c^{2} x^{4} + 1}}{75 c^{3}} + \frac{4 b \sqrt{- c^{2} x^{4} + 1}}{75 c^{5}} & \text{for}\: c \neq 0 \\\frac{a x^{10}}{10} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22419, size = 189, normalized size = 2.25 \begin{align*} \frac{15 \, a c x^{10} +{\left (\frac{15 \,{\left (c^{2} x^{4} - 1\right )}^{2} x^{2} \arcsin \left (c x^{2}\right )}{c^{3}} + \frac{30 \,{\left (c^{2} x^{4} - 1\right )} x^{2} \arcsin \left (c x^{2}\right )}{c^{3}} + \frac{15 \, x^{2} \arcsin \left (c x^{2}\right )}{c^{3}} + \frac{3 \,{\left (c^{2} x^{4} - 1\right )}^{2} \sqrt{-c^{2} x^{4} + 1}}{c^{4}} - \frac{10 \,{\left (-c^{2} x^{4} + 1\right )}^{\frac{3}{2}}}{c^{4}} + \frac{15 \, \sqrt{-c^{2} x^{4} + 1}}{c^{4}}\right )} b}{150 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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