Optimal. Leaf size=61 \[ \frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {2 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{9 c^{3/2}}+\frac {2 b x \sqrt {1-c^2 x^4}}{9 c} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, 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 = {4842, 12, 321, 221} \[ \frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^2\right )\right )+\frac {2 b x \sqrt {1-c^2 x^4}}{9 c}-\frac {2 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{9 c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 221
Rule 321
Rule 4842
Rubi steps
\begin {align*} \int x^2 \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{3} b \int \frac {2 c x^4}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{3} (2 b c) \int \frac {x^4}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {2 b x \sqrt {1-c^2 x^4}}{9 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {(2 b) \int \frac {1}{\sqrt {1-c^2 x^4}} \, dx}{9 c}\\ &=\frac {2 b x \sqrt {1-c^2 x^4}}{9 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {2 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{9 c^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 72, normalized size = 1.18 \[ \frac {1}{9} \left (3 a x^3+\frac {2 b x \sqrt {1-c^2 x^4}}{c}+3 b x^3 \sin ^{-1}\left (c x^2\right )-\frac {2 i b F\left (\left .i \sinh ^{-1}\left (\sqrt {-c} x\right )\right |-1\right )}{(-c)^{3/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b x^{2} \arcsin \left (c x^{2}\right ) + a x^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arcsin \left (c x^{2}\right ) + a\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 88, normalized size = 1.44 \[ \frac {x^{3} a}{3}+b \left (\frac {x^{3} \arcsin \left (c \,x^{2}\right )}{3}-\frac {2 c \left (-\frac {x \sqrt {-c^{2} x^{4}+1}}{3 c^{2}}+\frac {\sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \EllipticF \left (x \sqrt {c}, i\right )}{3 c^{\frac {5}{2}} \sqrt {-c^{2} x^{4}+1}}\right )}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a x^{3} + \frac {1}{3} \, {\left (x^{3} \arctan \left (c x^{2}, \sqrt {c x^{2} + 1} \sqrt {-c x^{2} + 1}\right ) + 6 \, c \int \frac {x^{4} e^{\left (\frac {1}{2} \, \log \left (c x^{2} + 1\right ) + \frac {1}{2} \, \log \left (-c x^{2} + 1\right )\right )}}{3 \, {\left (c^{4} x^{8} - c^{2} x^{4} + {\left (c^{2} x^{4} - 1\right )} e^{\left (\log \left (c x^{2} + 1\right ) + \log \left (-c x^{2} + 1\right )\right )}\right )}}\,{d x}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,\left (a+b\,\mathrm {asin}\left (c\,x^2\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.78, size = 58, normalized size = 0.95 \[ \frac {a x^{3}}{3} - \frac {b c x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {c^{2} x^{4} e^{2 i \pi }} \right )}}{6 \Gamma \left (\frac {9}{4}\right )} + \frac {b x^{3} \operatorname {asin}{\left (c x^{2} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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