Optimal. Leaf size=287 \[ -\frac{2 b \sqrt{1-c} (c+1) (3 c+1) \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{1-c}}\right ),-\frac{1-c}{c+1}\right )}{9 d^{3/2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )+\frac{2 b x \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{9 d}+\frac{8 b \sqrt{1-c} c (c+1) \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{1-c}}\right )|-\frac{1-c}{c+1}\right )}{9 d^{3/2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}} \]
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Rubi [A] time = 0.288101, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {4842, 12, 1122, 1202, 524, 424, 419} \[ \frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )+\frac{2 b x \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{9 d}-\frac{2 b \sqrt{1-c} (c+1) (3 c+1) \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{1-c}}\right )|-\frac{1-c}{c+1}\right )}{9 d^{3/2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}+\frac{8 b \sqrt{1-c} c (c+1) \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{1-c}}\right )|-\frac{1-c}{c+1}\right )}{9 d^{3/2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}} \]
Antiderivative was successfully verified.
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Rule 4842
Rule 12
Rule 1122
Rule 1202
Rule 524
Rule 424
Rule 419
Rubi steps
\begin{align*} \int x^2 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac{1}{3} b \int \frac{2 d x^4}{\sqrt{1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac{1}{3} (2 b d) \int \frac{x^4}{\sqrt{1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=\frac{2 b x \sqrt{1-c^2-2 c d x^2-d^2 x^4}}{9 d}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac{(2 b) \int \frac{1-c^2-4 c d x^2}{\sqrt{1-c^2-2 c d x^2-d^2 x^4}} \, dx}{9 d}\\ &=\frac{2 b x \sqrt{1-c^2-2 c d x^2-d^2 x^4}}{9 d}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac{\left (2 b \sqrt{1-\frac{2 d^2 x^2}{-2 d-2 c d}} \sqrt{1-\frac{2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac{1-c^2-4 c d x^2}{\sqrt{1-\frac{2 d^2 x^2}{-2 d-2 c d}} \sqrt{1-\frac{2 d^2 x^2}{2 d-2 c d}}} \, dx}{9 d \sqrt{1-c^2-2 c d x^2-d^2 x^4}}\\ &=\frac{2 b x \sqrt{1-c^2-2 c d x^2-d^2 x^4}}{9 d}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )+\frac{\left (8 b c (1+c) \sqrt{1-\frac{2 d^2 x^2}{-2 d-2 c d}} \sqrt{1-\frac{2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac{\sqrt{1-\frac{2 d^2 x^2}{-2 d-2 c d}}}{\sqrt{1-\frac{2 d^2 x^2}{2 d-2 c d}}} \, dx}{9 d \sqrt{1-c^2-2 c d x^2-d^2 x^4}}-\frac{\left (2 b (1+c) (1+3 c) \sqrt{1-\frac{2 d^2 x^2}{-2 d-2 c d}} \sqrt{1-\frac{2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac{1}{\sqrt{1-\frac{2 d^2 x^2}{-2 d-2 c d}} \sqrt{1-\frac{2 d^2 x^2}{2 d-2 c d}}} \, dx}{9 d \sqrt{1-c^2-2 c d x^2-d^2 x^4}}\\ &=\frac{2 b x \sqrt{1-c^2-2 c d x^2-d^2 x^4}}{9 d}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )+\frac{8 b \sqrt{1-c} c (1+c) \sqrt{1-\frac{d x^2}{1-c}} \sqrt{1+\frac{d x^2}{1+c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{1-c}}\right )|-\frac{1-c}{1+c}\right )}{9 d^{3/2} \sqrt{1-c^2-2 c d x^2-d^2 x^4}}-\frac{2 b \sqrt{1-c} (1+c) (1+3 c) \sqrt{1-\frac{d x^2}{1-c}} \sqrt{1+\frac{d x^2}{1+c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{1-c}}\right )|-\frac{1-c}{1+c}\right )}{9 d^{3/2} \sqrt{1-c^2-2 c d x^2-d^2 x^4}}\\ \end{align*}
Mathematica [F] time = 0.573444, size = 0, normalized size = 0. \[ \int x^2 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.01, size = 295, normalized size = 1. \begin{align*}{\frac{{x}^{3}a}{3}}+b \left ({\frac{{x}^{3}\arcsin \left ( d{x}^{2}+c \right ) }{3}}-{\frac{2\,d}{3} \left ( -{\frac{x}{3\,{d}^{2}}\sqrt{-{d}^{2}{x}^{4}-2\,cd{x}^{2}-{c}^{2}+1}}+{\frac{-{c}^{2}+1}{3\,{d}^{2}}\sqrt{1+{\frac{d{x}^{2}}{-1+c}}}\sqrt{1+{\frac{d{x}^{2}}{1+c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{-1+c}}},\sqrt{-1+2\,{\frac{c}{1+c}}} \right ){\frac{1}{\sqrt{-{\frac{d}{-1+c}}}}}{\frac{1}{\sqrt{-{d}^{2}{x}^{4}-2\,cd{x}^{2}-{c}^{2}+1}}}}+{\frac{8\,c \left ( -{c}^{2}+1 \right ) }{3\,d \left ( -2\,dc+2\,d \right ) }\sqrt{1+{\frac{d{x}^{2}}{-1+c}}}\sqrt{1+{\frac{d{x}^{2}}{1+c}}} \left ({\it EllipticF} \left ( x\sqrt{-{\frac{d}{-1+c}}},\sqrt{-1+2\,{\frac{c}{1+c}}} \right ) -{\it EllipticE} \left ( x\sqrt{-{\frac{d}{-1+c}}},\sqrt{-1+2\,{\frac{c}{1+c}}} \right ) \right ){\frac{1}{\sqrt{-{\frac{d}{-1+c}}}}}{\frac{1}{\sqrt{-{d}^{2}{x}^{4}-2\,cd{x}^{2}-{c}^{2}+1}}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b x^{2} \arcsin \left (d x^{2} + c\right ) + a x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \operatorname{asin}{\left (c + d x^{2} \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x^{2} + c\right ) + a\right )} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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