Optimal. Leaf size=69 \[ \frac {2 \text {Int}\left (\frac {x}{\left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b c}-\frac {x^2}{b c \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.20, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac {x^2}{b c \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}+\frac {2 \int \frac {x}{\left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ \end {align*}
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Mathematica [A] time = 8.57, size = 0, normalized size = 0.00 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{a^{2} c^{4} x^{4} - 2 \, a^{2} c^{2} x^{2} + {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arcsin \left (c x\right )^{2} + a^{2} + 2 \, {\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \arcsin \left (c x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.67, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {x^{2} + \frac {2 \, {\left (a b c^{3} x^{2} - a b c + {\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \int \frac {x}{{\left (c x + 1\right )}^{2} {\left (c x - 1\right )}^{2} {\left (b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a\right )}}\,{d x}}{b c}}{a b c^{3} x^{2} - a b c + {\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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