Optimal. Leaf size=25 \[ \text {Int}\left (\sqrt {d+e x^2} \left (a+b \sin ^{-1}(c x)\right )^2,x\right ) \]
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Rubi [A] time = 0.04, 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 = {} \[ \int \sqrt {d+e x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \sqrt {d+e x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int \sqrt {d+e x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ \end {align*}
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Mathematica [A] time = 18.09, size = 0, normalized size = 0.00 \[ \int \sqrt {d+e x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )} \sqrt {e x^{2} + d}, x\right ) \]
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 \[ \int \sqrt {e x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.70, size = 0, normalized size = 0.00 \[ \int \sqrt {e \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]
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 \[ \frac {1}{2} \, {\left (\sqrt {e x^{2} + d} x + \frac {d \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {e}}\right )} a^{2} + \int {\left (b^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {e x^{2} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {e\,x^2+d} \,d x \]
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 \[ \int \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \sqrt {d + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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