Optimal. Leaf size=55 \[ \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {c d x+d} \sqrt {e-c e x}} \]
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Rubi [A] time = 0.23, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4673, 4641} \[ \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {c d x+d} \sqrt {e-c e x}} \]
Antiderivative was successfully verified.
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Rule 4641
Rule 4673
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}\\ \end {align*}
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Mathematica [B] time = 0.68, size = 159, normalized size = 2.89 \[ \frac {-\frac {3 a^2 \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (c^2 x^2-1\right )}\right )}{\sqrt {d} \sqrt {e}}+\frac {3 a b \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{\sqrt {c d x+d} \sqrt {e-c e x}}+\frac {b^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^3}{\sqrt {c d x+d} \sqrt {e-c e x}}}{3 c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )} \sqrt {c d x + d} \sqrt {-c e x + e}}{c^{2} d e x^{2} - d e}, x\right ) \]
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 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {c d x + d} \sqrt {-c e x + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {c d x +d}\, \sqrt {-c e x +e}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 53, normalized size = 0.96 \[ \frac {b^{2} \arcsin \left (c x\right )^{3}}{3 \, \sqrt {d e} c} + \frac {a b \arcsin \left (c x\right )^{2}}{\sqrt {d e} c} + \frac {a^{2} \arcsin \left (c x\right )}{\sqrt {d e} c} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d+c\,d\,x}\,\sqrt {e-c\,e\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c x + 1\right )} \sqrt {- e \left (c x - 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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