Optimal. Leaf size=177 \[ \frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt {c d x+d} \sqrt {e-c e x}} \]
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Rubi [A] time = 0.38, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4739, 4677, 4619, 261} \[ \frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt {c d x+d} \sqrt {e-c e x}} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4619
Rule 4677
Rule 4739
Rubi steps
\begin {align*} \int \frac {x \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 {x \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 {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 150, normalized size = 0.85 \[ -\frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (a^2 \left (c^2 x^2-1\right )+2 a b c x \sqrt {1-c^2 x^2}+2 b \sin ^{-1}(c x) \left (a \left (c^2 x^2-1\right )+b c x \sqrt {1-c^2 x^2}\right )-2 b^2 \left (c^2 x^2-1\right )+b^2 \left (c^2 x^2-1\right ) \sin ^{-1}(c x)^2\right )}{c^2 d e (c x-1) (c x+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 138, normalized size = 0.78 \[ -\frac {{\left ({\left (a^{2} - 2 \, b^{2}\right )} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} - a^{2} + 2 \, b^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right ) + 2 \, {\left (b^{2} c x \arcsin \left (c x\right ) + a b c x\right )} \sqrt {-c^{2} x^{2} + 1}\right )} \sqrt {c d x + d} \sqrt {-c e x + e}}{c^{4} d e x^{2} - c^{2} d e} \]
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 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x}{\sqrt {c d x + d} \sqrt {-c e x + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.80, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {c d x +d}\, \sqrt {-c e x +e}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 157, normalized size = 0.89 \[ 2 \, b^{2} {\left (\frac {x \arcsin \left (c x\right )}{c \sqrt {d} \sqrt {e}} + \frac {\sqrt {-c^{2} x^{2} + 1}}{c^{2} \sqrt {d} \sqrt {e}}\right )} + \frac {2 \, a b x}{c \sqrt {d} \sqrt {e}} - \frac {\sqrt {-c^{2} d e x^{2} + d e} b^{2} \arcsin \left (c x\right )^{2}}{c^{2} d e} - \frac {2 \, \sqrt {-c^{2} d e x^{2} + d e} a b \arcsin \left (c x\right )}{c^{2} d e} - \frac {\sqrt {-c^{2} d e x^{2} + d e} a^{2}}{c^{2} d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d+c\,d\,x}\,\sqrt {e-c\,e\,x}} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c x + 1\right )} \sqrt {- e \left (c x - 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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