Optimal. Leaf size=250 \[ -\frac {x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {b^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {b^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt {c d x+d} \sqrt {e-c e x}} \]
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Rubi [A] time = 0.58, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4739, 4707, 4641, 4627, 321, 216} \[ \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {b^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {b^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt {c d x+d} \sqrt {e-c e x}} \]
Antiderivative was successfully verified.
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Rule 216
Rule 321
Rule 4627
Rule 4641
Rule 4707
Rule 4739
Rubi steps
\begin {align*} \int \frac {x^2 \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^2 \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 {x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\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}{2 c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \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}{6 b c^3 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {b^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \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}{6 b c^3 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c^2 \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {b^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {b^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \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}{6 b c^3 \sqrt {d+c d x} \sqrt {e-c e x}}\\ \end {align*}
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Mathematica [A] time = 1.31, size = 326, normalized size = 1.30 \[ \frac {-3 \sqrt {d} \sqrt {e} \left (a^2 \left (4 c x-4 c^3 x^3\right )+a b \sqrt {1-c^2 x^2}+a b \cos \left (3 \sin ^{-1}(c x)\right )+2 b^2 c x \left (c^2 x^2-1\right )\right )-12 a^2 \sqrt {c d x+d} \sqrt {e-c e x} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (c^2 x^2-1\right )}\right )+12 b \sqrt {d} \sqrt {e} \sin ^{-1}(c x)^2 \left (a \sqrt {1-c^2 x^2}+b c x \left (c^2 x^2-1\right )\right )-3 b \sqrt {d} \sqrt {e} \sin ^{-1}(c x) \left (2 a c x+2 a \sin \left (3 \sin ^{-1}(c x)\right )+b \sqrt {1-c^2 x^2}+b \cos \left (3 \sin ^{-1}(c x)\right )\right )+4 b^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^3}{24 c^3 \sqrt {d} \sqrt {e} \sqrt {c d x+d} \sqrt {e-c e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{2} x^{2} \arcsin \left (c x\right )^{2} + 2 \, a b x^{2} \arcsin \left (c x\right ) + a^{2} x^{2}\right )} \sqrt {c d x + d} \sqrt {-c e x + e}}{c^{2} d e x^{2} - d e}, x\right ) \]
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 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}{\sqrt {c d x + d} \sqrt {-c e x + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.67, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {c d x +d}\, \sqrt {-c e x +e}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, a^{2} {\left (\frac {\sqrt {-c^{2} d e x^{2} + d e} x}{c^{2} d e} - \frac {\arcsin \left (c x\right )}{\sqrt {d e} c^{3}}\right )} - \sqrt {d} \sqrt {e} \int \frac {{\left (b^{2} x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{2} d e x^{2} - d e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d+c\,d\,x}\,\sqrt {e-c\,e\,x}} \,d x \]
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 \[ \int \frac {x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c x + 1\right )} \sqrt {- e \left (c x - 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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