Optimal. Leaf size=78 \[ \frac {1}{18} \sqrt {1-x} x^{5/2}+\frac {5}{72} \sqrt {1-x} x^{3/2}+\frac {1}{3} x^3 \sin ^{-1}\left (\sqrt {x}\right )+\frac {5}{48} \sqrt {1-x} \sqrt {x}+\frac {5}{96} \sin ^{-1}(1-2 x) \]
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Rubi [A] time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4842, 12, 50, 53, 619, 216} \[ \frac {1}{18} \sqrt {1-x} x^{5/2}+\frac {5}{72} \sqrt {1-x} x^{3/2}+\frac {1}{3} x^3 \sin ^{-1}\left (\sqrt {x}\right )+\frac {5}{48} \sqrt {1-x} \sqrt {x}+\frac {5}{96} \sin ^{-1}(1-2 x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 53
Rule 216
Rule 619
Rule 4842
Rubi steps
\begin {align*} \int x^2 \sin ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{3} x^3 \sin ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \frac {x^{5/2}}{2 \sqrt {1-x}} \, dx\\ &=\frac {1}{3} x^3 \sin ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \int \frac {x^{5/2}}{\sqrt {1-x}} \, dx\\ &=\frac {1}{18} \sqrt {1-x} x^{5/2}+\frac {1}{3} x^3 \sin ^{-1}\left (\sqrt {x}\right )-\frac {5}{36} \int \frac {x^{3/2}}{\sqrt {1-x}} \, dx\\ &=\frac {5}{72} \sqrt {1-x} x^{3/2}+\frac {1}{18} \sqrt {1-x} x^{5/2}+\frac {1}{3} x^3 \sin ^{-1}\left (\sqrt {x}\right )-\frac {5}{48} \int \frac {\sqrt {x}}{\sqrt {1-x}} \, dx\\ &=\frac {5}{48} \sqrt {1-x} \sqrt {x}+\frac {5}{72} \sqrt {1-x} x^{3/2}+\frac {1}{18} \sqrt {1-x} x^{5/2}+\frac {1}{3} x^3 \sin ^{-1}\left (\sqrt {x}\right )-\frac {5}{96} \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx\\ &=\frac {5}{48} \sqrt {1-x} \sqrt {x}+\frac {5}{72} \sqrt {1-x} x^{3/2}+\frac {1}{18} \sqrt {1-x} x^{5/2}+\frac {1}{3} x^3 \sin ^{-1}\left (\sqrt {x}\right )-\frac {5}{96} \int \frac {1}{\sqrt {x-x^2}} \, dx\\ &=\frac {5}{48} \sqrt {1-x} \sqrt {x}+\frac {5}{72} \sqrt {1-x} x^{3/2}+\frac {1}{18} \sqrt {1-x} x^{5/2}+\frac {1}{3} x^3 \sin ^{-1}\left (\sqrt {x}\right )+\frac {5}{96} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,1-2 x\right )\\ &=\frac {5}{48} \sqrt {1-x} \sqrt {x}+\frac {5}{72} \sqrt {1-x} x^{3/2}+\frac {1}{18} \sqrt {1-x} x^{5/2}+\frac {5}{96} \sin ^{-1}(1-2 x)+\frac {1}{3} x^3 \sin ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 64, normalized size = 0.82 \[ \frac {1}{144} \left (8 \sqrt {1-x} x^{5/2}+10 \sqrt {1-x} x^{3/2}+3 \left (16 x^3-5\right ) \sin ^{-1}\left (\sqrt {x}\right )+15 \sqrt {-((x-1) x)}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 36, normalized size = 0.46 \[ \frac {1}{144} \, {\left (8 \, x^{2} + 10 \, x + 15\right )} \sqrt {x} \sqrt {-x + 1} + \frac {1}{48} \, {\left (16 \, x^{3} - 5\right )} \arcsin \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 77, normalized size = 0.99 \[ \frac {1}{3} \, {\left (x - 1\right )}^{3} \arcsin \left (\sqrt {x}\right ) + \frac {1}{18} \, {\left (x - 1\right )}^{2} \sqrt {x} \sqrt {-x + 1} + {\left (x - 1\right )}^{2} \arcsin \left (\sqrt {x}\right ) - \frac {13}{72} \, \sqrt {x} {\left (-x + 1\right )}^{\frac {3}{2}} + {\left (x - 1\right )} \arcsin \left (\sqrt {x}\right ) + \frac {11}{48} \, \sqrt {x} \sqrt {-x + 1} + \frac {11}{48} \, \arcsin \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 53, normalized size = 0.68 \[ \frac {x^{3} \arcsin \left (\sqrt {x}\right )}{3}+\frac {x^{\frac {5}{2}} \sqrt {1-x}}{18}+\frac {5 x^{\frac {3}{2}} \sqrt {1-x}}{72}+\frac {5 \sqrt {1-x}\, \sqrt {x}}{48}-\frac {5 \arcsin \left (\sqrt {x}\right )}{48} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 52, normalized size = 0.67 \[ \frac {1}{3} \, x^{3} \arcsin \left (\sqrt {x}\right ) + \frac {1}{18} \, x^{\frac {5}{2}} \sqrt {-x + 1} + \frac {5}{72} \, x^{\frac {3}{2}} \sqrt {-x + 1} + \frac {5}{48} \, \sqrt {x} \sqrt {-x + 1} - \frac {5}{48} \, \arcsin \left (\sqrt {x}\right ) \]
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 \[ \int x^2\,\mathrm {asin}\left (\sqrt {x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.98, size = 73, normalized size = 0.94 \[ \frac {x^{3} \operatorname {asin}{\left (\sqrt {x} \right )}}{3} - \frac {\begin {cases} \frac {x^{\frac {3}{2}} \left (1 - x\right )^{\frac {3}{2}}}{6} + \frac {3 \sqrt {x} \left (1 - 2 x\right ) \sqrt {1 - x}}{16} - \frac {\sqrt {x} \sqrt {1 - x}}{2} + \frac {5 \operatorname {asin}{\left (\sqrt {x} \right )}}{16} & \text {for}\: x \geq 0 \wedge x < 1 \end {cases}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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