Optimal. Leaf size=61 \[ -\frac {1}{3} \sqrt {1-x} (x+1)^{5/2}-\frac {1}{3} \sqrt {1-x} (x+1)^{3/2}-\sqrt {1-x} \sqrt {x+1}+\sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {80, 50, 41, 216} \[ -\frac {1}{3} \sqrt {1-x} (x+1)^{5/2}-\frac {1}{3} \sqrt {1-x} (x+1)^{3/2}-\sqrt {1-x} \sqrt {x+1}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 41
Rule 50
Rule 80
Rule 216
Rubi steps
\begin {align*} \int \frac {x (1+x)^{3/2}}{\sqrt {1-x}} \, dx &=-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {2}{3} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-\sqrt {1-x} \sqrt {1+x}-\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\sqrt {1-x} \sqrt {1+x}-\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\sqrt {1-x} \sqrt {1+x}-\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.69 \[ -\frac {1}{3} \sqrt {1-x^2} \left (x^2+3 x+5\right )-2 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 45, normalized size = 0.74 \[ -\frac {1}{3} \, {\left (x^{2} + 3 \, x + 5\right )} \sqrt {x + 1} \sqrt {-x + 1} - 2 \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 37, normalized size = 0.61 \[ -\frac {1}{3} \, {\left ({\left (x + 2\right )} {\left (x + 1\right )} + 3\right )} \sqrt {x + 1} \sqrt {-x + 1} + 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 1.08 \[ \frac {\sqrt {x +1}\, \sqrt {-x +1}\, \left (-\sqrt {-x^{2}+1}\, x^{2}-3 \sqrt {-x^{2}+1}\, x +3 \arcsin \relax (x )-5 \sqrt {-x^{2}+1}\right )}{3 \sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.97, size = 40, normalized size = 0.66 \[ -\frac {1}{3} \, \sqrt {-x^{2} + 1} x^{2} - \sqrt {-x^{2} + 1} x - \frac {5}{3} \, \sqrt {-x^{2} + 1} + \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,{\left (x+1\right )}^{3/2}}{\sqrt {1-x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 65.98, size = 129, normalized size = 2.11 \[ - 2 \left (\begin {cases} - \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} - \sqrt {1 - x} \sqrt {x + 1} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) + 2 \left (\begin {cases} - \frac {3 x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{6} - 2 \sqrt {1 - x} \sqrt {x + 1} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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