Optimal. Leaf size=88 \[ -\frac {1}{4} \sqrt {1-x} x (x+1)^{5/2}-\frac {1}{6} \sqrt {1-x} (x+1)^{5/2}-\frac {7}{24} \sqrt {1-x} (x+1)^{3/2}-\frac {7}{8} \sqrt {1-x} \sqrt {x+1}+\frac {7}{8} \sin ^{-1}(x) \]
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Rubi [A] time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {90, 80, 50, 41, 216} \begin {gather*} -\frac {1}{4} \sqrt {1-x} x (x+1)^{5/2}-\frac {1}{6} \sqrt {1-x} (x+1)^{5/2}-\frac {7}{24} \sqrt {1-x} (x+1)^{3/2}-\frac {7}{8} \sqrt {1-x} \sqrt {x+1}+\frac {7}{8} \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 50
Rule 80
Rule 90
Rule 216
Rubi steps
\begin {align*} \int \frac {x^2 (1+x)^{3/2}}{\sqrt {1-x}} \, dx &=-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}-\frac {1}{4} \int \frac {(-1-2 x) (1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{12} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-\frac {7}{8} \sqrt {1-x} \sqrt {1+x}-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {7}{8} \sqrt {1-x} \sqrt {1+x}-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {7}{8} \sqrt {1-x} \sqrt {1+x}-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.58 \begin {gather*} -\frac {1}{24} \sqrt {1-x^2} \left (6 x^3+16 x^2+21 x+32\right )-\frac {7}{4} \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 100, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {1-x} \left (\frac {21 (1-x)^3}{(x+1)^3}+\frac {77 (1-x)^2}{(x+1)^2}+\frac {83 (1-x)}{x+1}+75\right )}{12 \sqrt {x+1} \left (\frac {1-x}{x+1}+1\right )^4}-\frac {7}{4} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 52, normalized size = 0.59 \begin {gather*} -\frac {1}{24} \, {\left (6 \, x^{3} + 16 \, x^{2} + 21 \, x + 32\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {7}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 46, normalized size = 0.52 \begin {gather*} -\frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x + 2\right )} {\left (x + 1\right )} + 7\right )} {\left (x + 1\right )} + 21\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {7}{4} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.91 \begin {gather*} \frac {\sqrt {x +1}\, \sqrt {-x +1}\, \left (-6 \sqrt {-x^{2}+1}\, x^{3}-16 \sqrt {-x^{2}+1}\, x^{2}-21 \sqrt {-x^{2}+1}\, x +21 \arcsin \relax (x )-32 \sqrt {-x^{2}+1}\right )}{24 \sqrt {-x^{2}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.07, size = 56, normalized size = 0.64 \begin {gather*} -\frac {1}{4} \, \sqrt {-x^{2} + 1} x^{3} - \frac {2}{3} \, \sqrt {-x^{2} + 1} x^{2} - \frac {7}{8} \, \sqrt {-x^{2} + 1} x - \frac {4}{3} \, \sqrt {-x^{2} + 1} + \frac {7}{8} \, \arcsin \relax (x) \end {gather*}
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 \begin {gather*} \int \frac {x^2\,{\left (x+1\right )}^{3/2}}{\sqrt {1-x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 136.73, size = 240, normalized size = 2.73 \begin {gather*} 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 ) - 4 \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 ) + 2 \left (\begin {cases} - \frac {7 x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {2 \left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{3} + \frac {\sqrt {1 - x} \sqrt {x + 1} \left (- 5 x - 2 \left (x + 1\right )^{3} + 6 \left (x + 1\right )^{2} - 4\right )}{16} - 4 \sqrt {1 - x} \sqrt {x + 1} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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