Optimal. Leaf size=61 \[ \frac {(x+1)^{3/2}}{3 (1-x)^{3/2}}-\sqrt {1-x} \sqrt {x+1}-\frac {4 \sqrt {x+1}}{\sqrt {1-x}}+3 \sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {89, 21, 47, 50, 41, 216} \begin {gather*} -\frac {2 (x+1)^{3/2}}{\sqrt {1-x}}+\frac {(x+1)^{3/2}}{3 (1-x)^{3/2}}-3 \sqrt {1-x} \sqrt {x+1}+3 \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 41
Rule 47
Rule 50
Rule 89
Rule 216
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {1+x}}{(1-x)^{5/2}} \, dx &=\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\frac {1}{3} \int \frac {\sqrt {1+x} (3+3 x)}{(1-x)^{3/2}} \, dx\\ &=\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\int \frac {(1+x)^{3/2}}{(1-x)^{3/2}} \, dx\\ &=\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}+3 \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-3 \sqrt {1-x} \sqrt {1+x}+\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}+3 \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-3 \sqrt {1-x} \sqrt {1+x}+\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}+3 \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-3 \sqrt {1-x} \sqrt {1+x}+\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}+3 \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 0.80 \begin {gather*} -\frac {\sqrt {x+1} \left (3 x^2-19 x+14\right )}{3 (1-x)^{3/2}}-6 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 83, normalized size = 1.36 \begin {gather*} \frac {\sqrt {x+1} \left (\frac {(x+1)^2}{(1-x)^2}-\frac {11 (x+1)}{1-x}-18\right )}{3 \sqrt {1-x} \left (\frac {x+1}{1-x}+1\right )}+6 \tan ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 75, normalized size = 1.23 \begin {gather*} -\frac {14 \, x^{2} + {\left (3 \, x^{2} - 19 \, x + 14\right )} \sqrt {x + 1} \sqrt {-x + 1} + 18 \, {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 28 \, x + 14}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 44, normalized size = 0.72 \begin {gather*} -\frac {{\left ({\left (3 \, x - 22\right )} {\left (x + 1\right )} + 36\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} + 6 \, \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 = 83, normalized size = 1.36 \begin {gather*} \frac {\left (9 x^{2} \arcsin \relax (x )-3 \sqrt {-x^{2}+1}\, x^{2}-18 x \arcsin \relax (x )+19 \sqrt {-x^{2}+1}\, x +9 \arcsin \relax (x )-14 \sqrt {-x^{2}+1}\right ) \sqrt {-x +1}\, \sqrt {x +1}}{3 \left (x -1\right )^{2} \sqrt {-x^{2}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.02 \begin {gather*} \int \frac {x^2\,\sqrt {x+1}}{{\left (1-x\right )}^{5/2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {x^{2} \sqrt {x + 1}}{\left (1 - x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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