Optimal. Leaf size=78 \[ \frac {2 \sqrt {x+1} x^3}{3 (1-x)^{3/2}}-\frac {13 \sqrt {x+1} x^2}{3 \sqrt {1-x}}-\frac {1}{6} \sqrt {1-x} \sqrt {x+1} (33 x+52)+\frac {11}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {97, 150, 147, 41, 216} \[ \frac {2 \sqrt {x+1} x^3}{3 (1-x)^{3/2}}-\frac {13 \sqrt {x+1} x^2}{3 \sqrt {1-x}}-\frac {1}{6} \sqrt {1-x} \sqrt {x+1} (33 x+52)+\frac {11}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 41
Rule 97
Rule 147
Rule 150
Rule 216
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {1+x}}{(1-x)^{5/2}} \, dx &=\frac {2 x^3 \sqrt {1+x}}{3 (1-x)^{3/2}}-\frac {2}{3} \int \frac {x^2 \left (3+\frac {7 x}{2}\right )}{(1-x)^{3/2} \sqrt {1+x}} \, dx\\ &=-\frac {13 x^2 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 x^3 \sqrt {1+x}}{3 (1-x)^{3/2}}-\frac {2}{3} \int \frac {\left (-13-\frac {33 x}{2}\right ) x}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {13 x^2 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 x^3 \sqrt {1+x}}{3 (1-x)^{3/2}}-\frac {1}{6} \sqrt {1-x} \sqrt {1+x} (52+33 x)+\frac {11}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {13 x^2 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 x^3 \sqrt {1+x}}{3 (1-x)^{3/2}}-\frac {1}{6} \sqrt {1-x} \sqrt {1+x} (52+33 x)+\frac {11}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {13 x^2 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 x^3 \sqrt {1+x}}{3 (1-x)^{3/2}}-\frac {1}{6} \sqrt {1-x} \sqrt {1+x} (52+33 x)+\frac {11}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 54, normalized size = 0.69 \[ -\frac {\sqrt {x+1} \left (3 x^3+12 x^2-71 x+52\right )}{6 (1-x)^{3/2}}-11 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 80, normalized size = 1.03 \[ -\frac {52 \, x^{2} + {\left (3 \, x^{3} + 12 \, x^{2} - 71 \, x + 52\right )} \sqrt {x + 1} \sqrt {-x + 1} + 66 \, {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 104 \, x + 52}{6 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 49, normalized size = 0.63 \[ -\frac {{\left ({\left (3 \, {\left (x + 2\right )} {\left (x + 1\right )} - 86\right )} {\left (x + 1\right )} + 132\right )} \sqrt {x + 1} \sqrt {-x + 1}}{6 \, {\left (x - 1\right )}^{2}} + 11 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 97, normalized size = 1.24 \[ \frac {\left (-3 \sqrt {-x^{2}+1}\, x^{3}+33 x^{2} \arcsin \relax (x )-12 \sqrt {-x^{2}+1}\, x^{2}-66 x \arcsin \relax (x )+71 \sqrt {-x^{2}+1}\, x +33 \arcsin \relax (x )-52 \sqrt {-x^{2}+1}\right ) \sqrt {-x +1}\, \sqrt {x +1}}{6 \left (x -1\right )^{2} \sqrt {-x^{2}+1}} \]
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 \[ \text {Timed out} \]
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 \[ \int \frac {x^3\,\sqrt {x+1}}{{\left (1-x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{3} \sqrt {x + 1}}{\left (1 - x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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