Optimal. Leaf size=110 \[ -\frac {1}{5} \sqrt {1-x} x^2 (x+1)^{5/2}-\frac {1}{10} \sqrt {1-x} (x+1)^{7/2}-\frac {1}{10} \sqrt {1-x} (x+1)^{5/2}-\frac {1}{4} \sqrt {1-x} (x+1)^{3/2}-\frac {3}{4} \sqrt {1-x} \sqrt {x+1}+\frac {3}{4} \sin ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {100, 21, 80, 50, 41, 216} \[ -\frac {1}{5} \sqrt {1-x} x^2 (x+1)^{5/2}-\frac {1}{10} \sqrt {1-x} (x+1)^{7/2}-\frac {1}{10} \sqrt {1-x} (x+1)^{5/2}-\frac {1}{4} \sqrt {1-x} (x+1)^{3/2}-\frac {3}{4} \sqrt {1-x} \sqrt {x+1}+\frac {3}{4} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 21
Rule 41
Rule 50
Rule 80
Rule 100
Rule 216
Rubi steps
\begin {align*} \int \frac {x^3 (1+x)^{3/2}}{\sqrt {1-x}} \, dx &=-\frac {1}{5} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{5} \int \frac {(-2-2 x) x (1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{5} \sqrt {1-x} x^2 (1+x)^{5/2}+\frac {2}{5} \int \frac {x (1+x)^{5/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{5} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{10} \sqrt {1-x} (1+x)^{7/2}+\frac {3}{10} \int \frac {(1+x)^{5/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{10} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{5} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{10} \sqrt {1-x} (1+x)^{7/2}+\frac {1}{2} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{4} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{10} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{5} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{10} \sqrt {1-x} (1+x)^{7/2}+\frac {3}{4} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-\frac {3}{4} \sqrt {1-x} \sqrt {1+x}-\frac {1}{4} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{10} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{5} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{10} \sqrt {1-x} (1+x)^{7/2}+\frac {3}{4} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {3}{4} \sqrt {1-x} \sqrt {1+x}-\frac {1}{4} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{10} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{5} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{10} \sqrt {1-x} (1+x)^{7/2}+\frac {3}{4} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {3}{4} \sqrt {1-x} \sqrt {1+x}-\frac {1}{4} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{10} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{5} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{10} \sqrt {1-x} (1+x)^{7/2}+\frac {3}{4} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 56, normalized size = 0.51 \[ -\frac {1}{20} \sqrt {1-x^2} \left (4 x^4+10 x^3+12 x^2+15 x+24\right )-\frac {3}{2} \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 57, normalized size = 0.52 \[ -\frac {1}{20} \, {\left (4 \, x^{4} + 10 \, x^{3} + 12 \, x^{2} + 15 \, x + 24\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {3}{2} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.44, size = 52, normalized size = 0.47 \[ -\frac {1}{20} \, {\left ({\left (2 \, {\left ({\left (2 \, x - 1\right )} {\left (x + 1\right )} + 3\right )} {\left (x + 1\right )} + 5\right )} {\left (x + 1\right )} + 15\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {3}{2} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 94, normalized size = 0.85 \[ \frac {\sqrt {x +1}\, \sqrt {-x +1}\, \left (-4 \sqrt {-x^{2}+1}\, x^{4}-10 \sqrt {-x^{2}+1}\, x^{3}-12 \sqrt {-x^{2}+1}\, x^{2}-15 \sqrt {-x^{2}+1}\, x +15 \arcsin \relax (x )-24 \sqrt {-x^{2}+1}\right )}{20 \sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.92, size = 70, normalized size = 0.64 \[ -\frac {1}{5} \, \sqrt {-x^{2} + 1} x^{4} - \frac {1}{2} \, \sqrt {-x^{2} + 1} x^{3} - \frac {3}{5} \, \sqrt {-x^{2} + 1} x^{2} - \frac {3}{4} \, \sqrt {-x^{2} + 1} x - \frac {6}{5} \, \sqrt {-x^{2} + 1} + \frac {3}{4} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,{\left (x+1\right )}^{3/2}}{\sqrt {1-x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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