Optimal. Leaf size=118 \[ -\frac {1}{6} \sqrt {1-x} (x+1)^{5/2} x^3-\frac {1}{15} \sqrt {1-x} (x+1)^{5/2} x^2-\frac {11}{48} \sqrt {1-x} (x+1)^{3/2}-\frac {1}{120} \sqrt {1-x} (x+1)^{5/2} (19 x+18)-\frac {11}{16} \sqrt {1-x} \sqrt {x+1}+\frac {11}{16} \sin ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {100, 153, 147, 50, 41, 216} \begin {gather*} -\frac {1}{6} \sqrt {1-x} (x+1)^{5/2} x^3-\frac {1}{15} \sqrt {1-x} (x+1)^{5/2} x^2-\frac {11}{48} \sqrt {1-x} (x+1)^{3/2}-\frac {1}{120} \sqrt {1-x} (x+1)^{5/2} (19 x+18)-\frac {11}{16} \sqrt {1-x} \sqrt {x+1}+\frac {11}{16} \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 50
Rule 100
Rule 147
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \frac {x^4 (1+x)^{3/2}}{\sqrt {1-x}} \, dx &=-\frac {1}{6} \sqrt {1-x} x^3 (1+x)^{5/2}-\frac {1}{6} \int \frac {(-3-2 x) x^2 (1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{15} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{6} \sqrt {1-x} x^3 (1+x)^{5/2}+\frac {1}{30} \int \frac {x (1+x)^{3/2} (4+19 x)}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{15} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{6} \sqrt {1-x} x^3 (1+x)^{5/2}-\frac {1}{120} \sqrt {1-x} (1+x)^{5/2} (18+19 x)+\frac {11}{24} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {11}{48} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{15} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{6} \sqrt {1-x} x^3 (1+x)^{5/2}-\frac {1}{120} \sqrt {1-x} (1+x)^{5/2} (18+19 x)+\frac {11}{16} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-\frac {11}{16} \sqrt {1-x} \sqrt {1+x}-\frac {11}{48} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{15} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{6} \sqrt {1-x} x^3 (1+x)^{5/2}-\frac {1}{120} \sqrt {1-x} (1+x)^{5/2} (18+19 x)+\frac {11}{16} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {11}{16} \sqrt {1-x} \sqrt {1+x}-\frac {11}{48} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{15} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{6} \sqrt {1-x} x^3 (1+x)^{5/2}-\frac {1}{120} \sqrt {1-x} (1+x)^{5/2} (18+19 x)+\frac {11}{16} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {11}{16} \sqrt {1-x} \sqrt {1+x}-\frac {11}{48} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{15} \sqrt {1-x} x^2 (1+x)^{5/2}-\frac {1}{6} \sqrt {1-x} x^3 (1+x)^{5/2}-\frac {1}{120} \sqrt {1-x} (1+x)^{5/2} (18+19 x)+\frac {11}{16} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 0.60 \begin {gather*} \frac {\sqrt {x+1} \left (40 x^6+56 x^5+14 x^4+18 x^3+37 x^2+91 x-256\right )}{240 \sqrt {1-x}}-\frac {11}{8} \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 128, normalized size = 1.08 \begin {gather*} -\frac {\sqrt {1-x} \left (\frac {165 (1-x)^5}{(x+1)^5}+\frac {935 (1-x)^4}{(x+1)^4}+\frac {1986 (1-x)^3}{(x+1)^3}+\frac {3006 (1-x)^2}{(x+1)^2}+\frac {1305 (1-x)}{x+1}+795\right )}{120 \sqrt {x+1} \left (\frac {1-x}{x+1}+1\right )^6}-\frac {11}{8} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 62, normalized size = 0.53 \begin {gather*} -\frac {1}{240} \, {\left (40 \, x^{5} + 96 \, x^{4} + 110 \, x^{3} + 128 \, x^{2} + 165 \, x + 256\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {11}{8} \, \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 = 1.26, size = 59, normalized size = 0.50 \begin {gather*} -\frac {1}{240} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 8\right )} {\left (x + 1\right )} + 63\right )} {\left (x + 1\right )} - 13\right )} {\left (x + 1\right )} + 55\right )} {\left (x + 1\right )} + 165\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {11}{8} \, \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 = 0.02, size = 108, normalized size = 0.92 \begin {gather*} \frac {\sqrt {x +1}\, \sqrt {-x +1}\, \left (-40 \sqrt {-x^{2}+1}\, x^{5}-96 \sqrt {-x^{2}+1}\, x^{4}-110 \sqrt {-x^{2}+1}\, x^{3}-128 \sqrt {-x^{2}+1}\, x^{2}-165 \sqrt {-x^{2}+1}\, x +165 \arcsin \relax (x )-256 \sqrt {-x^{2}+1}\right )}{240 \sqrt {-x^{2}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.89, size = 84, normalized size = 0.71 \begin {gather*} -\frac {1}{6} \, \sqrt {-x^{2} + 1} x^{5} - \frac {2}{5} \, \sqrt {-x^{2} + 1} x^{4} - \frac {11}{24} \, \sqrt {-x^{2} + 1} x^{3} - \frac {8}{15} \, \sqrt {-x^{2} + 1} x^{2} - \frac {11}{16} \, \sqrt {-x^{2} + 1} x - \frac {16}{15} \, \sqrt {-x^{2} + 1} + \frac {11}{16} \, \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^4\,{\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 [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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