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 = 0.0303069, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 100
Rule 21
Rule 80
Rule 50
Rule 41
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*}
Mathematica [A] time = 0.057007, 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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Maple [A] time = 0.008, size = 94, normalized size = 0.9 \begin{align*}{\frac{1}{20}\sqrt{1-x}\sqrt{1+x} \left ( -4\,{x}^{4}\sqrt{-{x}^{2}+1}-10\,{x}^{3}\sqrt{-{x}^{2}+1}-12\,{x}^{2}\sqrt{-{x}^{2}+1}-15\,x\sqrt{-{x}^{2}+1}+15\,\arcsin \left ( x \right ) -24\,\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.72588, size = 95, normalized size = 0.86 \begin{align*} -\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 \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85683, size = 158, normalized size = 1.44 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28917, size = 70, normalized size = 0.64 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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