Optimal. Leaf size=108 \[ \frac{1}{6} (x+1)^{3/2} (1-x)^{9/2}+\frac{3}{10} (x+1)^{3/2} (1-x)^{7/2}+\frac{21}{40} (x+1)^{3/2} (1-x)^{5/2}+\frac{7}{8} (x+1)^{3/2} (1-x)^{3/2}+\frac{21}{16} x \sqrt{x+1} \sqrt{1-x}+\frac{21}{16} \sin ^{-1}(x) \]
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Rubi [A] time = 0.02088, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {49, 38, 41, 216} \[ \frac{1}{6} (x+1)^{3/2} (1-x)^{9/2}+\frac{3}{10} (x+1)^{3/2} (1-x)^{7/2}+\frac{21}{40} (x+1)^{3/2} (1-x)^{5/2}+\frac{7}{8} (x+1)^{3/2} (1-x)^{3/2}+\frac{21}{16} x \sqrt{x+1} \sqrt{1-x}+\frac{21}{16} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 49
Rule 38
Rule 41
Rule 216
Rubi steps
\begin{align*} \int (1-x)^{9/2} \sqrt{1+x} \, dx &=\frac{1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac{3}{2} \int (1-x)^{7/2} \sqrt{1+x} \, dx\\ &=\frac{3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac{1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac{21}{10} \int (1-x)^{5/2} \sqrt{1+x} \, dx\\ &=\frac{21}{40} (1-x)^{5/2} (1+x)^{3/2}+\frac{3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac{1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac{21}{8} \int (1-x)^{3/2} \sqrt{1+x} \, dx\\ &=\frac{7}{8} (1-x)^{3/2} (1+x)^{3/2}+\frac{21}{40} (1-x)^{5/2} (1+x)^{3/2}+\frac{3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac{1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac{21}{8} \int \sqrt{1-x} \sqrt{1+x} \, dx\\ &=\frac{21}{16} \sqrt{1-x} x \sqrt{1+x}+\frac{7}{8} (1-x)^{3/2} (1+x)^{3/2}+\frac{21}{40} (1-x)^{5/2} (1+x)^{3/2}+\frac{3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac{1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac{21}{16} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\frac{21}{16} \sqrt{1-x} x \sqrt{1+x}+\frac{7}{8} (1-x)^{3/2} (1+x)^{3/2}+\frac{21}{40} (1-x)^{5/2} (1+x)^{3/2}+\frac{3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac{1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac{21}{16} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{21}{16} \sqrt{1-x} x \sqrt{1+x}+\frac{7}{8} (1-x)^{3/2} (1+x)^{3/2}+\frac{21}{40} (1-x)^{5/2} (1+x)^{3/2}+\frac{3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac{1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac{21}{16} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0510854, size = 60, normalized size = 0.56 \[ \frac{1}{240} \left (\sqrt{1-x^2} \left (40 x^5-192 x^4+350 x^3-256 x^2-75 x+448\right )-630 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 113, normalized size = 1.1 \begin{align*}{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{9}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{3}{10} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{21}{40} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{7}{8} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{21}{16}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{21}{16}\sqrt{1-x}\sqrt{1+x}}+{\frac{21\,\arcsin \left ( x \right ) }{16}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57081, size = 92, normalized size = 0.85 \begin{align*} -\frac{1}{6} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{3} + \frac{4}{5} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{2} - \frac{13}{8} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{28}{15} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} + \frac{21}{16} \, \sqrt{-x^{2} + 1} x + \frac{21}{16} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6009, size = 178, normalized size = 1.65 \begin{align*} \frac{1}{240} \,{\left (40 \, x^{5} - 192 \, x^{4} + 350 \, x^{3} - 256 \, x^{2} - 75 \, x + 448\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{21}{8} \, \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 = 1.12889, size = 201, normalized size = 1.86 \begin{align*} -\frac{4}{15} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - \frac{4}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} + \frac{1}{48} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left (x + 1\right )}{\left (x - 4\right )} + 39\right )}{\left (x + 1\right )} - 37\right )}{\left (x + 1\right )} + 31\right )}{\left (x + 1\right )} - 3\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{3}{4} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{21}{8} \, \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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