Optimal. Leaf size=109 \[ \frac{1}{7} (x+1)^{5/2} (1-x)^{9/2}+\frac{3}{14} (x+1)^{5/2} (1-x)^{7/2}+\frac{3}{10} (x+1)^{5/2} (1-x)^{5/2}+\frac{3}{8} x (x+1)^{3/2} (1-x)^{3/2}+\frac{9}{16} x \sqrt{x+1} \sqrt{1-x}+\frac{9}{16} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0197551, antiderivative size = 109, 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} \[ \frac{1}{7} (x+1)^{5/2} (1-x)^{9/2}+\frac{3}{14} (x+1)^{5/2} (1-x)^{7/2}+\frac{3}{10} (x+1)^{5/2} (1-x)^{5/2}+\frac{3}{8} x (x+1)^{3/2} (1-x)^{3/2}+\frac{9}{16} x \sqrt{x+1} \sqrt{1-x}+\frac{9}{16} \sin ^{-1}(x) \]
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} (1+x)^{3/2} \, dx &=\frac{1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac{9}{7} \int (1-x)^{7/2} (1+x)^{3/2} \, dx\\ &=\frac{3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac{1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac{3}{2} \int (1-x)^{5/2} (1+x)^{3/2} \, dx\\ &=\frac{3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac{3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac{1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac{3}{2} \int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac{3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac{3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac{3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac{1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac{9}{8} \int \sqrt{1-x} \sqrt{1+x} \, dx\\ &=\frac{9}{16} \sqrt{1-x} x \sqrt{1+x}+\frac{3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac{3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac{3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac{1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac{9}{16} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\frac{9}{16} \sqrt{1-x} x \sqrt{1+x}+\frac{3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac{3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac{3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac{1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac{9}{16} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{9}{16} \sqrt{1-x} x \sqrt{1+x}+\frac{3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac{3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac{3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac{1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac{9}{16} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0572892, size = 66, normalized size = 0.61 \[ \frac{1}{560} \sqrt{1-x^2} \left (80 x^6-280 x^5+208 x^4+350 x^3-656 x^2+245 x+368\right )-\frac{9}{8} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 127, normalized size = 1.2 \begin{align*}{\frac{1}{7} \left ( 1-x \right ) ^{{\frac{9}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{3}{14} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{3}{10} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{3}{8} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{3}{8}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{3}{16}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{9}{16}\sqrt{1-x}\sqrt{1+x}}+{\frac{9\,\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 = 1.53441, size = 89, normalized size = 0.82 \begin{align*} \frac{1}{7} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x^{2} - \frac{1}{2} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x + \frac{23}{35} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} + \frac{3}{8} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{9}{16} \, \sqrt{-x^{2} + 1} x + \frac{9}{16} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5773, size = 192, normalized size = 1.76 \begin{align*} \frac{1}{560} \,{\left (80 \, x^{6} - 280 \, x^{5} + 208 \, x^{4} + 350 \, x^{3} - 656 \, x^{2} + 245 \, x + 368\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{9}{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 [B] time = 1.21059, size = 259, normalized size = 2.38 \begin{align*} \frac{1}{105} \,{\left ({\left (3 \,{\left ({\left (5 \,{\left (x + 1\right )}{\left (x - 5\right )} + 74\right )}{\left (x + 1\right )} - 96\right )}{\left (x + 1\right )} + 203\right )}{\left (x + 1\right )} - 70\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} + \frac{2}{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} -{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} - \frac{1}{16} \,{\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{1}{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{9}{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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