Optimal. Leaf size=90 \[ \frac{1}{7} (1-x)^{7/2} (x+1)^{7/2}+\frac{1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac{5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac{5}{16} \sqrt{1-x} x \sqrt{x+1}+\frac{5}{16} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0119287, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {49, 38, 41, 216} \[ \frac{1}{7} (1-x)^{7/2} (x+1)^{7/2}+\frac{1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac{5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac{5}{16} \sqrt{1-x} x \sqrt{x+1}+\frac{5}{16} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 49
Rule 38
Rule 41
Rule 216
Rubi steps
\begin{align*} \int (1-x)^{7/2} (1+x)^{5/2} \, dx &=\frac{1}{7} (1-x)^{7/2} (1+x)^{7/2}+\int (1-x)^{5/2} (1+x)^{5/2} \, dx\\ &=\frac{1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac{1}{7} (1-x)^{7/2} (1+x)^{7/2}+\frac{5}{6} \int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac{5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac{1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac{1}{7} (1-x)^{7/2} (1+x)^{7/2}+\frac{5}{8} \int \sqrt{1-x} \sqrt{1+x} \, dx\\ &=\frac{5}{16} \sqrt{1-x} x \sqrt{1+x}+\frac{5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac{1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac{1}{7} (1-x)^{7/2} (1+x)^{7/2}+\frac{5}{16} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\frac{5}{16} \sqrt{1-x} x \sqrt{1+x}+\frac{5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac{1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac{1}{7} (1-x)^{7/2} (1+x)^{7/2}+\frac{5}{16} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{5}{16} \sqrt{1-x} x \sqrt{1+x}+\frac{5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac{1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac{1}{7} (1-x)^{7/2} (1+x)^{7/2}+\frac{5}{16} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0630693, size = 66, normalized size = 0.73 \[ \frac{1}{336} \sqrt{1-x^2} \left (-48 x^6+56 x^5+144 x^4-182 x^3-144 x^2+231 x+48\right )-\frac{5}{8} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 127, normalized size = 1.4 \begin{align*}{\frac{1}{7} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{8}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{1}{24}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{5}{48}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{5}{16}\sqrt{1-x}\sqrt{1+x}}+{\frac{5\,\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 = 1.52674, size = 70, normalized size = 0.78 \begin{align*} \frac{1}{7} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} + \frac{1}{6} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x + \frac{5}{24} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{5}{16} \, \sqrt{-x^{2} + 1} x + \frac{5}{16} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56064, size = 190, normalized size = 2.11 \begin{align*} -\frac{1}{336} \,{\left (48 \, x^{6} - 56 \, x^{5} - 144 \, x^{4} + 182 \, x^{3} + 144 \, x^{2} - 231 \, x - 48\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{5}{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 = 1.17025, size = 259, normalized size = 2.88 \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} - \frac{1}{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{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{5}{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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