Optimal. Leaf size=130 \[ \frac{1}{9} (x+1)^{7/2} (1-x)^{11/2}+\frac{11}{72} (x+1)^{7/2} (1-x)^{9/2}+\frac{11}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac{11}{48} x (x+1)^{5/2} (1-x)^{5/2}+\frac{55}{192} x (x+1)^{3/2} (1-x)^{3/2}+\frac{55}{128} x \sqrt{x+1} \sqrt{1-x}+\frac{55}{128} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0254404, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {49, 38, 41, 216} \[ \frac{1}{9} (x+1)^{7/2} (1-x)^{11/2}+\frac{11}{72} (x+1)^{7/2} (1-x)^{9/2}+\frac{11}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac{11}{48} x (x+1)^{5/2} (1-x)^{5/2}+\frac{55}{192} x (x+1)^{3/2} (1-x)^{3/2}+\frac{55}{128} x \sqrt{x+1} \sqrt{1-x}+\frac{55}{128} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 49
Rule 38
Rule 41
Rule 216
Rubi steps
\begin{align*} \int (1-x)^{11/2} (1+x)^{5/2} \, dx &=\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{11}{9} \int (1-x)^{9/2} (1+x)^{5/2} \, dx\\ &=\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{11}{8} \int (1-x)^{7/2} (1+x)^{5/2} \, dx\\ &=\frac{11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{11}{8} \int (1-x)^{5/2} (1+x)^{5/2} \, dx\\ &=\frac{11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac{11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{55}{48} \int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac{55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac{11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac{11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{55}{64} \int \sqrt{1-x} \sqrt{1+x} \, dx\\ &=\frac{55}{128} \sqrt{1-x} x \sqrt{1+x}+\frac{55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac{11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac{11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{55}{128} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\frac{55}{128} \sqrt{1-x} x \sqrt{1+x}+\frac{55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac{11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac{11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{55}{128} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{55}{128} \sqrt{1-x} x \sqrt{1+x}+\frac{55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac{11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac{11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{55}{128} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0706632, size = 75, normalized size = 0.58 \[ \frac{\sqrt{1-x^2} \left (-896 x^8+3024 x^7-1024 x^6-7224 x^5+8448 x^4+3066 x^3-10240 x^2+4599 x+3712\right )-6930 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )}{8064} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 155, normalized size = 1.2 \begin{align*}{\frac{1}{9} \left ( 1-x \right ) ^{{\frac{11}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{72} \left ( 1-x \right ) ^{{\frac{9}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{56} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{48} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{48} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{64}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{11}{192}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{55}{384}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{55}{128}\sqrt{1-x}\sqrt{1+x}}+{\frac{55\,\arcsin \left ( x \right ) }{128}\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 = 1.55367, size = 105, normalized size = 0.81 \begin{align*} \frac{1}{9} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} x^{2} - \frac{3}{8} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} x + \frac{29}{63} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} + \frac{11}{48} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x + \frac{55}{192} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{55}{128} \, \sqrt{-x^{2} + 1} x + \frac{55}{128} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5462, size = 238, normalized size = 1.83 \begin{align*} -\frac{1}{8064} \,{\left (896 \, x^{8} - 3024 \, x^{7} + 1024 \, x^{6} + 7224 \, x^{5} - 8448 \, x^{4} - 3066 \, x^{3} + 10240 \, x^{2} - 4599 \, x - 3712\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{55}{64} \, \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 = 1.21076, size = 409, normalized size = 3.15 \begin{align*} -\frac{1}{315} \,{\left ({\left ({\left ({\left (5 \,{\left ({\left (7 \,{\left (x + 1\right )}{\left (x - 7\right )} + 195\right )}{\left (x + 1\right )} - 386\right )}{\left (x + 1\right )} + 2369\right )}{\left (x + 1\right )} - 1836\right )}{\left (x + 1\right )} + 861\right )}{\left (x + 1\right )} - 210\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - \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{1}{3} \,{\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}{128} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left ({\left (6 \,{\left (x + 1\right )}{\left (x - 6\right )} + 125\right )}{\left (x + 1\right )} - 205\right )}{\left (x + 1\right )} + 795\right )}{\left (x + 1\right )} - 449\right )}{\left (x + 1\right )} + 251\right )}{\left (x + 1\right )} - 15\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{5}{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}{8} \,{\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{55}{64} \, \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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