Optimal. Leaf size=88 \[ \frac{1}{5} (x+1)^{3/2} (1-x)^{7/2}+\frac{7}{20} (x+1)^{3/2} (1-x)^{5/2}+\frac{7}{12} (x+1)^{3/2} (1-x)^{3/2}+\frac{7}{8} x \sqrt{x+1} \sqrt{1-x}+\frac{7}{8} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0153782, antiderivative size = 88, 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} \[ \frac{1}{5} (x+1)^{3/2} (1-x)^{7/2}+\frac{7}{20} (x+1)^{3/2} (1-x)^{5/2}+\frac{7}{12} (x+1)^{3/2} (1-x)^{3/2}+\frac{7}{8} x \sqrt{x+1} \sqrt{1-x}+\frac{7}{8} \sin ^{-1}(x) \]
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} \sqrt{1+x} \, dx &=\frac{1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac{7}{5} \int (1-x)^{5/2} \sqrt{1+x} \, dx\\ &=\frac{7}{20} (1-x)^{5/2} (1+x)^{3/2}+\frac{1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac{7}{4} \int (1-x)^{3/2} \sqrt{1+x} \, dx\\ &=\frac{7}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac{7}{20} (1-x)^{5/2} (1+x)^{3/2}+\frac{1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac{7}{4} \int \sqrt{1-x} \sqrt{1+x} \, dx\\ &=\frac{7}{8} \sqrt{1-x} x \sqrt{1+x}+\frac{7}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac{7}{20} (1-x)^{5/2} (1+x)^{3/2}+\frac{1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac{7}{8} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\frac{7}{8} \sqrt{1-x} x \sqrt{1+x}+\frac{7}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac{7}{20} (1-x)^{5/2} (1+x)^{3/2}+\frac{1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac{7}{8} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{7}{8} \sqrt{1-x} x \sqrt{1+x}+\frac{7}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac{7}{20} (1-x)^{5/2} (1+x)^{3/2}+\frac{1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac{7}{8} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0528489, size = 56, normalized size = 0.64 \[ \frac{1}{120} \sqrt{1-x^2} \left (-24 x^4+90 x^3-112 x^2+15 x+136\right )-\frac{7}{4} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 99, normalized size = 1.1 \begin{align*}{\frac{1}{5} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{7}{20} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{7}{12} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{7}{8}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{7}{8}\sqrt{1-x}\sqrt{1+x}}+{\frac{7\,\arcsin \left ( x \right ) }{8}\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 = 1.5388, size = 73, normalized size = 0.83 \begin{align*} \frac{1}{5} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{2} - \frac{3}{4} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{17}{15} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} + \frac{7}{8} \, \sqrt{-x^{2} + 1} x + \frac{7}{8} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54134, size = 163, normalized size = 1.85 \begin{align*} -\frac{1}{120} \,{\left (24 \, x^{4} - 90 \, x^{3} + 112 \, x^{2} - 15 \, x - 136\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{7}{4} \, \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 [A] time = 74.6343, size = 253, normalized size = 2.88 \begin{align*} \begin{cases} - \frac{7 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{4} - \frac{i \left (x + 1\right )^{\frac{11}{2}}}{5 \sqrt{x - 1}} + \frac{39 i \left (x + 1\right )^{\frac{9}{2}}}{20 \sqrt{x - 1}} - \frac{449 i \left (x + 1\right )^{\frac{7}{2}}}{60 \sqrt{x - 1}} + \frac{1657 i \left (x + 1\right )^{\frac{5}{2}}}{120 \sqrt{x - 1}} - \frac{263 i \left (x + 1\right )^{\frac{3}{2}}}{24 \sqrt{x - 1}} + \frac{7 i \sqrt{x + 1}}{4 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\frac{7 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{4} + \frac{\left (x + 1\right )^{\frac{11}{2}}}{5 \sqrt{1 - x}} - \frac{39 \left (x + 1\right )^{\frac{9}{2}}}{20 \sqrt{1 - x}} + \frac{449 \left (x + 1\right )^{\frac{7}{2}}}{60 \sqrt{1 - x}} - \frac{1657 \left (x + 1\right )^{\frac{5}{2}}}{120 \sqrt{1 - x}} + \frac{263 \left (x + 1\right )^{\frac{3}{2}}}{24 \sqrt{1 - x}} - \frac{7 \sqrt{x + 1}}{4 \sqrt{1 - x}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10213, size = 143, normalized size = 1.62 \begin{align*} -\frac{1}{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{3}{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{7}{4} \, \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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