Optimal. Leaf size=63 \[ -\frac{2 (1-x)^{5/2}}{3 (x+1)^{3/2}}+\frac{10 (1-x)^{3/2}}{3 \sqrt{x+1}}+5 \sqrt{x+1} \sqrt{1-x}+5 \sin ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.010729, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {47, 50, 41, 216} \[ -\frac{2 (1-x)^{5/2}}{3 (x+1)^{3/2}}+\frac{10 (1-x)^{3/2}}{3 \sqrt{x+1}}+5 \sqrt{x+1} \sqrt{1-x}+5 \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{(1-x)^{5/2}}{(1+x)^{5/2}} \, dx &=-\frac{2 (1-x)^{5/2}}{3 (1+x)^{3/2}}-\frac{5}{3} \int \frac{(1-x)^{3/2}}{(1+x)^{3/2}} \, dx\\ &=-\frac{2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac{10 (1-x)^{3/2}}{3 \sqrt{1+x}}+5 \int \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac{10 (1-x)^{3/2}}{3 \sqrt{1+x}}+5 \sqrt{1-x} \sqrt{1+x}+5 \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac{10 (1-x)^{3/2}}{3 \sqrt{1+x}}+5 \sqrt{1-x} \sqrt{1+x}+5 \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac{10 (1-x)^{3/2}}{3 \sqrt{1+x}}+5 \sqrt{1-x} \sqrt{1+x}+5 \sin ^{-1}(x)\\ \end{align*}
Mathematica [C] time = 0.0102119, size = 37, normalized size = 0.59 \[ -\frac{(1-x)^{7/2} \, _2F_1\left (\frac{5}{2},\frac{7}{2};\frac{9}{2};\frac{1-x}{2}\right )}{14 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.019, size = 79, normalized size = 1.3 \begin{align*} -{\frac{3\,{x}^{3}+31\,{x}^{2}-11\,x-23}{3}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) } \left ( 1+x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}}+5\,{\frac{\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }\arcsin \left ( x \right ) }{\sqrt{1-x}\sqrt{1+x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.50109, size = 132, normalized size = 2.1 \begin{align*} \frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} - \frac{5 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} - \frac{10 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} + \frac{35 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x + 1\right )}} + 5 \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.85971, size = 204, normalized size = 3.24 \begin{align*} \frac{23 \, x^{2} +{\left (3 \, x^{2} + 34 \, x + 23\right )} \sqrt{x + 1} \sqrt{-x + 1} - 30 \,{\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 46 \, x + 23}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 12.9692, size = 160, normalized size = 2.54 \begin{align*} \begin{cases} \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) + \frac{28 \sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{8 \sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} + 5 i \log{\left (\frac{1}{x + 1} \right )} + 5 i \log{\left (x + 1 \right )} + 10 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) + \frac{28 i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{8 i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} + 5 i \log{\left (\frac{1}{x + 1} \right )} - 10 i \log{\left (\sqrt{1 - \frac{2}{x + 1}} + 1 \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.11284, size = 155, normalized size = 2.46 \begin{align*} \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{6 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \sqrt{x + 1} \sqrt{-x + 1} - \frac{9 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{2 \, \sqrt{x + 1}} + \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{27 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{6 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} + 10 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]