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 = 0.0457073, 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 \[ -\frac{2 (1-x)^{5/2}}{3 (x+1)^{3/2}}+\frac{10 (1-x)^{3/2}}{3 \sqrt{x+1}}+5 \sqrt{x+1} \sqrt{1-x}+5 \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(1 - x)^(5/2)/(1 + x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.19898, size = 53, normalized size = 0.84 \[ - \frac{2 \left (- x + 1\right )^{\frac{5}{2}}}{3 \left (x + 1\right )^{\frac{3}{2}}} + \frac{10 \left (- x + 1\right )^{\frac{3}{2}}}{3 \sqrt{x + 1}} + 5 \sqrt{- x + 1} \sqrt{x + 1} + 5 \operatorname{asin}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x)**(5/2)/(1+x)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.044941, size = 47, normalized size = 0.75 \[ \frac{\sqrt{1-x} \left (3 x^2+34 x+23\right )}{3 (x+1)^{3/2}}+10 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x)^(5/2)/(1 + x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.032, size = 79, normalized size = 1.3 \[ -{\frac{3\,{x}^{3}+31\,{x}^{2}-11\,x-23}{3}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) } \left ( 1+x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}}+5\,{\frac{\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }\arcsin \left ( x \right ) }{\sqrt{1-x}\sqrt{1+x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x)^(5/2)/(1+x)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.49698, size = 132, normalized size = 2.1 \[ \frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} - \frac{5 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} - \frac{10 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} + \frac{35 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x + 1\right )}} + 5 \, \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x + 1)^(5/2)/(x + 1)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.215367, size = 227, normalized size = 3.6 \[ \frac{3 \, x^{5} + 48 \, x^{4} + 7 \, x^{3} - 102 \, x^{2} +{\left (3 \, x^{4} + 17 \, x^{3} + 102 \, x^{2} + 48 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 30 \,{\left (x^{4} + 4 \, x^{3} + x^{2} -{\left (x^{3} - x^{2} - 6 \, x - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 6 \, x - 4\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 48 \, x}{3 \,{\left (x^{4} + 4 \, x^{3} + x^{2} -{\left (x^{3} - x^{2} - 6 \, x - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 6 \, x - 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x + 1)^(5/2)/(x + 1)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 62.0129, size = 162, normalized size = 2.57 \[ \begin{cases} \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) + \frac{28 \sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{8 \sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} + 5 i \log{\left (\frac{1}{x + 1} \right )} + 5 i \log{\left (x + 1 \right )} + 10 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) + \frac{28 i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{8 i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} + 5 i \log{\left (\frac{1}{x + 1} \right )} - 10 i \log{\left (\sqrt{1 - \frac{2}{x + 1}} + 1 \right )} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-x)**(5/2)/(1+x)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.238024, size = 155, normalized size = 2.46 \[ \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{6 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \sqrt{x + 1} \sqrt{-x + 1} - \frac{9 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{2 \, \sqrt{x + 1}} + \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{27 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{6 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} + 10 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x + 1)^(5/2)/(x + 1)^(5/2),x, algorithm="giac")
[Out]