3.1128 \(\int \frac{(1-x)^{5/2}}{(1+x)^{5/2}} \, dx\)

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]

(-2*(1 - x)^(5/2))/(3*(1 + x)^(3/2)) + (10*(1 - x)^(3/2))/(3*Sqrt[1 + x]) + 5*Sq
rt[1 - x]*Sqrt[1 + x] + 5*ArcSin[x]

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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]

(-2*(1 - x)^(5/2))/(3*(1 + x)^(3/2)) + (10*(1 - x)^(3/2))/(3*Sqrt[1 + x]) + 5*Sq
rt[1 - x]*Sqrt[1 + x] + 5*ArcSin[x]

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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]

-2*(-x + 1)**(5/2)/(3*(x + 1)**(3/2)) + 10*(-x + 1)**(3/2)/(3*sqrt(x + 1)) + 5*s
qrt(-x + 1)*sqrt(x + 1) + 5*asin(x)

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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]

(Sqrt[1 - x]*(23 + 34*x + 3*x^2))/(3*(1 + x)^(3/2)) + 10*ArcSin[Sqrt[1 + x]/Sqrt
[2]]

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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]

-1/3*(3*x^3+31*x^2-11*x-23)/(1+x)^(3/2)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2
)/(1-x)^(1/2)+5*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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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]

(-x^2 + 1)^(5/2)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) - 5/3*(-x^2 + 1)^(3/2)/(x^3 + 3
*x^2 + 3*x + 1) - 10/3*sqrt(-x^2 + 1)/(x^2 + 2*x + 1) + 35/3*sqrt(-x^2 + 1)/(x +
1) + 5*arcsin(x)

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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]

1/3*(3*x^5 + 48*x^4 + 7*x^3 - 102*x^2 + (3*x^4 + 17*x^3 + 102*x^2 + 48*x)*sqrt(x
+ 1)*sqrt(-x + 1) - 30*(x^4 + 4*x^3 + x^2 - (x^3 - x^2 - 6*x - 4)*sqrt(x + 1)*s
qrt(-x + 1) - 6*x - 4)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 48*x)/(x^4 + 4
*x^3 + x^2 - (x^3 - x^2 - 6*x - 4)*sqrt(x + 1)*sqrt(-x + 1) - 6*x - 4)

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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]

Piecewise((sqrt(-1 + 2/(x + 1))*(x + 1) + 28*sqrt(-1 + 2/(x + 1))/3 - 8*sqrt(-1
+ 2/(x + 1))/(3*(x + 1)) + 5*I*log(1/(x + 1)) + 5*I*log(x + 1) + 10*asin(sqrt(2)
*sqrt(x + 1)/2), 2*Abs(1/(x + 1)) > 1), (I*sqrt(1 - 2/(x + 1))*(x + 1) + 28*I*sq
rt(1 - 2/(x + 1))/3 - 8*I*sqrt(1 - 2/(x + 1))/(3*(x + 1)) + 5*I*log(1/(x + 1)) -
10*I*log(sqrt(1 - 2/(x + 1)) + 1), True))

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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]

1/6*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + sqrt(x + 1)*sqrt(-x + 1) - 9/2*(s
qrt(2) - sqrt(-x + 1))/sqrt(x + 1) + 1/6*(x + 1)^(3/2)*(27*(sqrt(2) - sqrt(-x +
1))^2/(x + 1) - 1)/(sqrt(2) - sqrt(-x + 1))^3 + 10*arcsin(1/2*sqrt(2)*sqrt(x + 1
))