3.1116 \(\int \frac{(1-x)^{7/2}}{(1+x)^{3/2}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{2 (1-x)^{7/2}}{\sqrt{x+1}}-\frac{7}{3} \sqrt{x+1} (1-x)^{5/2}-\frac{35}{6} \sqrt{x+1} (1-x)^{3/2}-\frac{35}{2} \sqrt{x+1} \sqrt{1-x}-\frac{35}{2} \sin ^{-1}(x) \]

[Out]

(-2*(1 - x)^(7/2))/Sqrt[1 + x] - (35*Sqrt[1 - x]*Sqrt[1 + x])/2 - (35*(1 - x)^(3
/2)*Sqrt[1 + x])/6 - (7*(1 - x)^(5/2)*Sqrt[1 + x])/3 - (35*ArcSin[x])/2

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Rubi [A]  time = 0.0608198, antiderivative size = 85, 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 \[ -\frac{2 (1-x)^{7/2}}{\sqrt{x+1}}-\frac{7}{3} \sqrt{x+1} (1-x)^{5/2}-\frac{35}{6} \sqrt{x+1} (1-x)^{3/2}-\frac{35}{2} \sqrt{x+1} \sqrt{1-x}-\frac{35}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(7/2)/(1 + x)^(3/2),x]

[Out]

(-2*(1 - x)^(7/2))/Sqrt[1 + x] - (35*Sqrt[1 - x]*Sqrt[1 + x])/2 - (35*(1 - x)^(3
/2)*Sqrt[1 + x])/6 - (7*(1 - x)^(5/2)*Sqrt[1 + x])/3 - (35*ArcSin[x])/2

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Rubi in Sympy [A]  time = 8.48451, size = 73, normalized size = 0.86 \[ - \frac{2 \left (- x + 1\right )^{\frac{7}{2}}}{\sqrt{x + 1}} - \frac{7 \left (- x + 1\right )^{\frac{5}{2}} \sqrt{x + 1}}{3} - \frac{35 \left (- x + 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{6} - \frac{35 \sqrt{- x + 1} \sqrt{x + 1}}{2} - \frac{35 \operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(7/2)/(1+x)**(3/2),x)

[Out]

-2*(-x + 1)**(7/2)/sqrt(x + 1) - 7*(-x + 1)**(5/2)*sqrt(x + 1)/3 - 35*(-x + 1)**
(3/2)*sqrt(x + 1)/6 - 35*sqrt(-x + 1)*sqrt(x + 1)/2 - 35*asin(x)/2

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Mathematica [A]  time = 0.047353, size = 52, normalized size = 0.61 \[ -\frac{\sqrt{1-x} \left (2 x^3-13 x^2+55 x+166\right )}{6 \sqrt{x+1}}-35 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^(7/2)/(1 + x)^(3/2),x]

[Out]

-(Sqrt[1 - x]*(166 + 55*x - 13*x^2 + 2*x^3))/(6*Sqrt[1 + x]) - 35*ArcSin[Sqrt[1
+ x]/Sqrt[2]]

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Maple [A]  time = 0.027, size = 84, normalized size = 1. \[{\frac{2\,{x}^{4}-15\,{x}^{3}+68\,{x}^{2}+111\,x-166}{6}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}-{\frac{35\,\arcsin \left ( x \right ) }{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(7/2)/(1+x)^(3/2),x)

[Out]

1/6*(2*x^4-15*x^3+68*x^2+111*x-166)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1
-x)^(1/2)/(1+x)^(1/2)-35/2*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.5161, size = 95, normalized size = 1.12 \[ \frac{x^{4}}{3 \, \sqrt{-x^{2} + 1}} - \frac{5 \, x^{3}}{2 \, \sqrt{-x^{2} + 1}} + \frac{34 \, x^{2}}{3 \, \sqrt{-x^{2} + 1}} + \frac{37 \, x}{2 \, \sqrt{-x^{2} + 1}} - \frac{83}{3 \, \sqrt{-x^{2} + 1}} - \frac{35}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(7/2)/(x + 1)^(3/2),x, algorithm="maxima")

[Out]

1/3*x^4/sqrt(-x^2 + 1) - 5/2*x^3/sqrt(-x^2 + 1) + 34/3*x^2/sqrt(-x^2 + 1) + 37/2
*x/sqrt(-x^2 + 1) - 83/3/sqrt(-x^2 + 1) - 35/2*arcsin(x)

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Fricas [A]  time = 0.213308, size = 251, normalized size = 2.95 \[ \frac{2 \, x^{7} - 7 \, x^{6} + 261 \, x^{4} + 624 \, x^{3} - 324 \, x^{2} -{\left (2 \, x^{6} - 21 \, x^{5} + 99 \, x^{4} + 180 \, x^{3} - 324 \, x^{2} - 888 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 210 \,{\left (x^{4} - 3 \, x^{3} - 8 \, x^{2} +{\left (x^{3} + 4 \, x^{2} - 4 \, x - 8\right )} \sqrt{x + 1} \sqrt{-x + 1} + 4 \, x + 8\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 888 \, x}{6 \,{\left (x^{4} - 3 \, x^{3} - 8 \, x^{2} +{\left (x^{3} + 4 \, x^{2} - 4 \, x - 8\right )} \sqrt{x + 1} \sqrt{-x + 1} + 4 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(7/2)/(x + 1)^(3/2),x, algorithm="fricas")

[Out]

1/6*(2*x^7 - 7*x^6 + 261*x^4 + 624*x^3 - 324*x^2 - (2*x^6 - 21*x^5 + 99*x^4 + 18
0*x^3 - 324*x^2 - 888*x)*sqrt(x + 1)*sqrt(-x + 1) + 210*(x^4 - 3*x^3 - 8*x^2 + (
x^3 + 4*x^2 - 4*x - 8)*sqrt(x + 1)*sqrt(-x + 1) + 4*x + 8)*arctan((sqrt(x + 1)*s
qrt(-x + 1) - 1)/x) - 888*x)/(x^4 - 3*x^3 - 8*x^2 + (x^3 + 4*x^2 - 4*x - 8)*sqrt
(x + 1)*sqrt(-x + 1) + 4*x + 8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-x)**(7/2)/(1+x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23653, size = 109, normalized size = 1.28 \[ -\frac{1}{6} \,{\left ({\left (2 \, x - 17\right )}{\left (x + 1\right )} + 87\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{8 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{\sqrt{x + 1}} - \frac{8 \, \sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}} - 35 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(7/2)/(x + 1)^(3/2),x, algorithm="giac")

[Out]

-1/6*((2*x - 17)*(x + 1) + 87)*sqrt(x + 1)*sqrt(-x + 1) + 8*(sqrt(2) - sqrt(-x +
1))/sqrt(x + 1) - 8*sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 35*arcsin(1/2*sqrt(2
)*sqrt(x + 1))