3.1127 \(\int \frac{(1-x)^{7/2}}{(1+x)^{5/2}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{2 (1-x)^{7/2}}{3 (x+1)^{3/2}}+\frac{14 (1-x)^{5/2}}{3 \sqrt{x+1}}+\frac{35}{6} \sqrt{x+1} (1-x)^{3/2}+\frac{35}{2} \sqrt{x+1} \sqrt{1-x}+\frac{35}{2} \sin ^{-1}(x) \]

[Out]

(-2*(1 - x)^(7/2))/(3*(1 + x)^(3/2)) + (14*(1 - x)^(5/2))/(3*Sqrt[1 + x]) + (35*
Sqrt[1 - x]*Sqrt[1 + x])/2 + (35*(1 - x)^(3/2)*Sqrt[1 + x])/6 + (35*ArcSin[x])/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0603658, antiderivative size = 87, 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 \[ -\frac{2 (1-x)^{7/2}}{3 (x+1)^{3/2}}+\frac{14 (1-x)^{5/2}}{3 \sqrt{x+1}}+\frac{35}{6} \sqrt{x+1} (1-x)^{3/2}+\frac{35}{2} \sqrt{x+1} \sqrt{1-x}+\frac{35}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(7/2)/(1 + x)^(5/2),x]

[Out]

(-2*(1 - x)^(7/2))/(3*(1 + x)^(3/2)) + (14*(1 - x)^(5/2))/(3*Sqrt[1 + x]) + (35*
Sqrt[1 - x]*Sqrt[1 + x])/2 + (35*(1 - x)^(3/2)*Sqrt[1 + x])/6 + (35*ArcSin[x])/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.77099, size = 73, normalized size = 0.84 \[ - \frac{2 \left (- x + 1\right )^{\frac{7}{2}}}{3 \left (x + 1\right )^{\frac{3}{2}}} + \frac{14 \left (- x + 1\right )^{\frac{5}{2}}}{3 \sqrt{x + 1}} + \frac{35 \left (- x + 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{6} + \frac{35 \sqrt{- x + 1} \sqrt{x + 1}}{2} + \frac{35 \operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(7/2)/(1+x)**(5/2),x)

[Out]

-2*(-x + 1)**(7/2)/(3*(x + 1)**(3/2)) + 14*(-x + 1)**(5/2)/(3*sqrt(x + 1)) + 35*
(-x + 1)**(3/2)*sqrt(x + 1)/6 + 35*sqrt(-x + 1)*sqrt(x + 1)/2 + 35*asin(x)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.0550758, size = 52, normalized size = 0.6 \[ \frac{\sqrt{1-x} \left (-3 x^3+30 x^2+229 x+164\right )}{6 (x+1)^{3/2}}+35 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^(7/2)/(1 + x)^(5/2),x]

[Out]

(Sqrt[1 - x]*(164 + 229*x + 30*x^2 - 3*x^3))/(6*(1 + x)^(3/2)) + 35*ArcSin[Sqrt[
1 + x]/Sqrt[2]]

_______________________________________________________________________________________

Maple [A]  time = 0.033, size = 84, normalized size = 1. \[{\frac{3\,{x}^{4}-33\,{x}^{3}-199\,{x}^{2}+65\,x+164}{6}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) } \left ( 1+x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}}+{\frac{35\,\arcsin \left ( x \right ) }{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(7/2)/(1+x)^(5/2),x)

[Out]

1/6*(3*x^4-33*x^3-199*x^2+65*x+164)/(1+x)^(3/2)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(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)

_______________________________________________________________________________________

Maxima [A]  time = 1.48719, size = 150, normalized size = 1.72 \[ -\frac{x^{5}}{2 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{6 \, x^{4}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{35}{6} \, x{\left (\frac{3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{2}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}\right )} - \frac{61 \, x}{6 \, \sqrt{-x^{2} + 1}} - \frac{44 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{16 \, x}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{82}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{35}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(7/2)/(x + 1)^(5/2),x, algorithm="maxima")

[Out]

-1/2*x^5/(-x^2 + 1)^(3/2) + 6*x^4/(-x^2 + 1)^(3/2) + 35/6*x*(3*x^2/(-x^2 + 1)^(3
/2) - 2/(-x^2 + 1)^(3/2)) - 61/6*x/sqrt(-x^2 + 1) - 44*x^2/(-x^2 + 1)^(3/2) + 16
/3*x/(-x^2 + 1)^(3/2) + 82/3/(-x^2 + 1)^(3/2) + 35/2*arcsin(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.211501, size = 271, normalized size = 3.11 \[ \frac{3 \, x^{7} - 21 \, x^{6} - 179 \, x^{5} - 951 \, x^{4} - 320 \, x^{3} + 1332 \, x^{2} -{\left (3 \, x^{6} - 42 \, x^{5} - 285 \, x^{4} + 76 \, x^{3} + 1332 \, x^{2} + 792 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 210 \,{\left (x^{5} - 2 \, x^{4} - 11 \, x^{3} - 4 \, x^{2} +{\left (x^{4} + 5 \, x^{3} - 12 \, x - 8\right )} \sqrt{x + 1} \sqrt{-x + 1} + 12 \, x + 8\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 792 \, x}{6 \,{\left (x^{5} - 2 \, x^{4} - 11 \, x^{3} - 4 \, x^{2} +{\left (x^{4} + 5 \, x^{3} - 12 \, x - 8\right )} \sqrt{x + 1} \sqrt{-x + 1} + 12 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(7/2)/(x + 1)^(5/2),x, algorithm="fricas")

[Out]

1/6*(3*x^7 - 21*x^6 - 179*x^5 - 951*x^4 - 320*x^3 + 1332*x^2 - (3*x^6 - 42*x^5 -
285*x^4 + 76*x^3 + 1332*x^2 + 792*x)*sqrt(x + 1)*sqrt(-x + 1) - 210*(x^5 - 2*x^
4 - 11*x^3 - 4*x^2 + (x^4 + 5*x^3 - 12*x - 8)*sqrt(x + 1)*sqrt(-x + 1) + 12*x +
8)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 792*x)/(x^5 - 2*x^4 - 11*x^3 - 4*x
^2 + (x^4 + 5*x^3 - 12*x - 8)*sqrt(x + 1)*sqrt(-x + 1) + 12*x + 8)

_______________________________________________________________________________________

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)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.247209, size = 161, normalized size = 1.85 \[ -\frac{1}{2} \, \sqrt{x + 1}{\left (x - 12\right )} \sqrt{-x + 1} + \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{3 \,{\left (x + 1\right )}^{\frac{3}{2}}} - \frac{13 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{\sqrt{x + 1}} + \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{39 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{3 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} + 35 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(7/2)/(x + 1)^(5/2),x, algorithm="giac")

[Out]

-1/2*sqrt(x + 1)*(x - 12)*sqrt(-x + 1) + 1/3*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^
(3/2) - 13*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + 1/3*(x + 1)^(3/2)*(39*(sqrt(2)
- sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) - sqrt(-x + 1))^3 + 35*arcsin(1/2*sqrt(
2)*sqrt(x + 1))