3.1126 \(\int \frac{(1-x)^{9/2}}{(1+x)^{5/2}} \, dx\)

Optimal. Leaf size=103 \[ -\frac{2 (1-x)^{9/2}}{3 (x+1)^{3/2}}+\frac{6 (1-x)^{7/2}}{\sqrt{x+1}}+7 \sqrt{x+1} (1-x)^{5/2}+\frac{35}{2} \sqrt{x+1} (1-x)^{3/2}+\frac{105}{2} \sqrt{x+1} \sqrt{1-x}+\frac{105}{2} \sin ^{-1}(x) \]

[Out]

(-2*(1 - x)^(9/2))/(3*(1 + x)^(3/2)) + (6*(1 - x)^(7/2))/Sqrt[1 + x] + (105*Sqrt
[1 - x]*Sqrt[1 + x])/2 + (35*(1 - x)^(3/2)*Sqrt[1 + x])/2 + 7*(1 - x)^(5/2)*Sqrt
[1 + x] + (105*ArcSin[x])/2

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Rubi [A]  time = 0.0760705, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{2 (1-x)^{9/2}}{3 (x+1)^{3/2}}+\frac{6 (1-x)^{7/2}}{\sqrt{x+1}}+7 \sqrt{x+1} (1-x)^{5/2}+\frac{35}{2} \sqrt{x+1} (1-x)^{3/2}+\frac{105}{2} \sqrt{x+1} \sqrt{1-x}+\frac{105}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(9/2)/(1 + x)^(5/2),x]

[Out]

(-2*(1 - x)^(9/2))/(3*(1 + x)^(3/2)) + (6*(1 - x)^(7/2))/Sqrt[1 + x] + (105*Sqrt
[1 - x]*Sqrt[1 + x])/2 + (35*(1 - x)^(3/2)*Sqrt[1 + x])/2 + 7*(1 - x)^(5/2)*Sqrt
[1 + x] + (105*ArcSin[x])/2

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Rubi in Sympy [A]  time = 10.4849, size = 87, normalized size = 0.84 \[ - \frac{2 \left (- x + 1\right )^{\frac{9}{2}}}{3 \left (x + 1\right )^{\frac{3}{2}}} + \frac{6 \left (- x + 1\right )^{\frac{7}{2}}}{\sqrt{x + 1}} + 7 \left (- x + 1\right )^{\frac{5}{2}} \sqrt{x + 1} + \frac{35 \left (- x + 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{2} + \frac{105 \sqrt{- x + 1} \sqrt{x + 1}}{2} + \frac{105 \operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(9/2)/(1+x)**(5/2),x)

[Out]

-2*(-x + 1)**(9/2)/(3*(x + 1)**(3/2)) + 6*(-x + 1)**(7/2)/sqrt(x + 1) + 7*(-x +
1)**(5/2)*sqrt(x + 1) + 35*(-x + 1)**(3/2)*sqrt(x + 1)/2 + 105*sqrt(-x + 1)*sqrt
(x + 1)/2 + 105*asin(x)/2

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Mathematica [A]  time = 0.0559897, size = 57, normalized size = 0.55 \[ \frac{\sqrt{1-x} \left (2 x^4-17 x^3+102 x^2+679 x+494\right )}{6 (x+1)^{3/2}}+105 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^(9/2)/(1 + x)^(5/2),x]

[Out]

(Sqrt[1 - x]*(494 + 679*x + 102*x^2 - 17*x^3 + 2*x^4))/(6*(1 + x)^(3/2)) + 105*A
rcSin[Sqrt[1 + x]/Sqrt[2]]

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Maple [A]  time = 0.033, size = 89, normalized size = 0.9 \[ -{\frac{2\,{x}^{5}-19\,{x}^{4}+119\,{x}^{3}+577\,{x}^{2}-185\,x-494}{6}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) } \left ( 1+x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}}+{\frac{105\,\arcsin \left ( x \right ) }{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(9/2)/(1+x)^(5/2),x)

[Out]

-1/6*(2*x^5-19*x^4+119*x^3+577*x^2-185*x-494)/(1+x)^(3/2)/(-(1+x)*(-1+x))^(1/2)*
((1+x)*(1-x))^(1/2)/(1-x)^(1/2)+105/2*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2
)*arcsin(x)

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Maxima [A]  time = 1.49945, size = 169, normalized size = 1.64 \[ \frac{x^{6}}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{7 \, x^{5}}{2 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{23 \, x^{4}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{35}{2} \, x{\left (\frac{3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{2}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}\right )} - \frac{143 \, x}{6 \, \sqrt{-x^{2} + 1}} - \frac{127 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{22 \, x}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{247}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{105}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(9/2)/(x + 1)^(5/2),x, algorithm="maxima")

[Out]

1/3*x^6/(-x^2 + 1)^(3/2) - 7/2*x^5/(-x^2 + 1)^(3/2) + 23*x^4/(-x^2 + 1)^(3/2) +
35/2*x*(3*x^2/(-x^2 + 1)^(3/2) - 2/(-x^2 + 1)^(3/2)) - 143/6*x/sqrt(-x^2 + 1) -
127*x^2/(-x^2 + 1)^(3/2) + 22/3*x/(-x^2 + 1)^(3/2) + 247/3/(-x^2 + 1)^(3/2) + 10
5/2*arcsin(x)

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Fricas [A]  time = 0.212936, size = 340, normalized size = 3.3 \[ \frac{2 \, x^{9} - 27 \, x^{8} + 171 \, x^{7} + 839 \, x^{6} - 1077 \, x^{5} - 7512 \, x^{4} - 3116 \, x^{3} + 7752 \, x^{2} +{\left (2 \, x^{8} - 9 \, x^{7} + 10 \, x^{6} + 781 \, x^{5} + 3636 \, x^{4} + 644 \, x^{3} - 7752 \, x^{2} - 4944 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 630 \,{\left (x^{6} + 6 \, x^{5} - 3 \, x^{4} - 28 \, x^{3} - 12 \, x^{2} -{\left (x^{5} - 3 \, x^{4} - 16 \, x^{3} - 4 \, x^{2} + 24 \, x + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 24 \, x + 16\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 4944 \, x}{6 \,{\left (x^{6} + 6 \, x^{5} - 3 \, x^{4} - 28 \, x^{3} - 12 \, x^{2} -{\left (x^{5} - 3 \, x^{4} - 16 \, x^{3} - 4 \, x^{2} + 24 \, x + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 24 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(9/2)/(x + 1)^(5/2),x, algorithm="fricas")

[Out]

1/6*(2*x^9 - 27*x^8 + 171*x^7 + 839*x^6 - 1077*x^5 - 7512*x^4 - 3116*x^3 + 7752*
x^2 + (2*x^8 - 9*x^7 + 10*x^6 + 781*x^5 + 3636*x^4 + 644*x^3 - 7752*x^2 - 4944*x
)*sqrt(x + 1)*sqrt(-x + 1) - 630*(x^6 + 6*x^5 - 3*x^4 - 28*x^3 - 12*x^2 - (x^5 -
3*x^4 - 16*x^3 - 4*x^2 + 24*x + 16)*sqrt(x + 1)*sqrt(-x + 1) + 24*x + 16)*arcta
n((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 4944*x)/(x^6 + 6*x^5 - 3*x^4 - 28*x^3 - 12
*x^2 - (x^5 - 3*x^4 - 16*x^3 - 4*x^2 + 24*x + 16)*sqrt(x + 1)*sqrt(-x + 1) + 24*
x + 16)

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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)**(9/2)/(1+x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.257099, size = 171, normalized size = 1.66 \[ \frac{1}{6} \,{\left ({\left (2 \, x - 23\right )}{\left (x + 1\right )} + 165\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{2 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{3 \,{\left (x + 1\right )}^{\frac{3}{2}}} - \frac{34 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{\sqrt{x + 1}} + \frac{2 \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{51 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{3 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} + 105 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(9/2)/(x + 1)^(5/2),x, algorithm="giac")

[Out]

1/6*((2*x - 23)*(x + 1) + 165)*sqrt(x + 1)*sqrt(-x + 1) + 2/3*(sqrt(2) - sqrt(-x
+ 1))^3/(x + 1)^(3/2) - 34*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + 2/3*(x + 1)^(
3/2)*(51*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) - sqrt(-x + 1))^3 + 10
5*arcsin(1/2*sqrt(2)*sqrt(x + 1))