3.1134 \(\int \frac{1}{(1-x)^{7/2} (1+x)^{5/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{8 x}{15 \sqrt{1-x} \sqrt{x+1}}+\frac{4 x}{15 (1-x)^{3/2} (x+1)^{3/2}}+\frac{1}{5 (1-x)^{5/2} (x+1)^{3/2}} \]

[Out]

1/(5*(1 - x)^(5/2)*(1 + x)^(3/2)) + (4*x)/(15*(1 - x)^(3/2)*(1 + x)^(3/2)) + (8*
x)/(15*Sqrt[1 - x]*Sqrt[1 + x])

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Rubi [A]  time = 0.0367401, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{8 x}{15 \sqrt{1-x} \sqrt{x+1}}+\frac{4 x}{15 (1-x)^{3/2} (x+1)^{3/2}}+\frac{1}{5 (1-x)^{5/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

1/(5*(1 - x)^(5/2)*(1 + x)^(3/2)) + (4*x)/(15*(1 - x)^(3/2)*(1 + x)^(3/2)) + (8*
x)/(15*Sqrt[1 - x]*Sqrt[1 + x])

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Rubi in Sympy [A]  time = 5.38036, size = 53, normalized size = 0.84 \[ \frac{8 x}{15 \sqrt{- x + 1} \sqrt{x + 1}} + \frac{4 x}{15 \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}} + \frac{1}{5 \left (- x + 1\right )^{\frac{5}{2}} \left (x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*x/(15*sqrt(-x + 1)*sqrt(x + 1)) + 4*x/(15*(-x + 1)**(3/2)*(x + 1)**(3/2)) + 1/
(5*(-x + 1)**(5/2)*(x + 1)**(3/2))

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Mathematica [A]  time = 0.0313996, size = 40, normalized size = 0.63 \[ \frac{8 x^4-8 x^3-12 x^2+12 x+3}{15 (1-x)^{5/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(3 + 12*x - 12*x^2 - 8*x^3 + 8*x^4)/(15*(1 - x)^(5/2)*(1 + x)^(3/2))

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Maple [A]  time = 0.004, size = 35, normalized size = 0.6 \[{\frac{8\,{x}^{4}-8\,{x}^{3}-12\,{x}^{2}+12\,x+3}{15} \left ( 1-x \right ) ^{-{\frac{5}{2}}} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(8*x^4-8*x^3-12*x^2+12*x+3)/(1+x)^(3/2)/(1-x)^(5/2)

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Maxima [A]  time = 1.34622, size = 70, normalized size = 1.11 \[ \frac{8 \, x}{15 \, \sqrt{-x^{2} + 1}} + \frac{4 \, x}{15 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{1}{5 \,{\left ({\left (-x^{2} + 1\right )}^{\frac{3}{2}} x -{\left (-x^{2} + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/15*x/sqrt(-x^2 + 1) + 4/15*x/(-x^2 + 1)^(3/2) - 1/5/((-x^2 + 1)^(3/2)*x - (-x^
2 + 1)^(3/2))

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Fricas [A]  time = 0.210478, size = 232, normalized size = 3.68 \[ -\frac{8 \, x^{8} - 20 \, x^{7} - 64 \, x^{6} + 124 \, x^{5} + 115 \, x^{4} - 220 \, x^{3} - 60 \, x^{2} +{\left (3 \, x^{7} + 29 \, x^{6} - 59 \, x^{5} - 85 \, x^{4} + 160 \, x^{3} + 60 \, x^{2} - 120 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 120 \, x}{15 \,{\left (4 \, x^{7} - 4 \, x^{6} - 16 \, x^{5} + 16 \, x^{4} + 20 \, x^{3} - 20 \, x^{2} -{\left (x^{7} - x^{6} - 9 \, x^{5} + 9 \, x^{4} + 16 \, x^{3} - 16 \, x^{2} - 8 \, x + 8\right )} \sqrt{x + 1} \sqrt{-x + 1} - 8 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(8*x^8 - 20*x^7 - 64*x^6 + 124*x^5 + 115*x^4 - 220*x^3 - 60*x^2 + (3*x^7 +
29*x^6 - 59*x^5 - 85*x^4 + 160*x^3 + 60*x^2 - 120*x)*sqrt(x + 1)*sqrt(-x + 1) +
120*x)/(4*x^7 - 4*x^6 - 16*x^5 + 16*x^4 + 20*x^3 - 20*x^2 - (x^7 - x^6 - 9*x^5
+ 9*x^4 + 16*x^3 - 16*x^2 - 8*x + 8)*sqrt(x + 1)*sqrt(-x + 1) - 8*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/(1-x)**(7/2)/(1+x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213489, size = 161, normalized size = 2.56 \[ \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{384 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{15 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{128 \, \sqrt{x + 1}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{45 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{384 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} - \frac{{\left ({\left (73 \, x - 247\right )}{\left (x + 1\right )} + 360\right )} \sqrt{x + 1} \sqrt{-x + 1}}{240 \,{\left (x - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 15/128*(sqrt(2) - sqrt(-x + 1))
/sqrt(x + 1) - 1/384*(x + 1)^(3/2)*(45*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(
sqrt(2) - sqrt(-x + 1))^3 - 1/240*((73*x - 247)*(x + 1) + 360)*sqrt(x + 1)*sqrt(
-x + 1)/(x - 1)^3