3.1123 \(\int \frac{1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=62 \[ \frac{2 x}{5 \sqrt{1-x} \sqrt{x+1}}+\frac{1}{5 (1-x)^{3/2} \sqrt{x+1}}+\frac{1}{5 (1-x)^{5/2} \sqrt{x+1}} \]

[Out]

1/(5*(1 - x)^(5/2)*Sqrt[1 + x]) + 1/(5*(1 - x)^(3/2)*Sqrt[1 + x]) + (2*x)/(5*Sqr
t[1 - x]*Sqrt[1 + x])

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Rubi [A]  time = 0.0369049, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 x}{5 \sqrt{1-x} \sqrt{x+1}}+\frac{1}{5 (1-x)^{3/2} \sqrt{x+1}}+\frac{1}{5 (1-x)^{5/2} \sqrt{x+1}} \]

Antiderivative was successfully verified.

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

[Out]

1/(5*(1 - x)^(5/2)*Sqrt[1 + x]) + 1/(5*(1 - x)^(3/2)*Sqrt[1 + x]) + (2*x)/(5*Sqr
t[1 - x]*Sqrt[1 + x])

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Rubi in Sympy [A]  time = 4.93796, size = 51, normalized size = 0.82 \[ \frac{2 x}{5 \sqrt{- x + 1} \sqrt{x + 1}} + \frac{1}{5 \left (- x + 1\right )^{\frac{3}{2}} \sqrt{x + 1}} + \frac{1}{5 \left (- x + 1\right )^{\frac{5}{2}} \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0251305, size = 33, normalized size = 0.53 \[ \frac{2 x^3-4 x^2+x+2}{5 (1-x)^{5/2} \sqrt{x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 28, normalized size = 0.5 \[{\frac{2\,{x}^{3}-4\,{x}^{2}+x+2}{5} \left ( 1-x \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34705, size = 107, normalized size = 1.73 \[ \frac{2 \, x}{5 \, \sqrt{-x^{2} + 1}} + \frac{1}{5 \,{\left (\sqrt{-x^{2} + 1} x^{2} - 2 \, \sqrt{-x^{2} + 1} x + \sqrt{-x^{2} + 1}\right )}} - \frac{1}{5 \,{\left (\sqrt{-x^{2} + 1} x - \sqrt{-x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*x/sqrt(-x^2 + 1) + 1/5/(sqrt(-x^2 + 1)*x^2 - 2*sqrt(-x^2 + 1)*x + sqrt(-x^2
+ 1)) - 1/5/(sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1))

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Fricas [A]  time = 0.20904, size = 178, normalized size = 2.87 \[ \frac{2 \, x^{6} + 2 \, x^{5} - 20 \, x^{4} + 15 \, x^{3} + 20 \, x^{2} -{\left (2 \, x^{5} - 10 \, x^{4} + 5 \, x^{3} + 20 \, x^{2} - 20 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 20 \, x}{5 \,{\left (x^{6} - 2 \, x^{5} - 4 \, x^{4} + 10 \, x^{3} - x^{2} +{\left (3 \, x^{4} - 6 \, x^{3} - x^{2} + 8 \, x - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 8 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(2*x^6 + 2*x^5 - 20*x^4 + 15*x^3 + 20*x^2 - (2*x^5 - 10*x^4 + 5*x^3 + 20*x^2
- 20*x)*sqrt(x + 1)*sqrt(-x + 1) - 20*x)/(x^6 - 2*x^5 - 4*x^4 + 10*x^3 - x^2 +
(3*x^4 - 6*x^3 - x^2 + 8*x - 4)*sqrt(x + 1)*sqrt(-x + 1) - 8*x + 4)

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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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.210014, size = 99, normalized size = 1.6 \[ \frac{\sqrt{2} - \sqrt{-x + 1}}{16 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1}}{16 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} - \frac{{\left ({\left (11 \, x - 39\right )}{\left (x + 1\right )} + 60\right )} \sqrt{x + 1} \sqrt{-x + 1}}{40 \,{\left (x - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/16*sqrt(x + 1)/(sqrt(2) - sqrt(-x
+ 1)) - 1/40*((11*x - 39)*(x + 1) + 60)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^3