3.1071 \(\int \frac{\sqrt{1+x}}{(1-x)^{7/2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{(x+1)^{3/2}}{15 (1-x)^{3/2}}+\frac{(x+1)^{3/2}}{5 (1-x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0241274, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{(x+1)^{3/2}}{15 (1-x)^{3/2}}+\frac{(x+1)^{3/2}}{5 (1-x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.01212, size = 29, normalized size = 0.71 \[ \frac{\left (x + 1\right )^{\frac{3}{2}}}{15 \left (- x + 1\right )^{\frac{3}{2}}} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{5 \left (- x + 1\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0159323, size = 28, normalized size = 0.68 \[ \frac{\sqrt{1-x^2} \left (x^2-3 x-4\right )}{15 (x-1)^3} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.006, size = 18, normalized size = 0.4 \[ -{\frac{x-4}{15} \left ( 1+x \right ) ^{{\frac{3}{2}}} \left ( 1-x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(1/2)/(1-x)^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.33895, size = 86, normalized size = 2.1 \[ -\frac{2 \, \sqrt{-x^{2} + 1}}{5 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{15 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{15 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.207174, size = 146, normalized size = 3.56 \[ \frac{3 \, x^{5} - 20 \, x^{4} + 35 \, x^{3} + 30 \, x^{2} + 5 \,{\left (x^{4} - x^{3} - 6 \, x^{2} + 12 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 60 \, x}{15 \,{\left (x^{5} - 5 \, x^{4} + 5 \, x^{3} + 5 \, x^{2} +{\left (x^{4} - 7 \, x^{2} + 10 \, x - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 10 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*x^5 - 20*x^4 + 35*x^3 + 30*x^2 + 5*(x^4 - x^3 - 6*x^2 + 12*x)*sqrt(x + 1
)*sqrt(-x + 1) - 60*x)/(x^5 - 5*x^4 + 5*x^3 + 5*x^2 + (x^4 - 7*x^2 + 10*x - 4)*s
qrt(x + 1)*sqrt(-x + 1) - 10*x + 4)

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Sympy [A]  time = 113.561, size = 173, normalized size = 4.22 \[ \begin{cases} \frac{i \left (x + 1\right )^{\frac{5}{2}}}{15 \sqrt{x - 1} \left (x + 1\right )^{2} - 60 \sqrt{x - 1} \left (x + 1\right ) + 60 \sqrt{x - 1}} - \frac{5 i \left (x + 1\right )^{\frac{3}{2}}}{15 \sqrt{x - 1} \left (x + 1\right )^{2} - 60 \sqrt{x - 1} \left (x + 1\right ) + 60 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- \frac{\left (x + 1\right )^{\frac{5}{2}}}{15 \sqrt{- x + 1} \left (x + 1\right )^{2} - 60 \sqrt{- x + 1} \left (x + 1\right ) + 60 \sqrt{- x + 1}} + \frac{5 \left (x + 1\right )^{\frac{3}{2}}}{15 \sqrt{- x + 1} \left (x + 1\right )^{2} - 60 \sqrt{- x + 1} \left (x + 1\right ) + 60 \sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((I*(x + 1)**(5/2)/(15*sqrt(x - 1)*(x + 1)**2 - 60*sqrt(x - 1)*(x + 1)
+ 60*sqrt(x - 1)) - 5*I*(x + 1)**(3/2)/(15*sqrt(x - 1)*(x + 1)**2 - 60*sqrt(x -
1)*(x + 1) + 60*sqrt(x - 1)), Abs(x + 1)/2 > 1), (-(x + 1)**(5/2)/(15*sqrt(-x +
1)*(x + 1)**2 - 60*sqrt(-x + 1)*(x + 1) + 60*sqrt(-x + 1)) + 5*(x + 1)**(3/2)/(1
5*sqrt(-x + 1)*(x + 1)**2 - 60*sqrt(-x + 1)*(x + 1) + 60*sqrt(-x + 1)), True))

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GIAC/XCAS [A]  time = 0.209052, size = 30, normalized size = 0.73 \[ \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 4\right )} \sqrt{-x + 1}}{15 \,{\left (x - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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