3.1113 \(\int \frac{1}{(1-x)^{7/2} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=61 \[ \frac{2 \sqrt{x+1}}{15 \sqrt{1-x}}+\frac{2 \sqrt{x+1}}{15 (1-x)^{3/2}}+\frac{\sqrt{x+1}}{5 (1-x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.038021, antiderivative size = 61, 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 \[ \frac{2 \sqrt{x+1}}{15 \sqrt{1-x}}+\frac{2 \sqrt{x+1}}{15 (1-x)^{3/2}}+\frac{\sqrt{x+1}}{5 (1-x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.87342, size = 48, normalized size = 0.79 \[ \frac{2 \sqrt{x + 1}}{15 \sqrt{- x + 1}} + \frac{2 \sqrt{x + 1}}{15 \left (- x + 1\right )^{\frac{3}{2}}} + \frac{\sqrt{x + 1}}{5 \left (- x + 1\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0216574, size = 30, normalized size = 0.49 \[ -\frac{\sqrt{1-x^2} \left (2 x^2-6 x+7\right )}{15 (x-1)^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - x^2]*(7 - 6*x + 2*x^2))/(15*(-1 + x)^3)

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Maple [A]  time = 0.005, size = 25, normalized size = 0.4 \[{\frac{2\,{x}^{2}-6\,x+7}{15}\sqrt{1+x} \left ( 1-x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(1+x)^(1/2)*(2*x^2-6*x+7)/(1-x)^(5/2)

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Maxima [A]  time = 1.48748, size = 86, normalized size = 1.41 \[ -\frac{\sqrt{-x^{2} + 1}}{5 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{15 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{15 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.20957, size = 146, normalized size = 2.39 \[ \frac{9 \, x^{5} - 35 \, x^{4} + 20 \, x^{3} + 60 \, x^{2} + 5 \,{\left (x^{4} + 2 \, x^{3} - 12 \, x^{2} + 12 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 60 \, x}{15 \,{\left (x^{5} - 5 \, x^{4} + 5 \, x^{3} + 5 \, x^{2} +{\left (x^{4} - 7 \, x^{2} + 10 \, x - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 10 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(9*x^5 - 35*x^4 + 20*x^3 + 60*x^2 + 5*(x^4 + 2*x^3 - 12*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
)*sqrt(x + 1)*sqrt(-x + 1) - 10*x + 4)

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Sympy [A]  time = 179.026, size = 303, normalized size = 4.97 \[ \begin{cases} \frac{2 \left (x + 1\right )^{2}}{15 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) + 60 \sqrt{-1 + \frac{2}{x + 1}}} - \frac{10 \left (x + 1\right )}{15 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) + 60 \sqrt{-1 + \frac{2}{x + 1}}} + \frac{15}{15 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) + 60 \sqrt{-1 + \frac{2}{x + 1}}} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\- \frac{2 i \left (x + 1\right )^{2}}{15 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) + 60 \sqrt{1 - \frac{2}{x + 1}}} + \frac{10 i \left (x + 1\right )}{15 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) + 60 \sqrt{1 - \frac{2}{x + 1}}} - \frac{15 i}{15 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) + 60 \sqrt{1 - \frac{2}{x + 1}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*(x + 1)**2/(15*sqrt(-1 + 2/(x + 1))*(x + 1)**2 - 60*sqrt(-1 + 2/(x
+ 1))*(x + 1) + 60*sqrt(-1 + 2/(x + 1))) - 10*(x + 1)/(15*sqrt(-1 + 2/(x + 1))*(
x + 1)**2 - 60*sqrt(-1 + 2/(x + 1))*(x + 1) + 60*sqrt(-1 + 2/(x + 1))) + 15/(15*
sqrt(-1 + 2/(x + 1))*(x + 1)**2 - 60*sqrt(-1 + 2/(x + 1))*(x + 1) + 60*sqrt(-1 +
2/(x + 1))), 2*Abs(1/(x + 1)) > 1), (-2*I*(x + 1)**2/(15*sqrt(1 - 2/(x + 1))*(x
+ 1)**2 - 60*sqrt(1 - 2/(x + 1))*(x + 1) + 60*sqrt(1 - 2/(x + 1))) + 10*I*(x +
1)/(15*sqrt(1 - 2/(x + 1))*(x + 1)**2 - 60*sqrt(1 - 2/(x + 1))*(x + 1) + 60*sqrt
(1 - 2/(x + 1))) - 15*I/(15*sqrt(1 - 2/(x + 1))*(x + 1)**2 - 60*sqrt(1 - 2/(x +
1))*(x + 1) + 60*sqrt(1 - 2/(x + 1))), True))

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GIAC/XCAS [A]  time = 0.21118, size = 39, normalized size = 0.64 \[ -\frac{{\left (2 \,{\left (x + 1\right )}{\left (x - 4\right )} + 15\right )} \sqrt{x + 1} \sqrt{-x + 1}}{15 \,{\left (x - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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