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

Optimal. Leaf size=61 \[ \frac{(x+1)^{3/2}}{3 (1-x)^{3/2}}-\sqrt{1-x} \sqrt{x+1}-\frac{4 \sqrt{x+1}}{\sqrt{1-x}}+3 \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0698379, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{2 (x+1)^{3/2}}{\sqrt{1-x}}+\frac{(x+1)^{3/2}}{3 (1-x)^{3/2}}-3 \sqrt{1-x} \sqrt{x+1}+3 \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.02049, size = 49, normalized size = 0.8 \[ - 3 \sqrt{- x + 1} \sqrt{x + 1} + 3 \operatorname{asin}{\left (x \right )} - \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{- x + 1}} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{3 \left (- x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0708282, size = 47, normalized size = 0.77 \[ 6 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-\frac{\sqrt{1-x^2} \left (3 x^2-19 x+14\right )}{3 (x-1)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - x^2]*(14 - 19*x + 3*x^2))/(3*(-1 + x)^2) + 6*ArcSin[Sqrt[1 + x]/Sqrt[
2]]

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Maple [A]  time = 0.014, size = 83, normalized size = 1.4 \[{\frac{1}{3\, \left ( -1+x \right ) ^{2}} \left ( 9\,\arcsin \left ( x \right ){x}^{2}-3\,{x}^{2}\sqrt{-{x}^{2}+1}-18\,\arcsin \left ( x \right ) x+19\,x\sqrt{-{x}^{2}+1}+9\,\arcsin \left ( x \right ) -14\,\sqrt{-{x}^{2}+1} \right ) \sqrt{1-x}\sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(9*arcsin(x)*x^2-3*x^2*(-x^2+1)^(1/2)-18*arcsin(x)*x+19*x*(-x^2+1)^(1/2)+9*a
rcsin(x)-14*(-x^2+1)^(1/2))*(1-x)^(1/2)*(1+x)^(1/2)/(-1+x)^2/(-x^2+1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Fricas [A]  time = 0.255379, size = 220, normalized size = 3.61 \[ \frac{3 \, x^{5} - 24 \, x^{4} + 7 \, x^{3} + 54 \, x^{2} -{\left (3 \, x^{4} - 11 \, x^{3} + 54 \, x^{2} - 36 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 18 \,{\left (x^{4} - 4 \, x^{3} + x^{2} +{\left (x^{3} + x^{2} - 6 \, x + 4\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6 \, x - 4\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 36 \, x}{3 \,{\left (x^{4} - 4 \, x^{3} + x^{2} +{\left (x^{3} + x^{2} - 6 \, x + 4\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6 \, x - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*x^5 - 24*x^4 + 7*x^3 + 54*x^2 - (3*x^4 - 11*x^3 + 54*x^2 - 36*x)*sqrt(x +
1)*sqrt(-x + 1) - 18*(x^4 - 4*x^3 + x^2 + (x^3 + x^2 - 6*x + 4)*sqrt(x + 1)*sqr
t(-x + 1) + 6*x - 4)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 36*x)/(x^4 - 4*x
^3 + x^2 + (x^3 + x^2 - 6*x + 4)*sqrt(x + 1)*sqrt(-x + 1) + 6*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(x**2*(1+x)**(1/2)/(1-x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219764, size = 59, normalized size = 0.97 \[ -\frac{{\left ({\left (3 \, x - 22\right )}{\left (x + 1\right )} + 36\right )} \sqrt{x + 1} \sqrt{-x + 1}}{3 \,{\left (x - 1\right )}^{2}} + 6 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*((3*x - 22)*(x + 1) + 36)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 + 6*arcsin(1/2
*sqrt(2)*sqrt(x + 1))