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

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

[Out]

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

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Rubi [A]  time = 0.0394299, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{(x+1)^{3/2}}{3 (1-x)^{3/2}}-\frac{2 \sqrt{x+1}}{\sqrt{1-x}}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.30974, size = 32, normalized size = 0.78 \[ \operatorname{asin}{\left (x \right )} - \frac{2 \sqrt{x + 1}}{\sqrt{- x + 1}} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{3 \left (- x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0423721, size = 42, normalized size = 1.02 \[ \frac{\sqrt{1-x^2} (7 x-5)}{3 (x-1)^2}+2 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-5 + 7*x)*Sqrt[1 - x^2])/(3*(-1 + x)^2) + 2*ArcSin[Sqrt[1 + x]/Sqrt[2]]

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Maple [B]  time = 0.016, size = 69, normalized size = 1.7 \[{\frac{1}{3\, \left ( -1+x \right ) ^{2}} \left ( 3\,\arcsin \left ( x \right ){x}^{2}-6\,\arcsin \left ( x \right ) x+7\,x\sqrt{-{x}^{2}+1}+3\,\arcsin \left ( x \right ) -5\,\sqrt{-{x}^{2}+1} \right ) \sqrt{1-x}\sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*arcsin(x)*x^2-6*arcsin(x)*x+7*x*(-x^2+1)^(1/2)+3*arcsin(x)-5*(-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. \[ \int \frac{\sqrt{x + 1} x}{{\left (-x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.245658, size = 163, normalized size = 3.98 \[ \frac{2 \,{\left (x^{3} - 6 \, x^{2} + 3 \,{\left (2 \, x^{2} - x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \,{\left (x^{3} -{\left (x^{2} - 3 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \, x + 2\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 3 \, x\right )}}{3 \,{\left (x^{3} -{\left (x^{2} - 3 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x^3 - 6*x^2 + 3*(2*x^2 - x)*sqrt(x + 1)*sqrt(-x + 1) - 3*(x^3 - (x^2 - 3*x
+ 2)*sqrt(x + 1)*sqrt(-x + 1) - 3*x + 2)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x
) + 3*x)/(x^3 - (x^2 - 3*x + 2)*sqrt(x + 1)*sqrt(-x + 1) - 3*x + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \sqrt{x + 1}}{\left (- x + 1\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt(x + 1)/(-x + 1)**(5/2), x)

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GIAC/XCAS [A]  time = 0.223238, size = 51, normalized size = 1.24 \[ \frac{{\left (7 \, x - 5\right )} \sqrt{x + 1} \sqrt{-x + 1}}{3 \,{\left (x - 1\right )}^{2}} + 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(7*x - 5)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 + 2*arcsin(1/2*sqrt(2)*sqrt(x +
1))