3.707 \(\int \frac{\sqrt{1-x} \sqrt{2+3 x}}{\sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=35 \[ \sqrt{x+1} \sqrt{3 x+2}-\frac{\sinh ^{-1}\left (\sqrt{3 x+2}\right )}{\sqrt{3}} \]

[Out]

Sqrt[1 + x]*Sqrt[2 + 3*x] - ArcSinh[Sqrt[2 + 3*x]]/Sqrt[3]

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Rubi [A]  time = 0.0287898, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \sqrt{x+1} \sqrt{3 x+2}-\frac{\sinh ^{-1}\left (\sqrt{3 x+2}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 + x]*Sqrt[2 + 3*x] - ArcSinh[Sqrt[2 + 3*x]]/Sqrt[3]

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Rubi in Sympy [A]  time = 2.95537, size = 31, normalized size = 0.89 \[ \sqrt{x + 1} \sqrt{3 x + 2} - \frac{\sqrt{3} \operatorname{asinh}{\left (\sqrt{3 x + 2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0748978, size = 79, normalized size = 2.26 \[ \frac{\sqrt{1-x} \left (3 \sqrt{3 x+2} (x+1)+\sqrt{3} \sqrt{-x-1} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{-x-1}}{\sqrt{3 x+2}}\right )\right )}{3 \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.014, size = 86, normalized size = 2.5 \[{\frac{1}{-6+6\,x}\sqrt{1-x}\sqrt{2+3\,x}\sqrt{-{x}^{2}+1} \left ( \ln \left ({\frac{5\,\sqrt{3}}{6}}+x\sqrt{3}+\sqrt{3\,{x}^{2}+5\,x+2} \right ) \sqrt{3}-6\,\sqrt{3\,{x}^{2}+5\,x+2} \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(1-x)^(1/2)*(2+3*x)^(1/2)*(-x^2+1)^(1/2)*(ln(5/6*3^(1/2)+x*3^(1/2)+(3*x^2+5*
x+2)^(1/2))*3^(1/2)-6*(3*x^2+5*x+2)^(1/2))/(-1+x)/(3*x^2+5*x+2)^(1/2)

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Maxima [A]  time = 0.780965, size = 55, normalized size = 1.57 \[ -\frac{1}{6} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + sqrt(3*x^2 + 5*x +
2)

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Fricas [A]  time = 0.302, size = 138, normalized size = 3.94 \[ -\frac{\sqrt{3}{\left (4 \, \sqrt{3} \sqrt{-x^{2} + 1} \sqrt{3 \, x + 2} \sqrt{-x + 1} -{\left (x - 1\right )} \log \left (-\frac{12 \, \sqrt{-x^{2} + 1}{\left (6 \, x + 5\right )} \sqrt{3 \, x + 2} \sqrt{-x + 1} + \sqrt{3}{\left (72 \, x^{3} + 48 \, x^{2} - 71 \, x - 49\right )}}{x - 1}\right )\right )}}{12 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*sqrt(3)*(4*sqrt(3)*sqrt(-x^2 + 1)*sqrt(3*x + 2)*sqrt(-x + 1) - (x - 1)*log
(-(12*sqrt(-x^2 + 1)*(6*x + 5)*sqrt(3*x + 2)*sqrt(-x + 1) + sqrt(3)*(72*x^3 + 48
*x^2 - 71*x - 49))/(x - 1)))/(x - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- x + 1} \sqrt{3 x + 2}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-x + 1)*sqrt(3*x + 2)/sqrt(-(x - 1)*(x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{3 \, x + 2} \sqrt{-x + 1}}{\sqrt{-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(3*x + 2)*sqrt(-x + 1)/sqrt(-x^2 + 1), x)