3.2445 \(\int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=50 \[ \frac{11 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2 \sqrt{10}}-\frac{1}{2} \sqrt{1-2 x} \sqrt{5 x+3} \]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2*Sqrt
[10])

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Rubi [A]  time = 0.0435897, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{11 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2 \sqrt{10}}-\frac{1}{2} \sqrt{1-2 x} \sqrt{5 x+3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[3 + 5*x]/Sqrt[1 - 2*x],x]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2*Sqrt
[10])

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Rubi in Sympy [A]  time = 4.76494, size = 42, normalized size = 0.84 \[ - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2} + \frac{11 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**(1/2)/(1-2*x)**(1/2),x)

[Out]

-sqrt(-2*x + 1)*sqrt(5*x + 3)/2 + 11*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/20

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Mathematica [A]  time = 0.0340056, size = 50, normalized size = 1. \[ -\frac{1}{2} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{11 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[3 + 5*x]/Sqrt[1 - 2*x],x]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2 - (11*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(2*Sqrt
[10])

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Maple [A]  time = 0.006, size = 56, normalized size = 1.1 \[ -{\frac{1}{2}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{11\,\sqrt{10}}{40}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^(1/2)/(1-2*x)^(1/2),x)

[Out]

-1/2*(1-2*x)^(1/2)*(3+5*x)^(1/2)+11/40*((1-2*x)*(3+5*x))^(1/2)/(3+5*x)^(1/2)/(1-
2*x)^(1/2)*10^(1/2)*arcsin(20/11*x+1/11)

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Maxima [A]  time = 1.50389, size = 39, normalized size = 0.78 \[ \frac{11}{40} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1}{2} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)/sqrt(-2*x + 1),x, algorithm="maxima")

[Out]

11/40*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/2*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.220745, size = 70, normalized size = 1.4 \[ -\frac{1}{40} \, \sqrt{10}{\left (2 \, \sqrt{10} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 11 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)/sqrt(-2*x + 1),x, algorithm="fricas")

[Out]

-1/40*sqrt(10)*(2*sqrt(10)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 11*arctan(1/20*sqrt(10
)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 3.09506, size = 141, normalized size = 2.82 \[ \begin{cases} - \frac{5 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{\sqrt{10 x - 5}} + \frac{11 i \sqrt{x + \frac{3}{5}}}{2 \sqrt{10 x - 5}} - \frac{11 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{20} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{11 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{20} + \frac{5 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{\sqrt{- 10 x + 5}} - \frac{11 \sqrt{x + \frac{3}{5}}}{2 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3+5*x)**(1/2)/(1-2*x)**(1/2),x)

[Out]

Piecewise((-5*I*(x + 3/5)**(3/2)/sqrt(10*x - 5) + 11*I*sqrt(x + 3/5)/(2*sqrt(10*
x - 5)) - 11*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/20, 10*Abs(x + 3/5)/11
> 1), (11*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/20 + 5*(x + 3/5)**(3/2)/sqr
t(-10*x + 5) - 11*sqrt(x + 3/5)/(2*sqrt(-10*x + 5)), True))

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GIAC/XCAS [A]  time = 0.228127, size = 54, normalized size = 1.08 \[ \frac{1}{20} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)/sqrt(-2*x + 1),x, algorithm="giac")

[Out]

1/20*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 2*sqrt(5*x + 3)*s
qrt(-10*x + 5))