3.2287 \(\int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.043501, 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 \[ \frac{1}{5} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{11 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.61384, size = 42, normalized size = 0.84 \[ \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{5} + \frac{11 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{50} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)*sqrt(5*x + 3)/5 + 11*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/50

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Mathematica [A]  time = 0.036887, size = 50, normalized size = 1. \[ \frac{1}{5} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{11 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 56, normalized size = 1.1 \[{\frac{1}{5}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{11\,\sqrt{10}}{100}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(1-2*x)^(1/2)*(3+5*x)^(1/2)+11/100*((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 = 1.4932, size = 39, normalized size = 0.78 \[ \frac{11}{100} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1}{5} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/100*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 1/5*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.218952, size = 70, normalized size = 1.4 \[ \frac{1}{100} \, \sqrt{10}{\left (2 \, \sqrt{10} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 11 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/100*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 = 2.7498, size = 141, normalized size = 2.82 \[ \begin{cases} \frac{2 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{\sqrt{10 x - 5}} - \frac{11 i \sqrt{x + \frac{3}{5}}}{5 \sqrt{10 x - 5}} - \frac{11 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{50} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{11 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{50} - \frac{2 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{\sqrt{- 10 x + 5}} + \frac{11 \sqrt{x + \frac{3}{5}}}{5 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*I*(x + 3/5)**(3/2)/sqrt(10*x - 5) - 11*I*sqrt(x + 3/5)/(5*sqrt(10*x
- 5)) - 11*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/50, 10*Abs(x + 3/5)/11
> 1), (11*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/50 - 2*(x + 3/5)**(3/2)/sqrt
(-10*x + 5) + 11*sqrt(x + 3/5)/(5*sqrt(-10*x + 5)), True))

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GIAC/XCAS [A]  time = 0.221138, size = 54, normalized size = 1.08 \[ \frac{1}{50} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/50*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*s
qrt(-10*x + 5))