3.2506 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{\sqrt{5 x+3}}{\sqrt{1-2 x}}-\sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0422528, antiderivative size = 47, 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 \[ \frac{\sqrt{5 x+3}}{\sqrt{1-2 x}}-\sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.20399, size = 39, normalized size = 0.83 \[ - \frac{\sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2} + \frac{\sqrt{5 x + 3}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0407409, size = 46, normalized size = 0.98 \[ \frac{\sqrt{5 x+3}}{\sqrt{1-2 x}}+\sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{1\sqrt{3+5\,x} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.48211, size = 49, normalized size = 1.04 \[ -\frac{1}{4} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{2 \, x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.231625, size = 93, normalized size = 1.98 \[ -\frac{\sqrt{2}{\left (\sqrt{5}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 2 \, \sqrt{2} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{4 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(2)*(sqrt(5)*(2*x - 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x
+ 3)*sqrt(-2*x + 1))) + 2*sqrt(2)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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Sympy [A]  time = 2.81075, size = 95, normalized size = 2.02 \[ \begin{cases} - \frac{5 i \sqrt{x + \frac{3}{5}}}{\sqrt{10 x - 5}} + \frac{\sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{2} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{\sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{2} + \frac{5 \sqrt{x + \frac{3}{5}}}{\sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223305, size = 61, normalized size = 1.3 \[ -\frac{1}{2} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{5 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/5*sqrt(5)*sqrt(5*x + 3)*sq
rt(-10*x + 5)/(2*x - 1)