3.2476 \(\int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=26 \[ \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0260399, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.17287, size = 22, normalized size = 0.85 \[ \frac{\sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0188169, size = 27, normalized size = 1.04 \[ -\sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.006, size = 39, normalized size = 1.5 \[{\frac{\sqrt{10}}{10}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*((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 = 1.49509, size = 15, normalized size = 0.58 \[ -\frac{1}{10} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*sqrt(10)*arcsin(-20/11*x - 1/11)

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Fricas [A]  time = 0.214564, size = 49, normalized size = 1.88 \[ \frac{1}{10} \, \sqrt{5} \sqrt{2} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.8459, size = 58, normalized size = 2.23 \[ \begin{cases} - \frac{\sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{5} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{\sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{5} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227863, size = 28, normalized size = 1.08 \[ \frac{1}{5} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sqrt(5)*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))