3.2475 \(\int \frac{2+3 x}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=50 \[ \frac{37 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10 \sqrt{10}}-\frac{3}{10} \sqrt{1-2 x} \sqrt{5 x+3} \]

[Out]

(-3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/10 + (37*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10*
Sqrt[10])

_______________________________________________________________________________________

Rubi [A]  time = 0.0556831, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{37 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10 \sqrt{10}}-\frac{3}{10} \sqrt{1-2 x} \sqrt{5 x+3} \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/10 + (37*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10*
Sqrt[10])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.13679, size = 44, normalized size = 0.88 \[ - \frac{3 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{10} + \frac{37 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{100} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-2*x + 1)*sqrt(5*x + 3)/10 + 37*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)
/100

_______________________________________________________________________________________

Mathematica [A]  time = 0.0324248, size = 50, normalized size = 1. \[ \frac{1}{100} \left (-30 \sqrt{1-2 x} \sqrt{5 x+3}-37 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x] - 37*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])
/100

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 55, normalized size = 1.1 \[{\frac{1}{200}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 37\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -60\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/200*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(37*10^(1/2)*arcsin(20/11*x+1/11)-60*(-10*x^2-
x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.49134, size = 35, normalized size = 0.7 \[ -\frac{37}{200} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{3}{10} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-37/200*sqrt(10)*arcsin(-20/11*x - 1/11) - 3/10*sqrt(-10*x^2 - x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.216072, size = 70, normalized size = 1.4 \[ -\frac{1}{200} \, \sqrt{10}{\left (6 \, \sqrt{10} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 37 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/200*sqrt(10)*(6*sqrt(10)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 37*arctan(1/20*sqrt(1
0)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{3 x + 2}{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)/(sqrt(-2*x + 1)*sqrt(5*x + 3)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227213, size = 54, normalized size = 1.08 \[ \frac{1}{100} \, \sqrt{5}{\left (37 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - 6 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/100*sqrt(5)*(37*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 6*sqrt(5*x + 3)*
sqrt(-10*x + 5))