3.2477 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=32 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}} \]

[Out]

(-2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7]

_______________________________________________________________________________________

Rubi [A]  time = 0.0488416, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.74363, size = 34, normalized size = 1.06 \[ - \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/7

_______________________________________________________________________________________

Mathematica [A]  time = 0.0443496, size = 35, normalized size = 1.09 \[ -\frac{\tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])]/Sqrt[7])

_______________________________________________________________________________________

Maple [B]  time = 0.017, size = 55, normalized size = 1.7 \[{\frac{\sqrt{7}}{7}\sqrt{1-2\,x}\sqrt{3+5\,x}\arctan \left ({\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{14}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-10*x^2-x+3)^(1/2)*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 1.4999, size = 38, normalized size = 1.19 \[ \frac{1}{7} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2))

_______________________________________________________________________________________

Fricas [A]  time = 0.224452, size = 41, normalized size = 1.28 \[ \frac{1}{7} \, \sqrt{7} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.231936, size = 99, normalized size = 3.09 \[ \frac{1}{70} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/70*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))
)