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

Optimal. Leaf size=35 \[ \sqrt{\frac{2}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} \sqrt{3 x-2}}{\sqrt{5 x+3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0420973, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \sqrt{\frac{2}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} \sqrt{3 x-2}}{\sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.74069, size = 31, normalized size = 0.89 \[ \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} \sqrt{3 x - 2}}{2 \sqrt{5 x + 3}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(6)*atan(sqrt(6)*sqrt(3*x - 2)/(2*sqrt(5*x + 3)))/3

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Mathematica [A]  time = 0.0370073, size = 38, normalized size = 1.09 \[ -\frac{\tan ^{-1}\left (\frac{x+12}{2 \sqrt{6} \sqrt{3 x-2} \sqrt{5 x+3}}\right )}{\sqrt{6}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTan[(12 + x)/(2*Sqrt[6]*Sqrt[-2 + 3*x]*Sqrt[3 + 5*x])]/Sqrt[6])

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Maple [B]  time = 0.028, size = 53, normalized size = 1.5 \[ -{\frac{\sqrt{6}}{6}\sqrt{-2+3\,x}\sqrt{3+5\,x}\arctan \left ({\frac{\sqrt{6} \left ( 12+x \right ) }{12}{\frac{1}{\sqrt{15\,{x}^{2}-x-6}}}} \right ){\frac{1}{\sqrt{15\,{x}^{2}-x-6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(-2+3*x)^(1/2)*(3+5*x)^(1/2)/(15*x^2-x-6)^(1/2)*6^(1/2)*arctan(1/12*6^(1/2)
*(12+x)/(15*x^2-x-6)^(1/2))

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Maxima [A]  time = 1.50568, size = 27, normalized size = 0.77 \[ -\frac{1}{6} \, \sqrt{6} \arcsin \left (\frac{x}{19 \,{\left | x \right |}} + \frac{12}{19 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(6)*arcsin(1/19*x/abs(x) + 12/19/abs(x))

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Fricas [A]  time = 0.238241, size = 46, normalized size = 1.31 \[ -\frac{1}{6} \, \sqrt{3} \sqrt{2} \arctan \left (\frac{\sqrt{3} \sqrt{2}{\left (x + 12\right )}}{12 \, \sqrt{5 \, x + 3} \sqrt{3 \, x - 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(3)*sqrt(2)*arctan(1/12*sqrt(3)*sqrt(2)*(x + 12)/(sqrt(5*x + 3)*sqrt(3*
x - 2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.22194, size = 57, normalized size = 1.63 \[ -\frac{1}{15} \, \sqrt{10} \sqrt{5} \sqrt{3} \arctan \left (\frac{1}{60} \, \sqrt{10}{\left ({\left (\sqrt{3} \sqrt{5 \, x + 3} - \sqrt{15 \, x - 10}\right )}^{2} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*sqrt(10)*sqrt(5)*sqrt(3)*arctan(1/60*sqrt(10)*((sqrt(3)*sqrt(5*x + 3) - sq
rt(15*x - 10))^2 + 1))