3.2657 \(\int \frac{\sqrt{1-2 x}}{\sqrt{-3-5 x} \sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=31 \[ \frac{2}{3} \sqrt{\frac{7}{5}} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right ) \]

[Out]

(2*Sqrt[7/5]*EllipticE[ArcSin[Sqrt[5]*Sqrt[2 + 3*x]], 2/35])/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0496607, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{2}{3} \sqrt{\frac{7}{5}} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[7/5]*EllipticE[ArcSin[Sqrt[5]*Sqrt[2 + 3*x]], 2/35])/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.65762, size = 26, normalized size = 0.84 \[ \frac{2 \sqrt{35} E\left (\operatorname{asin}{\left (\sqrt{5} \sqrt{3 x + 2} \right )}\middle | \frac{2}{35}\right )}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(35)*elliptic_e(asin(sqrt(5)*sqrt(3*x + 2)), 2/35)/15

_______________________________________________________________________________________

Mathematica [B]  time = 0.62316, size = 109, normalized size = 3.52 \[ -\frac{2 \left (\frac{3 \left (10 x^2+x-3\right )}{\sqrt{3 x+2}}+\sqrt{35} \sqrt{\frac{2 x-1}{3 x+2}} (3 x+2) \sqrt{\frac{5 x+3}{3 x+2}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{7}{2}}}{\sqrt{3 x+2}}\right )|\frac{2}{35}\right )\right )}{15 \sqrt{-5 x-3} \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*((3*(-3 + x + 10*x^2))/Sqrt[2 + 3*x] + Sqrt[35]*Sqrt[(-1 + 2*x)/(2 + 3*x)]*(
2 + 3*x)*Sqrt[(3 + 5*x)/(2 + 3*x)]*EllipticE[ArcSin[Sqrt[7/2]/Sqrt[2 + 3*x]], 2/
35]))/(15*Sqrt[-3 - 5*x]*Sqrt[1 - 2*x])

_______________________________________________________________________________________

Maple [C]  time = 0.046, size = 81, normalized size = 2.6 \[ -{\frac{\sqrt{2}}{15} \left ( 35\,{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2\,{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) \right ) \sqrt{-3-5\,x}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(35*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2
^(1/2))-2*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2
^(1/2)))/(3+5*x)^(1/2)*2^(1/2)*(-3-5*x)^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-2 \, x + 1}}{\sqrt{3 \, x + 2} \sqrt{-5 \, x - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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