3.2674 \(\int \frac{\sqrt{1-2 x}}{\sqrt{-3-5 x} \sqrt{2+3 x}} \, dx\)

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

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Rubi [A]  time = 0.0085949, 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, Rules used = {113} \[ \frac{2}{3} \sqrt{\frac{7}{5}} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right ) \]

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

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{1-2 x}}{\sqrt{-3-5 x} \sqrt{2+3 x}} \, dx &=\frac{2}{3} \sqrt{\frac{7}{5}} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{2+3 x}\right )|\frac{2}{35}\right )\\ \end{align*}

Mathematica [B]  time = 0.332576, 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])

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Maple [C]  time = 0.025, size = 57, normalized size = 1.8 \begin{align*} -{\frac{\sqrt{2}}{15} \left ( 35\,{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -2\,{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) \right ) \sqrt{-3-5\,x}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

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*(66+110*x)^(1/2),1/2*I*66^(1/2))-2*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2)))/(
3+5*x)^(1/2)*2^(1/2)*(-3-5*x)^(1/2)

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

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, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{3 \, x + 2} \sqrt{-2 \, x + 1} \sqrt{-5 \, x - 3}}{15 \, x^{2} + 19 \, x + 6}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

integral(-sqrt(3*x + 2)*sqrt(-2*x + 1)*sqrt(-5*x - 3)/(15*x^2 + 19*x + 6), x)

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

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

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

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, algorithm="giac")

[Out]

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