3.54 \(\int \frac{\sqrt{2-3 x}}{\sqrt{-5+2 x} \sqrt{1+4 x}} \, dx\)

Optimal. Leaf size=47 \[ \frac{\sqrt{\frac{11}{2}} \sqrt{5-2 x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{11}}\right )\right |3\right )}{2 \sqrt{2 x-5}} \]

[Out]

(Sqrt[11/2]*Sqrt[5 - 2*x]*EllipticE[ArcSin[Sqrt[1 + 4*x]/Sqrt[11]], 3])/(2*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.0148583, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {114, 113} \[ \frac{\sqrt{\frac{11}{2}} \sqrt{5-2 x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{11}}\right )\right |3\right )}{2 \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 - 3*x]/(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]),x]

[Out]

(Sqrt[11/2]*Sqrt[5 - 2*x]*EllipticE[ArcSin[Sqrt[1 + 4*x]/Sqrt[11]], 3])/(2*Sqrt[-5 + 2*x])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], 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]

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{2-3 x}}{\sqrt{-5+2 x} \sqrt{1+4 x}} \, dx &=\frac{\sqrt{5-2 x} \int \frac{\sqrt{\frac{8}{11}-\frac{12 x}{11}}}{\sqrt{\frac{10}{11}-\frac{4 x}{11}} \sqrt{1+4 x}} \, dx}{\sqrt{2} \sqrt{-5+2 x}}\\ &=\frac{\sqrt{\frac{11}{2}} \sqrt{5-2 x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{1+4 x}}{\sqrt{11}}\right )\right |3\right )}{2 \sqrt{-5+2 x}}\\ \end{align*}

Mathematica [B]  time = 0.340403, size = 111, normalized size = 2.36 \[ -\frac{\frac{2 (2 x-5) (3 x-2)}{\sqrt{2 x+\frac{1}{2}}}+\sqrt{11} \sqrt{\frac{2 x-5}{4 x+1}} \sqrt{\frac{3 x-2}{4 x+1}} (4 x+1) E\left (\left .\sin ^{-1}\left (\frac{\sqrt{\frac{11}{3}}}{\sqrt{4 x+1}}\right )\right |3\right )}{2 \sqrt{2-3 x} \sqrt{4 x-10}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 - 3*x]/(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]),x]

[Out]

-((2*(-5 + 2*x)*(-2 + 3*x))/Sqrt[1/2 + 2*x] + Sqrt[11]*Sqrt[(-5 + 2*x)/(1 + 4*x)]*Sqrt[(-2 + 3*x)/(1 + 4*x)]*(
1 + 4*x)*EllipticE[ArcSin[Sqrt[11/3]/Sqrt[1 + 4*x]], 3])/(2*Sqrt[2 - 3*x]*Sqrt[-10 + 4*x])

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Maple [C]  time = 0.011, size = 55, normalized size = 1.2 \begin{align*}{\frac{\sqrt{11}}{2} \left ({\it EllipticF} \left ({\frac{2}{11}\sqrt{22-33\,x}},{\frac{i}{2}}\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{2}{11}\sqrt{22-33\,x}},{\frac{i}{2}}\sqrt{2} \right ) \right ) \sqrt{5-2\,x}{\frac{1}{\sqrt{2\,x-5}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2-3*x)^(1/2)/(2*x-5)^(1/2)/(4*x+1)^(1/2),x)

[Out]

1/2*(EllipticF(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))-EllipticE(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2)))*(5-2*x)^(1/2
)*11^(1/2)/(2*x-5)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-3 \, x + 2}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{8 \, x^{2} - 18 \, x - 5}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(8*x^2 - 18*x - 5), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 - 3 x}}{\sqrt{2 x - 5} \sqrt{4 x + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-3*x)**(1/2)/(-5+2*x)**(1/2)/(1+4*x)**(1/2),x)

[Out]

Integral(sqrt(2 - 3*x)/(sqrt(2*x - 5)*sqrt(4*x + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-3 \, x + 2}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)), x)