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 = 0.101281, 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 \[ \frac{\sqrt{\frac{11}{2}} \sqrt{5-2 x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{11}}\right )\right |3\right )}{2 \sqrt{2 x-5}} \]

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])

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Rubi in Sympy [A]  time = 8.73329, size = 65, normalized size = 1.38 \[ \frac{\sqrt{11} \sqrt{- 3 x + 2} \sqrt{- \frac{4 x}{11} + \frac{10}{11}} E\left (\operatorname{asin}{\left (\frac{\sqrt{11} \sqrt{4 x + 1}}{11} \right )}\middle | 3\right )}{2 \sqrt{- \frac{12 x}{11} + \frac{8}{11}} \sqrt{2 x - 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2-3*x)**(1/2)/(-5+2*x)**(1/2)/(1+4*x)**(1/2),x)

[Out]

sqrt(11)*sqrt(-3*x + 2)*sqrt(-4*x/11 + 10/11)*elliptic_e(asin(sqrt(11)*sqrt(4*x
+ 1)/11), 3)/(2*sqrt(-12*x/11 + 8/11)*sqrt(2*x - 5))

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Mathematica [B]  time = 0.732478, 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 = 0.018, size = 61, normalized size = 1.3 \[{\frac{\sqrt{11}}{2} \left ({\it EllipticF} \left ({\frac{2\,\sqrt{11}}{11}\sqrt{2-3\,x}},{\frac{i}{2}}\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{2\,\sqrt{11}}{11}\sqrt{2-3\,x}},{\frac{i}{2}}\sqrt{2} \right ) \right ) \sqrt{5-2\,x}{\frac{1}{\sqrt{-5+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)),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. \[{\rm integral}\left (\frac{\sqrt{-3 \, x + 2}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)),x, algorithm="fricas")

[Out]

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

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

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-3 \, x + 2}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)),x, algorithm="giac")

[Out]

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