3.2801 \(\int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=31 \[ -\sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

-(Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])

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Rubi [A]  time = 0.0428812, 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 \[ -\sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])

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Rubi in Sympy [A]  time = 5.18788, size = 27, normalized size = 0.87 \[ - \frac{\sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3

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Mathematica [A]  time = 0.084367, size = 56, normalized size = 1.81 \[ \frac{1}{3} \sqrt{2} \left (E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2]*(EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - EllipticF[ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/3

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Maple [C]  time = 0.018, size = 65, normalized size = 2.1 \[ -{\frac{\sqrt{2}}{3} \left ({\it EllipticF} \left ({\frac{\sqrt{11}\sqrt{2}}{11}\sqrt{3+5\,x}},{\frac{i}{2}}\sqrt{11}\sqrt{3}\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{11}\sqrt{2}}{11}\sqrt{3+5\,x}},{\frac{i}{2}}\sqrt{11}\sqrt{3}\sqrt{2} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(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))-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)
))*2^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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