3.2845 \(\int \frac{\sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=31 \[ -\sqrt{\frac{7}{5}} E\left (\sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )|\frac{33}{35}\right ) \]

[Out]

-(Sqrt[7/5]*EllipticE[ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]], 33/35])

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Rubi [A]  time = 0.0470253, 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 \[ -\sqrt{\frac{7}{5}} E\left (\sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )|\frac{33}{35}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[7/5]*EllipticE[ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]], 33/35])

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Rubi in Sympy [A]  time = 5.18315, size = 27, normalized size = 0.87 \[ - \frac{\sqrt{35} E\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(35)*elliptic_e(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/5

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Mathematica [C]  time = 0.239119, size = 129, normalized size = 4.16 \[ \frac{\sqrt{3 x+2} \sqrt{\frac{2 x-1}{5 x+3}} \left (5 \sqrt{\frac{2 x-1}{5 x+3}} \sqrt{\frac{3 x+2}{5 x+3}} \sqrt{5 x+3}+i \sqrt{2} E\left (i \sinh ^{-1}\left (\frac{1}{\sqrt{15 x+9}}\right )|-\frac{33}{2}\right )\right )}{5 \sqrt{1-2 x} \sqrt{\frac{3 x+2}{5 x+3}}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 + 3*x]*Sqrt[(-1 + 2*x)/(3 + 5*x)]*(5*Sqrt[(-1 + 2*x)/(3 + 5*x)]*Sqrt[(2
+ 3*x)/(3 + 5*x)]*Sqrt[3 + 5*x] + I*Sqrt[2]*EllipticE[I*ArcSinh[1/Sqrt[9 + 15*x]
], -33/2]))/(5*Sqrt[1 - 2*x]*Sqrt[(2 + 3*x)/(3 + 5*x)])

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Maple [C]  time = 0.017, size = 34, normalized size = 1.1 \[{\frac{\sqrt{2}}{5}{\it EllipticE} \left ({\frac{\sqrt{11}\sqrt{2}}{11}\sqrt{3+5\,x}},{\frac{i}{2}}\sqrt{11}\sqrt{3}\sqrt{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*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. \[ \int \frac{\sqrt{3 \, x + 2}}{\sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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