3.2661 \(\int \frac{\sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=49 \[ \frac{2 \sqrt{\frac{7}{5}} \sqrt{-5 x-3} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right )}{3 \sqrt{5 x+3}} \]

[Out]

(2*Sqrt[7/5]*Sqrt[-3 - 5*x]*EllipticE[ArcSin[Sqrt[5]*Sqrt[2 + 3*x]], 2/35])/(3*S
qrt[3 + 5*x])

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Rubi [A]  time = 0.093612, antiderivative size = 49, 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 \[ \frac{2 \sqrt{\frac{7}{5}} \sqrt{-5 x-3} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right )}{3 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[7/5]*Sqrt[-3 - 5*x]*EllipticE[ArcSin[Sqrt[5]*Sqrt[2 + 3*x]], 2/35])/(3*S
qrt[3 + 5*x])

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Rubi in Sympy [A]  time = 9.14623, size = 65, normalized size = 1.33 \[ \frac{2 \sqrt{5} \sqrt{- 15 x - 9} \sqrt{- 2 x + 1} E\left (\operatorname{asin}{\left (\sqrt{5} \sqrt{3 x + 2} \right )}\middle | \frac{2}{35}\right )}{15 \sqrt{- \frac{6 x}{7} + \frac{3}{7}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(5)*sqrt(-15*x - 9)*sqrt(-2*x + 1)*elliptic_e(asin(sqrt(5)*sqrt(3*x + 2)),
2/35)/(15*sqrt(-6*x/7 + 3/7)*sqrt(5*x + 3))

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Mathematica [B]  time = 0.372234, size = 121, normalized size = 2.47 \[ \frac{2 \sqrt{1-2 x} \left (5 \left (6 x^2+x-2\right ) \sqrt{5 x+3}+\sqrt{33} \sqrt{\frac{2 x-1}{5 x+3}} \sqrt{\frac{3 x+2}{5 x+3}} (5 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{11}{2}}}{\sqrt{5 x+3}}\right )|-\frac{2}{33}\right )\right )}{15 \sqrt{3 x+2} \left (10 x^2+x-3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*(5*Sqrt[3 + 5*x]*(-2 + x + 6*x^2) + Sqrt[33]*Sqrt[(-1 + 2*x)/(3
+ 5*x)]*Sqrt[(2 + 3*x)/(3 + 5*x)]*(3 + 5*x)^2*EllipticE[ArcSin[Sqrt[11/2]/Sqrt[
3 + 5*x]], -2/33]))/(15*Sqrt[2 + 3*x]*(-3 + x + 10*x^2))

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Maple [C]  time = 0.018, size = 67, normalized size = 1.4 \[{\frac{\sqrt{2}}{15} \left ( 35\,{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2\,{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(35*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))-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. \[ \int \frac{\sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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