3.2631 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=98 \[ \frac{2}{9} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{2}{45} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{37}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 - (37*Sqrt[11/3]*EllipticE[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45 + (2*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/45

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Rubi [A]  time = 0.19372, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2}{9} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{2}{45} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{37}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 - (37*Sqrt[11/3]*EllipticE[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45 + (2*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/45

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Rubi in Sympy [A]  time = 16.7639, size = 85, normalized size = 0.87 \[ \frac{2 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{9} - \frac{37 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{135} + \frac{2 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{135} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/9 - 37*sqrt(33)*elliptic_e(asin(sqr
t(21)*sqrt(-2*x + 1)/7), 35/33)/135 + 2*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-
2*x + 1)/7), 35/33)/135

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Mathematica [A]  time = 0.165793, size = 92, normalized size = 0.94 \[ \frac{1}{135} \left (30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-70 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+37 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + 37*Sqrt[2]*EllipticE[ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]], -33/2] - 70*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2])/135

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Maple [C]  time = 0.031, size = 164, normalized size = 1.7 \[{\frac{1}{4050\,{x}^{3}+3105\,{x}^{2}-945\,x-810}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 70\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -37\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +900\,{x}^{3}+690\,{x}^{2}-210\,x-180 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/135*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(70*2^(1/2)*(3+5*x)^(1/2)*(2+3*x
)^(1/2)*(1-2*x)^(1/2)*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))-37*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipti
cE(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))+900*x^3+6
90*x^2-210*x-180)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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