3.2660 \(\int \frac{\sqrt{1-2 x} \sqrt{2+3 x}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=98 \[ \frac{2}{15} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{4}{75} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{31}{75} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/15 - (31*Sqrt[11/3]*EllipticE[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/75 - (4*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/
7]*Sqrt[1 - 2*x]], 35/33])/75

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Rubi [A]  time = 0.189784, 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 \[ \frac{2}{15} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{4}{75} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{31}{75} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/15 - (31*Sqrt[11/3]*EllipticE[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/75 - (4*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/
7]*Sqrt[1 - 2*x]], 35/33])/75

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Rubi in Sympy [A]  time = 19.4327, size = 85, normalized size = 0.87 \[ \frac{2 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{15} - \frac{31 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{225} - \frac{44 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2625} \]

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(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/15 - 31*sqrt(33)*elliptic_e(asin(sq
rt(21)*sqrt(-2*x + 1)/7), 35/33)/225 - 44*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt
(-2*x + 1)/11), 33/35)/2625

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Mathematica [A]  time = 0.160981, size = 92, normalized size = 0.94 \[ \frac{1}{225} \left (30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+35 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+31 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + 31*Sqrt[2]*EllipticE[ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]], -33/2] + 35*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2])/225

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Maple [C]  time = 0.014, size = 164, normalized size = 1.7 \[ -{\frac{1}{6750\,{x}^{3}+5175\,{x}^{2}-1575\,x-1350}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +31\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -900\,{x}^{3}-690\,{x}^{2}+210\,x+180 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/225*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(35*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))+31*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipt
icE(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-
690*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. \[ \int \frac{\sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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