3.2630 \(\int \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=129 \[ \frac{2}{25} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{31}{225} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{31 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125}-\frac{1159 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125} \]

[Out]

(-31*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/225 + (2*Sqrt[1 - 2*x]*Sqrt[2 +
3*x]*(3 + 5*x)^(3/2))/25 - (1159*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/1125 - (31*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/1125

_______________________________________________________________________________________

Rubi [A]  time = 0.259615, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{25} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{31}{225} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{31 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125}-\frac{1159 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125} \]

Antiderivative was successfully verified.

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

[Out]

(-31*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/225 + (2*Sqrt[1 - 2*x]*Sqrt[2 +
3*x]*(3 + 5*x)^(3/2))/25 - (1159*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/1125 - (31*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/1125

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.0735, size = 114, normalized size = 0.88 \[ \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{15} - \frac{37 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{225} - \frac{1159 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3375} - \frac{341 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{39375} \]

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)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/15 - 37*sqrt(-2*x + 1)*sqrt(3*x
+ 2)*sqrt(5*x + 3)/225 - 1159*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7
), 35/33)/3375 - 341*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35
)/39375

_______________________________________________________________________________________

Mathematica [A]  time = 0.245484, size = 97, normalized size = 0.75 \[ \frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} (90 x+23)-1295 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+2318 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{6750} \]

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]*(23 + 90*x) + 2318*Sqrt[2]*Ellipti
cE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 1295*Sqrt[2]*EllipticF[ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]], -33/2])/6750

_______________________________________________________________________________________

Maple [C]  time = 0.013, size = 169, normalized size = 1.3 \[{\frac{1}{202500\,{x}^{3}+155250\,{x}^{2}-47250\,x-40500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1295\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2318\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +81000\,{x}^{4}+82800\,{x}^{3}-3030\,{x}^{2}-21030\,x-4140 \right ) } \]

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]

1/6750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1295*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))-2318*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*El
lipticE(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))+8100
0*x^4+82800*x^3-3030*x^2-21030*x-4140)/(30*x^3+23*x^2-7*x-6)

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

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

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)