3.2846 \(\int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=29 \[ -\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}} \]

[Out]

(-2*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[33]

_______________________________________________________________________________________

Rubi [A]  time = 0.0459777, antiderivative size = 29, 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 \[ -\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[33]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.35424, size = 29, normalized size = 1. \[ - \frac{2 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{33} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/33

_______________________________________________________________________________________

Mathematica [C]  time = 0.177449, size = 74, normalized size = 2.55 \[ \frac{i \sqrt{3 x+2} \sqrt{\frac{4 x-2}{5 x+3}} F\left (i \sinh ^{-1}\left (\frac{1}{\sqrt{15 x+9}}\right )|-\frac{33}{2}\right )}{\sqrt{1-2 x} \sqrt{\frac{3 x+2}{5 x+3}}} \]

Antiderivative was successfully verified.

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

[Out]

(I*Sqrt[2 + 3*x]*Sqrt[(-2 + 4*x)/(3 + 5*x)]*EllipticF[I*ArcSinh[1/Sqrt[9 + 15*x]
], -33/2])/(Sqrt[1 - 2*x]*Sqrt[(2 + 3*x)/(3 + 5*x)])

_______________________________________________________________________________________

Maple [C]  time = 0.02, size = 33, normalized size = 1.1 \[{\it EllipticF} \left ({\frac{\sqrt{11}\sqrt{2}}{11}\sqrt{3+5\,x}},{\frac{i}{2}}\sqrt{11}\sqrt{3}\sqrt{2} \right ) \sqrt{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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^
(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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