3.2913 \(\int \frac{\sqrt{2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{20 \sqrt{1-2 x} \sqrt{3 x+2}}{121 \sqrt{5 x+3}}+\frac{2 \sqrt{3 x+2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{2}{11} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4}{11} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[2 + 3*x])/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (20*Sqrt[1 - 2*x]*Sqrt[2 +
3*x])/(121*Sqrt[3 + 5*x]) + (4*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*
x]], 35/33])/11 - (2*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/11

_______________________________________________________________________________________

Rubi [A]  time = 0.260769, 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 \[ -\frac{20 \sqrt{1-2 x} \sqrt{3 x+2}}{121 \sqrt{5 x+3}}+\frac{2 \sqrt{3 x+2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{2}{11} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4}{11} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 + 3*x])/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (20*Sqrt[1 - 2*x]*Sqrt[2 +
3*x])/(121*Sqrt[3 + 5*x]) + (4*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*
x]], 35/33])/11 - (2*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/11

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.3621, size = 114, normalized size = 0.88 \[ - \frac{20 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{121 \sqrt{5 x + 3}} + \frac{4 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{121} - \frac{6 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{385} + \frac{2 \sqrt{3 x + 2}}{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-20*sqrt(-2*x + 1)*sqrt(3*x + 2)/(121*sqrt(5*x + 3)) + 4*sqrt(33)*elliptic_e(asi
n(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/121 - 6*sqrt(35)*elliptic_f(asin(sqrt(55)*s
qrt(-2*x + 1)/11), 33/35)/385 + 2*sqrt(3*x + 2)/(11*sqrt(-2*x + 1)*sqrt(5*x + 3)
)

_______________________________________________________________________________________

Mathematica [A]  time = 0.183314, size = 122, normalized size = 0.95 \[ \frac{2 \sqrt{3 x+2} \sqrt{5 x+3} (20 x+1)+37 \sqrt{2-4 x} (5 x+3) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4 \sqrt{2-4 x} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{121 \sqrt{1-2 x} (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(1 + 20*x) - 4*Sqrt[2 - 4*x]*(3 + 5*x)*EllipticE[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 37*Sqrt[2 - 4*x]*(3 + 5*x)*EllipticF[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(121*Sqrt[1 - 2*x]*(3 + 5*x))

_______________________________________________________________________________________

Maple [C]  time = 0.029, size = 159, normalized size = 1.2 \[ -{\frac{1}{3630\,{x}^{3}+2783\,{x}^{2}-847\,x-726}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 37\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +120\,{x}^{2}+86\,x+4 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/121*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(37*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))-4*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))+120*x^2+8
6*x+4)/(30*x^3+23*x^2-7*x-6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{3 \, x + 2}}{{\left (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-sqrt(3*x + 2)/((10*x^2 + x - 3)*sqrt(5*x + 3)*sqrt(-2*x + 1)), x)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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