3.2886 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2} \sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=98 \[ \frac{11 \sqrt{3 x+2} \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\frac{1}{7} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{34}{7} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(11*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]) + (34*Sqrt[11/3]*EllipticE[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/7 + (Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/7

_______________________________________________________________________________________

Rubi [A]  time = 0.185797, 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 \[ \frac{11 \sqrt{3 x+2} \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\frac{1}{7} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{34}{7} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(11*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]) + (34*Sqrt[11/3]*EllipticE[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/7 + (Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/7

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.4436, size = 83, normalized size = 0.85 \[ \frac{34 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{21} + \frac{\sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{21} + \frac{11 \sqrt{3 x + 2} \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

34*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/21 + sqrt(33)*ell
iptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/21 + 11*sqrt(3*x + 2)*sqrt(5*x +
3)/(7*sqrt(-2*x + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.126796, size = 92, normalized size = 0.94 \[ \frac{1}{42} \left (\frac{66 \sqrt{3 x+2} \sqrt{5 x+3}}{\sqrt{1-2 x}}+35 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-68 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((66*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] - 68*Sqrt[2]*EllipticE[ArcSin[Sq
rt[2/11]*Sqrt[3 + 5*x]], -33/2] + 35*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2])/42

_______________________________________________________________________________________

Maple [C]  time = 0.025, size = 159, normalized size = 1.6 \[ -{\frac{1}{1260\,{x}^{3}+966\,{x}^{2}-294\,x-252}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -68\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +990\,{x}^{2}+1254\,x+396 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/42*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*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))-68*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))+990*x^2+1
254*x+396)/(30*x^3+23*x^2-7*x-6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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