3.2722 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{24}{125} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{5 \sqrt{5 x+3}}+\frac{458}{625} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{496 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125}-\frac{169 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125} \]

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2))/(5*Sqrt[3 + 5*x]) + (458*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x]*Sqrt[3 + 5*x])/625 - (24*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/
125 - (169*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3125 -
(496*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3125

_______________________________________________________________________________________

Rubi [A]  time = 0.322731, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{24}{125} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{5 \sqrt{5 x+3}}+\frac{458}{625} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{496 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125}-\frac{169 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2))/(5*Sqrt[3 + 5*x]) + (458*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x]*Sqrt[3 + 5*x])/625 - (24*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/
125 - (169*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3125 -
(496*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3125

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.2553, size = 143, normalized size = 0.89 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}}}{5 \sqrt{5 x + 3}} - \frac{24 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{125} + \frac{458 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{625} - \frac{169 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9375} - \frac{496 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)/(5*sqrt(5*x + 3)) - 24*sqrt(-2*x + 1)*(3*x
+ 2)**(3/2)*sqrt(5*x + 3)/125 + 458*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/
625 - 169*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/9375 - 496
*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/9375

_______________________________________________________________________________________

Mathematica [A]  time = 0.307114, size = 102, normalized size = 0.64 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (-150 x^2+130 x+77\right )}{\sqrt{5 x+3}}+8015 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+169 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{9375} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(77 + 130*x - 150*x^2))/Sqrt[3 + 5*x] + 169*Sqr
t[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 8015*Sqrt[2]*EllipticF
[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/9375

_______________________________________________________________________________________

Maple [C]  time = 0.027, size = 169, normalized size = 1.1 \[ -{\frac{1}{281250\,{x}^{3}+215625\,{x}^{2}-65625\,x-56250}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 8015\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +169\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +27000\,{x}^{4}-18900\,{x}^{3}-26760\,{x}^{2}+5490\,x+4620 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9375*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(8015*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))+169*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))+2700
0*x^4-18900*x^3-26760*x^2+5490*x+4620)/(30*x^3+23*x^2-7*x-6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (6 \, x^{2} + x - 2\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(6*x^2 + x - 2)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(5*x + 3)^(3/2), 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((1-2*x)**(3/2)*(2+3*x)**(3/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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