3.2694 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=160 \[ \frac{2}{21} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{74}{525} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{1847 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{4725}-\frac{1847 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{23625}-\frac{29933 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{23625} \]

[Out]

(-1847*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/4725 + (74*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*(3 + 5*x)^(3/2))/525 + (2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)
)/21 - (29933*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2362
5 - (1847*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/23625

_______________________________________________________________________________________

Rubi [A]  time = 0.323509, 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 \[ \frac{2}{21} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{74}{525} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{1847 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{4725}-\frac{1847 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{23625}-\frac{29933 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{23625} \]

Antiderivative was successfully verified.

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

[Out]

(-1847*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/4725 + (74*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*(3 + 5*x)^(3/2))/525 + (2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)
)/21 - (29933*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2362
5 - (1847*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/23625

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 30.9341, size = 143, normalized size = 0.89 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{21} - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{105} + \frac{1816 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{4725} - \frac{29933 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{70875} - \frac{20317 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{826875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*(5*x + 3)**(3/2)/21 - 37*(-2*x + 1)**(3/2)*sqr
t(3*x + 2)*sqrt(5*x + 3)/105 + 1816*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/4
725 - 29933*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/70875 -
20317*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/826875

_______________________________________________________________________________________

Mathematica [A]  time = 0.302236, size = 97, normalized size = 0.61 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (-4500 x^2+2880 x+1501\right )+1085 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+59866 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{70875 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(1501 + 2880*x - 4500*x^2) + 59866
*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 1085*EllipticF[ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]], -33/2])/(70875*Sqrt[2])

_______________________________________________________________________________________

Maple [C]  time = 0.017, size = 174, normalized size = 1.1 \[ -{\frac{1}{4252500\,{x}^{3}+3260250\,{x}^{2}-992250\,x-850500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1085\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +59866\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +4050000\,{x}^{5}+513000\,{x}^{4}-4283100\,{x}^{3}-1240890\,{x}^{2}+833610\,x+270180 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/141750*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1085*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))+59866*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2
)*EllipticE(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))+
4050000*x^5+513000*x^4-4283100*x^3-1240890*x^2+833610*x+270180)/(30*x^3+23*x^2-7
*x-6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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