3.2590 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{9/2}} \, dx\)

Optimal. Leaf size=175 \[ \frac{(347 x+358) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}+\frac{(6179 x+8561) \sqrt{3 x^2+5 x+2}}{1750 (2 x+3)^{3/2}}+\frac{1327 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{700 \sqrt{3 x^2+5 x+2}}-\frac{721 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{500 \sqrt{3 x^2+5 x+2}} \]

[Out]

((8561 + 6179*x)*Sqrt[2 + 5*x + 3*x^2])/(1750*(3 + 2*x)^(3/2)) + ((358 + 347*x)*
(2 + 5*x + 3*x^2)^(3/2))/(175*(3 + 2*x)^(7/2)) - (721*Sqrt[3]*Sqrt[-2 - 5*x - 3*
x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(500*Sqrt[2 + 5*x + 3*x^2]) +
(1327*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/
3])/(700*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.326709, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{(347 x+358) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}+\frac{(6179 x+8561) \sqrt{3 x^2+5 x+2}}{1750 (2 x+3)^{3/2}}+\frac{1327 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{700 \sqrt{3 x^2+5 x+2}}-\frac{721 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{500 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^(9/2),x]

[Out]

((8561 + 6179*x)*Sqrt[2 + 5*x + 3*x^2])/(1750*(3 + 2*x)^(3/2)) + ((358 + 347*x)*
(2 + 5*x + 3*x^2)^(3/2))/(175*(3 + 2*x)^(7/2)) - (721*Sqrt[3]*Sqrt[-2 - 5*x - 3*
x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(500*Sqrt[2 + 5*x + 3*x^2]) +
(1327*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/
3])/(700*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 47.5053, size = 165, normalized size = 0.94 \[ - \frac{721 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{500 \sqrt{3 x^{2} + 5 x + 2}} + \frac{1327 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{700 \sqrt{3 x^{2} + 5 x + 2}} + \frac{\left (6179 x + 8561\right ) \sqrt{3 x^{2} + 5 x + 2}}{1750 \left (2 x + 3\right )^{\frac{3}{2}}} + \frac{\left (347 x + 358\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{175 \left (2 x + 3\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**(3/2)/(3+2*x)**(9/2),x)

[Out]

-721*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(5
00*sqrt(3*x**2 + 5*x + 2)) + 1327*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(
2)*sqrt(6*x + 6)/2), -2/3)/(700*sqrt(3*x**2 + 5*x + 2)) + (6179*x + 8561)*sqrt(3
*x**2 + 5*x + 2)/(1750*(2*x + 3)**(3/2)) + (347*x + 358)*(3*x**2 + 5*x + 2)**(3/
2)/(175*(2*x + 3)**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.48603, size = 192, normalized size = 1.1 \[ -\frac{31500 x^5+323760 x^4+1009230 x^3+1386750 x^2+878020 x-1066 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{9/2} \sqrt{\frac{3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+5047 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{9/2} \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+208240}{3500 (2 x+3)^{7/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^(9/2),x]

[Out]

-(208240 + 878020*x + 1386750*x^2 + 1009230*x^3 + 323760*x^4 + 31500*x^5 + 5047*
Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(9/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*Ellipt
icE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 1066*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]
*(3 + 2*x)^(9/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2
*x]], 3/5])/(3500*(3 + 2*x)^(7/2)*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.026, size = 413, normalized size = 2.4 \[{\frac{1}{35000} \left ( 12704\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+40376\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+57168\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+181692\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+85752\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+272538\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+42876\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +136269\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +2107560\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+3368360 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}} \left ( 3+2\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(9/2),x)

[Out]

1/35000*(12704*15^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^3*(
3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)+40376*15^(1/2)*EllipticE(1/5*15^(1/
2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^3*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)
+57168*15^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^2*(-30*x-20
)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)+181692*15^(1/2)*EllipticE(1/5*15^(1/2)*(3+2
*x)^(1/2),1/3*15^(1/2))*x^2*(-30*x-20)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)+85752*
15^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x*(-2-2*x)^(1/2)*(-3
0*x-20)^(1/2)*(3+2*x)^(1/2)+272538*15^(1/2)*EllipticE(1/5*15^(1/2)*(3+2*x)^(1/2)
,1/3*15^(1/2))*x*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)*(3+2*x)^(1/2)+42876*(3+2*x)^(1/
2)*15^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2)
,1/3*15^(1/2))+136269*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)*Ell
ipticE(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))+2107560*x^5+11701520*x^4+2604422
0*x^3+28830120*x^2+15748220*x+3368360)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(7/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(9/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(9/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \sqrt{2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(9/2),x, algorithm="fricas")

[Out]

integral(-(3*x^3 - 10*x^2 - 23*x - 10)*sqrt(3*x^2 + 5*x + 2)/((16*x^4 + 96*x^3 +
216*x^2 + 216*x + 81)*sqrt(2*x + 3)), 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((5-x)*(3*x**2+5*x+2)**(3/2)/(3+2*x)**(9/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(9/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(9/2), x)