3.2620 \(\int \frac{(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=202 \[ -\frac{2 (139 x+121) (2 x+3)^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 (2571 x+2164) (2 x+3)^{3/2}}{9 \sqrt{3 x^2+5 x+2}}-\frac{59512}{81} \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}+\frac{148780 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{81 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{110516 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{81 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-2*(3 + 2*x)^(7/2)*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (4*(3 + 2*x)^(3
/2)*(2164 + 2571*x))/(9*Sqrt[2 + 5*x + 3*x^2]) - (59512*Sqrt[3 + 2*x]*Sqrt[2 + 5
*x + 3*x^2])/81 - (110516*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1
+ x]], -2/3])/(81*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (148780*Sqrt[-2 - 5*x - 3*x^
2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(81*Sqrt[3]*Sqrt[2 + 5*x + 3*x^
2])

_______________________________________________________________________________________

Rubi [A]  time = 0.418504, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{2 (139 x+121) (2 x+3)^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 (2571 x+2164) (2 x+3)^{3/2}}{9 \sqrt{3 x^2+5 x+2}}-\frac{59512}{81} \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}+\frac{148780 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{81 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{110516 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{81 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(3 + 2*x)^(7/2)*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (4*(3 + 2*x)^(3
/2)*(2164 + 2571*x))/(9*Sqrt[2 + 5*x + 3*x^2]) - (59512*Sqrt[3 + 2*x]*Sqrt[2 + 5
*x + 3*x^2])/81 - (110516*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1
+ x]], -2/3])/(81*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (148780*Sqrt[-2 - 5*x - 3*x^
2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(81*Sqrt[3]*Sqrt[2 + 5*x + 3*x^
2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 55.1903, size = 194, normalized size = 0.96 \[ - \frac{2 \left (2 x + 3\right )^{\frac{7}{2}} \left (139 x + 121\right )}{9 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{4 \left (2 x + 3\right )^{\frac{3}{2}} \left (7713 x + 6492\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} - \frac{59512 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}{81} - \frac{110516 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{243 \sqrt{3 x^{2} + 5 x + 2}} + \frac{148780 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{243 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(2*x + 3)**(7/2)*(139*x + 121)/(9*(3*x**2 + 5*x + 2)**(3/2)) + 4*(2*x + 3)**(
3/2)*(7713*x + 6492)/(27*sqrt(3*x**2 + 5*x + 2)) - 59512*sqrt(2*x + 3)*sqrt(3*x*
*2 + 5*x + 2)/81 - 110516*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(
6*x + 6)/2), -2/3)/(243*sqrt(3*x**2 + 5*x + 2)) + 148780*sqrt(-9*x**2 - 15*x - 6
)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(243*sqrt(3*x**2 + 5*x + 2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.874484, size = 220, normalized size = 1.09 \[ -\frac{2 \left (2 \left (3 x^2+5 x+2\right ) \left (55258 \left (3 x^2+5 x+2\right )-5312 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+27629 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )+3 (2 x+3) \left (144 x^4-166566 x^3-411640 x^2-330053 x-85285\right )\right )}{243 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(3*(3 + 2*x)*(-85285 - 330053*x - 411640*x^2 - 166566*x^3 + 144*x^4) + 2*(2
+ 5*x + 3*x^2)*(55258*(2 + 5*x + 3*x^2) + 27629*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*
(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*
x]], 3/5] - 5312*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/
(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])))/(243*Sqrt[3 + 2*x]
*(2 + 5*x + 3*x^2)^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.069, size = 343, normalized size = 1.7 \[{\frac{2}{ \left ( 1215+1215\,x \right ) \left ( 2\,{x}^{2}+5\,x+3 \right ) \left ( 2+3\,x \right ) ^{2}} \left ( 82887\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+28698\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+138145\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+47830\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+55258\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +19132\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -4320\,{x}^{5}+4990500\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+3837825 \right ) \sqrt{3\,{x}^{2}+5\,x+2}\sqrt{3+2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1215*(82887*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)+28698*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)+
138145*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)+47830*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)*(-30*x-20)^(1/2)*(3+2*x)^(1/2)+55258*(3+2*
x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)*EllipticE(1/5*15^(1/2)*(3+2*x)
^(1/2),1/3*15^(1/2))+19132*(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))-4320*x^5+4990500*x^4+198446
70*x^3+28425390*x^2+17410935*x+3837825)*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)/(1+x)/
(2*x^2+5*x+3)/(2+3*x)^2

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (16 \, x^{5} + 16 \, x^{4} - 264 \, x^{3} - 864 \, x^{2} - 999 \, x - 405\right )} \sqrt{2 \, x + 3}}{{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(16*x^5 + 16*x^4 - 264*x^3 - 864*x^2 - 999*x - 405)*sqrt(2*x + 3)/((9*
x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*sqrt(3*x^2 + 5*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((5-x)*(3+2*x)**(9/2)/(3*x**2+5*x+2)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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