3.2591 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx\)

Optimal. Leaf size=202 \[ \frac{(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}-\frac{(211 x+189) \sqrt{3 x^2+5 x+2}}{2250 (2 x+3)^{5/2}}-\frac{23 \sqrt{3 x^2+5 x+2}}{11250 \sqrt{2 x+3}}+\frac{7 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1500 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{23 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{7500 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-23*Sqrt[2 + 5*x + 3*x^2])/(11250*Sqrt[3 + 2*x]) - ((189 + 211*x)*Sqrt[2 + 5*x
+ 3*x^2])/(2250*(3 + 2*x)^(5/2)) + ((44 + 51*x)*(2 + 5*x + 3*x^2)^(3/2))/(45*(3
+ 2*x)^(9/2)) + (23*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]]
, -2/3])/(7500*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (7*Sqrt[-2 - 5*x - 3*x^2]*Ellipt
icF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(1500*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.396475, 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 \[ \frac{(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}-\frac{(211 x+189) \sqrt{3 x^2+5 x+2}}{2250 (2 x+3)^{5/2}}-\frac{23 \sqrt{3 x^2+5 x+2}}{11250 \sqrt{2 x+3}}+\frac{7 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1500 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{23 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{7500 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^(11/2),x]

[Out]

(-23*Sqrt[2 + 5*x + 3*x^2])/(11250*Sqrt[3 + 2*x]) - ((189 + 211*x)*Sqrt[2 + 5*x
+ 3*x^2])/(2250*(3 + 2*x)^(5/2)) + ((44 + 51*x)*(2 + 5*x + 3*x^2)^(3/2))/(45*(3
+ 2*x)^(9/2)) + (23*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]]
, -2/3])/(7500*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (7*Sqrt[-2 - 5*x - 3*x^2]*Ellipt
icF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(1500*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 54.8214, size = 190, normalized size = 0.94 \[ \frac{23 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{22500 \sqrt{3 x^{2} + 5 x + 2}} + \frac{7 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{4500 \sqrt{3 x^{2} + 5 x + 2}} - \frac{23 \sqrt{3 x^{2} + 5 x + 2}}{11250 \sqrt{2 x + 3}} - \frac{\left (1477 x + 1323\right ) \sqrt{3 x^{2} + 5 x + 2}}{15750 \left (2 x + 3\right )^{\frac{5}{2}}} + \frac{\left (357 x + 308\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{315 \left (2 x + 3\right )^{\frac{9}{2}}} \]

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)**(11/2),x)

[Out]

23*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(225
00*sqrt(3*x**2 + 5*x + 2)) + 7*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*
sqrt(6*x + 6)/2), -2/3)/(4500*sqrt(3*x**2 + 5*x + 2)) - 23*sqrt(3*x**2 + 5*x + 2
)/(11250*sqrt(2*x + 3)) - (1477*x + 1323)*sqrt(3*x**2 + 5*x + 2)/(15750*(2*x + 3
)**(5/2)) + (357*x + 308)*(3*x**2 + 5*x + 2)**(3/2)/(315*(2*x + 3)**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.720896, size = 192, normalized size = 0.95 \[ \frac{204180 x^5+822160 x^4+1297210 x^3+998860 x^2+373610 x-44 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{11/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+23 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{11/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+53980}{22500 (2 x+3)^{9/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^(11/2),x]

[Out]

(53980 + 373610*x + 998860*x^2 + 1297210*x^3 + 822160*x^4 + 204180*x^5 + 23*Sqrt
[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(11/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE
[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 44*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 +
2*x)^(11/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]]
, 3/5])/(22500*(3 + 2*x)^(9/2)*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.051, size = 512, normalized size = 2.5 \[{\frac{1}{225000} \left ( 928\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{4}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}-368\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{4}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+5568\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}-2208\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+12528\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}-4968\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+12528\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}-4968\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}-22080\,{x}^{6}+4698\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -1863\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +1872520\,{x}^{5}+7688000\,{x}^{4}[phone]\,{x}^{3}+9181300\,{x}^{2}+3351080\,x+465280 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}} \left ( 3+2\,x \right ) ^{-{\frac{9}{2}}}} \]

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)^(11/2),x)

[Out]

1/225000*(928*15^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^4*(3
+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)-368*15^(1/2)*EllipticE(1/5*15^(1/2)*
(3+2*x)^(1/2),1/3*15^(1/2))*x^4*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)+55
68*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)-2208*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)+12528*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)-4968*15^(1/2)*EllipticE(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*1
5^(1/2))*x^2*(-30*x-20)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)+12528*15^(1/2)*Ellipt
icF(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)-4968*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)-22080*x^6+4698*(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))-1863*(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))+1872520*x^5+7688000*x^4+12088900*x^3+91813
00*x^2+3351080*x+465280)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(9/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{11}{2}}}\,{d x} \]

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)^(11/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(11/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \sqrt{2 \, x + 3}}, x\right ) \]

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)^(11/2),x, algorithm="fricas")

[Out]

integral(-(3*x^3 - 10*x^2 - 23*x - 10)*sqrt(3*x^2 + 5*x + 2)/((32*x^5 + 240*x^4
+ 720*x^3 + 1080*x^2 + 810*x + 243)*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)**(11/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{11}{2}}}\,{d x} \]

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)^(11/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(11/2), x)