3.2584 \(\int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=229 \[ -\frac{2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{886 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{1287}+\frac{\sqrt{2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}}{27027}-\frac{\sqrt{2 x+3} (783711 x+486863) \sqrt{3 x^2+5 x+2}}{2432430}+\frac{332459 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{972972 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{152657 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{694980 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

-(Sqrt[3 + 2*x]*(486863 + 783711*x)*Sqrt[2 + 5*x + 3*x^2])/2432430 + (Sqrt[3 + 2
*x]*(43822 + 50771*x)*(2 + 5*x + 3*x^2)^(3/2))/27027 + (886*Sqrt[3 + 2*x]*(2 + 5
*x + 3*x^2)^(5/2))/1287 - (2*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(5/2))/39 - (1526
57*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(694980*
Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (332459*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin
[Sqrt[3]*Sqrt[1 + x]], -2/3])/(972972*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.484467, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{886 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{1287}+\frac{\sqrt{2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}}{27027}-\frac{\sqrt{2 x+3} (783711 x+486863) \sqrt{3 x^2+5 x+2}}{2432430}+\frac{332459 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{972972 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{152657 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{694980 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[3 + 2*x]*(486863 + 783711*x)*Sqrt[2 + 5*x + 3*x^2])/2432430 + (Sqrt[3 + 2
*x]*(43822 + 50771*x)*(2 + 5*x + 3*x^2)^(3/2))/27027 + (886*Sqrt[3 + 2*x]*(2 + 5
*x + 3*x^2)^(5/2))/1287 - (2*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(5/2))/39 - (1526
57*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(694980*
Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (332459*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin
[Sqrt[3]*Sqrt[1 + x]], -2/3])/(972972*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 63.4353, size = 226, normalized size = 0.99 \[ - \frac{2 \left (2 x + 3\right )^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{39} + \frac{4 \sqrt{2 x + 3} \left (\frac{456939 x}{4} + \frac{197199}{2}\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{243243} - \frac{2 \sqrt{2 x + 3} \left (\frac{2351133 x}{4} + \frac{1460589}{4}\right ) \sqrt{3 x^{2} + 5 x + 2}}{3648645} + \frac{886 \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{1287} - \frac{152657 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{2084940 \sqrt{3 x^{2} + 5 x + 2}} + \frac{332459 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{2918916 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(2*x + 3)**(3/2)*(3*x**2 + 5*x + 2)**(5/2)/39 + 4*sqrt(2*x + 3)*(456939*x/4 +
197199/2)*(3*x**2 + 5*x + 2)**(3/2)/243243 - 2*sqrt(2*x + 3)*(2351133*x/4 + 146
0589/4)*sqrt(3*x**2 + 5*x + 2)/3648645 + 886*sqrt(2*x + 3)*(3*x**2 + 5*x + 2)**(
5/2)/1287 - 152657*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6
)/2), -2/3)/(2084940*sqrt(3*x**2 + 5*x + 2)) + 332459*sqrt(-9*x**2 - 15*x - 6)*e
lliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(2918916*sqrt(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.550693, size = 213, normalized size = 0.93 \[ -\frac{2 \left (40415760 x^8+52050600 x^7-895236300 x^6-4079217510 x^5-7944858702 x^4-8470029969 x^3-5141306625 x^2-1668494576 x-224705588\right ) \sqrt{2 x+3}-71222 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+1068599 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{14594580 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[3 + 2*x]*(-224705588 - 1668494576*x - 5141306625*x^2 - 8470029969*x^3 -
7944858702*x^4 - 4079217510*x^5 - 895236300*x^6 + 52050600*x^7 + 40415760*x^8)
+ 1068599*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*
EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 71222*Sqrt[5]*Sqrt[(1 + x)/(3
+ 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3
+ 2*x]], 3/5])/(14594580*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.016, size = 167, normalized size = 0.7 \[{\frac{1}{875674800\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+875674800}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -808315200\,{x}^{8}-1041012000\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}+593696\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +1068599\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+4536855720 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/145945800*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(-808315200*x^8-1041012000*x^7+179
04726000*x^6+81584350200*x^5+593696*(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))+1068599*(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))[phone]*x^4+169400599380*x^3+102890248440*x^2+33476751420*x
[phone])/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (6 \, x^{4} - 11 \, x^{3} - 76 \, x^{2} - 89 \, x - 30\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(6*x^4 - 11*x^3 - 76*x^2 - 89*x - 30)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x +
3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- 30 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 89 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 76 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 11 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 6 x^{4} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-30*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-89*x*sqrt(2*x
+ 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-76*x**2*sqrt(2*x + 3)*sqrt(3*x**2 +
5*x + 2), x) - Integral(-11*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Int
egral(6*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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