3.2625 \(\int \frac{5-x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=175 \[ -\frac{2 \sqrt{2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 \sqrt{2 x+3} (2607 x+2152)}{25 \sqrt{3 x^2+5 x+2}}+\frac{916 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3 x^2+5 x+2}}-\frac{3476 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3 x^2+5 x+2}} \]

[Out]

(-2*Sqrt[3 + 2*x]*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)^(3/2)) + (4*Sqrt[3 + 2*x]*(2
152 + 2607*x))/(25*Sqrt[2 + 5*x + 3*x^2]) - (3476*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]
*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(25*Sqrt[2 + 5*x + 3*x^2]) + (916
*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5
*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.343664, 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 \[ -\frac{2 \sqrt{2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 \sqrt{2 x+3} (2607 x+2152)}{25 \sqrt{3 x^2+5 x+2}}+\frac{916 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3 x^2+5 x+2}}-\frac{3476 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[3 + 2*x]*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)^(3/2)) + (4*Sqrt[3 + 2*x]*(2
152 + 2607*x))/(25*Sqrt[2 + 5*x + 3*x^2]) - (3476*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]
*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(25*Sqrt[2 + 5*x + 3*x^2]) + (916
*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5
*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 47.5815, size = 168, normalized size = 0.96 \[ - \frac{2 \sqrt{2 x + 3} \left (141 x + 111\right )}{15 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{4 \sqrt{2 x + 3} \left (7821 x + 6456\right )}{75 \sqrt{3 x^{2} + 5 x + 2}} - \frac{3476 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{25 \sqrt{3 x^{2} + 5 x + 2}} + \frac{916 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{5 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(2*x + 3)*(141*x + 111)/(15*(3*x**2 + 5*x + 2)**(3/2)) + 4*sqrt(2*x + 3)*
(7821*x + 6456)/(75*sqrt(3*x**2 + 5*x + 2)) - 3476*sqrt(-9*x**2 - 15*x - 6)*elli
ptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(25*sqrt(3*x**2 + 5*x + 2)) + 916*sq
rt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(5*sqrt(3
*x**2 + 5*x + 2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.543661, size = 196, normalized size = 1.12 \[ \frac{-\frac{6952 \left (3 x^2+5 x+2\right )}{\sqrt{2 x+3}}+\frac{2 \sqrt{2 x+3} \left (15642 x^3+38982 x^2+31713 x+8423\right )}{3 x^2+5 x+2}+\frac{728 (x+1) \sqrt{\frac{3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{\sqrt{\frac{x+1}{10 x+15}}}-\frac{3476 (x+1) \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{\sqrt{\frac{x+1}{10 x+15}}}}{25 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

((-6952*(2 + 5*x + 3*x^2))/Sqrt[3 + 2*x] + (2*Sqrt[3 + 2*x]*(8423 + 31713*x + 38
982*x^2 + 15642*x^3))/(2 + 5*x + 3*x^2) - (3476*(1 + x)*Sqrt[(2 + 3*x)/(3 + 2*x)
]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/Sqrt[(1 + x)/(15 + 10*x)] + (
728*(1 + x)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]],
3/5])/Sqrt[(1 + x)/(15 + 10*x)])/(25*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.033, size = 326, normalized size = 1.9 \[{\frac{2}{125\, \left ( 1+x \right ) ^{2} \left ( 2+3\,x \right ) ^{2}}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 2607\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+828\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+4345\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+1380\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+1738\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +552\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +156420\,{x}^{4}+624450\,{x}^{3}+901860\,{x}^{2}+559925\,x+126345 \right ){\frac{1}{\sqrt{3+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/125*(3*x^2+5*x+2)^(1/2)*(2607*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)+828*15^(1/2)*Ellip
ticF(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)+4345*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)+1380*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
)+1738*(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))+552*(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))+156420*x^4+624450
*x^3+901860*x^2+559925*x+126345)/(1+x)^2/(2+3*x)^2/(3+2*x)^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(x - 5)/((9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*sqrt(3*x^2 + 5*x + 2)*sq
rt(2*x + 3)), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{9 x^{4} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{5}{9 x^{4} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(9*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(2*x + 3)
*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 20*x*sq
rt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x)
- Integral(-5/(9*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(2*x +
3)*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 20*x
*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)),
x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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