3.2518 \(\int \frac{(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{1124 (6 x+5)}{9 \sqrt{3 x^2+5 x+2}}-\frac{2 (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}} \]

[Out]

(-2*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (1124*(5 + 6*x))/(9*Sqrt[2 + 5*
x + 3*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.0592359, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{1124 (6 x+5)}{9 \sqrt{3 x^2+5 x+2}}-\frac{2 (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (1124*(5 + 6*x))/(9*Sqrt[2 + 5*
x + 3*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.33083, size = 39, normalized size = 0.83 \[ \frac{562 \left (12 x + 10\right )}{9 \sqrt{3 x^{2} + 5 x + 2}} - \frac{2 \left (139 x + 121\right )}{9 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

562*(12*x + 10)/(9*sqrt(3*x**2 + 5*x + 2)) - 2*(139*x + 121)/(9*(3*x**2 + 5*x +
2)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0314921, size = 31, normalized size = 0.66 \[ \frac{2 \left (1124 x^3+2810 x^2+2295 x+611\right )}{\left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(611 + 2295*x + 2810*x^2 + 1124*x^3))/(2 + 5*x + 3*x^2)^(3/2)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 38, normalized size = 0.8 \[ 2\,{\frac{ \left ( 1124\,{x}^{3}+2810\,{x}^{2}+2295\,x+611 \right ) \left ( 1+x \right ) \left ( 2+3\,x \right ) }{ \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{5/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(1124*x^3+2810*x^2+2295*x+611)*(1+x)*(2+3*x)/(3*x^2+5*x+2)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.710899, size = 80, normalized size = 1.7 \[ \frac{2248 \, x}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{5620}{9 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{278 \, x}{9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{242}{9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2248/3*x/sqrt(3*x^2 + 5*x + 2) + 5620/9/sqrt(3*x^2 + 5*x + 2) - 278/9*x/(3*x^2 +
5*x + 2)^(3/2) - 242/9/(3*x^2 + 5*x + 2)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.276964, size = 69, normalized size = 1.47 \[ \frac{2 \,{\left (1124 \, x^{3} + 2810 \, x^{2} + 2295 \, x + 611\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(1124*x^3 + 2810*x^2 + 2295*x + 611)*sqrt(3*x^2 + 5*x + 2)/(9*x^4 + 30*x^3 + 3
7*x^2 + 20*x + 4)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{7 x}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{2 x^{2}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{15}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-7*x/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) +
37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 +
5*x + 2)), x) - Integral(2*x**2/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*
x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) +
4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-15/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 3
0*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**
2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.277025, size = 38, normalized size = 0.81 \[ \frac{2 \,{\left ({\left (562 \,{\left (2 \, x + 5\right )} x + 2295\right )} x + 611\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*((562*(2*x + 5)*x + 2295)*x + 611)/(3*x^2 + 5*x + 2)^(3/2)