3.2519 \(\int \frac{(5-x) (3+2 x)}{(2+5 x+3 x^2)^{5/2}} \, dx\)

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])

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Rubi [A]  time = 0.0156649, 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, Rules used = {777, 613} \[ \frac{1124 (6 x+5)}{9 \sqrt{3 x^2+5 x+2}}-\frac{2 (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}} \]

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])

Rule 777

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{562}{9} \int \frac{1}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{1124 (5+6 x)}{9 \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0627073, 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)

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Maple [A]  time = 0.004, size = 38, normalized size = 0.8 \begin{align*} 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}}} \end{align*}

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)

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Maxima [A]  time = 1.19307, size = 80, normalized size = 1.7 \begin{align*} \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}}} \end{align*}

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, 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)

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Fricas [A]  time = 1.84629, size = 134, normalized size = 2.85 \begin{align*} \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} \end{align*}

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, 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 + 37*x^2 + 20*x + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \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}}\, 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 - \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}}\, dx \end{align*}

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) + 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)

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Giac [A]  time = 1.13379, size = 38, normalized size = 0.81 \begin{align*} \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}}} \end{align*}

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, algorithm="giac")

[Out]

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