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

Optimal. Leaf size=54 \[ \frac{376 (8 x+7)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 (2 x+3)^2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]

[Out]

(-2*(3 + 2*x)^2*(29 + 35*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (376*(7 + 8*x))/(3*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.0216529, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {804, 636} \[ \frac{376 (8 x+7)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 (2 x+3)^2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3 + 2*x)^2*(29 + 35*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (376*(7 + 8*x))/(3*Sqrt[2 + 5*x + 3*x^2])

Rule 804

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

Rule 636

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

Rubi steps

\begin{align*} \int \frac{(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (3+2 x)^2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{188}{3} \int \frac{3+2 x}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (3+2 x)^2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{376 (7+8 x)}{3 \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0333362, size = 33, normalized size = 0.61 \[ \frac{2 \left (4372 x^3+10932 x^2+8925 x+2371\right )}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2371 + 8925*x + 10932*x^2 + 4372*x^3))/(3*(2 + 5*x + 3*x^2)^(3/2))

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Maple [A]  time = 0.004, size = 38, normalized size = 0.7 \begin{align*}{\frac{ \left ( 8744\,{x}^{3}+21864\,{x}^{2}+17850\,x+4742 \right ) \left ( 1+x \right ) \left ( 2+3\,x \right ) }{3} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(4372*x^3+10932*x^2+8925*x+2371)*(1+x)*(2+3*x)/(3*x^2+5*x+2)^(5/2)

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Maxima [A]  time = 1.14355, size = 103, normalized size = 1.91 \begin{align*} \frac{8744 \, x}{9 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{4 \, x^{2}}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} + \frac{21860}{27 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{1114 \, x}{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{1042}{27 \,{\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)^2/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")

[Out]

8744/9*x/sqrt(3*x^2 + 5*x + 2) + 4/3*x^2/(3*x^2 + 5*x + 2)^(3/2) + 21860/27/sqrt(3*x^2 + 5*x + 2) - 1114/27*x/
(3*x^2 + 5*x + 2)^(3/2) - 1042/27/(3*x^2 + 5*x + 2)^(3/2)

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Fricas [A]  time = 1.94344, size = 139, normalized size = 2.57 \begin{align*} \frac{2 \,{\left (4372 \, x^{3} + 10932 \, x^{2} + 8925 \, x + 2371\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{3 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(4372*x^3 + 10932*x^2 + 8925*x + 2371)*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{51 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{8 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{4 x^{3}}{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{45}{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)**2/(3*x**2+5*x+2)**(5/2),x)

[Out]

-Integral(-51*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(-8*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(4*x**3/(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(-45/(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 = 1.12009, size = 38, normalized size = 0.7 \begin{align*} \frac{2 \,{\left ({\left (4 \,{\left (1093 \, x + 2733\right )} x + 8925\right )} x + 2371\right )}}{3 \,{\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)^2/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")

[Out]

2/3*((4*(1093*x + 2733)*x + 8925)*x + 2371)/(3*x^2 + 5*x + 2)^(3/2)