3.2520 \(\int \frac{5-x}{(2+5 x+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{280 (6 x+5)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]

[Out]

(-2*(29 + 35*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (280*(5 + 6*x))/(3*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.01041, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {638, 613} \[ \frac{280 (6 x+5)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(29 + 35*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (280*(5 + 6*x))/(3*Sqrt[2 + 5*x + 3*x^2])

Rule 638

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

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}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{140}{3} \int \frac{1}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{280 (5+6 x)}{3 \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0453393, size = 31, normalized size = 0.66 \[ \frac{2 \left (840 x^3+2100 x^2+1715 x+457\right )}{\left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(457 + 1715*x + 2100*x^2 + 840*x^3))/(2 + 5*x + 3*x^2)^(3/2)

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Maple [A]  time = 0.003, size = 38, normalized size = 0.8 \begin{align*} 2\,{\frac{ \left ( 840\,{x}^{3}+2100\,{x}^{2}+1715\,x+457 \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*x^2+5*x+2)^(5/2),x)

[Out]

2*(840*x^3+2100*x^2+1715*x+457)*(1+x)*(2+3*x)/(3*x^2+5*x+2)^(5/2)

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Maxima [A]  time = 1.21875, size = 80, normalized size = 1.7 \begin{align*} \frac{560 \, x}{\sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{1400}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{70 \, x}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{58}{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*x^2+5*x+2)^(5/2),x, algorithm="maxima")

[Out]

560*x/sqrt(3*x^2 + 5*x + 2) + 1400/3/sqrt(3*x^2 + 5*x + 2) - 70/3*x/(3*x^2 + 5*x + 2)^(3/2) - 58/3/(3*x^2 + 5*
x + 2)^(3/2)

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Fricas [A]  time = 1.85477, size = 132, normalized size = 2.81 \begin{align*} \frac{2 \,{\left (840 \, x^{3} + 2100 \, x^{2} + 1715 \, x + 457\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*x^2+5*x+2)^(5/2),x, algorithm="fricas")

[Out]

2*(840*x^3 + 2100*x^2 + 1715*x + 457)*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{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{5}{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*x**2+5*x+2)**(5/2),x)

[Out]

-Integral(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(-5/(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 = 1.1187, size = 39, normalized size = 0.83 \begin{align*} \frac{2 \,{\left (35 \,{\left (12 \,{\left (2 \, x + 5\right )} x + 49\right )} x + 457\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*x^2+5*x+2)^(5/2),x, algorithm="giac")

[Out]

2*(35*(12*(2*x + 5)*x + 49)*x + 457)/(3*x^2 + 5*x + 2)^(3/2)