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]

-2/9*(121+139*x)/(3*x^2+5*x+2)^(3/2)+1124/9*(5+6*x)/(3*x^2+5*x+2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 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]

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]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [A]  time = 0.65, size = 51, normalized size = 1.09 \[ \frac {2 \, {\left (1124 \, x^{3} + 2810 \, x^{2} + 2295 \, x + 611\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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giac [A]  time = 0.24, size = 28, normalized size = 0.60 \[ \frac {2 \, {\left ({\left (562 \, {\left (2 \, x + 5\right )} x + 2295\right )} x + 611\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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maple [A]  time = 0.00, size = 38, normalized size = 0.81 \[ \frac {2 \left (1124 x^{3}+2810 x^{2}+2295 x +611\right ) \left (x +1\right ) \left (3 x +2\right )}{\left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}} \]

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)*(x+1)*(3*x+2)/(3*x^2+5*x+2)^(5/2)

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maxima [A]  time = 0.53, size = 59, normalized size = 1.26 \[ \frac {2248 \, x}{3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {5620}{9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {278 \, x}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {242}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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mupad [B]  time = 2.44, size = 48, normalized size = 1.02 \[ \frac {9274\,x+2248\,x\,\left (3\,x^2+5\,x+2\right )+5620\,x^2+3666}{\sqrt {3\,x^2+5\,x+2}\,\left (9\,x^2+15\,x+6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(9274*x + 2248*x*(5*x + 3*x^2 + 2) + 5620*x^2 + 3666)/((5*x + 3*x^2 + 2)^(1/2)*(15*x + 9*x^2 + 6))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {7 x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {2 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {15}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]

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) + 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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