3.1422 \(\int \frac{(5-x) (3+2 x)^2}{(2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=48 \[ \frac{(2 x+3)^2 (15 x+2)}{18 \left (3 x^2+2\right )^{3/2}}-\frac{41 (4-9 x)}{54 \sqrt{3 x^2+2}} \]

[Out]

((3 + 2*x)^2*(2 + 15*x))/(18*(2 + 3*x^2)^(3/2)) - (41*(4 - 9*x))/(54*Sqrt[2 + 3*x^2])

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Rubi [A]  time = 0.0181029, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {805, 637} \[ \frac{(2 x+3)^2 (15 x+2)}{18 \left (3 x^2+2\right )^{3/2}}-\frac{41 (4-9 x)}{54 \sqrt{3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3 + 2*x)^2*(2 + 15*x))/(18*(2 + 3*x^2)^(3/2)) - (41*(4 - 9*x))/(54*Sqrt[2 + 3*x^2])

Rule 805

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

Rule 637

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

Rubi steps

\begin{align*} \int \frac{(5-x) (3+2 x)^2}{\left (2+3 x^2\right )^{5/2}} \, dx &=\frac{(3+2 x)^2 (2+15 x)}{18 \left (2+3 x^2\right )^{3/2}}+\frac{41}{9} \int \frac{3+2 x}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{(3+2 x)^2 (2+15 x)}{18 \left (2+3 x^2\right )^{3/2}}-\frac{41 (4-9 x)}{54 \sqrt{2+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.110465, size = 30, normalized size = 0.62 \[ -\frac{-1287 x^3-72 x^2-1215 x+274}{54 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(274 - 1215*x - 72*x^2 - 1287*x^3)/(54*(2 + 3*x^2)^(3/2))

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Maple [A]  time = 0.004, size = 27, normalized size = 0.6 \begin{align*}{\frac{1287\,{x}^{3}+72\,{x}^{2}+1215\,x-274}{54} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(1287*x^3+72*x^2+1215*x-274)/(3*x^2+2)^(3/2)

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Maxima [A]  time = 0.990341, size = 68, normalized size = 1.42 \begin{align*} \frac{143 \, x}{18 \, \sqrt{3 \, x^{2} + 2}} + \frac{4 \, x^{2}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{119 \, x}{18 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{137}{27 \,{\left (3 \, x^{2} + 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+2)^(5/2),x, algorithm="maxima")

[Out]

143/18*x/sqrt(3*x^2 + 2) + 4/3*x^2/(3*x^2 + 2)^(3/2) + 119/18*x/(3*x^2 + 2)^(3/2) - 137/27/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 1.50824, size = 105, normalized size = 2.19 \begin{align*} \frac{{\left (1287 \, x^{3} + 72 \, x^{2} + 1215 \, x - 274\right )} \sqrt{3 \, x^{2} + 2}}{54 \,{\left (9 \, x^{4} + 12 \, x^{2} + 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+2)^(5/2),x, algorithm="fricas")

[Out]

1/54*(1287*x^3 + 72*x^2 + 1215*x - 274)*sqrt(3*x^2 + 2)/(9*x^4 + 12*x^2 + 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} + 2} + 12 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{8 x^{2}}{9 x^{4} \sqrt{3 x^{2} + 2} + 12 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}}\, dx - \int \frac{4 x^{3}}{9 x^{4} \sqrt{3 x^{2} + 2} + 12 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{45}{9 x^{4} \sqrt{3 x^{2} + 2} + 12 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 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+2)**(5/2),x)

[Out]

-Integral(-51*x/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-8*x*
*2/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(4*x**3/(9*x**4*sqr
t(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-45/(9*x**4*sqrt(3*x**2 + 2) + 1
2*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x)

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Giac [A]  time = 1.10715, size = 34, normalized size = 0.71 \begin{align*} \frac{9 \,{\left ({\left (143 \, x + 8\right )} x + 135\right )} x - 274}{54 \,{\left (3 \, x^{2} + 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+2)^(5/2),x, algorithm="giac")

[Out]

1/54*(9*((143*x + 8)*x + 135)*x - 274)/(3*x^2 + 2)^(3/2)