3.1423 \(\int \frac{(5-x) (3+2 x)}{(2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=37 \[ \frac{43 x}{18 \sqrt{3 x^2+2}}-\frac{7 (2-7 x)}{18 \left (3 x^2+2\right )^{3/2}} \]

[Out]

(-7*(2 - 7*x))/(18*(2 + 3*x^2)^(3/2)) + (43*x)/(18*Sqrt[2 + 3*x^2])

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Rubi [A]  time = 0.0099111, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {778, 191} \[ \frac{43 x}{18 \sqrt{3 x^2+2}}-\frac{7 (2-7 x)}{18 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 - 7*x))/(18*(2 + 3*x^2)^(3/2)) + (43*x)/(18*Sqrt[2 + 3*x^2])

Rule 778

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{(5-x) (3+2 x)}{\left (2+3 x^2\right )^{5/2}} \, dx &=-\frac{7 (2-7 x)}{18 \left (2+3 x^2\right )^{3/2}}+\frac{43}{9} \int \frac{1}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{7 (2-7 x)}{18 \left (2+3 x^2\right )^{3/2}}+\frac{43 x}{18 \sqrt{2+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0154025, size = 25, normalized size = 0.68 \[ -\frac{-129 x^3-135 x+14}{18 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(14 - 135*x - 129*x^3)/(18*(2 + 3*x^2)^(3/2))

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Maple [A]  time = 0.003, size = 22, normalized size = 0.6 \begin{align*}{\frac{129\,{x}^{3}+135\,x-14}{18} \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)/(3*x^2+2)^(5/2),x)

[Out]

1/18*(129*x^3+135*x-14)/(3*x^2+2)^(3/2)

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Maxima [A]  time = 0.982255, size = 49, normalized size = 1.32 \begin{align*} \frac{43 \, x}{18 \, \sqrt{3 \, x^{2} + 2}} + \frac{49 \, x}{18 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{7}{9 \,{\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)/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

43/18*x/sqrt(3*x^2 + 2) + 49/18*x/(3*x^2 + 2)^(3/2) - 7/9/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 1.47716, size = 89, normalized size = 2.41 \begin{align*} \frac{{\left (129 \, x^{3} + 135 \, x - 14\right )} \sqrt{3 \, x^{2} + 2}}{18 \,{\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)/(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

1/18*(129*x^3 + 135*x - 14)*sqrt(3*x^2 + 2)/(9*x^4 + 12*x^2 + 4)

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Sympy [B]  time = 91.9838, size = 122, normalized size = 3.3 \begin{align*} - \frac{2 x^{3}}{18 x^{2} \sqrt{3 x^{2} + 2} + 12 \sqrt{3 x^{2} + 2}} + \frac{15 x^{3}}{6 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}} + \frac{15 x}{6 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}} - \frac{7}{27 x^{2} \sqrt{3 x^{2} + 2} + 18 \sqrt{3 x^{2} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)/(3*x**2+2)**(5/2),x)

[Out]

-2*x**3/(18*x**2*sqrt(3*x**2 + 2) + 12*sqrt(3*x**2 + 2)) + 15*x**3/(6*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 +
2)) + 15*x/(6*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)) - 7/(27*x**2*sqrt(3*x**2 + 2) + 18*sqrt(3*x**2 + 2))

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Giac [A]  time = 1.15975, size = 31, normalized size = 0.84 \begin{align*} \frac{3 \,{\left (43 \, x^{2} + 45\right )} x - 14}{18 \,{\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)/(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

1/18*(3*(43*x^2 + 45)*x - 14)/(3*x^2 + 2)^(3/2)