3.1424 \(\int \frac{5-x}{(2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=37 \[ \frac{5 x}{6 \sqrt{3 x^2+2}}+\frac{15 x+2}{18 \left (3 x^2+2\right )^{3/2}} \]

[Out]

(2 + 15*x)/(18*(2 + 3*x^2)^(3/2)) + (5*x)/(6*Sqrt[2 + 3*x^2])

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Rubi [A]  time = 0.0065831, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {639, 191} \[ \frac{5 x}{6 \sqrt{3 x^2+2}}+\frac{15 x+2}{18 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)/(2 + 3*x^2)^(5/2),x]

[Out]

(2 + 15*x)/(18*(2 + 3*x^2)^(3/2)) + (5*x)/(6*Sqrt[2 + 3*x^2])

Rule 639

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

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}{\left (2+3 x^2\right )^{5/2}} \, dx &=\frac{2+15 x}{18 \left (2+3 x^2\right )^{3/2}}+\frac{5}{3} \int \frac{1}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{2+15 x}{18 \left (2+3 x^2\right )^{3/2}}+\frac{5 x}{6 \sqrt{2+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0103427, size = 25, normalized size = 0.68 \[ \frac{45 x^3+45 x+2}{18 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2 + 45*x + 45*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{45\,{x}^{3}+45\,x+2}{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*x^2+2)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.00243, size = 49, normalized size = 1.32 \begin{align*} \frac{5 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} + \frac{5 \, x}{6 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{1}{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*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

5/6*x/sqrt(3*x^2 + 2) + 5/6*x/(3*x^2 + 2)^(3/2) + 1/9/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 1.50362, size = 85, normalized size = 2.3 \begin{align*} \frac{{\left (45 \, x^{3} + 45 \, x + 2\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*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

1/18*(45*x^3 + 45*x + 2)*sqrt(3*x^2 + 2)/(9*x^4 + 12*x^2 + 4)

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Sympy [B]  time = 51.7392, size = 90, normalized size = 2.43 \begin{align*} \frac{5 x^{3}}{6 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}} + \frac{5 x}{6 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}} + \frac{1}{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*x**2+2)**(5/2),x)

[Out]

5*x**3/(6*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)) + 5*x/(6*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)) + 1
/(27*x**2*sqrt(3*x**2 + 2) + 18*sqrt(3*x**2 + 2))

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Giac [A]  time = 1.2109, size = 28, normalized size = 0.76 \begin{align*} \frac{45 \,{\left (x^{2} + 1\right )} x + 2}{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*x^2+2)^(5/2),x, algorithm="giac")

[Out]

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