3.1412 \(\int \frac{5-x}{(2+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=20 \[ \frac{15 x+2}{6 \sqrt{3 x^2+2}} \]

[Out]

(2 + 15*x)/(6*Sqrt[2 + 3*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0041599, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {637} \[ \frac{15 x+2}{6 \sqrt{3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)/(2 + 3*x^2)^(3/2),x]

[Out]

(2 + 15*x)/(6*Sqrt[2 + 3*x^2])

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

Mathematica [A]  time = 0.0091496, size = 20, normalized size = 1. \[ \frac{15 x+2}{6 \sqrt{3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2 + 15*x)/(6*Sqrt[2 + 3*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 17, normalized size = 0.9 \begin{align*}{\frac{2+15\,x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2+15*x)/(3*x^2+2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 0.974665, size = 32, normalized size = 1.6 \begin{align*} \frac{5 \, x}{2 \, \sqrt{3 \, x^{2} + 2}} + \frac{1}{3 \, \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)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.46404, size = 43, normalized size = 2.15 \begin{align*} \frac{15 \, x + 2}{6 \, \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)^(3/2),x, algorithm="fricas")

[Out]

1/6*(15*x + 2)/sqrt(3*x^2 + 2)

________________________________________________________________________________________

Sympy [A]  time = 8.052, size = 27, normalized size = 1.35 \begin{align*} \frac{5 x}{2 \sqrt{3 x^{2} + 2}} + \frac{1}{3 \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)**(3/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.17358, size = 22, normalized size = 1.1 \begin{align*} \frac{15 \, x + 2}{6 \, \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)^(3/2),x, algorithm="giac")

[Out]

1/6*(15*x + 2)/sqrt(3*x^2 + 2)