3.924 \(\int \frac{3+x}{\sqrt [3]{6 x+x^2}} \, dx\)

Optimal. Leaf size=15 \[ \frac{3}{4} \left (x^2+6 x\right )^{2/3} \]

[Out]

(3*(6*x + x^2)^(2/3))/4

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Rubi [A]  time = 0.0033134, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {629} \[ \frac{3}{4} \left (x^2+6 x\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[(3 + x)/(6*x + x^2)^(1/3),x]

[Out]

(3*(6*x + x^2)^(2/3))/4

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{3+x}{\sqrt [3]{6 x+x^2}} \, dx &=\frac{3}{4} \left (6 x+x^2\right )^{2/3}\\ \end{align*}

Mathematica [A]  time = 0.0065614, size = 13, normalized size = 0.87 \[ \frac{3}{4} (x (x+6))^{2/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + x)/(6*x + x^2)^(1/3),x]

[Out]

(3*(x*(6 + x))^(2/3))/4

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Maple [A]  time = 0.005, size = 16, normalized size = 1.1 \begin{align*}{\frac{3\,x \left ( x+6 \right ) }{4}{\frac{1}{\sqrt [3]{{x}^{2}+6\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+x)/(x^2+6*x)^(1/3),x)

[Out]

3/4*x*(x+6)/(x^2+6*x)^(1/3)

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Maxima [A]  time = 1.04785, size = 15, normalized size = 1. \begin{align*} \frac{3}{4} \,{\left (x^{2} + 6 \, x\right )}^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(x^2+6*x)^(1/3),x, algorithm="maxima")

[Out]

3/4*(x^2 + 6*x)^(2/3)

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Fricas [A]  time = 1.41599, size = 31, normalized size = 2.07 \begin{align*} \frac{3}{4} \,{\left (x^{2} + 6 \, x\right )}^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(x^2+6*x)^(1/3),x, algorithm="fricas")

[Out]

3/4*(x^2 + 6*x)^(2/3)

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Sympy [A]  time = 0.147264, size = 12, normalized size = 0.8 \begin{align*} \frac{3 \left (x^{2} + 6 x\right )^{\frac{2}{3}}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(x**2+6*x)**(1/3),x)

[Out]

3*(x**2 + 6*x)**(2/3)/4

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Giac [A]  time = 1.09174, size = 15, normalized size = 1. \begin{align*} \frac{3}{4} \,{\left (x^{2} + 6 \, x\right )}^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(x^2+6*x)^(1/3),x, algorithm="giac")

[Out]

3/4*(x^2 + 6*x)^(2/3)