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]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.65, size = 11, normalized size = 0.73 \[ \frac {3}{4} \, {\left (x^{2} + 6 \, x\right )}^{\frac {2}{3}} \]

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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giac [A]  time = 0.43, size = 11, normalized size = 0.73 \[ \frac {3}{4} \, {\left (x^{2} + 6 \, x\right )}^{\frac {2}{3}} \]

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)

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maple [A]  time = 0.00, size = 16, normalized size = 1.07 \[ \frac {3 \left (x +6\right ) x}{4 \left (x^{2}+6 x \right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+3)/(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 = 0.86, size = 11, normalized size = 0.73 \[ \frac {3}{4} \, {\left (x^{2} + 6 \, x\right )}^{\frac {2}{3}} \]

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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mupad [B]  time = 3.56, size = 9, normalized size = 0.60 \[ \frac {3\,{\left (x\,\left (x+6\right )\right )}^{2/3}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 3)/(6*x + x^2)^(1/3),x)

[Out]

(3*(x*(x + 6))^(2/3))/4

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sympy [A]  time = 0.16, size = 12, normalized size = 0.80 \[ \frac {3 \left (x^{2} + 6 x\right )^{\frac {2}{3}}}{4} \]

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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