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

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 = 0.0075356, 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 \[ \frac{3}{4} \left (x^2+6 x\right )^{2/3} \]

Antiderivative was successfully verified.

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

[Out]

(3*(6*x + x^2)^(2/3))/4

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Rubi in Sympy [A]  time = 1.13665, size = 12, normalized size = 0.8 \[ \frac{3 \left (x^{2} + 6 x\right )^{\frac{2}{3}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+x)/(x**2+6*x)**(1/3),x)

[Out]

3*(x**2 + 6*x)**(2/3)/4

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Mathematica [A]  time = 0.0170807, 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 \[{\frac{3\,x \left ( x+6 \right ) }{4}{\frac{1}{\sqrt [3]{{x}^{2}+6\,x}}}} \]

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 = 0.674427, size = 15, normalized size = 1. \[ \frac{3}{4} \,{\left (x^{2} + 6 \, x\right )}^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 3)/(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 = 0.259433, size = 15, normalized size = 1. \[ \frac{3}{4} \,{\left (x^{2} + 6 \, x\right )}^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 3)/(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 = 0.381422, size = 12, normalized size = 0.8 \[ \frac{3 \left (x^{2} + 6 x\right )^{\frac{2}{3}}}{4} \]

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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GIAC/XCAS [A]  time = 0.258715, size = 15, normalized size = 1. \[ \frac{3}{4} \,{\left (x^{2} + 6 \, x\right )}^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*(x^2 + 6*x)^(2/3)