3.233 \(\int \frac{1}{\left (9+3 x-5 x^2+x^3\right )^{2/3}} \, dx\)

Optimal. Leaf size=29 \[ \frac{3 (3-x) (x+1)}{4 \left (x^3-5 x^2+3 x+9\right )^{2/3}} \]

[Out]

(3*(3 - x)*(1 + x))/(4*(9 + 3*x - 5*x^2 + x^3)^(2/3))

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Rubi [A]  time = 0.0678521, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{3 (3-x) (x+1)}{4 \left (x^3-5 x^2+3 x+9\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(9 + 3*x - 5*x^2 + x^3)^(-2/3),x]

[Out]

(3*(3 - x)*(1 + x))/(4*(9 + 3*x - 5*x^2 + x^3)^(2/3))

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Rubi in Sympy [A]  time = 1.8465, size = 26, normalized size = 0.9 \[ \frac{3 \left (- x + 3\right ) \left (x + 1\right )}{4 \left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**3-5*x**2+3*x+9)**(2/3),x)

[Out]

3*(-x + 3)*(x + 1)/(4*(x**3 - 5*x**2 + 3*x + 9)**(2/3))

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Mathematica [A]  time = 0.0170167, size = 23, normalized size = 0.79 \[ -\frac{3 (x-3) (x+1)}{4 \left ((x-3)^2 (x+1)\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(9 + 3*x - 5*x^2 + x^3)^(-2/3),x]

[Out]

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

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Maple [A]  time = 0.005, size = 24, normalized size = 0.8 \[ -{\frac{ \left ( 3+3\,x \right ) \left ( -3+x \right ) }{4} \left ({x}^{3}-5\,{x}^{2}+3\,x+9 \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^3-5*x^2+3*x+9)^(2/3),x)

[Out]

-3/4*(1+x)*(-3+x)/(x^3-5*x^2+3*x+9)^(2/3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 5*x^2 + 3*x + 9)^(-2/3),x, algorithm="maxima")

[Out]

integrate((x^3 - 5*x^2 + 3*x + 9)^(-2/3), x)

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Fricas [A]  time = 0.210786, size = 30, normalized size = 1.03 \[ -\frac{3 \,{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac{1}{3}}}{4 \,{\left (x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 5*x^2 + 3*x + 9)^(-2/3),x, algorithm="fricas")

[Out]

-3/4*(x^3 - 5*x^2 + 3*x + 9)^(1/3)/(x - 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**3-5*x**2+3*x+9)**(2/3),x)

[Out]

Integral((x**3 - 5*x**2 + 3*x + 9)**(-2/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 5*x^2 + 3*x + 9)^(-2/3),x, algorithm="giac")

[Out]

integrate((x^3 - 5*x^2 + 3*x + 9)^(-2/3), x)