3.234 \(\int \frac{1}{\left (9+3 x-5 x^2+x^3\right )^{4/3}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{27 (x+1) (3-x)^3}{320 \left (x^3-5 x^2+3 x+9\right )^{4/3}}+\frac{9 (x+1) (3-x)^2}{80 \left (x^3-5 x^2+3 x+9\right )^{4/3}}+\frac{3 (x+1) (3-x)}{20 \left (x^3-5 x^2+3 x+9\right )^{4/3}} \]

[Out]

(3*(3 - x)*(1 + x))/(20*(9 + 3*x - 5*x^2 + x^3)^(4/3)) + (9*(3 - x)^2*(1 + x))/(
80*(9 + 3*x - 5*x^2 + x^3)^(4/3)) - (27*(3 - x)^3*(1 + x))/(320*(9 + 3*x - 5*x^2
+ x^3)^(4/3))

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Rubi [A]  time = 0.187232, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{27 (x+1) (3-x)^3}{320 \left (x^3-5 x^2+3 x+9\right )^{4/3}}+\frac{9 (x+1) (3-x)^2}{80 \left (x^3-5 x^2+3 x+9\right )^{4/3}}+\frac{3 (x+1) (3-x)}{20 \left (x^3-5 x^2+3 x+9\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]  Int[(9 + 3*x - 5*x^2 + x^3)^(-4/3),x]

[Out]

(3*(3 - x)*(1 + x))/(20*(9 + 3*x - 5*x^2 + x^3)^(4/3)) + (9*(3 - x)^2*(1 + x))/(
80*(9 + 3*x - 5*x^2 + x^3)^(4/3)) - (27*(3 - x)^3*(1 + x))/(320*(9 + 3*x - 5*x^2
+ x^3)^(4/3))

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Rubi in Sympy [A]  time = 3.32098, size = 85, normalized size = 0.92 \[ \frac{27 \left (- x + 3\right )^{2} \left (x + 1\right )^{2}}{320 \left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac{4}{3}}} + \frac{9 \left (- x + 3\right ) \left (x + 1\right )^{2}}{40 \left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac{4}{3}}} - \frac{3 \left (- x + 3\right ) \left (x + 1\right )}{4 \left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**3-5*x**2+3*x+9)**(4/3),x)

[Out]

27*(-x + 3)**2*(x + 1)**2/(320*(x**3 - 5*x**2 + 3*x + 9)**(4/3)) + 9*(-x + 3)*(x
+ 1)**2/(40*(x**3 - 5*x**2 + 3*x + 9)**(4/3)) - 3*(-x + 3)*(x + 1)/(4*(x**3 - 5
*x**2 + 3*x + 9)**(4/3))

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Mathematica [A]  time = 0.0192313, size = 32, normalized size = 0.35 \[ \frac{3 \left (9 x^2-42 x+29\right )}{320 (x-3) \sqrt [3]{(x-3)^2 (x+1)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(9 + 3*x - 5*x^2 + x^3)^(-4/3),x]

[Out]

(3*(29 - 42*x + 9*x^2))/(320*(-3 + x)*((-3 + x)^2*(1 + x))^(1/3))

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Maple [A]  time = 0.006, size = 34, normalized size = 0.4 \[{\frac{ \left ( 3+3\,x \right ) \left ( -3+x \right ) \left ( 9\,{x}^{2}-42\,x+29 \right ) }{320} \left ({x}^{3}-5\,{x}^{2}+3\,x+9 \right ) ^{-{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^3-5*x^2+3*x+9)^(4/3),x)

[Out]

3/320*(1+x)*(-3+x)*(9*x^2-42*x+29)/(x^3-5*x^2+3*x+9)^(4/3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 5*x^2 + 3*x + 9)^(-4/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.200946, size = 43, normalized size = 0.47 \[ \frac{3 \,{\left (9 \, x^{2} - 42 \, x + 29\right )}}{320 \,{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac{1}{3}}{\left (x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 5*x^2 + 3*x + 9)^(-4/3),x, algorithm="fricas")

[Out]

3/320*(9*x^2 - 42*x + 29)/((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. \[ \int \frac{1}{\left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac{4}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**3-5*x**2+3*x+9)**(4/3),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 5*x^2 + 3*x + 9)^(-4/3),x, algorithm="giac")

[Out]

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