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

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2067, 2064, 37} \[ \frac {3 (3-x) (x+1)}{4 \left (x^3-5 x^2+3 x+9\right )^{2/3}} \]

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))

Rule 37

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

Rule 2064

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> Dist[(a + b*x + d*x^3)^p/((3*a - b*x)^p*(3*a + 2*b*
x)^(2*p)), Int[(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0]
&&  !IntegerQ[p]

Rule 2067

Int[(P3_)^(p_), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3
, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x,
x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[p, x] && PolyQ[P3, x, 3]

Rubi steps

\begin {align*} \int \frac {1}{\left (9+3 x-5 x^2+x^3\right )^{2/3}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (\frac {128}{27}-\frac {16 x}{3}+x^3\right )^{2/3}} \, dx,x,-\frac {5}{3}+x\right )\\ &=\frac {\left (512 \sqrt [3]{2} (3-x)^{4/3} (1+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {128}{9}-\frac {32 x}{3}\right )^{4/3} \left (\frac {128}{9}+\frac {16 x}{3}\right )^{2/3}} \, dx,x,-\frac {5}{3}+x\right )}{9 \left (9+3 x-5 x^2+x^3\right )^{2/3}}\\ &=\frac {3 (3-x) (1+x)}{4 \left (9+3 x-5 x^2+x^3\right )^{2/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, 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))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 3.22, size = 26, normalized size = 0.90 \[ -\frac {3 \sqrt [3]{x^3-5 x^2+3 x+9}}{4 (x-3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.88, size = 22, normalized size = 0.76 \[ -\frac {3 \, {\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {1}{3}}}{4 \, {\left (x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 20, normalized size = 0.69

method	result	size
risch	\(-\frac {3 \left (-3+x \right ) \left (1+x \right )}{4 \left (\left (1+x \right ) \left (-3+x \right )^{2}\right )^{\frac {2}{3}}}\)	\(20\)
trager	\(-\frac {3 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}}{4 \left (-3+x \right )}\)	\(23\)
gosper	\(-\frac {3 \left (1+x \right ) \left (-3+x \right )}{4 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3-5*x^2+3*x+9)^(2/3),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 24, normalized size = 0.83 \[ -\frac {3\,{\left (x^3-5\,x^2+3\,x+9\right )}^{1/3}}{4\,\left (x-3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac {2}{3}}}\, dx \]

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)

________________________________________________________________________________________