3.140 \(\int \frac{x (9-9 x+2 x^2)}{\sqrt [3]{-3 x+x^2}} \, dx\)

Optimal. Leaf size=15 \[ \frac{3}{5} \left (x^2-3 x\right )^{5/3} \]

[Out]

(3*(-3*x + x^2)^(5/3))/5

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Rubi [A]  time = 0.0282344, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1631, 629} \[ \frac{3}{5} \left (x^2-3 x\right )^{5/3} \]

Antiderivative was successfully verified.

[In]

Int[(x*(9 - 9*x + 2*x^2))/(-3*x + x^2)^(1/3),x]

[Out]

(3*(-3*x + x^2)^(5/3))/5

Rule 1631

Int[(Pq_)*((e_.)*(x_))^(m_.)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[e, Int[(e*x)^(m - 1)*Polynom
ialQuotient[Pq, b + c*x, x]*(b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{b, c, e, m, p}, x] && PolyQ[Pq, x] && EqQ[
PolynomialRemainder[Pq, b + c*x, x], 0]

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{x \left (9-9 x+2 x^2\right )}{\sqrt [3]{-3 x+x^2}} \, dx &=\int (-3+2 x) \left (-3 x+x^2\right )^{2/3} \, dx\\ &=\frac{3}{5} \left (-3 x+x^2\right )^{5/3}\\ \end{align*}

Mathematica [A]  time = 0.0057753, size = 13, normalized size = 0.87 \[ \frac{3}{5} ((x-3) x)^{5/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(9 - 9*x + 2*x^2))/(-3*x + x^2)^(1/3),x]

[Out]

(3*((-3 + x)*x)^(5/3))/5

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Maple [A]  time = 0.046, size = 20, normalized size = 1.3 \begin{align*}{\frac{3\, \left ( -3+x \right ) ^{2}{x}^{2}}{5}{\frac{1}{\sqrt [3]{{x}^{2}-3\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*x^2-9*x+9)/(x^2-3*x)^(1/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47822, size = 31, normalized size = 2.07 \begin{align*} \frac{3}{5} \,{\left (x^{2} - 3 \, x\right )}^{\frac{5}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*x^2-9*x+9)/(x^2-3*x)^(1/3),x, algorithm="fricas")

[Out]

3/5*(x^2 - 3*x)^(5/3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (x - 3\right ) \left (2 x - 3\right )}{\sqrt [3]{x \left (x - 3\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*x**2-9*x+9)/(x**2-3*x)**(1/3),x)

[Out]

Integral(x*(x - 3)*(2*x - 3)/(x*(x - 3))**(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*x^2-9*x+9)/(x^2-3*x)^(1/3),x, algorithm="giac")

[Out]

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