3.138 \(\int (-3+2 x) (-3 x+x^2)^{2/3} \, dx\)

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 = 0.0039147, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {629} \[ \frac{3}{5} \left (x^2-3 x\right )^{5/3} \]

Antiderivative was successfully verified.

[In]

Int[(-3 + 2*x)*(-3*x + x^2)^(2/3),x]

[Out]

(3*(-3*x + x^2)^(5/3))/5

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 (-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 = 0.0067621, size = 13, normalized size = 0.87 \[ \frac{3}{5} ((x-3) x)^{5/3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*((-3 + x)*x)^(5/3))/5

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Maple [A]  time = 0.046, size = 16, normalized size = 1.1 \begin{align*}{\frac{ \left ( -9+3\,x \right ) x}{5} \left ({x}^{2}-3\,x \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.988348, size = 15, normalized size = 1. \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((-3+2*x)*(x^2-3*x)^(2/3),x, algorithm="maxima")

[Out]

3/5*(x^2 - 3*x)^(5/3)

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Fricas [A]  time = 1.49361, 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((-3+2*x)*(x^2-3*x)^(2/3),x, algorithm="fricas")

[Out]

3/5*(x^2 - 3*x)^(5/3)

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Sympy [B]  time = 0.448597, size = 31, normalized size = 2.07 \begin{align*} \frac{3 x^{2} \left (x^{2} - 3 x\right )^{\frac{2}{3}}}{5} - \frac{9 x \left (x^{2} - 3 x\right )^{\frac{2}{3}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x**2*(x**2 - 3*x)**(2/3)/5 - 9*x*(x**2 - 3*x)**(2/3)/5

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Giac [A]  time = 1.15488, size = 15, normalized size = 1. \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((-3+2*x)*(x^2-3*x)^(2/3),x, algorithm="giac")

[Out]

3/5*(x^2 - 3*x)^(5/3)