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

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {3}{5} \left (x^2-3 x\right )^{5/3} \end {gather*}

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 = 0.01, size = 13, normalized size = 0.87 \begin {gather*} \frac {3}{5} ((x-3) x)^{5/3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*((-3 + x)*x)^(5/3))/5

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.02, size = 13, normalized size = 0.87 \begin {gather*} \frac {3}{5} ((x-3) x)^{5/3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*((-3 + x)*x)^(5/3))/5

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 11, normalized size = 0.73 \begin {gather*} \frac {3}{5} \, {\left (x^{2} - 3 \, x\right )}^{\frac {5}{3}} \end {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 11, normalized size = 0.73 \begin {gather*} \frac {3}{5} \, {\left (x^{2} - 3 \, x\right )}^{\frac {5}{3}} \end {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 16, normalized size = 1.07 \begin {gather*} \frac {3 \left (x -3\right ) \left (x^{2}-3 x \right )^{\frac {2}{3}} x}{5} \end {gather*}

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

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 11, normalized size = 0.73 \begin {gather*} \frac {3}{5} \, {\left (x^{2} - 3 \, x\right )}^{\frac {5}{3}} \end {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 3.72, size = 15, normalized size = 1.00 \begin {gather*} \frac {3\,x\,{\left (x^2-3\,x\right )}^{2/3}\,\left (x-3\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 3)*(x^2 - 3*x)^(2/3),x)

[Out]

(3*x*(x^2 - 3*x)^(2/3)*(x - 3))/5

________________________________________________________________________________________

sympy [B]  time = 0.31, size = 31, normalized size = 2.07 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________