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

Optimal. Leaf size=16 \[ \frac{3}{5} (-(3-x) x)^{5/3} \]

[Out]

(3*(-((3 - x)*x))^(5/3))/5

________________________________________________________________________________________

Rubi [A]  time = 0.006378, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1588} \[ \frac{3}{5} (-(3-x) x)^{5/3} \]

Antiderivative was successfully verified.

[In]

Int[((-3 + x)*x)^(2/3)*(-3 + 2*x),x]

[Out]

(3*(-((3 - x)*x))^(5/3))/5

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int ((-3+x) x)^{2/3} (-3+2 x) \, dx &=\frac{3}{5} (-(3-x) x)^{5/3}\\ \end{align*}

Mathematica [A]  time = 0.0031555, size = 13, normalized size = 0.81 \[ \frac{3}{5} ((x-3) x)^{5/3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*((-3 + x)*x)^(5/3))/5

________________________________________________________________________________________

Maple [A]  time = 0.045, size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ( -9+3\,x \right ) x}{5} \left ( \left ( -3+x \right ) x \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.01458, size = 12, normalized size = 0.75 \begin{align*} \frac{3}{5} \, \left ({\left (x - 3\right )} x\right )^{\frac{5}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*((x - 3)*x)^(5/3)

________________________________________________________________________________________

Fricas [A]  time = 1.66149, size = 31, normalized size = 1.94 \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+x)*x)^(2/3)*(-3+2*x),x, algorithm="fricas")

[Out]

3/5*(x^2 - 3*x)^(5/3)

________________________________________________________________________________________

Sympy [A]  time = 10.0711, size = 10, normalized size = 0.62 \begin{align*} \frac{3 \left (x \left (x - 3\right )\right )^{\frac{5}{3}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(x*(x - 3))**(5/3)/5

________________________________________________________________________________________

Giac [A]  time = 1.18262, size = 15, normalized size = 0.94 \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+x)*x)^(2/3)*(-3+2*x),x, algorithm="giac")

[Out]

3/5*(x^2 - 3*x)^(5/3)