3.2.38 \(\int (-3+2 x) (-3 x+x^2)^{2/3} \, dx\) [138]

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

[Out]

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

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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 = {643} \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 643

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*}

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Mathematica [A]
time = 9.31, size = 13, normalized size = 0.87 \begin {gather*} \frac {3}{5} ((-3+x) 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

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Maple [A]
time = 0.11, size = 12, normalized size = 0.80

method	result	size
derivativedivides	\(\frac {3 \left (x^{2}-3 x \right )^{\frac {5}{3}}}{5}\)	\(12\)
default	\(\frac {3 \left (x^{2}-3 x \right )^{\frac {5}{3}}}{5}\)	\(12\)
gosper	\(\frac {3 \left (x -3\right ) x \left (x^{2}-3 x \right )^{\frac {2}{3}}}{5}\)	\(16\)
trager	\(\frac {3 \left (x -3\right ) x \left (x^{2}-3 x \right )^{\frac {2}{3}}}{5}\)	\(16\)
risch	\(\frac {3 \left (x -3\right )^{2} x^{2}}{5 \left (\left (x -3\right ) x \right )^{\frac {1}{3}}}\)	\(18\)
meijerg	\(-\frac {9 \,3^{\frac {2}{3}} \mathrm {signum}\left (x -3\right )^{\frac {2}{3}} x^{\frac {5}{3}} \hypergeom \left (\left [-\frac {2}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], \frac {x}{3}\right )}{5 \left (-\mathrm {signum}\left (x -3\right )\right )^{\frac {2}{3}}}+\frac {3 \,3^{\frac {2}{3}} \mathrm {signum}\left (x -3\right )^{\frac {2}{3}} x^{\frac {8}{3}} \hypergeom \left (\left [-\frac {2}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], \frac {x}{3}\right )}{4 \left (-\mathrm {signum}\left (x -3\right )\right )^{\frac {2}{3}}}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x-3)*(x^2-3*x)^(2/3),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, 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)

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Fricas [A]
time = 0.33, 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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
time = 0.07, 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

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Giac [A]
time = 3.68, 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)

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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

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