3.9.37 \(\int \frac {(-3+2 \sqrt {x}) (-3 \sqrt {x}+x)^{2/3}}{\sqrt {x}} \, dx\) [837]

Optimal. Leaf size=17 \[ \frac {6}{5} \left (-3 \sqrt {x}+x\right )^{5/3} \]

[Out]

6/5*(x-3*x^(1/2))^(5/3)

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2059, 643} \begin {gather*} \frac {6}{5} \left (x-3 \sqrt {x}\right )^{5/3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(-3*Sqrt[x] + x)^(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]

Rule 2059

Int[(x_)^(m_.)*((b_.)*(x_)^(k_.) + (a_.)*(x_)^(j_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n
, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n], x]
/; FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] &&  !IntegerQ[p] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && Integ
erQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rubi steps

\begin {align*} \int \frac {\left (-3+2 \sqrt {x}\right ) \left (-3 \sqrt {x}+x\right )^{2/3}}{\sqrt {x}} \, dx &=2 \text {Subst}\left (\int (-3+2 x) \left (-3 x+x^2\right )^{2/3} \, dx,x,\sqrt {x}\right )\\ &=\frac {6}{5} \left (-3 \sqrt {x}+x\right )^{5/3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 17, normalized size = 1.00 \begin {gather*} \frac {6}{5} \left (-3 \sqrt {x}+x\right )^{5/3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(-3*Sqrt[x] + x)^(5/3))/5

________________________________________________________________________________________

Maple [A]
time = 0.54, size = 12, normalized size = 0.71

method	result	size
derivativedivides	\(\frac {6 \left (x -3 \sqrt {x}\right )^{\frac {5}{3}}}{5}\)	\(12\)
default	\(\frac {6 \left (x -3 \sqrt {x}\right )^{\frac {5}{3}}}{5}\)	\(12\)
meijerg	\(-\frac {18 \,3^{\frac {2}{3}} \mathrm {signum}\left (-1+\frac {\sqrt {x}}{3}\right )^{\frac {2}{3}} x^{\frac {5}{6}} \hypergeom \left (\left [-\frac {2}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], \frac {\sqrt {x}}{3}\right )}{5 \left (-\mathrm {signum}\left (-1+\frac {\sqrt {x}}{3}\right )\right )^{\frac {2}{3}}}+\frac {3 \,3^{\frac {2}{3}} \mathrm {signum}\left (-1+\frac {\sqrt {x}}{3}\right )^{\frac {2}{3}} x^{\frac {4}{3}} \hypergeom \left (\left [-\frac {2}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], \frac {\sqrt {x}}{3}\right )}{2 \left (-\mathrm {signum}\left (-1+\frac {\sqrt {x}}{3}\right )\right )^{\frac {2}{3}}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-3*x^(1/2))^(2/3)*(-3+2*x^(1/2))/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

6/5*(x-3*x^(1/2))^(5/3)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x - 3*sqrt(x))^(2/3)*(2*sqrt(x) - 3)/sqrt(x), x)

________________________________________________________________________________________

Fricas [A]
time = 0.41, size = 11, normalized size = 0.65 \begin {gather*} \frac {6}{5} \, {\left (x - 3 \, \sqrt {x}\right )}^{\frac {5}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/5*(x - 3*sqrt(x))^(5/3)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).
time = 0.27, size = 36, normalized size = 2.12 \begin {gather*} - \frac {18 \sqrt {x} \left (- 3 \sqrt {x} + x\right )^{\frac {2}{3}}}{5} + \frac {6 x \left (- 3 \sqrt {x} + x\right )^{\frac {2}{3}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-18*sqrt(x)*(-3*sqrt(x) + x)**(2/3)/5 + 6*x*(-3*sqrt(x) + x)**(2/3)/5

________________________________________________________________________________________

Giac [A]
time = 3.26, size = 11, normalized size = 0.65 \begin {gather*} \frac {6}{5} \, {\left (x - 3 \, \sqrt {x}\right )}^{\frac {5}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/5*(x - 3*sqrt(x))^(5/3)

________________________________________________________________________________________

Mupad [B]
time = 3.70, size = 11, normalized size = 0.65 \begin {gather*} \frac {6\,{\left (x-3\,\sqrt {x}\right )}^{5/3}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*(x - 3*x^(1/2))^(5/3))/5

________________________________________________________________________________________