3.837 \(\int \frac{(-3+2 \sqrt{x}) (-3 \sqrt{x}+x)^{2/3}}{\sqrt{x}} \, dx\)

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

[Out]

(6*(-3*Sqrt[x] + x)^(5/3))/5

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Rubi [A]  time = 0.0613476, antiderivative size = 17, normalized size of antiderivative = 1., 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 = {2034, 629} \[ \frac{6}{5} \left (x-3 \sqrt{x}\right )^{5/3} \]

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 2034

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]

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 \frac{\left (-3+2 \sqrt{x}\right ) \left (-3 \sqrt{x}+x\right )^{2/3}}{\sqrt{x}} \, dx &=2 \operatorname{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 = 0.0158369, size = 17, normalized size = 1. \[ \frac{6}{5} \left (x-3 \sqrt{x}\right )^{5/3} \]

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

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Maple [A]  time = 0.006, size = 12, normalized size = 0.7 \begin{align*}{\frac{6}{5} \left ( x-3\,\sqrt{x} \right ) ^{{\frac{5}{3}}}} \end{align*}

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)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x - 3 \, \sqrt{x}\right )}^{\frac{2}{3}}{\left (2 \, \sqrt{x} - 3\right )}}{\sqrt{x}}\,{d x} \end{align*}

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)

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Fricas [A]  time = 2.07424, size = 36, normalized size = 2.12 \begin{align*} \frac{6}{5} \,{\left (x - 3 \, \sqrt{x}\right )}^{\frac{5}{3}} \end{align*}

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)

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Sympy [B]  time = 1.23684, size = 36, normalized size = 2.12 \begin{align*} - \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{align*}

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

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Giac [A]  time = 1.09796, size = 15, normalized size = 0.88 \begin{align*} \frac{6}{5} \,{\left (x - 3 \, \sqrt{x}\right )}^{\frac{5}{3}} \end{align*}

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)