### 3.2332 $$\int \frac{3-x+x^2}{\sqrt {x}} \, dx$$

Optimal. Leaf size=28 $\frac{3 x^{8/3}}{8}-\frac{3 x^{5/3}}{5}+\frac{9 x^{2/3}}{2}$

[Out]

(9*x^(2/3))/2 - (3*x^(5/3))/5 + (3*x^(8/3))/8

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Rubi [A]  time = 0.0044893, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.071, Rules used = {14} $\frac{3 x^{8/3}}{8}-\frac{3 x^{5/3}}{5}+\frac{9 x^{2/3}}{2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(3 - x + x^2)/x^(1/3),x]

[Out]

(9*x^(2/3))/2 - (3*x^(5/3))/5 + (3*x^(8/3))/8

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{3-x+x^2}{\sqrt {x}} \, dx &=\int \left (\frac{3}{\sqrt {x}}-x^{2/3}+x^{5/3}\right ) \, dx\\ &=\frac{9 x^{2/3}}{2}-\frac{3 x^{5/3}}{5}+\frac{3 x^{8/3}}{8}\\ \end{align*}

Mathematica [A]  time = 0.0049692, size = 19, normalized size = 0.68 $\frac{3}{40} x^{2/3} \left (5 x^2-8 x+60\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 - x + x^2)/x^(1/3),x]

[Out]

(3*x^(2/3)*(60 - 8*x + 5*x^2))/40

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Maple [A]  time = 0.038, size = 16, normalized size = 0.6 \begin{align*}{\frac{15\,{x}^{2}-24\,x+180}{40}{x}^{{\frac{2}{3}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/40*x^(2/3)*(5*x^2-8*x+60)

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Maxima [A]  time = 1.0002, size = 22, normalized size = 0.79 \begin{align*} \frac{3}{8} \, x^{\frac{8}{3}} - \frac{3}{5} \, x^{\frac{5}{3}} + \frac{9}{2} \, x^{\frac{2}{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*x^(8/3) - 3/5*x^(5/3) + 9/2*x^(2/3)

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Fricas [A]  time = 2.182, size = 45, normalized size = 1.61 \begin{align*} \frac{3}{40} \,{\left (5 \, x^{2} - 8 \, x + 60\right )} x^{\frac{2}{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/40*(5*x^2 - 8*x + 60)*x^(2/3)

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Sympy [A]  time = 1.96746, size = 24, normalized size = 0.86 \begin{align*} \frac{3 x^{\frac{8}{3}}}{8} - \frac{3 x^{\frac{5}{3}}}{5} + \frac{9 x^{\frac{2}{3}}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x**(8/3)/8 - 3*x**(5/3)/5 + 9*x**(2/3)/2

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Giac [A]  time = 1.891, size = 22, normalized size = 0.79 \begin{align*} \frac{3}{8} \, x^{\frac{8}{3}} - \frac{3}{5} \, x^{\frac{5}{3}} + \frac{9}{2} \, x^{\frac{2}{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*x^(8/3) - 3/5*x^(5/3) + 9/2*x^(2/3)