3.3 \(\int \frac{x+x^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=19 \[ \frac{2 x^{5/2}}{5}+\frac{2 x^{3/2}}{3} \]

[Out]

(2*x^(3/2))/3 + (2*x^(5/2))/5

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Rubi [A]  time = 0.002675, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ \frac{2 x^{5/2}}{5}+\frac{2 x^{3/2}}{3} \]

Antiderivative was successfully verified.

[In]

Int[(x + x^2)/Sqrt[x],x]

[Out]

(2*x^(3/2))/3 + (2*x^(5/2))/5

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

Mathematica [A]  time = 0.0029317, size = 14, normalized size = 0.74 \[ \frac{2}{15} x^{3/2} (3 x+5) \]

Antiderivative was successfully verified.

[In]

Integrate[(x + x^2)/Sqrt[x],x]

[Out]

(2*x^(3/2)*(5 + 3*x))/15

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Maple [A]  time = 0.001, size = 11, normalized size = 0.6 \begin{align*}{\frac{10+6\,x}{15}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*x^(3/2)*(5+3*x)

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Maxima [A]  time = 0.935709, size = 15, normalized size = 0.79 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} + \frac{2}{3} \, x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2) + 2/3*x^(3/2)

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Fricas [A]  time = 1.49986, size = 38, normalized size = 2. \begin{align*} \frac{2}{15} \,{\left (3 \, x^{2} + 5 \, x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*x^2 + 5*x)*sqrt(x)

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Sympy [A]  time = 0.223083, size = 15, normalized size = 0.79 \begin{align*} \frac{2 x^{\frac{5}{2}}}{5} + \frac{2 x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**(5/2)/5 + 2*x**(3/2)/3

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Giac [A]  time = 1.09026, size = 15, normalized size = 0.79 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} + \frac{2}{3} \, x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2) + 2/3*x^(3/2)