3.89 \(\int \frac{1+x^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=17 \[ \frac{2 x^{5/2}}{5}+2 \sqrt{x} \]

[Out]

2*Sqrt[x] + (2*x^(5/2))/5

________________________________________________________________________________________

Rubi [A]  time = 0.0032497, antiderivative size = 17, 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} \[ \frac{2 x^{5/2}}{5}+2 \sqrt{x} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/Sqrt[x],x]

[Out]

2*Sqrt[x] + (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{1+x^2}{\sqrt{x}} \, dx &=\int \left (\frac{1}{\sqrt{x}}+x^{3/2}\right ) \, dx\\ &=2 \sqrt{x}+\frac{2 x^{5/2}}{5}\\ \end{align*}

Mathematica [A]  time = 0.0028835, size = 14, normalized size = 0.82 \[ \frac{2}{5} \sqrt{x} \left (x^2+5\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(5 + x^2))/5

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 11, normalized size = 0.7 \begin{align*}{\frac{2\,{x}^{2}+10}{5}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(1/2)*(x^2+5)

________________________________________________________________________________________

Maxima [A]  time = 0.923614, size = 15, normalized size = 0.88 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} + 2 \, \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2) + 2*sqrt(x)

________________________________________________________________________________________

Fricas [A]  time = 1.77471, size = 31, normalized size = 1.82 \begin{align*} \frac{2}{5} \,{\left (x^{2} + 5\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(x^2 + 5)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 0.202541, size = 14, normalized size = 0.82 \begin{align*} \frac{2 x^{\frac{5}{2}}}{5} + 2 \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**(5/2)/5 + 2*sqrt(x)

________________________________________________________________________________________

Giac [A]  time = 1.05947, size = 15, normalized size = 0.88 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} + 2 \, \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2) + 2*sqrt(x)