3.1650 \(\int \frac{a+\frac{b}{x}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=17 \[ 2 a \sqrt{x}-\frac{2 b}{\sqrt{x}} \]

[Out]

(-2*b)/Sqrt[x] + 2*a*Sqrt[x]

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Rubi [A]  time = 0.0042461, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ 2 a \sqrt{x}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)/Sqrt[x],x]

[Out]

(-2*b)/Sqrt[x] + 2*a*Sqrt[x]

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

Mathematica [A]  time = 0.0050942, size = 14, normalized size = 0.82 \[ \frac{2 (a x-b)}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)/Sqrt[x],x]

[Out]

(2*(-b + a*x))/Sqrt[x]

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Maple [A]  time = 0.001, size = 13, normalized size = 0.8 \begin{align*} 2\,{\frac{ax-b}{\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a*x-b)/x^(1/2)

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Maxima [A]  time = 1.00165, size = 18, normalized size = 1.06 \begin{align*} 2 \, a \sqrt{x} - \frac{2 \, b}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(x) - 2*b/sqrt(x)

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Fricas [A]  time = 1.99839, size = 28, normalized size = 1.65 \begin{align*} \frac{2 \,{\left (a x - b\right )}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a*x - b)/sqrt(x)

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Sympy [A]  time = 0.342251, size = 15, normalized size = 0.88 \begin{align*} 2 a \sqrt{x} - \frac{2 b}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(x) - 2*b/sqrt(x)

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Giac [A]  time = 1.09194, size = 18, normalized size = 1.06 \begin{align*} 2 \, a \sqrt{x} - \frac{2 \, b}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(x) - 2*b/sqrt(x)