3.2117 \(\int \frac{a+b \sqrt{x}}{x^2} \, dx\)

Optimal. Leaf size=15 \[ -\frac{a}{x}-\frac{2 b}{\sqrt{x}} \]

[Out]

-(a/x) - (2*b)/Sqrt[x]

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Rubi [A]  time = 0.0047636, antiderivative size = 15, 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} \[ -\frac{a}{x}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])/x^2,x]

[Out]

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

Mathematica [A]  time = 0.0058011, size = 15, normalized size = 1. \[ -\frac{a}{x}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])/x^2,x]

[Out]

-(a/x) - (2*b)/Sqrt[x]

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Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \begin{align*} -{\frac{a}{x}}-2\,{\frac{b}{\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.951387, size = 18, normalized size = 1.2 \begin{align*} -\frac{2 \, b \sqrt{x} + a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*sqrt(x) + a)/x

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Fricas [A]  time = 1.51455, size = 30, normalized size = 2. \begin{align*} -\frac{2 \, b \sqrt{x} + a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*sqrt(x) + a)/x

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Sympy [A]  time = 0.38967, size = 12, normalized size = 0.8 \begin{align*} - \frac{a}{x} - \frac{2 b}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/x - 2*b/sqrt(x)

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Giac [A]  time = 1.1144, size = 18, normalized size = 1.2 \begin{align*} -\frac{2 \, b \sqrt{x} + a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*sqrt(x) + a)/x