\(\int \frac {a+b \sqrt {x}}{x^2} \, dx\) [2117]

Optimal result

Integrand size = 13, antiderivative size = 15 \[ \int \frac {a+b \sqrt {x}}{x^2} \, dx=-\frac {a}{x}-\frac {2 b}{\sqrt {x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \[ \int \frac {a+b \sqrt {x}}{x^2} \, dx=-\frac {a}{x}-\frac {2 b}{\sqrt {x}} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \sqrt {x}}{x^2} \, dx=\frac {-a-2 b \sqrt {x}}{x} \]

[In]

Integrate[(a + b*Sqrt[x])/x^2,x]

[Out]

(-a - 2*b*Sqrt[x])/x

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(-\frac {a}{x}-\frac {2 b}{\sqrt {x}}\)	\(14\)
default	\(-\frac {a}{x}-\frac {2 b}{\sqrt {x}}\)	\(14\)
trager	\(\frac {a \left (-1+x \right )}{x}-\frac {2 b}{\sqrt {x}}\)	\(16\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \sqrt {x}}{x^2} \, dx=-\frac {2 \, b \sqrt {x} + a}{x} \]

[In]

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

[Out]

-(2*b*sqrt(x) + a)/x

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \sqrt {x}}{x^2} \, dx=- \frac {a}{x} - \frac {2 b}{\sqrt {x}} \]

[In]

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

[Out]

-a/x - 2*b/sqrt(x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \sqrt {x}}{x^2} \, dx=-\frac {2 \, b \sqrt {x} + a}{x} \]

[In]

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

[Out]

-(2*b*sqrt(x) + a)/x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \sqrt {x}}{x^2} \, dx=-\frac {2 \, b \sqrt {x} + a}{x} \]

[In]

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

[Out]

-(2*b*sqrt(x) + a)/x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \sqrt {x}}{x^2} \, dx=-\frac {a}{x}-\frac {2\,b}{\sqrt {x}} \]

[In]

int((a + b*x^(1/2))/x^2,x)

[Out]

- a/x - (2*b)/x^(1/2)