\(\int \frac {a+b \sqrt {x}}{x^3} \, dx\) [2118]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \frac {a+b \sqrt {x}}{x^3} \, dx=-\frac {a}{2 x^2}-\frac {2 b}{3 x^{3/2}} \]

[Out]

-1/2*a/x^2-2/3*b/x^(3/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, 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} \[ \int \frac {a+b \sqrt {x}}{x^3} \, dx=-\frac {a}{2 x^2}-\frac {2 b}{3 x^{3/2}} \]

[In]

Int[(a + b*Sqrt[x])/x^3,x]

[Out]

-1/2*a/x^2 - (2*b)/(3*x^(3/2))

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^3}+\frac {b}{x^{5/2}}\right ) \, dx \\ & = -\frac {a}{2 x^2}-\frac {2 b}{3 x^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \sqrt {x}}{x^3} \, dx=\frac {-3 a-4 b \sqrt {x}}{6 x^2} \]

[In]

Integrate[(a + b*Sqrt[x])/x^3,x]

[Out]

(-3*a - 4*b*Sqrt[x])/(6*x^2)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(-\frac {a}{2 x^{2}}-\frac {2 b}{3 x^{\frac {3}{2}}}\)	\(14\)
default	\(-\frac {a}{2 x^{2}}-\frac {2 b}{3 x^{\frac {3}{2}}}\)	\(14\)
trager	\(\frac {\left (-1+x \right ) a \left (1+x \right )}{2 x^{2}}-\frac {2 b}{3 x^{\frac {3}{2}}}\)	\(20\)

[In]

int((a+b*x^(1/2))/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*a/x^2-2/3*b/x^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \sqrt {x}}{x^3} \, dx=-\frac {4 \, b \sqrt {x} + 3 \, a}{6 \, x^{2}} \]

[In]

integrate((a+b*x^(1/2))/x^3,x, algorithm="fricas")

[Out]

-1/6*(4*b*sqrt(x) + 3*a)/x^2

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \sqrt {x}}{x^3} \, dx=- \frac {a}{2 x^{2}} - \frac {2 b}{3 x^{\frac {3}{2}}} \]

[In]

integrate((a+b*x**(1/2))/x**3,x)

[Out]

-a/(2*x**2) - 2*b/(3*x**(3/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \sqrt {x}}{x^3} \, dx=-\frac {4 \, b \sqrt {x} + 3 \, a}{6 \, x^{2}} \]

[In]

integrate((a+b*x^(1/2))/x^3,x, algorithm="maxima")

[Out]

-1/6*(4*b*sqrt(x) + 3*a)/x^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \sqrt {x}}{x^3} \, dx=-\frac {4 \, b \sqrt {x} + 3 \, a}{6 \, x^{2}} \]

[In]

integrate((a+b*x^(1/2))/x^3,x, algorithm="giac")

[Out]

-1/6*(4*b*sqrt(x) + 3*a)/x^2

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {a+b \sqrt {x}}{x^3} \, dx=-\frac {a}{2\,x^2}-\frac {2\,b}{3\,x^{3/2}} \]

[In]

int((a + b*x^(1/2))/x^3,x)

[Out]

- a/(2*x^2) - (2*b)/(3*x^(3/2))