\(\int \frac {a+b x}{x^3} \, dx\) [49]

Optimal result

Integrand size = 9, antiderivative size = 17 \[ \int \frac {a+b x}{x^3} \, dx=-\frac {(a+b x)^2}{2 a x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {37} \[ \int \frac {a+b x}{x^3} \, dx=-\frac {(a+b x)^2}{2 a x^2} \]

[In]

Int[(a + b*x)/x^3,x]

[Out]

-1/2*(a + b*x)^2/(a*x^2)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^2}{2 a x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x}{x^3} \, dx=-\frac {a}{2 x^2}-\frac {b}{x} \]

[In]

Integrate[(a + b*x)/x^3,x]

[Out]

-1/2*a/x^2 - b/x

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71

method	result	size
gosper	\(-\frac {2 b x +a}{2 x^{2}}\)	\(12\)
norman	\(\frac {-b x -\frac {a}{2}}{x^{2}}\)	\(13\)
risch	\(\frac {-b x -\frac {a}{2}}{x^{2}}\)	\(13\)
default	\(-\frac {b}{x}-\frac {a}{2 x^{2}}\)	\(14\)
parallelrisch	\(\frac {-2 b x -a}{2 x^{2}}\)	\(14\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {a+b x}{x^3} \, dx=-\frac {2 \, b x + a}{2 \, x^{2}} \]

[In]

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

[Out]

-1/2*(2*b*x + a)/x^2

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \frac {a+b x}{x^3} \, dx=\frac {- a - 2 b x}{2 x^{2}} \]

[In]

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

[Out]

(-a - 2*b*x)/(2*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {a+b x}{x^3} \, dx=-\frac {2 \, b x + a}{2 \, x^{2}} \]

[In]

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

[Out]

-1/2*(2*b*x + a)/x^2

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {a+b x}{x^3} \, dx=-\frac {2 \, b x + a}{2 \, x^{2}} \]

[In]

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

[Out]

-1/2*(2*b*x + a)/x^2

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {a+b x}{x^3} \, dx=-\frac {a+2\,b\,x}{2\,x^2} \]

[In]

int((a + b*x)/x^3,x)

[Out]

-(a + 2*b*x)/(2*x^2)