\(\int \frac {a+b x}{x^5} \, dx\) [51]

Optimal result

Integrand size = 9, antiderivative size = 17 \[ \int \frac {a+b x}{x^5} \, dx=-\frac {a}{4 x^4}-\frac {b}{3 x^3} \]

[Out]

-1/4*a/x^4-1/3*b/x^3

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \frac {a+b x}{x^5} \, dx=-\frac {a}{4 x^4}-\frac {b}{3 x^3} \]

[In]

Int[(a + b*x)/x^5,x]

[Out]

-1/4*a/x^4 - b/(3*x^3)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{x^5}+\frac {b}{x^4}\right ) \, dx \\ & = -\frac {a}{4 x^4}-\frac {b}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{x^5} \, dx=-\frac {a}{4 x^4}-\frac {b}{3 x^3} \]

[In]

Integrate[(a + b*x)/x^5,x]

[Out]

-1/4*a/x^4 - b/(3*x^3)

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76

method	result	size
norman	\(\frac {-\frac {b x}{3}-\frac {a}{4}}{x^{4}}\)	\(13\)
risch	\(\frac {-\frac {b x}{3}-\frac {a}{4}}{x^{4}}\)	\(13\)
gosper	\(-\frac {4 b x +3 a}{12 x^{4}}\)	\(14\)
default	\(-\frac {a}{4 x^{4}}-\frac {b}{3 x^{3}}\)	\(14\)
parallelrisch	\(\frac {-4 b x -3 a}{12 x^{4}}\)	\(14\)

[In]

int((b*x+a)/x^5,x,method=_RETURNVERBOSE)

[Out]

1/x^4*(-1/3*b*x-1/4*a)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x}{x^5} \, dx=-\frac {4 \, b x + 3 \, a}{12 \, x^{4}} \]

[In]

integrate((b*x+a)/x^5,x, algorithm="fricas")

[Out]

-1/12*(4*b*x + 3*a)/x^4

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {a+b x}{x^5} \, dx=\frac {- 3 a - 4 b x}{12 x^{4}} \]

[In]

integrate((b*x+a)/x**5,x)

[Out]

(-3*a - 4*b*x)/(12*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x}{x^5} \, dx=-\frac {4 \, b x + 3 \, a}{12 \, x^{4}} \]

[In]

integrate((b*x+a)/x^5,x, algorithm="maxima")

[Out]

-1/12*(4*b*x + 3*a)/x^4

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x}{x^5} \, dx=-\frac {4 \, b x + 3 \, a}{12 \, x^{4}} \]

[In]

integrate((b*x+a)/x^5,x, algorithm="giac")

[Out]

-1/12*(4*b*x + 3*a)/x^4

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x}{x^5} \, dx=-\frac {3\,a+4\,b\,x}{12\,x^4} \]

[In]

int((a + b*x)/x^5,x)

[Out]

-(3*a + 4*b*x)/(12*x^4)