\(\int \frac {a+b x}{x^{10}} \, dx\) [250]

Optimal result

Integrand size = 9, antiderivative size = 17 \[ \int \frac {a+b x}{x^{10}} \, dx=-\frac {a}{9 x^9}-\frac {b}{8 x^8} \]

[Out]

-1/9*a/x^9-1/8*b/x^8

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} \[ \int \frac {a+b x}{x^{10}} \, dx=-\frac {a}{9 x^9}-\frac {b}{8 x^8} \]

[In]

Int[(a + b*x)/x^10,x]

[Out]

-1/9*a/x^9 - b/(8*x^8)

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^{10}}+\frac {b}{x^9}\right ) \, dx \\ & = -\frac {a}{9 x^9}-\frac {b}{8 x^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{x^{10}} \, dx=-\frac {a}{9 x^9}-\frac {b}{8 x^8} \]

[In]

Integrate[(a + b*x)/x^10,x]

[Out]

-1/9*a/x^9 - b/(8*x^8)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76

method	result	size
norman	\(\frac {-\frac {b x}{8}-\frac {a}{9}}{x^{9}}\)	\(13\)
risch	\(\frac {-\frac {b x}{8}-\frac {a}{9}}{x^{9}}\)	\(13\)
gosper	\(-\frac {9 b x +8 a}{72 x^{9}}\)	\(14\)
default	\(-\frac {a}{9 x^{9}}-\frac {b}{8 x^{8}}\)	\(14\)
parallelrisch	\(\frac {-9 b x -8 a}{72 x^{9}}\)	\(14\)

[In]

int((b*x+a)/x^10,x,method=_RETURNVERBOSE)

[Out]

1/x^9*(-1/8*b*x-1/9*a)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x}{x^{10}} \, dx=-\frac {9 \, b x + 8 \, a}{72 \, x^{9}} \]

[In]

integrate((b*x+a)/x^10,x, algorithm="fricas")

[Out]

-1/72*(9*b*x + 8*a)/x^9

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {a+b x}{x^{10}} \, dx=\frac {- 8 a - 9 b x}{72 x^{9}} \]

[In]

integrate((b*x+a)/x**10,x)

[Out]

(-8*a - 9*b*x)/(72*x**9)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x}{x^{10}} \, dx=-\frac {9 \, b x + 8 \, a}{72 \, x^{9}} \]

[In]

integrate((b*x+a)/x^10,x, algorithm="maxima")

[Out]

-1/72*(9*b*x + 8*a)/x^9

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x}{x^{10}} \, dx=-\frac {9 \, b x + 8 \, a}{72 \, x^{9}} \]

[In]

integrate((b*x+a)/x^10,x, algorithm="giac")

[Out]

-1/72*(9*b*x + 8*a)/x^9

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x}{x^{10}} \, dx=-\frac {8\,a+9\,b\,x}{72\,x^9} \]

[In]

int((a + b*x)/x^10,x)

[Out]

-(8*a + 9*b*x)/(72*x^9)