\(\int \frac {a+b x^4}{x^9} \, dx\) [618]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {a+b x^4}{x^9} \, dx=-\frac {a}{8 x^8}-\frac {b}{4 x^4} \]

[Out]

-1/8*a/x^8-1/4*b/x^4

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \int \frac {a+b x^4}{x^9} \, dx=-\frac {a}{8 x^8}-\frac {b}{4 x^4} \]

[In]

Int[(a + b*x^4)/x^9,x]

[Out]

-1/8*a/x^8 - b/(4*x^4)

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^9}+\frac {b}{x^5}\right ) \, dx \\ & = -\frac {a}{8 x^8}-\frac {b}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^4}{x^9} \, dx=-\frac {a}{8 x^8}-\frac {b}{4 x^4} \]

[In]

Integrate[(a + b*x^4)/x^9,x]

[Out]

-1/8*a/x^8 - b/(4*x^4)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
gosper	\(-\frac {2 b \,x^{4}+a}{8 x^{8}}\)	\(14\)
default	\(-\frac {a}{8 x^{8}}-\frac {b}{4 x^{4}}\)	\(14\)
norman	\(\frac {-\frac {b \,x^{4}}{4}-\frac {a}{8}}{x^{8}}\)	\(15\)
risch	\(\frac {-\frac {b \,x^{4}}{4}-\frac {a}{8}}{x^{8}}\)	\(15\)
parallelrisch	\(\frac {-2 b \,x^{4}-a}{8 x^{8}}\)	\(16\)

[In]

int((b*x^4+a)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/8*(2*b*x^4+a)/x^8

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x^4}{x^9} \, dx=-\frac {2 \, b x^{4} + a}{8 \, x^{8}} \]

[In]

integrate((b*x^4+a)/x^9,x, algorithm="fricas")

[Out]

-1/8*(2*b*x^4 + a)/x^8

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {a+b x^4}{x^9} \, dx=\frac {- a - 2 b x^{4}}{8 x^{8}} \]

[In]

integrate((b*x**4+a)/x**9,x)

[Out]

(-a - 2*b*x**4)/(8*x**8)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x^4}{x^9} \, dx=-\frac {2 \, b x^{4} + a}{8 \, x^{8}} \]

[In]

integrate((b*x^4+a)/x^9,x, algorithm="maxima")

[Out]

-1/8*(2*b*x^4 + a)/x^8

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x^4}{x^9} \, dx=-\frac {2 \, b x^{4} + a}{8 \, x^{8}} \]

[In]

integrate((b*x^4+a)/x^9,x, algorithm="giac")

[Out]

-1/8*(2*b*x^4 + a)/x^8

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x^4}{x^9} \, dx=-\frac {2\,b\,x^4+a}{8\,x^8} \]

[In]

int((a + b*x^4)/x^9,x)

[Out]

-(a + 2*b*x^4)/(8*x^8)