\(\int \frac {a+b x^4}{x^8} \, dx\) [617]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {a+b x^4}{x^8} \, dx=-\frac {a}{7 x^7}-\frac {b}{3 x^3} \]

[Out]

-1/7*a/x^7-1/3*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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \int \frac {a+b x^4}{x^8} \, dx=-\frac {a}{7 x^7}-\frac {b}{3 x^3} \]

[In]

Int[(a + b*x^4)/x^8,x]

[Out]

-1/7*a/x^7 - b/(3*x^3)

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^8}+\frac {b}{x^4}\right ) \, dx \\ & = -\frac {a}{7 x^7}-\frac {b}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^4}{x^8} \, dx=-\frac {a}{7 x^7}-\frac {b}{3 x^3} \]

[In]

Integrate[(a + b*x^4)/x^8,x]

[Out]

-1/7*a/x^7 - b/(3*x^3)

Maple [A] (verified)

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

method	result	size
default	\(-\frac {a}{7 x^{7}}-\frac {b}{3 x^{3}}\)	\(14\)
norman	\(\frac {-\frac {b \,x^{4}}{3}-\frac {a}{7}}{x^{7}}\)	\(15\)
risch	\(\frac {-\frac {b \,x^{4}}{3}-\frac {a}{7}}{x^{7}}\)	\(15\)
gosper	\(-\frac {7 b \,x^{4}+3 a}{21 x^{7}}\)	\(16\)
parallelrisch	\(\frac {-7 b \,x^{4}-3 a}{21 x^{7}}\)	\(16\)

[In]

int((b*x^4+a)/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/7*a/x^7-1/3*b/x^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x^4}{x^8} \, dx=-\frac {7 \, b x^{4} + 3 \, a}{21 \, x^{7}} \]

[In]

integrate((b*x^4+a)/x^8,x, algorithm="fricas")

[Out]

-1/21*(7*b*x^4 + 3*a)/x^7

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x^4}{x^8} \, dx=\frac {- 3 a - 7 b x^{4}}{21 x^{7}} \]

[In]

integrate((b*x**4+a)/x**8,x)

[Out]

(-3*a - 7*b*x**4)/(21*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x^4}{x^8} \, dx=-\frac {7 \, b x^{4} + 3 \, a}{21 \, x^{7}} \]

[In]

integrate((b*x^4+a)/x^8,x, algorithm="maxima")

[Out]

-1/21*(7*b*x^4 + 3*a)/x^7

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x^4}{x^8} \, dx=-\frac {7 \, b x^{4} + 3 \, a}{21 \, x^{7}} \]

[In]

integrate((b*x^4+a)/x^8,x, algorithm="giac")

[Out]

-1/21*(7*b*x^4 + 3*a)/x^7

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x^4}{x^8} \, dx=-\frac {7\,b\,x^4+3\,a}{21\,x^7} \]

[In]

int((a + b*x^4)/x^8,x)

[Out]

-(3*a + 7*b*x^4)/(21*x^7)