\(\int \frac {a+b x^2+c x^4}{x^8} \, dx\) [824]

Optimal result

Integrand size = 16, antiderivative size = 25 \[ \int \frac {a+b x^2+c x^4}{x^8} \, dx=-\frac {a}{7 x^7}-\frac {b}{5 x^5}-\frac {c}{3 x^3} \]

[Out]

-1/7*a/x^7-1/5*b/x^5-1/3*c/x^3

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14} \[ \int \frac {a+b x^2+c x^4}{x^8} \, dx=-\frac {a}{7 x^7}-\frac {b}{5 x^5}-\frac {c}{3 x^3} \]

[In]

Int[(a + b*x^2 + c*x^4)/x^8,x]

[Out]

-1/7*a/x^7 - b/(5*x^5) - c/(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^6}+\frac {c}{x^4}\right ) \, dx \\ & = -\frac {a}{7 x^7}-\frac {b}{5 x^5}-\frac {c}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2+c x^4}{x^8} \, dx=-\frac {a}{7 x^7}-\frac {b}{5 x^5}-\frac {c}{3 x^3} \]

[In]

Integrate[(a + b*x^2 + c*x^4)/x^8,x]

[Out]

-1/7*a/x^7 - b/(5*x^5) - c/(3*x^3)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {a}{7 x^{7}}-\frac {b}{5 x^{5}}-\frac {c}{3 x^{3}}\)	\(20\)
norman	\(\frac {-\frac {1}{3} c \,x^{4}-\frac {1}{5} b \,x^{2}-\frac {1}{7} a}{x^{7}}\)	\(21\)
risch	\(\frac {-\frac {1}{3} c \,x^{4}-\frac {1}{5} b \,x^{2}-\frac {1}{7} a}{x^{7}}\)	\(21\)
gosper	\(-\frac {35 c \,x^{4}+21 b \,x^{2}+15 a}{105 x^{7}}\)	\(22\)
parallelrisch	\(\frac {-35 c \,x^{4}-21 b \,x^{2}-15 a}{105 x^{7}}\)	\(22\)

[In]

int((c*x^4+b*x^2+a)/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/7*a/x^7-1/5*b/x^5-1/3*c/x^3

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {a+b x^2+c x^4}{x^8} \, dx=-\frac {35 \, c x^{4} + 21 \, b x^{2} + 15 \, a}{105 \, x^{7}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^8,x, algorithm="fricas")

[Out]

-1/105*(35*c*x^4 + 21*b*x^2 + 15*a)/x^7

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x^2+c x^4}{x^8} \, dx=\frac {- 15 a - 21 b x^{2} - 35 c x^{4}}{105 x^{7}} \]

[In]

integrate((c*x**4+b*x**2+a)/x**8,x)

[Out]

(-15*a - 21*b*x**2 - 35*c*x**4)/(105*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {a+b x^2+c x^4}{x^8} \, dx=-\frac {35 \, c x^{4} + 21 \, b x^{2} + 15 \, a}{105 \, x^{7}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^8,x, algorithm="maxima")

[Out]

-1/105*(35*c*x^4 + 21*b*x^2 + 15*a)/x^7

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {a+b x^2+c x^4}{x^8} \, dx=-\frac {35 \, c x^{4} + 21 \, b x^{2} + 15 \, a}{105 \, x^{7}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^8,x, algorithm="giac")

[Out]

-1/105*(35*c*x^4 + 21*b*x^2 + 15*a)/x^7

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {a+b x^2+c x^4}{x^8} \, dx=-\frac {\frac {c\,x^4}{3}+\frac {b\,x^2}{5}+\frac {a}{7}}{x^7} \]

[In]

int((a + b*x^2 + c*x^4)/x^8,x)

[Out]

-(a/7 + (b*x^2)/5 + (c*x^4)/3)/x^7