\(\int \frac {a+b x^2+c x^4}{x^6} \, dx\) [822]

Optimal result

Integrand size = 16, antiderivative size = 23 \[ \int \frac {a+b x^2+c x^4}{x^6} \, dx=-\frac {a}{5 x^5}-\frac {b}{3 x^3}-\frac {c}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, 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} \[ \int \frac {a+b x^2+c x^4}{x^6} \, dx=-\frac {a}{5 x^5}-\frac {b}{3 x^3}-\frac {c}{x} \]

[In]

Int[(a + b*x^2 + c*x^4)/x^6,x]

[Out]

-1/5*a/x^5 - b/(3*x^3) - c/x

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^6}+\frac {b}{x^4}+\frac {c}{x^2}\right ) \, dx \\ & = -\frac {a}{5 x^5}-\frac {b}{3 x^3}-\frac {c}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2+c x^4}{x^6} \, dx=-\frac {a}{5 x^5}-\frac {b}{3 x^3}-\frac {c}{x} \]

[In]

Integrate[(a + b*x^2 + c*x^4)/x^6,x]

[Out]

-1/5*a/x^5 - b/(3*x^3) - c/x

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87

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

[In]

int((c*x^4+b*x^2+a)/x^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {a+b x^2+c x^4}{x^6} \, dx=-\frac {15 \, c x^{4} + 5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^6,x, algorithm="fricas")

[Out]

-1/15*(15*c*x^4 + 5*b*x^2 + 3*a)/x^5

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {a+b x^2+c x^4}{x^6} \, dx=\frac {- 3 a - 5 b x^{2} - 15 c x^{4}}{15 x^{5}} \]

[In]

integrate((c*x**4+b*x**2+a)/x**6,x)

[Out]

(-3*a - 5*b*x**2 - 15*c*x**4)/(15*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {a+b x^2+c x^4}{x^6} \, dx=-\frac {15 \, c x^{4} + 5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^6,x, algorithm="maxima")

[Out]

-1/15*(15*c*x^4 + 5*b*x^2 + 3*a)/x^5

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {a+b x^2+c x^4}{x^6} \, dx=-\frac {15 \, c x^{4} + 5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^6,x, algorithm="giac")

[Out]

-1/15*(15*c*x^4 + 5*b*x^2 + 3*a)/x^5

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x^2+c x^4}{x^6} \, dx=-\frac {c\,x^4+\frac {b\,x^2}{3}+\frac {a}{5}}{x^5} \]

[In]

int((a + b*x^2 + c*x^4)/x^6,x)

[Out]

-(a/5 + (b*x^2)/3 + c*x^4)/x^5