\(\int \frac {a+b x^2+c x^4}{x^4} \, dx\) [820]

Optimal result

Integrand size = 16, antiderivative size = 18 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {b}{x}+c x \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, 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} \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {b}{x}+c x \]

[In]

Int[(a + b*x^2 + c*x^4)/x^4,x]

[Out]

-1/3*a/x^3 - b/x + 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 (c+\frac {a}{x^4}+\frac {b}{x^2}\right ) \, dx \\ & = -\frac {a}{3 x^3}-\frac {b}{x}+c x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {b}{x}+c x \]

[In]

Integrate[(a + b*x^2 + c*x^4)/x^4,x]

[Out]

-1/3*a/x^3 - b/x + c*x

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {a}{3 x^{3}}-\frac {b}{x}+c x\)	\(17\)
risch	\(c x +\frac {-b \,x^{2}-\frac {a}{3}}{x^{3}}\)	\(19\)
gosper	\(-\frac {-3 c \,x^{4}+3 b \,x^{2}+a}{3 x^{3}}\)	\(20\)
norman	\(\frac {c \,x^{4}-b \,x^{2}-\frac {1}{3} a}{x^{3}}\)	\(20\)
parallelrisch	\(\frac {3 c \,x^{4}-3 b \,x^{2}-a}{3 x^{3}}\)	\(22\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=\frac {3 \, c x^{4} - 3 \, b x^{2} - a}{3 \, x^{3}} \]

[In]

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

[Out]

1/3*(3*c*x^4 - 3*b*x^2 - a)/x^3

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=c x + \frac {- a - 3 b x^{2}}{3 x^{3}} \]

[In]

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

[Out]

c*x + (-a - 3*b*x**2)/(3*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=c x - \frac {3 \, b x^{2} + a}{3 \, x^{3}} \]

[In]

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

[Out]

c*x - 1/3*(3*b*x^2 + a)/x^3

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=c x - \frac {3 \, b x^{2} + a}{3 \, x^{3}} \]

[In]

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

[Out]

c*x - 1/3*(3*b*x^2 + a)/x^3

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=c\,x-\frac {b\,x^2+\frac {a}{3}}{x^3} \]

[In]

int((a + b*x^2 + c*x^4)/x^4,x)

[Out]

c*x - (a/3 + b*x^2)/x^3