\(\int \frac {b x^2+c x^4}{x^4} \, dx\) [140]

Optimal result

Integrand size = 15, antiderivative size = 10 \[ \int \frac {b x^2+c x^4}{x^4} \, dx=-\frac {b}{x}+c x \]

[Out]

-b/x+c*x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \[ \int \frac {b x^2+c x^4}{x^4} \, dx=c x-\frac {b}{x} \]

[In]

Int[(b*x^2 + c*x^4)/x^4,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {b x^2+c x^4}{x^4} \, dx=-\frac {b}{x}+c x \]

[In]

Integrate[(b*x^2 + c*x^4)/x^4,x]

[Out]

-(b/x) + c*x

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10

method	result	size
default	\(-\frac {b}{x}+c x\)	\(11\)
risch	\(-\frac {b}{x}+c x\)	\(11\)
gosper	\(-\frac {-c \,x^{2}+b}{x}\)	\(14\)
parallelrisch	\(\frac {c \,x^{2}-b}{x}\)	\(14\)
norman	\(\frac {c \,x^{4}-b \,x^{2}}{x^{3}}\)	\(17\)

[In]

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

[Out]

-b/x+c*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.30 \[ \int \frac {b x^2+c x^4}{x^4} \, dx=\frac {c x^{2} - b}{x} \]

[In]

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

[Out]

(c*x^2 - b)/x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.50 \[ \int \frac {b x^2+c x^4}{x^4} \, dx=- \frac {b}{x} + c x \]

[In]

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

[Out]

-b/x + c*x

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {b x^2+c x^4}{x^4} \, dx=c x - \frac {b}{x} \]

[In]

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

[Out]

c*x - b/x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {b x^2+c x^4}{x^4} \, dx=c x - \frac {b}{x} \]

[In]

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

[Out]

c*x - b/x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {b x^2+c x^4}{x^4} \, dx=c\,x-\frac {b}{x} \]

[In]

int((b*x^2 + c*x^4)/x^4,x)

[Out]

c*x - b/x