\(\int \frac {-486-243 x-18 x^3-18 x^4-3 x^5}{x^3} \, dx\) [3355]

Optimal result

Integrand size = 24, antiderivative size = 22 \[ \int \frac {-486-243 x-18 x^3-18 x^4-3 x^5}{x^3} \, dx=5 e+(3-x) \left (-6-\frac {9}{x}-x\right )^2 \]

[Out]

(-6-x-9/x)^2*(-x+3)+5*exp(1)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {14} \[ \int \frac {-486-243 x-18 x^3-18 x^4-3 x^5}{x^3} \, dx=-x^3-9 x^2+\frac {243}{x^2}-18 x+\frac {243}{x} \]

[In]

Int[(-486 - 243*x - 18*x^3 - 18*x^4 - 3*x^5)/x^3,x]

[Out]

243/x^2 + 243/x - 18*x - 9*x^2 - 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 (-18-\frac {486}{x^3}-\frac {243}{x^2}-18 x-3 x^2\right ) \, dx \\ & = \frac {243}{x^2}+\frac {243}{x}-18 x-9 x^2-x^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {-486-243 x-18 x^3-18 x^4-3 x^5}{x^3} \, dx=\frac {243}{x^2}+\frac {243}{x}-18 x-9 x^2-x^3 \]

[In]

Integrate[(-486 - 243*x - 18*x^3 - 18*x^4 - 3*x^5)/x^3,x]

[Out]

243/x^2 + 243/x - 18*x - 9*x^2 - x^3

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09

method	result	size
gosper	\(-\frac {x^{5}+9 x^{4}+18 x^{3}-243 x -243}{x^{2}}\)	\(24\)
risch	\(-x^{3}-9 x^{2}-18 x +\frac {243 x +243}{x^{2}}\)	\(24\)
parallelrisch	\(-\frac {x^{5}+9 x^{4}+18 x^{3}-243 x -243}{x^{2}}\)	\(24\)
default	\(-x^{3}-9 x^{2}-18 x +\frac {243}{x}+\frac {243}{x^{2}}\)	\(25\)
norman	\(\frac {-x^{5}-9 x^{4}-18 x^{3}+243 x +243}{x^{2}}\)	\(25\)

[In]

int((-3*x^5-18*x^4-18*x^3-243*x-486)/x^3,x,method=_RETURNVERBOSE)

[Out]

-(x^5+9*x^4+18*x^3-243*x-243)/x^2

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-486-243 x-18 x^3-18 x^4-3 x^5}{x^3} \, dx=-\frac {x^{5} + 9 \, x^{4} + 18 \, x^{3} - 243 \, x - 243}{x^{2}} \]

[In]

integrate((-3*x^5-18*x^4-18*x^3-243*x-486)/x^3,x, algorithm="fricas")

[Out]

-(x^5 + 9*x^4 + 18*x^3 - 243*x - 243)/x^2

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-486-243 x-18 x^3-18 x^4-3 x^5}{x^3} \, dx=- x^{3} - 9 x^{2} - 18 x - \frac {- 243 x - 243}{x^{2}} \]

[In]

integrate((-3*x**5-18*x**4-18*x**3-243*x-486)/x**3,x)

[Out]

-x**3 - 9*x**2 - 18*x - (-243*x - 243)/x**2

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-486-243 x-18 x^3-18 x^4-3 x^5}{x^3} \, dx=-x^{3} - 9 \, x^{2} - 18 \, x + \frac {243 \, {\left (x + 1\right )}}{x^{2}} \]

[In]

integrate((-3*x^5-18*x^4-18*x^3-243*x-486)/x^3,x, algorithm="maxima")

[Out]

-x^3 - 9*x^2 - 18*x + 243*(x + 1)/x^2

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-486-243 x-18 x^3-18 x^4-3 x^5}{x^3} \, dx=-x^{3} - 9 \, x^{2} - 18 \, x + \frac {243 \, {\left (x + 1\right )}}{x^{2}} \]

[In]

integrate((-3*x^5-18*x^4-18*x^3-243*x-486)/x^3,x, algorithm="giac")

[Out]

-x^3 - 9*x^2 - 18*x + 243*(x + 1)/x^2

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-486-243 x-18 x^3-18 x^4-3 x^5}{x^3} \, dx=\frac {243\,x+243}{x^2}-18\,x-9\,x^2-x^3 \]

[In]

int(-(243*x + 18*x^3 + 18*x^4 + 3*x^5 + 486)/x^3,x)

[Out]

(243*x + 243)/x^2 - 18*x - 9*x^2 - x^3