\(\int \frac {-196608-49152 e-4608 e^2-192 e^3-3 e^4-x^2}{x^2} \, dx\) [9096]

Optimal result

Integrand size = 29, antiderivative size = 23 \[ \int \frac {-196608-49152 e-4608 e^2-192 e^3-3 e^4-x^2}{x^2} \, dx=\frac {11}{5}+2 \left (\frac {3 (16+e)^4}{2 x}-x\right )+x \]

[Out]

3/x*(16+exp(1))^4-x+11/5

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {14} \[ \int \frac {-196608-49152 e-4608 e^2-192 e^3-3 e^4-x^2}{x^2} \, dx=\frac {3 (16+e)^4}{x}-x \]

[In]

Int[(-196608 - 49152*E - 4608*E^2 - 192*E^3 - 3*E^4 - x^2)/x^2,x]

[Out]

(3*(16 + E)^4)/x - 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 (-1-\frac {3 (16+e)^4}{x^2}\right ) \, dx \\ & = \frac {3 (16+e)^4}{x}-x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {-196608-49152 e-4608 e^2-192 e^3-3 e^4-x^2}{x^2} \, dx=\frac {3 (16+e)^4}{x}-x \]

[In]

Integrate[(-196608 - 49152*E - 4608*E^2 - 192*E^3 - 3*E^4 - x^2)/x^2,x]

[Out]

(3*(16 + E)^4)/x - x

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22

method	result	size
default	\(-x -\frac {-4608 \,{\mathrm e}^{2}-49152 \,{\mathrm e}-3 \,{\mathrm e}^{4}-192 \,{\mathrm e}^{3}-196608}{x}\)	\(28\)
gosper	\(\frac {-x^{2}+4608 \,{\mathrm e}^{2}+49152 \,{\mathrm e}+3 \,{\mathrm e}^{4}+192 \,{\mathrm e}^{3}+196608}{x}\)	\(34\)
parallelrisch	\(\frac {-x^{2}+4608 \,{\mathrm e}^{2}+49152 \,{\mathrm e}+3 \,{\mathrm e}^{4}+192 \,{\mathrm e}^{3}+196608}{x}\)	\(34\)
risch	\(-x +\frac {4608 \,{\mathrm e}^{2}}{x}+\frac {49152 \,{\mathrm e}}{x}+\frac {3 \,{\mathrm e}^{4}}{x}+\frac {192 \,{\mathrm e}^{3}}{x}+\frac {196608}{x}\)	\(38\)

[In]

int((-3*exp(1)^4-192*exp(1)^3-4608*exp(1)^2-49152*exp(1)-x^2-196608)/x^2,x,method=_RETURNVERBOSE)

[Out]

-x-(-4608*exp(2)-49152*exp(1)-3*exp(4)-192*exp(3)-196608)/x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-196608-49152 e-4608 e^2-192 e^3-3 e^4-x^2}{x^2} \, dx=-\frac {x^{2} - 3 \, e^{4} - 192 \, e^{3} - 4608 \, e^{2} - 49152 \, e - 196608}{x} \]

[In]

integrate((-3*exp(1)^4-192*exp(1)^3-4608*exp(1)^2-49152*exp(1)-x^2-196608)/x^2,x, algorithm="fricas")

[Out]

-(x^2 - 3*e^4 - 192*e^3 - 4608*e^2 - 49152*e - 196608)/x

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {-196608-49152 e-4608 e^2-192 e^3-3 e^4-x^2}{x^2} \, dx=- x - \frac {-196608 - 49152 e - 4608 e^{2} - 192 e^{3} - 3 e^{4}}{x} \]

[In]

integrate((-3*exp(1)**4-192*exp(1)**3-4608*exp(1)**2-49152*exp(1)-x**2-196608)/x**2,x)

[Out]

-x - (-196608 - 49152*E - 4608*exp(2) - 192*exp(3) - 3*exp(4))/x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-196608-49152 e-4608 e^2-192 e^3-3 e^4-x^2}{x^2} \, dx=-x + \frac {3 \, {\left (e^{4} + 64 \, e^{3} + 1536 \, e^{2} + 16384 \, e + 65536\right )}}{x} \]

[In]

integrate((-3*exp(1)^4-192*exp(1)^3-4608*exp(1)^2-49152*exp(1)-x^2-196608)/x^2,x, algorithm="maxima")

[Out]

-x + 3*(e^4 + 64*e^3 + 1536*e^2 + 16384*e + 65536)/x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-196608-49152 e-4608 e^2-192 e^3-3 e^4-x^2}{x^2} \, dx=-x + \frac {3 \, {\left (e^{4} + 64 \, e^{3} + 1536 \, e^{2} + 16384 \, e + 65536\right )}}{x} \]

[In]

integrate((-3*exp(1)^4-192*exp(1)^3-4608*exp(1)^2-49152*exp(1)-x^2-196608)/x^2,x, algorithm="giac")

[Out]

-x + 3*(e^4 + 64*e^3 + 1536*e^2 + 16384*e + 65536)/x

Mupad [B] (verification not implemented)

Time = 13.74 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \frac {-196608-49152 e-4608 e^2-192 e^3-3 e^4-x^2}{x^2} \, dx=\frac {3\,{\left (\mathrm {e}+16\right )}^4}{x}-x \]

[In]

int(-(49152*exp(1) + 4608*exp(2) + 192*exp(3) + 3*exp(4) + x^2 + 196608)/x^2,x)

[Out]

(3*(exp(1) + 16)^4)/x - x