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

Optimal. Leaf size=23 \[ \frac {11}{5}+2 \left (\frac {3 (16+e)^4}{2 x}-x\right )+x \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.61, number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {14} \begin {gather*} \frac {3 (16+e)^4}{x}-x \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \text {integral} &=\int \left (-1-\frac {3 (16+e)^4}{x^2}\right ) \, dx\\ &=\frac {3 (16+e)^4}{x}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.61 \begin {gather*} \frac {3 (16+e)^4}{x}-x \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.51, size = 26, normalized size = 1.13 \begin {gather*} -\frac {x^{2} - 3 \, e^{4} - 192 \, e^{3} - 4608 \, e^{2} - 49152 \, e - 196608}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.16, size = 25, normalized size = 1.09 \begin {gather*} -x + \frac {3 \, {\left (e^{4} + 64 \, e^{3} + 1536 \, e^{2} + 16384 \, e + 65536\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.05, size = 28, normalized size = 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	\(\frac {3 \,{\mathrm e}^{4}+192 \,{\mathrm e}^{3}+4608 \,{\mathrm e}^{2}-x^{2}+49152 \,{\mathrm e}+196608}{x}\)	\(34\)
risch	\(-x +\frac {4608 \,{\mathrm e}^{2}}{x}+\frac {49152 \,{\mathrm e}}{x}+\frac {3 \,{\mathrm e}^{4}}{x}+\frac {192 \,{\mathrm e}^{3}}{x}+\frac {196608}{x}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [A]  time = 0.34, size = 25, normalized size = 1.09 \begin {gather*} -x + \frac {3 \, {\left (e^{4} + 64 \, e^{3} + 1536 \, e^{2} + 16384 \, e + 65536\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 6.78, size = 15, normalized size = 0.65 \begin {gather*} \frac {3\,{\left (\mathrm {e}+16\right )}^4}{x}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.14, size = 27, normalized size = 1.17 \begin {gather*} - x - \frac {-196608 - 49152 e - 4608 e^{2} - 192 e^{3} - 3 e^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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