3.30.4 \(\int \frac {-4 e^2+118098 x-16 x^2+e (-39366+16 x)}{e^2 x^3-4 e x^4+4 x^5} \, dx\)

Optimal. Leaf size=23 \[ \frac {\frac {19683}{e-2 x}+\frac {1}{2} \left (4+x^2\right )}{x^2} \]

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Rubi [A]  time = 0.08, antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps used = 4, number of rules used = 3, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {1594, 27, 1820} \begin {gather*} \frac {19683+2 e}{e x^2}+\frac {39366}{e^2 x}+\frac {78732}{e^2 (e-2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*E^2 + 118098*x - 16*x^2 + E*(-39366 + 16*x))/(E^2*x^3 - 4*E*x^4 + 4*x^5),x]

[Out]

78732/(E^2*(E - 2*x)) + (19683 + 2*E)/(E*x^2) + 39366/(E^2*x)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1820

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 e^2+118098 x-16 x^2+e (-39366+16 x)}{x^3 \left (e^2-4 e x+4 x^2\right )} \, dx\\ &=\int \frac {-4 e^2+118098 x-16 x^2+e (-39366+16 x)}{(e-2 x)^2 x^3} \, dx\\ &=\int \left (\frac {157464}{e^2 (e-2 x)^2}-\frac {2 (19683+2 e)}{e x^3}-\frac {39366}{e^2 x^2}\right ) \, dx\\ &=\frac {78732}{e^2 (e-2 x)}+\frac {19683+2 e}{e x^2}+\frac {39366}{e^2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.87 \begin {gather*} -\frac {-19683-2 e+4 x}{(e-2 x) x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*E^2 + 118098*x - 16*x^2 + E*(-39366 + 16*x))/(E^2*x^3 - 4*E*x^4 + 4*x^5),x]

[Out]

-((-19683 - 2*E + 4*x)/((E - 2*x)*x^2))

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fricas [A]  time = 0.78, size = 25, normalized size = 1.09 \begin {gather*} \frac {4 \, x - 2 \, e - 19683}{2 \, x^{3} - x^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(1)^2+(16*x-39366)*exp(1)-16*x^2+118098*x)/(x^3*exp(1)^2-4*x^4*exp(1)+4*x^5),x, algorithm="fr
icas")

[Out]

(4*x - 2*e - 19683)/(2*x^3 - x^2*e)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(1)^2+(16*x-39366)*exp(1)-16*x^2+118098*x)/(x^3*exp(1)^2-4*x^4*exp(1)+4*x^5),x, algorithm="gi
ac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -2*((-157464*exp(1)^3+157464*exp(1)*exp(
2))/exp(2)^3*ln(4*sageVARx^2-4*sageVARx*exp(1)+exp(2))+(1259712*exp(1)^4-1889568*exp(1)^2*exp(2)+472392*exp(2)
^2)*1/4/exp(2)^3/sqrt

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maple [A]  time = 0.08, size = 22, normalized size = 0.96

method	result	size
gosper	\(\frac {19683-4 x +2 \,{\mathrm e}}{x^{2} \left ({\mathrm e}-2 x \right )}\)	\(22\)
norman	\(\frac {19683-4 x +2 \,{\mathrm e}}{x^{2} \left ({\mathrm e}-2 x \right )}\)	\(22\)
risch	\(\frac {19683-4 x +2 \,{\mathrm e}}{x^{2} \left ({\mathrm e}-2 x \right )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*exp(1)^2+(16*x-39366)*exp(1)-16*x^2+118098*x)/(x^3*exp(1)^2-4*x^4*exp(1)+4*x^5),x,method=_RETURNVERBOS
E)

[Out]

1/x^2*(19683-4*x+2*exp(1))/(exp(1)-2*x)

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maxima [A]  time = 0.37, size = 25, normalized size = 1.09 \begin {gather*} \frac {4 \, x - 2 \, e - 19683}{2 \, x^{3} - x^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(1)^2+(16*x-39366)*exp(1)-16*x^2+118098*x)/(x^3*exp(1)^2-4*x^4*exp(1)+4*x^5),x, algorithm="ma
xima")

[Out]

(4*x - 2*e - 19683)/(2*x^3 - x^2*e)

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mupad [B]  time = 0.14, size = 24, normalized size = 1.04 \begin {gather*} \frac {2\,\mathrm {e}-4\,x+19683}{x^2\,\mathrm {e}-2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((118098*x - 4*exp(2) - 16*x^2 + exp(1)*(16*x - 39366))/(x^3*exp(2) - 4*x^4*exp(1) + 4*x^5),x)

[Out]

(2*exp(1) - 4*x + 19683)/(x^2*exp(1) - 2*x^3)

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sympy [A]  time = 0.37, size = 22, normalized size = 0.96 \begin {gather*} - \frac {- 4 x + 2 e + 19683}{2 x^{3} - e x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(1)**2+(16*x-39366)*exp(1)-16*x**2+118098*x)/(x**3*exp(1)**2-4*x**4*exp(1)+4*x**5),x)

[Out]

-(-4*x + 2*E + 19683)/(2*x**3 - E*x**2)

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