3.96.80 \(\int \frac {-27+e^6+e^4 (-9-3 x)-27 x-9 x^2-x^3+15 x^4+3 x^5+e^2 (27+18 x+3 x^2-5 x^4)}{-27+e^6+e^4 (-9-3 x)-27 x-9 x^2-x^3+e^2 (27+18 x+3 x^2)} \, dx\)

Optimal. Leaf size=17 \[ x-\frac {x^5}{\left (3-e^2+x\right )^2} \]

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Rubi [B]  time = 0.17, antiderivative size = 74, normalized size of antiderivative = 4.35, number of steps used = 2, number of rules used = 1, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {2074} \begin {gather*} -x^3+2 \left (3-e^2\right ) x^2-\left (26-18 e^2+3 e^4\right ) x-\frac {5 \left (3-e^2\right )^4}{x-e^2+3}+\frac {\left (3-e^2\right )^5}{\left (x-e^2+3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-27 + E^6 + E^4*(-9 - 3*x) - 27*x - 9*x^2 - x^3 + 15*x^4 + 3*x^5 + E^2*(27 + 18*x + 3*x^2 - 5*x^4))/(-27
+ E^6 + E^4*(-9 - 3*x) - 27*x - 9*x^2 - x^3 + E^2*(27 + 18*x + 3*x^2)),x]

[Out]

-((26 - 18*E^2 + 3*E^4)*x) + 2*(3 - E^2)*x^2 - x^3 + (3 - E^2)^5/(3 - E^2 + x)^2 - (5*(3 - E^2)^4)/(3 - E^2 +
x)

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-26 \left (1+\frac {3}{26} e^2 \left (-6+e^2\right )\right )-\frac {2 \left (-3+e^2\right )^5}{\left (-3+e^2-x\right )^3}+\frac {5 \left (-3+e^2\right )^4}{\left (-3+e^2-x\right )^2}-4 \left (-3+e^2\right ) x-3 x^2\right ) \, dx\\ &=-\left (\left (26-18 e^2+3 e^4\right ) x\right )+2 \left (3-e^2\right ) x^2-x^3+\frac {\left (3-e^2\right )^5}{\left (3-e^2+x\right )^2}-\frac {5 \left (3-e^2\right )^4}{3-e^2+x}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.04, size = 88, normalized size = 5.18 \begin {gather*} -\frac {\left (-3+e^2\right )^5}{\left (-3+e^2-x\right )^2}+\frac {5 \left (-3+e^2\right )^4}{-3+e^2-x}-\left (-89+60 e^2-10 e^4\right ) \left (-3+e^2-x\right )-5 \left (-3+e^2\right ) \left (-3+e^2-x\right )^2+\left (-3+e^2-x\right )^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-27 + E^6 + E^4*(-9 - 3*x) - 27*x - 9*x^2 - x^3 + 15*x^4 + 3*x^5 + E^2*(27 + 18*x + 3*x^2 - 5*x^4))
/(-27 + E^6 + E^4*(-9 - 3*x) - 27*x - 9*x^2 - x^3 + E^2*(27 + 18*x + 3*x^2)),x]

[Out]

-((-3 + E^2)^5/(-3 + E^2 - x)^2) + (5*(-3 + E^2)^4)/(-3 + E^2 - x) - (-89 + 60*E^2 - 10*E^4)*(-3 + E^2 - x) -
5*(-3 + E^2)*(-3 + E^2 - x)^2 + (-3 + E^2 - x)^3

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fricas [B]  time = 0.55, size = 91, normalized size = 5.35 \begin {gather*} -\frac {x^{5} - x^{3} + 102 \, x^{2} + 4 \, {\left (2 \, x + 15\right )} e^{8} - 4 \, {\left (x^{2} + 24 \, x + 90\right )} e^{6} + {\left (36 \, x^{2} + 431 \, x + 1080\right )} e^{4} - 2 \, {\left (53 \, x^{2} + 429 \, x + 810\right )} e^{2} + 639 \, x - 4 \, e^{10} + 972}{x^{2} - 2 \, {\left (x + 3\right )} e^{2} + 6 \, x + e^{4} + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2)^3+(-3*x-9)*exp(2)^2+(-5*x^4+3*x^2+18*x+27)*exp(2)+3*x^5+15*x^4-x^3-9*x^2-27*x-27)/(exp(2)^3+
(-3*x-9)*exp(2)^2+(3*x^2+18*x+27)*exp(2)-x^3-9*x^2-27*x-27),x, algorithm="fricas")

[Out]

-(x^5 - x^3 + 102*x^2 + 4*(2*x + 15)*e^8 - 4*(x^2 + 24*x + 90)*e^6 + (36*x^2 + 431*x + 1080)*e^4 - 2*(53*x^2 +
429*x + 810)*e^2 + 639*x - 4*e^10 + 972)/(x^2 - 2*(x + 3)*e^2 + 6*x + e^4 + 9)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2)^3+(-3*x-9)*exp(2)^2+(-5*x^4+3*x^2+18*x+27)*exp(2)+3*x^5+15*x^4-x^3-9*x^2-27*x-27)/(exp(2)^3+
(-3*x-9)*exp(2)^2+(3*x^2+18*x+27)*exp(2)-x^3-9*x^2-27*x-27),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.10, size = 51, normalized size = 3.00

method	result	size
norman	\(\frac {-x^{5}-54+x^{3}+\left (18 \,{\mathrm e}^{2}-3 \,{\mathrm e}^{4}-27\right ) x +2 \,{\mathrm e}^{6}-18 \,{\mathrm e}^{4}+54 \,{\mathrm e}^{2}}{\left ({\mathrm e}^{2}-3-x \right )^{2}}\)	\(51\)
gosper	\(\frac {-x^{5}+2 \,{\mathrm e}^{6}-3 x \,{\mathrm e}^{4}+x^{3}-18 \,{\mathrm e}^{4}+18 \,{\mathrm e}^{2} x +54 \,{\mathrm e}^{2}-27 x -54}{{\mathrm e}^{4}-2 \,{\mathrm e}^{2} x +x^{2}-6 \,{\mathrm e}^{2}+6 x +9}\)	\(66\)
risch	\(-3 x \,{\mathrm e}^{4}-2 x^{2} {\mathrm e}^{2}-x^{3}+18 \,{\mathrm e}^{2} x +6 x^{2}-26 x +\frac {\left (60 \,{\mathrm e}^{6}-5 \,{\mathrm e}^{8}-270 \,{\mathrm e}^{4}+540 \,{\mathrm e}^{2}-405\right ) x +4 \,{\mathrm e}^{10}-60 \,{\mathrm e}^{8}+360 \,{\mathrm e}^{6}-1080 \,{\mathrm e}^{4}+1620 \,{\mathrm e}^{2}-972}{{\mathrm e}^{4}-2 \,{\mathrm e}^{2} x +x^{2}-6 \,{\mathrm e}^{2}+6 x +9}\)	\(96\)
default	\(-3 x \,{\mathrm e}^{4}-2 x^{2} {\mathrm e}^{2}-x^{3}+18 \,{\mathrm e}^{2} x +6 x^{2}-26 x -\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+\left (-3 \,{\mathrm e}^{2}+9\right ) \textit {\_Z}^{2}+\left (-18 \,{\mathrm e}^{2}+3 \,{\mathrm e}^{4}+27\right ) \textit {\_Z} -27 \,{\mathrm e}^{2}+9 \,{\mathrm e}^{4}-{\mathrm e}^{6}+27\right )}{\sum }\frac {\left (540 \,{\mathrm e}^{2} \textit {\_R} -270 \textit {\_R} \,{\mathrm e}^{4}-5 \textit {\_R} \,{\mathrm e}^{8}+60 \textit {\_R} \,{\mathrm e}^{6}+1215 \,{\mathrm e}^{2}-810 \,{\mathrm e}^{4}-45 \,{\mathrm e}^{8}+3 \,{\mathrm e}^{10}+270 \,{\mathrm e}^{6}-405 \textit {\_R} -729\right ) \ln \left (x -\textit {\_R} \right )}{9+{\mathrm e}^{4}-2 \,{\mathrm e}^{2} \textit {\_R} +\textit {\_R}^{2}-6 \,{\mathrm e}^{2}+6 \textit {\_R}}\right )}{3}\)	\(157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2)^3+(-3*x-9)*exp(2)^2+(-5*x^4+3*x^2+18*x+27)*exp(2)+3*x^5+15*x^4-x^3-9*x^2-27*x-27)/(exp(2)^3+(-3*x-
9)*exp(2)^2+(3*x^2+18*x+27)*exp(2)-x^3-9*x^2-27*x-27),x,method=_RETURNVERBOSE)

[Out]

(-x^5-54+x^3+(-3*exp(2)^2+18*exp(2)-27)*x+2*exp(2)^3-18*exp(2)^2+54*exp(2))/(exp(2)-3-x)^2

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maxima [B]  time = 0.36, size = 91, normalized size = 5.35 \begin {gather*} -x^{3} - 2 \, x^{2} {\left (e^{2} - 3\right )} - x {\left (3 \, e^{4} - 18 \, e^{2} + 26\right )} - \frac {5 \, x {\left (e^{8} - 12 \, e^{6} + 54 \, e^{4} - 108 \, e^{2} + 81\right )} - 4 \, e^{10} + 60 \, e^{8} - 360 \, e^{6} + 1080 \, e^{4} - 1620 \, e^{2} + 972}{x^{2} - 2 \, x {\left (e^{2} - 3\right )} + e^{4} - 6 \, e^{2} + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2)^3+(-3*x-9)*exp(2)^2+(-5*x^4+3*x^2+18*x+27)*exp(2)+3*x^5+15*x^4-x^3-9*x^2-27*x-27)/(exp(2)^3+
(-3*x-9)*exp(2)^2+(3*x^2+18*x+27)*exp(2)-x^3-9*x^2-27*x-27),x, algorithm="maxima")

[Out]

-x^3 - 2*x^2*(e^2 - 3) - x*(3*e^4 - 18*e^2 + 26) - (5*x*(e^8 - 12*e^6 + 54*e^4 - 108*e^2 + 81) - 4*e^10 + 60*e
^8 - 360*e^6 + 1080*e^4 - 1620*e^2 + 972)/(x^2 - 2*x*(e^2 - 3) + e^4 - 6*e^2 + 9)

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mupad [B]  time = 9.33, size = 109, normalized size = 6.41 \begin {gather*} x\,\left (9\,{\left ({\mathrm {e}}^2-3\right )}^2-\left (3\,{\mathrm {e}}^2-9\right )\,\left (4\,{\mathrm {e}}^2-12\right )+1\right )-x^2\,\left (2\,{\mathrm {e}}^2-6\right )-\frac {1080\,{\mathrm {e}}^4-1620\,{\mathrm {e}}^2-360\,{\mathrm {e}}^6+60\,{\mathrm {e}}^8-4\,{\mathrm {e}}^{10}+x\,\left (270\,{\mathrm {e}}^4-540\,{\mathrm {e}}^2-60\,{\mathrm {e}}^6+5\,{\mathrm {e}}^8+405\right )+972}{x^2+\left (6-2\,{\mathrm {e}}^2\right )\,x-6\,{\mathrm {e}}^2+{\mathrm {e}}^4+9}-x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((27*x - exp(6) - exp(2)*(18*x + 3*x^2 - 5*x^4 + 27) + 9*x^2 + x^3 - 15*x^4 - 3*x^5 + exp(4)*(3*x + 9) + 27
)/(27*x - exp(6) - exp(2)*(18*x + 3*x^2 + 27) + 9*x^2 + x^3 + exp(4)*(3*x + 9) + 27),x)

[Out]

x*(9*(exp(2) - 3)^2 - (3*exp(2) - 9)*(4*exp(2) - 12) + 1) - x^2*(2*exp(2) - 6) - (1080*exp(4) - 1620*exp(2) -
360*exp(6) + 60*exp(8) - 4*exp(10) + x*(270*exp(4) - 540*exp(2) - 60*exp(6) + 5*exp(8) + 405) + 972)/(exp(4) -
6*exp(2) + x^2 - x*(2*exp(2) - 6) + 9) - x^3

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sympy [B]  time = 0.77, size = 100, normalized size = 5.88 \begin {gather*} - x^{3} - x^{2} \left (-6 + 2 e^{2}\right ) - x \left (- 18 e^{2} + 26 + 3 e^{4}\right ) - \frac {x \left (- 60 e^{6} - 540 e^{2} + 405 + 270 e^{4} + 5 e^{8}\right ) - 360 e^{6} - 4 e^{10} - 1620 e^{2} + 972 + 1080 e^{4} + 60 e^{8}}{x^{2} + x \left (6 - 2 e^{2}\right ) - 6 e^{2} + 9 + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2)**3+(-3*x-9)*exp(2)**2+(-5*x**4+3*x**2+18*x+27)*exp(2)+3*x**5+15*x**4-x**3-9*x**2-27*x-27)/(e
xp(2)**3+(-3*x-9)*exp(2)**2+(3*x**2+18*x+27)*exp(2)-x**3-9*x**2-27*x-27),x)

[Out]

-x**3 - x**2*(-6 + 2*exp(2)) - x*(-18*exp(2) + 26 + 3*exp(4)) - (x*(-60*exp(6) - 540*exp(2) + 405 + 270*exp(4)
+ 5*exp(8)) - 360*exp(6) - 4*exp(10) - 1620*exp(2) + 972 + 1080*exp(4) + 60*exp(8))/(x**2 + x*(6 - 2*exp(2))
- 6*exp(2) + 9 + exp(4))

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