3.93.44 \(\int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 (x^3+2 x^4+x^5)+e^{2 e^5} (x^3+2 x^4+x^5)+e^2 (-50 x^3-98 x^4-46 x^5+2 x^6)+e^{e^5} (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 (2 x^3+4 x^4+2 x^5))} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{x \left (-25+e^2+e^{e^5}+x\right ) \left (x+x^2\right )} \]

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Rubi [B]  time = 0.31, antiderivative size = 132, normalized size of antiderivative = 5.50, number of steps used = 4, number of rules used = 2, integrand size = 165, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6, 2074} \begin {gather*} -\frac {1}{\left (25-e^2-e^{e^5}\right ) x^2}+\frac {24-e^2-e^{e^5}}{\left (25-e^2-e^{e^5}\right )^2 x}-\frac {1}{\left (26-e^2-e^{e^5}\right ) (x+1)}-\frac {1}{\left (25-e^2-e^{e^5}\right )^2 \left (26-e^2-e^{e^5}\right ) \left (-x-e^{e^5}-e^2+25\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(50 + E^2*(-2 - 3*x) + E^E^5*(-2 - 3*x) + 72*x - 4*x^2)/(625*x^3 + 1200*x^4 + 526*x^5 - 48*x^6 + x^7 + E^4
*(x^3 + 2*x^4 + x^5) + E^(2*E^5)*(x^3 + 2*x^4 + x^5) + E^2*(-50*x^3 - 98*x^4 - 46*x^5 + 2*x^6) + E^E^5*(-50*x^
3 - 98*x^4 - 46*x^5 + 2*x^6 + E^2*(2*x^3 + 4*x^4 + 2*x^5))),x]

[Out]

-(1/((25 - E^2 - E^E^5)^2*(26 - E^2 - E^E^5)*(25 - E^2 - E^E^5 - x))) - 1/((25 - E^2 - E^E^5)*x^2) + (24 - E^2
- E^E^5)/((25 - E^2 - E^E^5)^2*x) - 1/((26 - E^2 - E^E^5)*(1 + x))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 \frac {50+\left (e^2+e^{e^5}\right ) (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx\\ &=\int \frac {50+\left (e^2+e^{e^5}\right ) (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+\left (e^4+e^{2 e^5}\right ) \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx\\ &=\int \left (-\frac {2}{\left (-25+e^2+e^{e^5}\right ) x^3}+\frac {-24+e^2+e^{e^5}}{\left (-25+e^2+e^{e^5}\right )^2 x^2}-\frac {1}{\left (-26+e^2+e^{e^5}\right ) (1+x)^2}+\frac {1}{\left (-26+e^2+e^{e^5}\right ) \left (-25+e^2+e^{e^5}\right )^2 \left (-25+e^2+e^{e^5}+x\right )^2}\right ) \, dx\\ &=-\frac {1}{\left (25-e^2-e^{e^5}\right )^2 \left (26-e^2-e^{e^5}\right ) \left (25-e^2-e^{e^5}-x\right )}-\frac {1}{\left (25-e^2-e^{e^5}\right ) x^2}+\frac {24-e^2-e^{e^5}}{\left (25-e^2-e^{e^5}\right )^2 x}-\frac {1}{\left (26-e^2-e^{e^5}\right ) (1+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{x^2 (1+x) \left (-25+e^2+e^{e^5}+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(50 + E^2*(-2 - 3*x) + E^E^5*(-2 - 3*x) + 72*x - 4*x^2)/(625*x^3 + 1200*x^4 + 526*x^5 - 48*x^6 + x^7
+ E^4*(x^3 + 2*x^4 + x^5) + E^(2*E^5)*(x^3 + 2*x^4 + x^5) + E^2*(-50*x^3 - 98*x^4 - 46*x^5 + 2*x^6) + E^E^5*(
-50*x^3 - 98*x^4 - 46*x^5 + 2*x^6 + E^2*(2*x^3 + 4*x^4 + 2*x^5))),x]

[Out]

1/(x^2*(1 + x)*(-25 + E^2 + E^E^5 + x))

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fricas [A]  time = 0.62, size = 37, normalized size = 1.54 \begin {gather*} \frac {1}{x^{4} - 24 \, x^{3} - 25 \, x^{2} + {\left (x^{3} + x^{2}\right )} e^{2} + {\left (x^{3} + x^{2}\right )} e^{\left (e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4+x^3)*exp(exp(5))^2+((2*x^5+4*x^4+2*
x^3)*exp(2)+2*x^6-46*x^5-98*x^4-50*x^3)*exp(exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(
2)+x^7-48*x^6+526*x^5+1200*x^4+625*x^3),x, algorithm="fricas")

[Out]

1/(x^4 - 24*x^3 - 25*x^2 + (x^3 + x^2)*e^2 + (x^3 + x^2)*e^(e^5))

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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(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4+x^3)*exp(exp(5))^2+((2*x^5+4*x^4+2*
x^3)*exp(2)+2*x^6-46*x^5-98*x^4-50*x^3)*exp(exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(
2)+x^7-48*x^6+526*x^5+1200*x^4+625*x^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -(2*exp(4)^3-14*exp(4)^2*exp(2)^2-16*exp
(4)^2*exp(2)*exp(exp(5))+404*exp(4)^2*exp(2)+6*exp(4)^2*exp(2*exp(5))-14*exp(4)^2*exp(exp(5))^2+404*exp(4)^2*e
xp(exp(5))-5099*exp(4

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maple [A]  time = 2.28, size = 20, normalized size = 0.83

method	result	size
norman	\(\frac {1}{x^{2} \left (x +1\right ) \left ({\mathrm e}^{2}+{\mathrm e}^{{\mathrm e}^{5}}+x -25\right )}\)	\(20\)
gosper	\(\frac {1}{x^{2} \left ({\mathrm e}^{2} x +x \,{\mathrm e}^{{\mathrm e}^{5}}+x^{2}+{\mathrm e}^{2}+{\mathrm e}^{{\mathrm e}^{5}}-24 x -25\right )}\)	\(29\)
risch	\(\frac {1}{x^{2} \left ({\mathrm e}^{2} x +x \,{\mathrm e}^{{\mathrm e}^{5}}+x^{2}+{\mathrm e}^{2}+{\mathrm e}^{{\mathrm e}^{5}}-24 x -25\right )}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4+x^3)*exp(exp(5))^2+((2*x^5+4*x^4+2*x^3)*e
xp(2)+2*x^6-46*x^5-98*x^4-50*x^3)*exp(exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(2)+x^7
-48*x^6+526*x^5+1200*x^4+625*x^3),x,method=_RETURNVERBOSE)

[Out]

1/x^2/(x+1)/(exp(2)+exp(exp(5))+x-25)

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maxima [A]  time = 0.36, size = 28, normalized size = 1.17 \begin {gather*} \frac {1}{x^{4} + x^{3} {\left (e^{2} + e^{\left (e^{5}\right )} - 24\right )} + x^{2} {\left (e^{2} + e^{\left (e^{5}\right )} - 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4+x^3)*exp(exp(5))^2+((2*x^5+4*x^4+2*
x^3)*exp(2)+2*x^6-46*x^5-98*x^4-50*x^3)*exp(exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(
2)+x^7-48*x^6+526*x^5+1200*x^4+625*x^3),x, algorithm="maxima")

[Out]

1/(x^4 + x^3*(e^2 + e^(e^5) - 24) + x^2*(e^2 + e^(e^5) - 25))

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(5))*(3*x + 2) - 72*x + 4*x^2 + exp(2)*(3*x + 2) - 50)/(exp(4)*(x^3 + 2*x^4 + x^5) - exp(exp(5))*
(50*x^3 - exp(2)*(2*x^3 + 4*x^4 + 2*x^5) + 98*x^4 + 46*x^5 - 2*x^6) + exp(2*exp(5))*(x^3 + 2*x^4 + x^5) + 625*
x^3 + 1200*x^4 + 526*x^5 - 48*x^6 + x^7 - exp(2)*(50*x^3 + 98*x^4 + 46*x^5 - 2*x^6)),x)

[Out]

\text{Hanged}

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sympy [A]  time = 5.69, size = 31, normalized size = 1.29 \begin {gather*} \frac {1}{x^{4} + x^{3} \left (-24 + e^{2} + e^{e^{5}}\right ) + x^{2} \left (-25 + e^{2} + e^{e^{5}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x**2+72*x+50)/((x**5+2*x**4+x**3)*exp(exp(5))**2+((2*x**5+4*
x**4+2*x**3)*exp(2)+2*x**6-46*x**5-98*x**4-50*x**3)*exp(exp(5))+(x**5+2*x**4+x**3)*exp(2)**2+(2*x**6-46*x**5-9
8*x**4-50*x**3)*exp(2)+x**7-48*x**6+526*x**5+1200*x**4+625*x**3),x)

[Out]

1/(x**4 + x**3*(-24 + exp(2) + exp(exp(5))) + x**2*(-25 + exp(2) + exp(exp(5))))

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