3.69.58 \(\int \frac {-12+6 x-2 x^6+e^2 (-20 x^4-40 x^5-20 x^6)+e^4 (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6)+e^5 (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6)+e (14-6 x+10 x^5+10 x^6)+e^3 (20 x^3+60 x^4+60 x^5+20 x^6)}{-x^5+e^2 (-10 x^3-20 x^4-10 x^5)+e^4 (-5 x-20 x^2-30 x^3-20 x^4-5 x^5)+e^5 (1+5 x+10 x^2+10 x^3+5 x^4+x^5)+e (5 x^4+5 x^5)+e^3 (10 x^2+30 x^3+30 x^4+10 x^5)} \, dx\)

Optimal. Leaf size=22 \[ 11+x^2-\frac {3-2 x}{(x-e (1+x))^4} \]

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Rubi [B]  time = 0.18, antiderivative size = 51, normalized size of antiderivative = 2.32, number of steps used = 2, number of rules used = 1, integrand size = 257, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.004, Rules used = {2074} \begin {gather*} x^2-\frac {2}{(1-e) (e-(1-e) x)^3}-\frac {3-5 e}{(1-e) (e-(1-e) x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12 + 6*x - 2*x^6 + E^2*(-20*x^4 - 40*x^5 - 20*x^6) + E^4*(-10*x^2 - 40*x^3 - 60*x^4 - 40*x^5 - 10*x^6) +
E^5*(2*x + 10*x^2 + 20*x^3 + 20*x^4 + 10*x^5 + 2*x^6) + E*(14 - 6*x + 10*x^5 + 10*x^6) + E^3*(20*x^3 + 60*x^4
+ 60*x^5 + 20*x^6))/(-x^5 + E^2*(-10*x^3 - 20*x^4 - 10*x^5) + E^4*(-5*x - 20*x^2 - 30*x^3 - 20*x^4 - 5*x^5) +
E^5*(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5) + E*(5*x^4 + 5*x^5) + E^3*(10*x^2 + 30*x^3 + 30*x^4 + 10*x^5)),
x]

[Out]

x^2 - (3 - 5*E)/((1 - E)*(E - (1 - E)*x)^4) - 2/((1 - E)*(E - (1 - E)*x)^3)

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 (2 x+\frac {4 (-3+5 e)}{(e-(1-e) x)^5}-\frac {6}{(e-(1-e) x)^4}\right ) \, dx\\ &=x^2-\frac {3-5 e}{(1-e) (e-(1-e) x)^4}-\frac {2}{(1-e) (e-(1-e) x)^3}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 47, normalized size = 2.14 \begin {gather*} -2 x+(1+x)^2+\frac {3-5 e}{(-1+e) (e-x+e x)^4}+\frac {2}{(-1+e) (e-x+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 + 6*x - 2*x^6 + E^2*(-20*x^4 - 40*x^5 - 20*x^6) + E^4*(-10*x^2 - 40*x^3 - 60*x^4 - 40*x^5 - 10*
x^6) + E^5*(2*x + 10*x^2 + 20*x^3 + 20*x^4 + 10*x^5 + 2*x^6) + E*(14 - 6*x + 10*x^5 + 10*x^6) + E^3*(20*x^3 +
60*x^4 + 60*x^5 + 20*x^6))/(-x^5 + E^2*(-10*x^3 - 20*x^4 - 10*x^5) + E^4*(-5*x - 20*x^2 - 30*x^3 - 20*x^4 - 5*
x^5) + E^5*(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5) + E*(5*x^4 + 5*x^5) + E^3*(10*x^2 + 30*x^3 + 30*x^4 + 10*
x^5)),x]

[Out]

-2*x + (1 + x)^2 + (3 - 5*E)/((-1 + E)*(E - x + E*x)^4) + 2/((-1 + E)*(E - x + E*x)^3)

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fricas [B]  time = 0.53, size = 155, normalized size = 7.05 \begin {gather*} \frac {x^{6} + {\left (x^{6} + 4 \, x^{5} + 6 \, x^{4} + 4 \, x^{3} + x^{2}\right )} e^{4} - 4 \, {\left (x^{6} + 3 \, x^{5} + 3 \, x^{4} + x^{3}\right )} e^{3} + 6 \, {\left (x^{6} + 2 \, x^{5} + x^{4}\right )} e^{2} - 4 \, {\left (x^{6} + x^{5}\right )} e + 2 \, x - 3}{x^{4} + {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )} e^{4} - 4 \, {\left (x^{4} + 3 \, x^{3} + 3 \, x^{2} + x\right )} e^{3} + 6 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{2} - 4 \, {\left (x^{4} + x^{3}\right )} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^5-60*x^4-40*x^3-10*x^2)*exp(1)^4+(20
*x^6+60*x^5+60*x^4+20*x^3)*exp(1)^3+(-20*x^6-40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-1
2)/((x^5+5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5*x)*exp(1)^4+(10*x^5+30*x^4+30*x^3+
10*x^2)*exp(1)^3+(-10*x^5-20*x^4-10*x^3)*exp(1)^2+(5*x^5+5*x^4)*exp(1)-x^5),x, algorithm="fricas")

[Out]

(x^6 + (x^6 + 4*x^5 + 6*x^4 + 4*x^3 + x^2)*e^4 - 4*(x^6 + 3*x^5 + 3*x^4 + x^3)*e^3 + 6*(x^6 + 2*x^5 + x^4)*e^2
- 4*(x^6 + x^5)*e + 2*x - 3)/(x^4 + (x^4 + 4*x^3 + 6*x^2 + 4*x + 1)*e^4 - 4*(x^4 + 3*x^3 + 3*x^2 + x)*e^3 + 6
*(x^4 + 2*x^3 + x^2)*e^2 - 4*(x^4 + x^3)*e)

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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(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^5-60*x^4-40*x^3-10*x^2)*exp(1)^4+(20
*x^6+60*x^5+60*x^4+20*x^3)*exp(1)^3+(-20*x^6-40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-1
2)/((x^5+5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5*x)*exp(1)^4+(10*x^5+30*x^4+30*x^3+
10*x^2)*exp(1)^3+(-10*x^5-20*x^4-10*x^3)*exp(1)^2+(5*x^5+5*x^4)*exp(1)-x^5),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.38, size = 105, normalized size = 4.77

method	result	size
risch	\(x^{2}+\frac {2 x -3}{x^{4} {\mathrm e}^{4}+4 x^{3} {\mathrm e}^{4}-4 x^{4} {\mathrm e}^{3}+6 x^{2} {\mathrm e}^{4}-12 x^{3} {\mathrm e}^{3}+6 x^{4} {\mathrm e}^{2}+4 x \,{\mathrm e}^{4}-12 x^{2} {\mathrm e}^{3}+12 x^{3} {\mathrm e}^{2}-4 x^{4} {\mathrm e}+{\mathrm e}^{4}-4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}-4 x^{3} {\mathrm e}+x^{4}}\)	\(105\)
norman	\(\frac {\left ({\mathrm e}^{4}-4 \,{\mathrm e}^{3}+6 \,{\mathrm e}^{2}-4 \,{\mathrm e}+1\right ) x^{6}+{\mathrm e}^{-2} \left ({\mathrm e}^{6}+18 \,{\mathrm e}^{2}-36 \,{\mathrm e}+18\right ) x^{2}+\left (6 \,{\mathrm e}^{8}-12 \,{\mathrm e}^{7}+6 \,{\mathrm e}^{6}+3+3 \,{\mathrm e}^{4}-12 \,{\mathrm e}^{3}+18 \,{\mathrm e}^{2}-12 \,{\mathrm e}\right ) {\mathrm e}^{-4} x^{4}+2 \,{\mathrm e}^{-1} \left (7 \,{\mathrm e}-6\right ) x +4 \left ({\mathrm e}^{7}-{\mathrm e}^{6}+3 \,{\mathrm e}^{3}-9 \,{\mathrm e}^{2}+9 \,{\mathrm e}-3\right ) {\mathrm e}^{-3} x^{3}+4 \,{\mathrm e} \left ({\mathrm e}^{3}-3 \,{\mathrm e}^{2}+3 \,{\mathrm e}-1\right ) x^{5}}{\left (x \,{\mathrm e}+{\mathrm e}-x \right )^{4}}\)	\(188\)
default	\(x^{2}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (\left ({\mathrm e}^{5}-10 \,{\mathrm e}^{2}+5 \,{\mathrm e}-5 \,{\mathrm e}^{4}+10 \,{\mathrm e}^{3}-1\right ) \textit {\_Z}^{5}+\left (5 \,{\mathrm e}^{5}-20 \,{\mathrm e}^{2}+5 \,{\mathrm e}-20 \,{\mathrm e}^{4}+30 \,{\mathrm e}^{3}\right ) \textit {\_Z}^{4}+\left (10 \,{\mathrm e}^{5}-10 \,{\mathrm e}^{2}-30 \,{\mathrm e}^{4}+30 \,{\mathrm e}^{3}\right ) \textit {\_Z}^{3}+\left (10 \,{\mathrm e}^{5}-20 \,{\mathrm e}^{4}+10 \,{\mathrm e}^{3}\right ) \textit {\_Z}^{2}+\left (5 \,{\mathrm e}^{5}-5 \,{\mathrm e}^{4}\right ) \textit {\_Z} +{\mathrm e}^{5}\right )}{\sum }\frac {\left (-6+3 \left (1-{\mathrm e}\right ) \textit {\_R} +7 \,{\mathrm e}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{4} {\mathrm e}^{5}+4 \textit {\_R}^{3} {\mathrm e}^{5}-5 \textit {\_R}^{4} {\mathrm e}^{4}+6 \textit {\_R}^{2} {\mathrm e}^{5}-16 \textit {\_R}^{3} {\mathrm e}^{4}+10 \textit {\_R}^{4} {\mathrm e}^{3}+4 \textit {\_R} \,{\mathrm e}^{5}-18 \textit {\_R}^{2} {\mathrm e}^{4}+24 \textit {\_R}^{3} {\mathrm e}^{3}-10 \textit {\_R}^{4} {\mathrm e}^{2}+{\mathrm e}^{5}-8 \textit {\_R} \,{\mathrm e}^{4}+18 \textit {\_R}^{2} {\mathrm e}^{3}-16 \textit {\_R}^{3} {\mathrm e}^{2}+5 \textit {\_R}^{4} {\mathrm e}-{\mathrm e}^{4}+4 \textit {\_R} \,{\mathrm e}^{3}-6 \textit {\_R}^{2} {\mathrm e}^{2}+4 \textit {\_R}^{3} {\mathrm e}-\textit {\_R}^{4}}\right )}{5}\)	\(260\)
gosper	\(\frac {x \left (18 \,{\mathrm e}^{2} x +x \,{\mathrm e}^{8}+6 x^{3} {\mathrm e}^{6}+4 x^{2} {\mathrm e}^{8}+18 x \,{\mathrm e}^{4}+14 \,{\mathrm e}^{4}-12 \,{\mathrm e}^{3}+3 x^{3}+3 x^{3} {\mathrm e}^{4}+12 x^{2} {\mathrm e}^{4}-4 x^{5} {\mathrm e}^{5}+18 x^{3} {\mathrm e}^{2}+36 x^{2} {\mathrm e}^{2}-4 x^{4} {\mathrm e}^{5}-36 x \,{\mathrm e}^{3}-12 x^{2} {\mathrm e}-12 x^{3} {\mathrm e}-12 x^{3} {\mathrm e}^{3}-36 x^{2} {\mathrm e}^{3}+4 x^{4} {\mathrm e}^{8}+6 x^{3} {\mathrm e}^{8}-4 x^{2} {\mathrm e}^{7}+x^{5} {\mathrm e}^{4}+6 \,{\mathrm e}^{6} x^{5}+12 \,{\mathrm e}^{6} x^{4}+{\mathrm e}^{8} x^{5}-4 \,{\mathrm e}^{7} x^{5}-12 \,{\mathrm e}^{7} x^{4}-12 \,{\mathrm e}^{7} x^{3}\right ) {\mathrm e}^{-4}}{x^{4} {\mathrm e}^{4}+4 x^{3} {\mathrm e}^{4}-4 x^{4} {\mathrm e}^{3}+6 x^{2} {\mathrm e}^{4}-12 x^{3} {\mathrm e}^{3}+6 x^{4} {\mathrm e}^{2}+4 x \,{\mathrm e}^{4}-12 x^{2} {\mathrm e}^{3}+12 x^{3} {\mathrm e}^{2}-4 x^{4} {\mathrm e}+{\mathrm e}^{4}-4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}-4 x^{3} {\mathrm e}+x^{4}}\)	\(362\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^5-60*x^4-40*x^3-10*x^2)*exp(1)^4+(20*x^6+6
0*x^5+60*x^4+20*x^3)*exp(1)^3+(-20*x^6-40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-12)/((x
^5+5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5*x)*exp(1)^4+(10*x^5+30*x^4+30*x^3+10*x^2
)*exp(1)^3+(-10*x^5-20*x^4-10*x^3)*exp(1)^2+(5*x^5+5*x^4)*exp(1)-x^5),x,method=_RETURNVERBOSE)

[Out]

x^2+(2*x-3)/(x^4*exp(4)+4*x^3*exp(4)-4*x^4*exp(3)+6*x^2*exp(4)-12*x^3*exp(3)+6*x^4*exp(2)+4*x*exp(4)-12*x^2*ex
p(3)+12*x^3*exp(2)-4*x^4*exp(1)+exp(4)-4*x*exp(3)+6*x^2*exp(2)-4*x^3*exp(1)+x^4)

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maxima [B]  time = 0.37, size = 79, normalized size = 3.59 \begin {gather*} x^{2} + \frac {2 \, x - 3}{x^{4} {\left (e^{4} - 4 \, e^{3} + 6 \, e^{2} - 4 \, e + 1\right )} + 4 \, x^{3} {\left (e^{4} - 3 \, e^{3} + 3 \, e^{2} - e\right )} + 6 \, x^{2} {\left (e^{4} - 2 \, e^{3} + e^{2}\right )} + 4 \, x {\left (e^{4} - e^{3}\right )} + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^5-60*x^4-40*x^3-10*x^2)*exp(1)^4+(20
*x^6+60*x^5+60*x^4+20*x^3)*exp(1)^3+(-20*x^6-40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-1
2)/((x^5+5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5*x)*exp(1)^4+(10*x^5+30*x^4+30*x^3+
10*x^2)*exp(1)^3+(-10*x^5-20*x^4-10*x^3)*exp(1)^2+(5*x^5+5*x^4)*exp(1)-x^5),x, algorithm="maxima")

[Out]

x^2 + (2*x - 3)/(x^4*(e^4 - 4*e^3 + 6*e^2 - 4*e + 1) + 4*x^3*(e^4 - 3*e^3 + 3*e^2 - e) + 6*x^2*(e^4 - 2*e^3 +
e^2) + 4*x*(e^4 - e^3) + e^4)

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mupad [B]  time = 4.43, size = 48, normalized size = 2.18 \begin {gather*} \frac {2}{{\left (\mathrm {e}+x\,\left (\mathrm {e}-1\right )\right )}^3\,\left (\mathrm {e}-1\right )}+x^2-\frac {5\,\mathrm {e}-3}{{\left (\mathrm {e}+x\,\left (\mathrm {e}-1\right )\right )}^4\,\left (\mathrm {e}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + exp(5)*(2*x + 10*x^2 + 20*x^3 + 20*x^4 + 10*x^5 + 2*x^6) - exp(4)*(10*x^2 + 40*x^3 + 60*x^4 + 40*x^
5 + 10*x^6) + exp(1)*(10*x^5 - 6*x + 10*x^6 + 14) - exp(2)*(20*x^4 + 40*x^5 + 20*x^6) - 2*x^6 + exp(3)*(20*x^3
+ 60*x^4 + 60*x^5 + 20*x^6) - 12)/(exp(5)*(5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5 + 1) + exp(1)*(5*x^4 + 5*x^5)
- exp(2)*(10*x^3 + 20*x^4 + 10*x^5) - exp(4)*(5*x + 20*x^2 + 30*x^3 + 20*x^4 + 5*x^5) - x^5 + exp(3)*(10*x^2 +
30*x^3 + 30*x^4 + 10*x^5)),x)

[Out]

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

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sympy [B]  time = 4.74, size = 88, normalized size = 4.00 \begin {gather*} x^{2} + \frac {2 x - 3}{x^{4} \left (- 4 e^{3} - 4 e + 1 + 6 e^{2} + e^{4}\right ) + x^{3} \left (- 12 e^{3} - 4 e + 12 e^{2} + 4 e^{4}\right ) + x^{2} \left (- 12 e^{3} + 6 e^{2} + 6 e^{4}\right ) + x \left (- 4 e^{3} + 4 e^{4}\right ) + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**6+10*x**5+20*x**4+20*x**3+10*x**2+2*x)*exp(1)**5+(-10*x**6-40*x**5-60*x**4-40*x**3-10*x**2)*e
xp(1)**4+(20*x**6+60*x**5+60*x**4+20*x**3)*exp(1)**3+(-20*x**6-40*x**5-20*x**4)*exp(1)**2+(10*x**6+10*x**5-6*x
+14)*exp(1)-2*x**6+6*x-12)/((x**5+5*x**4+10*x**3+10*x**2+5*x+1)*exp(1)**5+(-5*x**5-20*x**4-30*x**3-20*x**2-5*x
)*exp(1)**4+(10*x**5+30*x**4+30*x**3+10*x**2)*exp(1)**3+(-10*x**5-20*x**4-10*x**3)*exp(1)**2+(5*x**5+5*x**4)*e
xp(1)-x**5),x)

[Out]

x**2 + (2*x - 3)/(x**4*(-4*exp(3) - 4*E + 1 + 6*exp(2) + exp(4)) + x**3*(-12*exp(3) - 4*E + 12*exp(2) + 4*exp(
4)) + x**2*(-12*exp(3) + 6*exp(2) + 6*exp(4)) + x*(-4*exp(3) + 4*exp(4)) + exp(4))

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