∫ 108241+658e3+e6+6561e4x+e2x(158274+162e3+648x)+e3x(52488+162x−81x2)+ex(213192+658x+329x2+e3(648+2x+x2))_________________________________________________________________________________________ 108241+658e3+e6+52488e3x+6561e4x+e2x(158274+162e3)+ex(213192+648e3) dx

Optimal. Leaf size=26 \[ -2+x+\frac {e^x x^2}{5+e^3+81 \left (2+e^x\right )^2} \]

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Rubi [F] time = 2.93, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {108241+658 e^3+e^6+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3+648 x\right )+e^{3 x} \left (52488+162 x-81 x^2\right )+e^x \left (213192+658 x+329 x^2+e^3 \left (648+2 x+x^2\right )\right )}{108241+658 e^3+e^6+52488 e^{3 x}+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3\right )+e^x \left (213192+648 e^3\right )} \, dx \end {gather*}

Int[(108241 + 658*E^3 + E^6 + 6561*E^(4*x) + E^(2*x)*(158274 + 162*E^3 + 648*x) + E^(3*x)*(52488 + 162*x - 81*

x^2) + E^x*(213192 + 658*x + 329*x^2 + E^3*(648 + 2*x + x^2)))/(108241 + 658*E^3 + E^6 + 52488*E^(3*x) + 6561*

x + (2*x^3)/(3*(5 + E^3 - (18*I)*Sqrt[5 + E^3])) + (2*x^3)/(3*(5 + E^3 + (18*I)*Sqrt[5 + E^3])) - ((I/9)*x*Log

[1 + (9*E^x)/(18 - I*Sqrt[5 + E^3])])/Sqrt[5 + E^3] + ((I/18)*x^2*Log[1 + (9*E^x)/(18 - I*Sqrt[5 + E^3])])/Sqr

t[5 + E^3] - (2*x^2*Log[1 + (9*E^x)/(18 - I*Sqrt[5 + E^3])])/(5 + E^3 + (18*I)*Sqrt[5 + E^3]) + ((I/9)*x*Log[1

+ (9*E^x)/(18 + I*Sqrt[5 + E^3])])/Sqrt[5 + E^3] - ((I/18)*x^2*Log[1 + (9*E^x)/(18 + I*Sqrt[5 + E^3])])/Sqrt[

5 + E^3] - (2*x^2*Log[1 + (9*E^x)/(18 + I*Sqrt[5 + E^3])])/(5 + E^3 - (18*I)*Sqrt[5 + E^3]) - ((I/9)*PolyLog[2

, (-9*E^x)/(18 - I*Sqrt[5 + E^3])])/Sqrt[5 + E^3] + ((I/9)*x*PolyLog[2, (-9*E^x)/(18 - I*Sqrt[5 + E^3])])/Sqrt

[5 + E^3] - (4*x*PolyLog[2, (-9*E^x)/(18 - I*Sqrt[5 + E^3])])/(5 + E^3 + (18*I)*Sqrt[5 + E^3]) + ((I/9)*PolyLo

g[2, (-9*E^x)/(18 + I*Sqrt[5 + E^3])])/Sqrt[5 + E^3] - ((I/9)*x*PolyLog[2, (-9*E^x)/(18 + I*Sqrt[5 + E^3])])/S

qrt[5 + E^3] - (4*x*PolyLog[2, (-9*E^x)/(18 + I*Sqrt[5 + E^3])])/(5 + E^3 - (18*I)*Sqrt[5 + E^3]) - ((I/9)*Pol

yLog[3, (-9*E^x)/(18 - I*Sqrt[5 + E^3])])/Sqrt[5 + E^3] + (4*PolyLog[3, (-9*E^x)/(18 - I*Sqrt[5 + E^3])])/(5 +

E^3 + (18*I)*Sqrt[5 + E^3]) + ((I/9)*PolyLog[3, (-9*E^x)/(18 + I*Sqrt[5 + E^3])])/Sqrt[5 + E^3] + (4*PolyLog[

3, (-9*E^x)/(18 + I*Sqrt[5 + E^3])])/(5 + E^3 - (18*I)*Sqrt[5 + E^3]) - 4*(329 + E^3)*Defer[Int][x^2/(324*E^x

+ 81*E^(2*x) + 329*(1 + E^3/329))^2, x] - 2*(319 - E^3)*Defer[Int][(E^x*x^2)/(324*E^x + 81*E^(2*x) + 329*(1 +

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6561 e^{4 x}+108241 \left (1+\frac {e^3 \left (658+e^3\right )}{108241}\right )+e^{2 x} \left (158274+162 e^3+648 x\right )+e^{3 x} \left (52488+162 x-81 x^2\right )+e^x \left (213192+658 x+329 x^2+e^3 \left (648+2 x+x^2\right )\right )}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx\\ &=\int \left (1+\frac {2 \left (-319 e^x \left (1-\frac {e^3}{319}\right )-658 \left (1+\frac {e^3}{329}\right )\right ) x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2}+\frac {x \left (2 e^x+4 x-e^x x\right )}{324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )}\right ) \, dx\\ &=x+2 \int \frac {\left (-319 e^x \left (1-\frac {e^3}{319}\right )-658 \left (1+\frac {e^3}{329}\right )\right ) x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx+\int \frac {x \left (2 e^x+4 x-e^x x\right )}{324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )} \, dx\\ &=x+2 \int \left (\frac {2 \left (-329-e^3\right ) x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2}+\frac {e^x \left (-319+e^3\right ) x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2}\right ) \, dx+\int \left (\frac {2 e^x x}{324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )}+\frac {e^x x^2}{-324 e^x-81 e^{2 x}-329 \left (1+\frac {e^3}{329}\right )}+\frac {4 x^2}{324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )}\right ) \, dx\\ &=x+2 \int \frac {e^x x}{324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )} \, dx+4 \int \frac {x^2}{324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )} \, dx-\left (2 \left (319-e^3\right )\right ) \int \frac {e^x x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx-\left (4 \left (329+e^3\right )\right ) \int \frac {x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx+\int \frac {e^x x^2}{-324 e^x-81 e^{2 x}-329 \left (1+\frac {e^3}{329}\right )} \, dx\\ &=x-\left (2 \left (319-e^3\right )\right ) \int \frac {e^x x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx+\frac {(9 i) \int \frac {e^x x^2}{-324-162 e^x-18 i \sqrt {5+e^3}} \, dx}{\sqrt {5+e^3}}-\frac {(9 i) \int \frac {e^x x^2}{-324-162 e^x+18 i \sqrt {5+e^3}} \, dx}{\sqrt {5+e^3}}-\frac {(18 i) \int \frac {e^x x}{324+162 e^x-18 i \sqrt {5+e^3}} \, dx}{\sqrt {5+e^3}}+\frac {(18 i) \int \frac {e^x x}{324+162 e^x+18 i \sqrt {5+e^3}} \, dx}{\sqrt {5+e^3}}-\frac {(36 i) \int \frac {x^2}{324+162 e^x-18 i \sqrt {5+e^3}} \, dx}{\sqrt {5+e^3}}+\frac {(36 i) \int \frac {x^2}{324+162 e^x+18 i \sqrt {5+e^3}} \, dx}{\sqrt {5+e^3}}-\left (4 \left (329+e^3\right )\right ) \int \frac {x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx\\ &=x+\frac {2 x^3}{3 \left (5+e^3-18 i \sqrt {5+e^3}\right )}+\frac {2 x^3}{3 \left (5+e^3+18 i \sqrt {5+e^3}\right )}-\frac {i x \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {i x^2 \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{18 \sqrt {5+e^3}}+\frac {i x \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {i x^2 \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{18 \sqrt {5+e^3}}-\left (2 \left (319-e^3\right )\right ) \int \frac {e^x x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx+\frac {i \int x \log \left (1-\frac {162 e^x}{-324-18 i \sqrt {5+e^3}}\right ) \, dx}{9 \sqrt {5+e^3}}+\frac {i \int \log \left (1+\frac {162 e^x}{324-18 i \sqrt {5+e^3}}\right ) \, dx}{9 \sqrt {5+e^3}}-\frac {i \int x \log \left (1-\frac {162 e^x}{-324+18 i \sqrt {5+e^3}}\right ) \, dx}{9 \sqrt {5+e^3}}-\frac {i \int \log \left (1+\frac {162 e^x}{324+18 i \sqrt {5+e^3}}\right ) \, dx}{9 \sqrt {5+e^3}}-\left (4 \left (329+e^3\right )\right ) \int \frac {x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx-\frac {324 \int \frac {e^x x^2}{324+162 e^x+18 i \sqrt {5+e^3}} \, dx}{5+e^3-18 i \sqrt {5+e^3}}-\frac {324 \int \frac {e^x x^2}{324+162 e^x-18 i \sqrt {5+e^3}} \, dx}{5+e^3+18 i \sqrt {5+e^3}}\\ &=x+\frac {2 x^3}{3 \left (5+e^3-18 i \sqrt {5+e^3}\right )}+\frac {2 x^3}{3 \left (5+e^3+18 i \sqrt {5+e^3}\right )}-\frac {i x \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {i x^2 \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{18 \sqrt {5+e^3}}-\frac {2 x^2 \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{5+e^3+18 i \sqrt {5+e^3}}+\frac {i x \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {i x^2 \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{18 \sqrt {5+e^3}}-\frac {2 x^2 \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{5+e^3-18 i \sqrt {5+e^3}}+\frac {i x \text {Li}_2\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {i x \text {Li}_2\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\left (2 \left (319-e^3\right )\right ) \int \frac {e^x x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx+\frac {i \int \text {Li}_2\left (\frac {162 e^x}{-324-18 i \sqrt {5+e^3}}\right ) \, dx}{9 \sqrt {5+e^3}}-\frac {i \int \text {Li}_2\left (\frac {162 e^x}{-324+18 i \sqrt {5+e^3}}\right ) \, dx}{9 \sqrt {5+e^3}}+\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {162 x}{324-18 i \sqrt {5+e^3}}\right )}{x} \, dx,x,e^x\right )}{9 \sqrt {5+e^3}}-\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {162 x}{324+18 i \sqrt {5+e^3}}\right )}{x} \, dx,x,e^x\right )}{9 \sqrt {5+e^3}}-\left (4 \left (329+e^3\right )\right ) \int \frac {x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx+\frac {4 \int x \log \left (1+\frac {162 e^x}{324+18 i \sqrt {5+e^3}}\right ) \, dx}{5+e^3-18 i \sqrt {5+e^3}}+\frac {4 \int x \log \left (1+\frac {162 e^x}{324-18 i \sqrt {5+e^3}}\right ) \, dx}{5+e^3+18 i \sqrt {5+e^3}}\\ &=x+\frac {2 x^3}{3 \left (5+e^3-18 i \sqrt {5+e^3}\right )}+\frac {2 x^3}{3 \left (5+e^3+18 i \sqrt {5+e^3}\right )}-\frac {i x \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {i x^2 \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{18 \sqrt {5+e^3}}-\frac {2 x^2 \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{5+e^3+18 i \sqrt {5+e^3}}+\frac {i x \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {i x^2 \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{18 \sqrt {5+e^3}}-\frac {2 x^2 \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{5+e^3-18 i \sqrt {5+e^3}}-\frac {i \text {Li}_2\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {i x \text {Li}_2\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {4 x \text {Li}_2\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{5+e^3+18 i \sqrt {5+e^3}}+\frac {i \text {Li}_2\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {i x \text {Li}_2\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {4 x \text {Li}_2\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{5+e^3-18 i \sqrt {5+e^3}}-\left (2 \left (319-e^3\right )\right ) \int \frac {e^x x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx+\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {9 i x}{-18 i+\sqrt {5+e^3}}\right )}{x} \, dx,x,e^x\right )}{9 \sqrt {5+e^3}}-\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {9 i x}{18 i+\sqrt {5+e^3}}\right )}{x} \, dx,x,e^x\right )}{9 \sqrt {5+e^3}}-\left (4 \left (329+e^3\right )\right ) \int \frac {x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx+\frac {4 \int \text {Li}_2\left (-\frac {162 e^x}{324+18 i \sqrt {5+e^3}}\right ) \, dx}{5+e^3-18 i \sqrt {5+e^3}}+\frac {4 \int \text {Li}_2\left (-\frac {162 e^x}{324-18 i \sqrt {5+e^3}}\right ) \, dx}{5+e^3+18 i \sqrt {5+e^3}}\\ &=x+\frac {2 x^3}{3 \left (5+e^3-18 i \sqrt {5+e^3}\right )}+\frac {2 x^3}{3 \left (5+e^3+18 i \sqrt {5+e^3}\right )}-\frac {i x \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {i x^2 \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{18 \sqrt {5+e^3}}-\frac {2 x^2 \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{5+e^3+18 i \sqrt {5+e^3}}+\frac {i x \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {i x^2 \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{18 \sqrt {5+e^3}}-\frac {2 x^2 \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{5+e^3-18 i \sqrt {5+e^3}}-\frac {i \text {Li}_2\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {i x \text {Li}_2\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {4 x \text {Li}_2\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{5+e^3+18 i \sqrt {5+e^3}}+\frac {i \text {Li}_2\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {i x \text {Li}_2\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {4 x \text {Li}_2\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{5+e^3-18 i \sqrt {5+e^3}}-\frac {i \text {Li}_3\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {i \text {Li}_3\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\left (2 \left (319-e^3\right )\right ) \int \frac {e^x x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx-\left (4 \left (329+e^3\right )\right ) \int \frac {x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx+\frac {4 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {9 i x}{-18 i+\sqrt {5+e^3}}\right )}{x} \, dx,x,e^x\right )}{5+e^3-18 i \sqrt {5+e^3}}+\frac {4 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {9 i x}{18 i+\sqrt {5+e^3}}\right )}{x} \, dx,x,e^x\right )}{5+e^3+18 i \sqrt {5+e^3}}\\ &=x+\frac {2 x^3}{3 \left (5+e^3-18 i \sqrt {5+e^3}\right )}+\frac {2 x^3}{3 \left (5+e^3+18 i \sqrt {5+e^3}\right )}-\frac {i x \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {i x^2 \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{18 \sqrt {5+e^3}}-\frac {2 x^2 \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{5+e^3+18 i \sqrt {5+e^3}}+\frac {i x \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {i x^2 \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{18 \sqrt {5+e^3}}-\frac {2 x^2 \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{5+e^3-18 i \sqrt {5+e^3}}-\frac {i \text {Li}_2\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {i x \text {Li}_2\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {4 x \text {Li}_2\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{5+e^3+18 i \sqrt {5+e^3}}+\frac {i \text {Li}_2\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {i x \text {Li}_2\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-\frac {4 x \text {Li}_2\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{5+e^3-18 i \sqrt {5+e^3}}-\frac {i \text {Li}_3\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {4 \text {Li}_3\left (-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{5+e^3+18 i \sqrt {5+e^3}}+\frac {i \text {Li}_3\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {4 \text {Li}_3\left (-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{5+e^3-18 i \sqrt {5+e^3}}-\left (2 \left (319-e^3\right )\right ) \int \frac {e^x x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx-\left (4 \left (329+e^3\right )\right ) \int \frac {x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.08, size = 28, normalized size = 1.08 \begin {gather*} x+\frac {e^x x^2}{329+e^3+324 e^x+81 e^{2 x}} \end {gather*}

Integrate[(108241 + 658*E^3 + E^6 + 6561*E^(4*x) + E^(2*x)*(158274 + 162*E^3 + 648*x) + E^(3*x)*(52488 + 162*x

- 81*x^2) + E^x*(213192 + 658*x + 329*x^2 + E^3*(648 + 2*x + x^2)))/(108241 + 658*E^3 + E^6 + 52488*E^(3*x) +

6561*E^(4*x) + E^(2*x)*(158274 + 162*E^3) + E^x*(213192 + 648*E^3)),x]

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fricas [A] time = 0.66, size = 42, normalized size = 1.62 \begin {gather*} \frac {x e^{3} + 81 \, x e^{\left (2 \, x\right )} + {\left (x^{2} + 324 \, x\right )} e^{x} + 329 \, x}{e^{3} + 81 \, e^{\left (2 \, x\right )} + 324 \, e^{x} + 329} \end {gather*}

integrate((6561*exp(x)^4+(-81*x^2+162*x+52488)*exp(x)^3+(162*exp(3)+648*x+158274)*exp(x)^2+((x^2+2*x+648)*exp(

3)+329*x^2+658*x+213192)*exp(x)+exp(3)^2+658*exp(3)+108241)/(6561*exp(x)^4+52488*exp(x)^3+(162*exp(3)+158274)*

exp(x)^2+(648*exp(3)+213192)*exp(x)+exp(3)^2+658*exp(3)+108241),x, algorithm="fricas")

(x*e^3 + 81*x*e^(2*x) + (x^2 + 324*x)*e^x + 329*x)/(e^3 + 81*e^(2*x) + 324*e^x + 329)

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giac [A] time = 0.31, size = 44, normalized size = 1.69 \begin {gather*} \frac {2 \, x^{2} e^{x} + x e^{3} + 81 \, x e^{\left (2 \, x\right )} + 324 \, x e^{x} + 329 \, x}{e^{3} + 81 \, e^{\left (2 \, x\right )} + 324 \, e^{x} + 329} \end {gather*}

integrate((6561*exp(x)^4+(-81*x^2+162*x+52488)*exp(x)^3+(162*exp(3)+648*x+158274)*exp(x)^2+((x^2+2*x+648)*exp(

3)+329*x^2+658*x+213192)*exp(x)+exp(3)^2+658*exp(3)+108241)/(6561*exp(x)^4+52488*exp(x)^3+(162*exp(3)+158274)*

exp(x)^2+(648*exp(3)+213192)*exp(x)+exp(3)^2+658*exp(3)+108241),x, algorithm="giac")

(2*x^2*e^x + x*e^3 + 81*x*e^(2*x) + 324*x*e^x + 329*x)/(e^3 + 81*e^(2*x) + 324*e^x + 329)

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method	result	size

risch	\(x +\frac {x^{2} {\mathrm e}^{x}}{81 \,{\mathrm e}^{2 x}+{\mathrm e}^{3}+324 \,{\mathrm e}^{x}+329}\)	\(25\)
norman	\(\frac {\left ({\mathrm e}^{3}+329\right ) x +{\mathrm e}^{x} x^{2}+81 x \,{\mathrm e}^{2 x}+324 \,{\mathrm e}^{x} x}{81 \,{\mathrm e}^{2 x}+{\mathrm e}^{3}+324 \,{\mathrm e}^{x}+329}\)	\(43\)

int((6561*exp(x)^4+(-81*x^2+162*x+52488)*exp(x)^3+(162*exp(3)+648*x+158274)*exp(x)^2+((x^2+2*x+648)*exp(3)+329

*x^2+658*x+213192)*exp(x)+exp(3)^2+658*exp(3)+108241)/(6561*exp(x)^4+52488*exp(x)^3+(162*exp(3)+158274)*exp(x)

^2+(648*exp(3)+213192)*exp(x)+exp(3)^2+658*exp(3)+108241),x,method=_RETURNVERBOSE)

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maxima [B] time = 0.53, size = 585, normalized size = 22.50 \begin {gather*} -\frac {1}{2} \, {\left (\frac {54 \, {\left (e^{3} + 113\right )} \arctan \left (\frac {9 \, {\left (e^{x} + 2\right )}}{\sqrt {e^{3} + 5}}\right )}{{\left (e^{9} + 663 \, e^{6} + 111531 \, e^{3} + 541205\right )} \sqrt {e^{3} + 5}} - \frac {2 \, x}{e^{6} + 658 \, e^{3} + 108241} - \frac {e^{3} - 162 \, e^{x} - 319}{81 \, {\left (e^{6} + 334 \, e^{3} + 1645\right )} e^{\left (2 \, x\right )} + 324 \, {\left (e^{6} + 334 \, e^{3} + 1645\right )} e^{x} + e^{9} + 663 \, e^{6} + 111531 \, e^{3} + 541205} + \frac {\log \left (e^{3} + 81 \, e^{\left (2 \, x\right )} + 324 \, e^{x} + 329\right )}{e^{6} + 658 \, e^{3} + 108241}\right )} e^{6} - 329 \, {\left (\frac {54 \, {\left (e^{3} + 113\right )} \arctan \left (\frac {9 \, {\left (e^{x} + 2\right )}}{\sqrt {e^{3} + 5}}\right )}{{\left (e^{9} + 663 \, e^{6} + 111531 \, e^{3} + 541205\right )} \sqrt {e^{3} + 5}} - \frac {2 \, x}{e^{6} + 658 \, e^{3} + 108241} - \frac {e^{3} - 162 \, e^{x} - 319}{81 \, {\left (e^{6} + 334 \, e^{3} + 1645\right )} e^{\left (2 \, x\right )} + 324 \, {\left (e^{6} + 334 \, e^{3} + 1645\right )} e^{x} + e^{9} + 663 \, e^{6} + 111531 \, e^{3} + 541205} + \frac {\log \left (e^{3} + 81 \, e^{\left (2 \, x\right )} + 324 \, e^{x} + 329\right )}{e^{6} + 658 \, e^{3} + 108241}\right )} e^{3} - x + \frac {9 \, {\left (3 \, e^{3} - 977\right )} \arctan \left (\frac {9 \, {\left (e^{x} + 2\right )}}{\sqrt {e^{3} + 5}}\right )}{{\left (e^{3} + 5\right )}^{\frac {3}{2}}} - \frac {2922507 \, {\left (e^{3} + 113\right )} \arctan \left (\frac {9 \, {\left (e^{x} + 2\right )}}{\sqrt {e^{3} + 5}}\right )}{{\left (e^{9} + 663 \, e^{6} + 111531 \, e^{3} + 541205\right )} \sqrt {e^{3} + 5}} + \frac {162 \, x {\left (e^{3} + 5\right )} e^{\left (2 \, x\right )} + 2 \, x {\left (e^{6} + 334 \, e^{3} + 1645\right )} + 2 \, {\left (x^{2} {\left (e^{3} + 5\right )} + 324 \, x {\left (e^{3} + 5\right )} + 81 \, e^{3} - 79947\right )} e^{x} - e^{6} - 10 \, e^{3} - 321433}{2 \, {\left (81 \, {\left (e^{3} + 5\right )} e^{\left (2 \, x\right )} + 324 \, {\left (e^{3} + 5\right )} e^{x} + e^{6} + 334 \, e^{3} + 1645\right )}} + \frac {108241 \, x}{e^{6} + 658 \, e^{3} + 108241} + \frac {108241 \, {\left (e^{3} - 162 \, e^{x} - 319\right )}}{2 \, {\left (81 \, {\left (e^{6} + 334 \, e^{3} + 1645\right )} e^{\left (2 \, x\right )} + 324 \, {\left (e^{6} + 334 \, e^{3} + 1645\right )} e^{x} + e^{9} + 663 \, e^{6} + 111531 \, e^{3} + 541205\right )}} + \frac {106596 \, {\left (e^{x} + 2\right )}}{81 \, {\left (e^{3} + 5\right )} e^{\left (2 \, x\right )} + 324 \, {\left (e^{3} + 5\right )} e^{x} + e^{6} + 334 \, e^{3} + 1645} - \frac {108241 \, \log \left (e^{3} + 81 \, e^{\left (2 \, x\right )} + 324 \, e^{x} + 329\right )}{2 \, {\left (e^{6} + 658 \, e^{3} + 108241\right )}} + \frac {11844 \, \arctan \left (\frac {9 \, {\left (e^{x} + 2\right )}}{\sqrt {e^{3} + 5}}\right )}{{\left (e^{3} + 5\right )}^{\frac {3}{2}}} + \frac {1}{2} \, \log \left (e^{3} + 81 \, e^{\left (2 \, x\right )} + 324 \, e^{x} + 329\right ) \end {gather*}

integrate((6561*exp(x)^4+(-81*x^2+162*x+52488)*exp(x)^3+(162*exp(3)+648*x+158274)*exp(x)^2+((x^2+2*x+648)*exp(

3)+329*x^2+658*x+213192)*exp(x)+exp(3)^2+658*exp(3)+108241)/(6561*exp(x)^4+52488*exp(x)^3+(162*exp(3)+158274)*

exp(x)^2+(648*exp(3)+213192)*exp(x)+exp(3)^2+658*exp(3)+108241),x, algorithm="maxima")

-1/2*(54*(e^3 + 113)*arctan(9*(e^x + 2)/sqrt(e^3 + 5))/((e^9 + 663*e^6 + 111531*e^3 + 541205)*sqrt(e^3 + 5)) -

2*x/(e^6 + 658*e^3 + 108241) - (e^3 - 162*e^x - 319)/(81*(e^6 + 334*e^3 + 1645)*e^(2*x) + 324*(e^6 + 334*e^3

+ 1645)*e^x + e^9 + 663*e^6 + 111531*e^3 + 541205) + log(e^3 + 81*e^(2*x) + 324*e^x + 329)/(e^6 + 658*e^3 + 10

8241))*e^6 - 329*(54*(e^3 + 113)*arctan(9*(e^x + 2)/sqrt(e^3 + 5))/((e^9 + 663*e^6 + 111531*e^3 + 541205)*sqrt

(e^3 + 5)) - 2*x/(e^6 + 658*e^3 + 108241) - (e^3 - 162*e^x - 319)/(81*(e^6 + 334*e^3 + 1645)*e^(2*x) + 324*(e^

6 + 334*e^3 + 1645)*e^x + e^9 + 663*e^6 + 111531*e^3 + 541205) + log(e^3 + 81*e^(2*x) + 324*e^x + 329)/(e^6 +

658*e^3 + 108241))*e^3 - x + 9*(3*e^3 - 977)*arctan(9*(e^x + 2)/sqrt(e^3 + 5))/(e^3 + 5)^(3/2) - 2922507*(e^3

+ 113)*arctan(9*(e^x + 2)/sqrt(e^3 + 5))/((e^9 + 663*e^6 + 111531*e^3 + 541205)*sqrt(e^3 + 5)) + 1/2*(162*x*(e

^3 + 5)*e^(2*x) + 2*x*(e^6 + 334*e^3 + 1645) + 2*(x^2*(e^3 + 5) + 324*x*(e^3 + 5) + 81*e^3 - 79947)*e^x - e^6

- 10*e^3 - 321433)/(81*(e^3 + 5)*e^(2*x) + 324*(e^3 + 5)*e^x + e^6 + 334*e^3 + 1645) + 108241*x/(e^6 + 658*e^3

+ 108241) + 108241/2*(e^3 - 162*e^x - 319)/(81*(e^6 + 334*e^3 + 1645)*e^(2*x) + 324*(e^6 + 334*e^3 + 1645)*e^

x + e^9 + 663*e^6 + 111531*e^3 + 541205) + 106596*(e^x + 2)/(81*(e^3 + 5)*e^(2*x) + 324*(e^3 + 5)*e^x + e^6 +

334*e^3 + 1645) - 108241/2*log(e^3 + 81*e^(2*x) + 324*e^x + 329)/(e^6 + 658*e^3 + 108241) + 11844*arctan(9*(e^

x + 2)/sqrt(e^3 + 5))/(e^3 + 5)^(3/2) + 1/2*log(e^3 + 81*e^(2*x) + 324*e^x + 329)

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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {6561\,{\mathrm {e}}^{4\,x}+658\,{\mathrm {e}}^3+{\mathrm {e}}^6+{\mathrm {e}}^{3\,x}\,\left (-81\,x^2+162\,x+52488\right )+{\mathrm {e}}^x\,\left (658\,x+{\mathrm {e}}^3\,\left (x^2+2\,x+648\right )+329\,x^2+213192\right )+{\mathrm {e}}^{2\,x}\,\left (648\,x+162\,{\mathrm {e}}^3+158274\right )+108241}{52488\,{\mathrm {e}}^{3\,x}+6561\,{\mathrm {e}}^{4\,x}+658\,{\mathrm {e}}^3+{\mathrm {e}}^6+{\mathrm {e}}^x\,\left (648\,{\mathrm {e}}^3+213192\right )+{\mathrm {e}}^{2\,x}\,\left (162\,{\mathrm {e}}^3+158274\right )+108241} \,d x \end {gather*}

int((6561*exp(4*x) + 658*exp(3) + exp(6) + exp(3*x)*(162*x - 81*x^2 + 52488) + exp(x)*(658*x + exp(3)*(2*x + x

^2 + 648) + 329*x^2 + 213192) + exp(2*x)*(648*x + 162*exp(3) + 158274) + 108241)/(52488*exp(3*x) + 6561*exp(4*

x) + 658*exp(3) + exp(6) + exp(x)*(648*exp(3) + 213192) + exp(2*x)*(162*exp(3) + 158274) + 108241),x)

int((6561*exp(4*x) + 658*exp(3) + exp(6) + exp(3*x)*(162*x - 81*x^2 + 52488) + exp(x)*(658*x + exp(3)*(2*x + x

^2 + 648) + 329*x^2 + 213192) + exp(2*x)*(648*x + 162*exp(3) + 158274) + 108241)/(52488*exp(3*x) + 6561*exp(4*

x) + 658*exp(3) + exp(6) + exp(x)*(648*exp(3) + 213192) + exp(2*x)*(162*exp(3) + 158274) + 108241), x)

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sympy [A] time = 0.20, size = 24, normalized size = 0.92 \begin {gather*} \frac {x^{2} e^{x}}{81 e^{2 x} + 324 e^{x} + e^{3} + 329} + x \end {gather*}

integrate((6561*exp(x)**4+(-81*x**2+162*x+52488)*exp(x)**3+(162*exp(3)+648*x+158274)*exp(x)**2+((x**2+2*x+648)

*exp(3)+329*x**2+658*x+213192)*exp(x)+exp(3)**2+658*exp(3)+108241)/(6561*exp(x)**4+52488*exp(x)**3+(162*exp(3)

+158274)*exp(x)**2+(648*exp(3)+213192)*exp(x)+exp(3)**2+658*exp(3)+108241),x)

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3.64.16 \(\int \frac {108241+658 e^3+e^6+6561 e^{4 x}+e^{2 x} (158274+162 e^3+648 x)+e^{3 x} (52488+162 x-81 x^2)+e^x (213192+658 x+329 x^2+e^3 (648+2 x+x^2))}{108241+658 e^3+e^6+52488 e^{3 x}+6561 e^{4 x}+e^{2 x} (158274+162 e^3)+e^x (213192+648 e^3)} \, dx\)