\(\int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x (-6 x^2-2 e^3 x^2-2 x^3)+e^{\frac {1-3 x+x \log (5)}{x}} (2 e^3-6 x^2+2 e^x x^2-2 x^3)}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x (-6 x^2-2 x^3)+e^{\frac {1-3 x+x \log (5)}{x}} (-6 x^2+2 e^x x^2-2 x^3)} \, dx\) [1431]

Optimal result

Integrand size = 210, antiderivative size = 26 \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=x-\frac {2 e^3}{3-5 e^{-3+\frac {1}{x}}-e^x+x} \]

[Out]

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

Rubi [F]

\[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=\int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx \]

[In]

Int[(9*x^2 + 2*E^3*x^2 + E^(2*x)*x^2 + E^((2*(1 - 3*x + x*Log[5]))/x)*x^2 + 6*x^3 + x^4 + E^x*(-6*x^2 - 2*E^3*
x^2 - 2*x^3) + E^((1 - 3*x + x*Log[5])/x)*(2*E^3 - 6*x^2 + 2*E^x*x^2 - 2*x^3))/(9*x^2 + E^(2*x)*x^2 + E^((2*(1
- 3*x + x*Log[5]))/x)*x^2 + 6*x^3 + x^4 + E^x*(-6*x^2 - 2*x^3) + E^((1 - 3*x + x*Log[5])/x)*(-6*x^2 + 2*E^x*x
^2 - 2*x^3)),x]

[Out]

x - 4*E^9*Defer[Int][(-3*E^3 + 5*E^x^(-1) + E^(3 + x) - E^3*x)^(-2), x] + 10*E^6*Defer[Int][E^x^(-1)/(-3*E^3 +
5*E^x^(-1) + E^(3 + x) - E^3*x)^2, x] + 10*E^6*Defer[Int][E^x^(-1)/(x^2*(-3*E^3 + 5*E^x^(-1) + E^(3 + x) - E^
3*x)^2), x] - 2*E^6*Defer[Int][(-3*E^3 + 5*E^x^(-1) + E^(3 + x) - E^3*x)^(-1), x] - 2*E^9*Defer[Int][x/(3*E^3
- 5*E^x^(-1) - E^(3 + x) + E^3*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+\left (9+2 e^3\right ) x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx \\ & = \int \frac {10 e^{6+\frac {1}{x}}+2 e^9 x^2+25 e^{2/x} x^2-2 e^{9+x} x^2+10 e^{3+\frac {1}{x}+x} x^2+e^{6+2 x} x^2-10 e^{3+\frac {1}{x}} x^2 (3+x)-2 e^{6+x} x^2 (3+x)+e^6 x^2 (3+x)^2}{x^2 \left (5 e^{\frac {1}{x}}+e^{3+x}-e^3 (3+x)\right )^2} \, dx \\ & = \int \left (1-\frac {2 e^6}{-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x}-\frac {2 e^6 \left (-5 e^{\frac {1}{x}}+2 e^3 x^2-5 e^{\frac {1}{x}} x^2+e^3 x^3\right )}{x^2 \left (3 e^3-5 e^{\frac {1}{x}}-e^{3+x}+e^3 x\right )^2}\right ) \, dx \\ & = x-\left (2 e^6\right ) \int \frac {1}{-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x} \, dx-\left (2 e^6\right ) \int \frac {-5 e^{\frac {1}{x}}+2 e^3 x^2-5 e^{\frac {1}{x}} x^2+e^3 x^3}{x^2 \left (3 e^3-5 e^{\frac {1}{x}}-e^{3+x}+e^3 x\right )^2} \, dx \\ & = x-\left (2 e^6\right ) \int \frac {1}{-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x} \, dx-\left (2 e^6\right ) \int \left (\frac {2 e^3}{\left (-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x\right )^2}-\frac {5 e^{\frac {1}{x}}}{\left (-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x\right )^2}-\frac {5 e^{\frac {1}{x}}}{x^2 \left (-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x\right )^2}+\frac {e^3 x}{\left (3 e^3-5 e^{\frac {1}{x}}-e^{3+x}+e^3 x\right )^2}\right ) \, dx \\ & = x-\left (2 e^6\right ) \int \frac {1}{-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x} \, dx+\left (10 e^6\right ) \int \frac {e^{\frac {1}{x}}}{\left (-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x\right )^2} \, dx+\left (10 e^6\right ) \int \frac {e^{\frac {1}{x}}}{x^2 \left (-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x\right )^2} \, dx-\left (2 e^9\right ) \int \frac {x}{\left (3 e^3-5 e^{\frac {1}{x}}-e^{3+x}+e^3 x\right )^2} \, dx-\left (4 e^9\right ) \int \frac {1}{\left (-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=x+\frac {2 e^6}{5 e^{\frac {1}{x}}+e^{3+x}-e^3 (3+x)} \]

[In]

Integrate[(9*x^2 + 2*E^3*x^2 + E^(2*x)*x^2 + E^((2*(1 - 3*x + x*Log[5]))/x)*x^2 + 6*x^3 + x^4 + E^x*(-6*x^2 -
2*E^3*x^2 - 2*x^3) + E^((1 - 3*x + x*Log[5])/x)*(2*E^3 - 6*x^2 + 2*E^x*x^2 - 2*x^3))/(9*x^2 + E^(2*x)*x^2 + E^
((2*(1 - 3*x + x*Log[5]))/x)*x^2 + 6*x^3 + x^4 + E^x*(-6*x^2 - 2*x^3) + E^((1 - 3*x + x*Log[5])/x)*(-6*x^2 + 2
*E^x*x^2 - 2*x^3)),x]

[Out]

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

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12

method	result	size
risch	\(x -\frac {2 \,{\mathrm e}^{3}}{x -{\mathrm e}^{x}-5 \,{\mathrm e}^{-\frac {-1+3 x}{x}}+3}\)	\(29\)
parallelrisch	\(-\frac {-x^{2}+{\mathrm e}^{x} x +x \,{\mathrm e}^{\frac {x \ln \left (5\right )-3 x +1}{x}}+2 \,{\mathrm e}^{3}-3 x}{x -{\mathrm e}^{x}-{\mathrm e}^{\frac {x \ln \left (5\right )-3 x +1}{x}}+3}\)	\(61\)

[In]

int((x^2*exp((x*ln(5)-3*x+1)/x)^2+(2*exp(x)*x^2+2*exp(3)-2*x^3-6*x^2)*exp((x*ln(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*
x^2*exp(3)-2*x^3-6*x^2)*exp(x)+2*x^2*exp(3)+x^4+6*x^3+9*x^2)/(x^2*exp((x*ln(5)-3*x+1)/x)^2+(2*exp(x)*x^2-2*x^3
-6*x^2)*exp((x*ln(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^3-6*x^2)*exp(x)+x^4+6*x^3+9*x^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.27 \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=\frac {x^{2} - x e^{x} - x e^{\left (\frac {x \log \left (5\right ) - 3 \, x + 1}{x}\right )} + 3 \, x - 2 \, e^{3}}{x - e^{x} - e^{\left (\frac {x \log \left (5\right ) - 3 \, x + 1}{x}\right )} + 3} \]

[In]

integrate((x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*x^2+2*exp(3)-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*
x^2+(-2*x^2*exp(3)-2*x^3-6*x^2)*exp(x)+2*x^2*exp(3)+x^4+6*x^3+9*x^2)/(x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*
x^2-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^3-6*x^2)*exp(x)+x^4+6*x^3+9*x^2),x, algorithm="fri
cas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=x + \frac {2 e^{3}}{- x + e^{x} + e^{\frac {- 3 x + x \log {\left (5 \right )} + 1}{x}} - 3} \]

[In]

integrate((x**2*exp((x*ln(5)-3*x+1)/x)**2+(2*exp(x)*x**2+2*exp(3)-2*x**3-6*x**2)*exp((x*ln(5)-3*x+1)/x)+exp(x)
**2*x**2+(-2*x**2*exp(3)-2*x**3-6*x**2)*exp(x)+2*x**2*exp(3)+x**4+6*x**3+9*x**2)/(x**2*exp((x*ln(5)-3*x+1)/x)*
*2+(2*exp(x)*x**2-2*x**3-6*x**2)*exp((x*ln(5)-3*x+1)/x)+exp(x)**2*x**2+(-2*x**3-6*x**2)*exp(x)+x**4+6*x**3+9*x
**2),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).

Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=\frac {x^{2} e^{3} + 3 \, x e^{3} - x e^{\left (x + 3\right )} - 5 \, x e^{\frac {1}{x}} - 2 \, e^{6}}{x e^{3} + 3 \, e^{3} - e^{\left (x + 3\right )} - 5 \, e^{\frac {1}{x}}} \]

[In]

integrate((x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*x^2+2*exp(3)-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*
x^2+(-2*x^2*exp(3)-2*x^3-6*x^2)*exp(x)+2*x^2*exp(3)+x^4+6*x^3+9*x^2)/(x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*
x^2-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^3-6*x^2)*exp(x)+x^4+6*x^3+9*x^2),x, algorithm="max
ima")

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=\text {Timed out} \]

[In]

integrate((x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*x^2+2*exp(3)-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*
x^2+(-2*x^2*exp(3)-2*x^3-6*x^2)*exp(x)+2*x^2*exp(3)+x^4+6*x^3+9*x^2)/(x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*
x^2-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^3-6*x^2)*exp(x)+x^4+6*x^3+9*x^2),x, algorithm="gia
c")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 8.52 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=x-\frac {2\,{\mathrm {e}}^3}{x-{\mathrm {e}}^x-5\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^{-3}+3} \]

[In]

int((exp((x*log(5) - 3*x + 1)/x)*(2*exp(3) + 2*x^2*exp(x) - 6*x^2 - 2*x^3) + x^2*exp((2*(x*log(5) - 3*x + 1))/
x) + x^2*exp(2*x) + 2*x^2*exp(3) - exp(x)*(2*x^2*exp(3) + 6*x^2 + 2*x^3) + 9*x^2 + 6*x^3 + x^4)/(x^2*exp((2*(x
*log(5) - 3*x + 1))/x) - exp(x)*(6*x^2 + 2*x^3) - exp((x*log(5) - 3*x + 1)/x)*(6*x^2 - 2*x^2*exp(x) + 2*x^3) +
x^2*exp(2*x) + 9*x^2 + 6*x^3 + x^4),x)

[Out]

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