Integrand size = 80, antiderivative size = 19 \[ \int \frac {e^x \left (e^{e^{e^x} x^{5/4}}+x\right ) \left (4+4 x+e^{e^{e^x} x^{5/4}} \left (4+e^{e^x} \sqrt [4]{x} \left (5+4 e^x x\right )\right )\right )}{4 e^{e^{e^x} x^{5/4}}+4 x} \, dx=e^x \left (e^{e^{e^x} x^{5/4}}+x\right ) \] Output:
exp(ln(exp(x^(5/4)*exp(exp(x)))+x)+x)
Time = 0.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (e^{e^{e^x} x^{5/4}}+x\right ) \left (4+4 x+e^{e^{e^x} x^{5/4}} \left (4+e^{e^x} \sqrt [4]{x} \left (5+4 e^x x\right )\right )\right )}{4 e^{e^{e^x} x^{5/4}}+4 x} \, dx=e^x \left (e^{e^{e^x} x^{5/4}}+x\right ) \] Input:
Integrate[(E^x*(E^(E^E^x*x^(5/4)) + x)*(4 + 4*x + E^(E^E^x*x^(5/4))*(4 + E ^E^x*x^(1/4)*(5 + 4*E^x*x))))/(4*E^(E^E^x*x^(5/4)) + 4*x),x]
Output:
E^x*(E^(E^E^x*x^(5/4)) + x)
Time = 2.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {7267, 2726}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^x \left (e^{e^{e^x} x^{5/4}}+x\right ) \left (e^{e^{e^x} x^{5/4}} \left (e^{e^x} \sqrt [4]{x} \left (4 e^x x+5\right )+4\right )+4 x+4\right )}{4 e^{e^{e^x} x^{5/4}}+4 x} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle \int e^x x^{3/4} \left (4 e^{e^{e^x} x^{5/4}+x+e^x} x^{5/4}+5 e^{e^{e^x} x^{5/4}+e^x} \sqrt [4]{x}+4 e^{e^{e^x} x^{5/4}}+4 x+4\right )d\sqrt [4]{x}\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle e^x \left (e^{e^{e^x} x^{5/4}}+x\right )\) |
Input:
Int[(E^x*(E^(E^E^x*x^(5/4)) + x)*(4 + 4*x + E^(E^E^x*x^(5/4))*(4 + E^E^x*x ^(1/4)*(5 + 4*E^x*x))))/(4*E^(E^E^x*x^(5/4)) + 4*x),x]
Output:
E^x*(E^(E^E^x*x^(5/4)) + x)
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
\[\int \frac {\left (\left (\left (4 \,{\mathrm e}^{x} x +5\right ) {\mathrm e}^{{\mathrm e}^{x}} x^{\frac {1}{4}}+4\right ) {\mathrm e}^{x^{\frac {5}{4}} {\mathrm e}^{{\mathrm e}^{x}}}+4 x +4\right ) {\mathrm e}^{\ln \left ({\mathrm e}^{x^{\frac {5}{4}} {\mathrm e}^{{\mathrm e}^{x}}}+x \right )+x}}{4 \,{\mathrm e}^{x^{\frac {5}{4}} {\mathrm e}^{{\mathrm e}^{x}}}+4 x}d x\]
Input:
int((((4*exp(x)*x+5)*exp(exp(x))*x^(1/4)+4)*exp(x^(5/4)*exp(exp(x)))+4*x+4 )*exp(ln(exp(x^(5/4)*exp(exp(x)))+x)+x)/(4*exp(x^(5/4)*exp(exp(x)))+4*x),x )
Output:
int((((4*exp(x)*x+5)*exp(exp(x))*x^(1/4)+4)*exp(x^(5/4)*exp(exp(x)))+4*x+4 )*exp(ln(exp(x^(5/4)*exp(exp(x)))+x)+x)/(4*exp(x^(5/4)*exp(exp(x)))+4*x),x )
Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {e^x \left (e^{e^{e^x} x^{5/4}}+x\right ) \left (4+4 x+e^{e^{e^x} x^{5/4}} \left (4+e^{e^x} \sqrt [4]{x} \left (5+4 e^x x\right )\right )\right )}{4 e^{e^{e^x} x^{5/4}}+4 x} \, dx=x e^{x} + e^{\left (x^{\frac {5}{4}} e^{\left (e^{x}\right )} + x\right )} \] Input:
integrate((((4*exp(x)*x+5)*exp(exp(x))*x^(1/4)+4)*exp(x^(5/4)*exp(exp(x))) +4*x+4)*exp(log(exp(x^(5/4)*exp(exp(x)))+x)+x)/(4*exp(x^(5/4)*exp(exp(x))) +4*x),x, algorithm="fricas")
Output:
x*e^x + e^(x^(5/4)*e^(e^x) + x)
Timed out. \[ \int \frac {e^x \left (e^{e^{e^x} x^{5/4}}+x\right ) \left (4+4 x+e^{e^{e^x} x^{5/4}} \left (4+e^{e^x} \sqrt [4]{x} \left (5+4 e^x x\right )\right )\right )}{4 e^{e^{e^x} x^{5/4}}+4 x} \, dx=\text {Timed out} \] Input:
integrate((((4*exp(x)*x+5)*exp(exp(x))*x**(1/4)+4)*exp(x**(5/4)*exp(exp(x) ))+4*x+4)*exp(ln(exp(x**(5/4)*exp(exp(x)))+x)+x)/(4*exp(x**(5/4)*exp(exp(x )))+4*x),x)
Output:
Timed out
Time = 0.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (e^{e^{e^x} x^{5/4}}+x\right ) \left (4+4 x+e^{e^{e^x} x^{5/4}} \left (4+e^{e^x} \sqrt [4]{x} \left (5+4 e^x x\right )\right )\right )}{4 e^{e^{e^x} x^{5/4}}+4 x} \, dx={\left (x - 1\right )} e^{x} + e^{\left (x^{\frac {5}{4}} e^{\left (e^{x}\right )} + x\right )} + e^{x} \] Input:
integrate((((4*exp(x)*x+5)*exp(exp(x))*x^(1/4)+4)*exp(x^(5/4)*exp(exp(x))) +4*x+4)*exp(log(exp(x^(5/4)*exp(exp(x)))+x)+x)/(4*exp(x^(5/4)*exp(exp(x))) +4*x),x, algorithm="maxima")
Output:
(x - 1)*e^x + e^(x^(5/4)*e^(e^x) + x) + e^x
\[ \int \frac {e^x \left (e^{e^{e^x} x^{5/4}}+x\right ) \left (4+4 x+e^{e^{e^x} x^{5/4}} \left (4+e^{e^x} \sqrt [4]{x} \left (5+4 e^x x\right )\right )\right )}{4 e^{e^{e^x} x^{5/4}}+4 x} \, dx=\int { \frac {{\left ({\left ({\left (4 \, x e^{x} + 5\right )} x^{\frac {1}{4}} e^{\left (e^{x}\right )} + 4\right )} e^{\left (x^{\frac {5}{4}} e^{\left (e^{x}\right )}\right )} + 4 \, x + 4\right )} e^{\left (x + \log \left (x + e^{\left (x^{\frac {5}{4}} e^{\left (e^{x}\right )}\right )}\right )\right )}}{4 \, {\left (x + e^{\left (x^{\frac {5}{4}} e^{\left (e^{x}\right )}\right )}\right )}} \,d x } \] Input:
integrate((((4*exp(x)*x+5)*exp(exp(x))*x^(1/4)+4)*exp(x^(5/4)*exp(exp(x))) +4*x+4)*exp(log(exp(x^(5/4)*exp(exp(x)))+x)+x)/(4*exp(x^(5/4)*exp(exp(x))) +4*x),x, algorithm="giac")
Output:
integrate(1/4*(((4*x*e^x + 5)*x^(1/4)*e^(e^x) + 4)*e^(x^(5/4)*e^(e^x)) + 4 *x + 4)*e^(x + log(x + e^(x^(5/4)*e^(e^x))))/(x + e^(x^(5/4)*e^(e^x))), x)
Timed out. \[ \int \frac {e^x \left (e^{e^{e^x} x^{5/4}}+x\right ) \left (4+4 x+e^{e^{e^x} x^{5/4}} \left (4+e^{e^x} \sqrt [4]{x} \left (5+4 e^x x\right )\right )\right )}{4 e^{e^{e^x} x^{5/4}}+4 x} \, dx=\int \frac {{\mathrm {e}}^{x+\ln \left (x+{\mathrm {e}}^{x^{5/4}\,{\mathrm {e}}^{{\mathrm {e}}^x}}\right )}\,\left (4\,x+{\mathrm {e}}^{x^{5/4}\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,\left (x^{1/4}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (4\,x\,{\mathrm {e}}^x+5\right )+4\right )+4\right )}{4\,x+4\,{\mathrm {e}}^{x^{5/4}\,{\mathrm {e}}^{{\mathrm {e}}^x}}} \,d x \] Input:
int((exp(x + log(x + exp(x^(5/4)*exp(exp(x)))))*(4*x + exp(x^(5/4)*exp(exp (x)))*(x^(1/4)*exp(exp(x))*(4*x*exp(x) + 5) + 4) + 4))/(4*x + 4*exp(x^(5/4 )*exp(exp(x)))),x)
Output:
int((exp(x + log(x + exp(x^(5/4)*exp(exp(x)))))*(4*x + exp(x^(5/4)*exp(exp (x)))*(x^(1/4)*exp(exp(x))*(4*x*exp(x) + 5) + 4) + 4))/(4*x + 4*exp(x^(5/4 )*exp(exp(x)))), x)
Time = 0.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {e^x \left (e^{e^{e^x} x^{5/4}}+x\right ) \left (4+4 x+e^{e^{e^x} x^{5/4}} \left (4+e^{e^x} \sqrt [4]{x} \left (5+4 e^x x\right )\right )\right )}{4 e^{e^{e^x} x^{5/4}}+4 x} \, dx=e^{x} \left (e^{x^{\frac {5}{4}} e^{e^{x}}}+x \right ) \] Input:
int((((4*exp(x)*x+5)*exp(exp(x))*x^(1/4)+4)*exp(x^(5/4)*exp(exp(x)))+4*x+4 )*exp(log(exp(x^(5/4)*exp(exp(x)))+x)+x)/(4*exp(x^(5/4)*exp(exp(x)))+4*x), x)
Output:
e**x*(e**(x**(1/4)*e**(e**x)*x) + x)