\(\int e^{2+e^x+x+4 x^3+4 x^4 \log (x)} (1+e^x+12 x^2+4 x^3+16 x^3 \log (x)) \, dx\) [5892]

Optimal result

Integrand size = 43, antiderivative size = 19 \[ \int e^{2+e^x+x+4 x^3+4 x^4 \log (x)} \left (1+e^x+12 x^2+4 x^3+16 x^3 \log (x)\right ) \, dx=e^{2+e^x+x+x^3 (4+4 x \log (x))} \]

[Out]

exp(x^3*(4*x*ln(x)+4)+x+exp(x)+2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6838} \[ \int e^{2+e^x+x+4 x^3+4 x^4 \log (x)} \left (1+e^x+12 x^2+4 x^3+16 x^3 \log (x)\right ) \, dx=e^{4 x^3+x+e^x+2} x^{4 x^4} \]

[In]

Int[E^(2 + E^x + x + 4*x^3 + 4*x^4*Log[x])*(1 + E^x + 12*x^2 + 4*x^3 + 16*x^3*Log[x]),x]

[Out]

E^(2 + E^x + x + 4*x^3)*x^(4*x^4)

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = e^{2+e^x+x+4 x^3} x^{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int e^{2+e^x+x+4 x^3+4 x^4 \log (x)} \left (1+e^x+12 x^2+4 x^3+16 x^3 \log (x)\right ) \, dx=e^{2+e^x+x+4 x^3} x^{4 x^4} \]

[In]

Integrate[E^(2 + E^x + x + 4*x^3 + 4*x^4*Log[x])*(1 + E^x + 12*x^2 + 4*x^3 + 16*x^3*Log[x]),x]

[Out]

E^(2 + E^x + x + 4*x^3)*x^(4*x^4)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\({\mathrm e}^{4 x^{4} \ln \left (x \right )+{\mathrm e}^{x}+4 x^{3}+x +2}\)	\(19\)
default	\({\mathrm e}^{4 x^{4} \ln \left (x \right )+{\mathrm e}^{x}+4 x^{3}+x +2}\)	\(19\)
parallelrisch	\({\mathrm e}^{4 x^{4} \ln \left (x \right )+{\mathrm e}^{x}+4 x^{3}+x +2}\)	\(19\)
risch	\(x^{4 x^{4}} {\mathrm e}^{2+{\mathrm e}^{x}+4 x^{3}+x}\)	\(20\)

[In]

int((16*x^3*ln(x)+exp(x)+4*x^3+12*x^2+1)*exp(4*x^4*ln(x)+exp(x)+4*x^3+x+2),x,method=_RETURNVERBOSE)

[Out]

exp(4*x^4*ln(x)+exp(x)+4*x^3+x+2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int e^{2+e^x+x+4 x^3+4 x^4 \log (x)} \left (1+e^x+12 x^2+4 x^3+16 x^3 \log (x)\right ) \, dx=e^{\left (4 \, x^{4} \log \left (x\right ) + 4 \, x^{3} + x + e^{x} + 2\right )} \]

[In]

integrate((16*x^3*log(x)+exp(x)+4*x^3+12*x^2+1)*exp(4*x^4*log(x)+exp(x)+4*x^3+x+2),x, algorithm="fricas")

[Out]

e^(4*x^4*log(x) + 4*x^3 + x + e^x + 2)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{2+e^x+x+4 x^3+4 x^4 \log (x)} \left (1+e^x+12 x^2+4 x^3+16 x^3 \log (x)\right ) \, dx=e^{4 x^{4} \log {\left (x \right )} + 4 x^{3} + x + e^{x} + 2} \]

[In]

integrate((16*x**3*ln(x)+exp(x)+4*x**3+12*x**2+1)*exp(4*x**4*ln(x)+exp(x)+4*x**3+x+2),x)

[Out]

exp(4*x**4*log(x) + 4*x**3 + x + exp(x) + 2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int e^{2+e^x+x+4 x^3+4 x^4 \log (x)} \left (1+e^x+12 x^2+4 x^3+16 x^3 \log (x)\right ) \, dx=e^{\left (4 \, x^{4} \log \left (x\right ) + 4 \, x^{3} + x + e^{x} + 2\right )} \]

[In]

integrate((16*x^3*log(x)+exp(x)+4*x^3+12*x^2+1)*exp(4*x^4*log(x)+exp(x)+4*x^3+x+2),x, algorithm="maxima")

[Out]

e^(4*x^4*log(x) + 4*x^3 + x + e^x + 2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int e^{2+e^x+x+4 x^3+4 x^4 \log (x)} \left (1+e^x+12 x^2+4 x^3+16 x^3 \log (x)\right ) \, dx=e^{\left (4 \, x^{4} \log \left (x\right ) + 4 \, x^{3} + x + e^{x} + 2\right )} \]

[In]

integrate((16*x^3*log(x)+exp(x)+4*x^3+12*x^2+1)*exp(4*x^4*log(x)+exp(x)+4*x^3+x+2),x, algorithm="giac")

[Out]

e^(4*x^4*log(x) + 4*x^3 + x + e^x + 2)

Mupad [B] (verification not implemented)

Time = 11.66 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int e^{2+e^x+x+4 x^3+4 x^4 \log (x)} \left (1+e^x+12 x^2+4 x^3+16 x^3 \log (x)\right ) \, dx=x^{4\,x^4}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{4\,x^3}\,{\mathrm {e}}^x \]

[In]

int(exp(x + exp(x) + 4*x^4*log(x) + 4*x^3 + 2)*(exp(x) + 16*x^3*log(x) + 12*x^2 + 4*x^3 + 1),x)

[Out]

x^(4*x^4)*exp(exp(x))*exp(2)*exp(4*x^3)*exp(x)