\(\int (6 x^5+e^{\frac {2 (e^2+x^3+2 x^2 \log (5)+x \log ^2(5))}{x}} (-2 e^2+2 x+4 x^3+4 x^2 \log (5))+e^{\frac {e^2+x^3+2 x^2 \log (5)+x \log ^2(5)}{x}} (-2 e^2 x^2+8 x^3+4 x^5+4 x^4 \log (5))) \, dx\) [1600]

Optimal result

Integrand size = 108, antiderivative size = 35 \[ \int \left (6 x^5+e^{\frac {2 \left (e^2+x^3+2 x^2 \log (5)+x \log ^2(5)\right )}{x}} \left (-2 e^2+2 x+4 x^3+4 x^2 \log (5)\right )+e^{\frac {e^2+x^3+2 x^2 \log (5)+x \log ^2(5)}{x}} \left (-2 e^2 x^2+8 x^3+4 x^5+4 x^4 \log (5)\right )\right ) \, dx=x^2 \left (e^{\frac {e^2}{x}+(x+\log (5))^2}-x+\left (1+\frac {1}{x}\right ) x^2\right )^2 \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(187\) vs. \(2(35)=70\).

Time = 0.20 (sec) , antiderivative size = 187, normalized size of antiderivative = 5.34, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {2326} \[ \int \left (6 x^5+e^{\frac {2 \left (e^2+x^3+2 x^2 \log (5)+x \log ^2(5)\right )}{x}} \left (-2 e^2+2 x+4 x^3+4 x^2 \log (5)\right )+e^{\frac {e^2+x^3+2 x^2 \log (5)+x \log ^2(5)}{x}} \left (-2 e^2 x^2+8 x^3+4 x^5+4 x^4 \log (5)\right )\right ) \, dx=x^6-\frac {5^{4 x} e^{\frac {2 \left (x^3+x \log ^2(5)+e^2\right )}{x}} \left (-2 x^3-2 x^2 \log (5)+e^2\right )}{\frac {3 x^2+4 x \log (5)+\log ^2(5)}{x}-\frac {x^3+2 x^2 \log (5)+x \log ^2(5)+e^2}{x^2}}-\frac {2\ 5^{2 x} e^{\frac {x^3+x \log ^2(5)+e^2}{x}} \left (-2 x^5-2 x^4 \log (5)+e^2 x^2\right )}{\frac {3 x^2+4 x \log (5)+\log ^2(5)}{x}-\frac {x^3+2 x^2 \log (5)+x \log ^2(5)+e^2}{x^2}} \]

[In]

Int[6*x^5 + E^((2*(E^2 + x^3 + 2*x^2*Log[5] + x*Log[5]^2))/x)*(-2*E^2 + 2*x + 4*x^3 + 4*x^2*Log[5]) + E^((E^2
+ x^3 + 2*x^2*Log[5] + x*Log[5]^2)/x)*(-2*E^2*x^2 + 8*x^3 + 4*x^5 + 4*x^4*Log[5]),x]

[Out]

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

Rule 2326

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]

Rubi steps \begin{align*} \text {integral}& = x^6+\int e^{\frac {2 \left (e^2+x^3+2 x^2 \log (5)+x \log ^2(5)\right )}{x}} \left (-2 e^2+2 x+4 x^3+4 x^2 \log (5)\right ) \, dx+\int e^{\frac {e^2+x^3+2 x^2 \log (5)+x \log ^2(5)}{x}} \left (-2 e^2 x^2+8 x^3+4 x^5+4 x^4 \log (5)\right ) \, dx \\ & = x^6-\frac {5^{4 x} e^{\frac {2 \left (e^2+x^3+x \log ^2(5)\right )}{x}} \left (e^2-2 x^3-2 x^2 \log (5)\right )}{\frac {3 x^2+4 x \log (5)+\log ^2(5)}{x}-\frac {e^2+x^3+2 x^2 \log (5)+x \log ^2(5)}{x^2}}-\frac {2\ 5^{2 x} e^{\frac {e^2+x^3+x \log ^2(5)}{x}} \left (e^2 x^2-2 x^5-2 x^4 \log (5)\right )}{\frac {3 x^2+4 x \log (5)+\log ^2(5)}{x}-\frac {e^2+x^3+2 x^2 \log (5)+x \log ^2(5)}{x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.83 \[ \int \left (6 x^5+e^{\frac {2 \left (e^2+x^3+2 x^2 \log (5)+x \log ^2(5)\right )}{x}} \left (-2 e^2+2 x+4 x^3+4 x^2 \log (5)\right )+e^{\frac {e^2+x^3+2 x^2 \log (5)+x \log ^2(5)}{x}} \left (-2 e^2 x^2+8 x^3+4 x^5+4 x^4 \log (5)\right )\right ) \, dx=625^x e^{\frac {2 e^2}{x}+2 x^2+2 \log ^2(5)} x^2+x^6+\frac {4\ 25^x e^{\frac {e^2}{x}+x^2+\log ^2(5)} x^4 \log (5)}{\log (25)} \]

[In]

Integrate[6*x^5 + E^((2*(E^2 + x^3 + 2*x^2*Log[5] + x*Log[5]^2))/x)*(-2*E^2 + 2*x + 4*x^3 + 4*x^2*Log[5]) + E^
((E^2 + x^3 + 2*x^2*Log[5] + x*Log[5]^2)/x)*(-2*E^2*x^2 + 8*x^3 + 4*x^5 + 4*x^4*Log[5]),x]

[Out]

625^x*E^((2*E^2)/x + 2*x^2 + 2*Log[5]^2)*x^2 + x^6 + (4*25^x*E^(E^2/x + x^2 + Log[5]^2)*x^4*Log[5])/Log[25]

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.63

method	result	size
risch	\(25^{2 x} {\mathrm e}^{\frac {2 x \ln \left (5\right )^{2}+2 x^{3}+2 \,{\mathrm e}^{2}}{x}} x^{2}+2 \,25^{x} {\mathrm e}^{\frac {x \ln \left (5\right )^{2}+x^{3}+{\mathrm e}^{2}}{x}} x^{4}+x^{6}\)	\(57\)
default	\({\mathrm e}^{\frac {2 x \ln \left (5\right )^{2}+4 x^{2} \ln \left (5\right )+2 \,{\mathrm e}^{2}+2 x^{3}}{x}} x^{2}+2 \,{\mathrm e}^{\frac {x \ln \left (5\right )^{2}+2 x^{2} \ln \left (5\right )+{\mathrm e}^{2}+x^{3}}{x}} x^{4}+x^{6}\)	\(64\)
norman	\({\mathrm e}^{\frac {2 x \ln \left (5\right )^{2}+4 x^{2} \ln \left (5\right )+2 \,{\mathrm e}^{2}+2 x^{3}}{x}} x^{2}+2 \,{\mathrm e}^{\frac {x \ln \left (5\right )^{2}+2 x^{2} \ln \left (5\right )+{\mathrm e}^{2}+x^{3}}{x}} x^{4}+x^{6}\)	\(64\)
parallelrisch	\({\mathrm e}^{\frac {2 x \ln \left (5\right )^{2}+4 x^{2} \ln \left (5\right )+2 \,{\mathrm e}^{2}+2 x^{3}}{x}} x^{2}+2 \,{\mathrm e}^{\frac {x \ln \left (5\right )^{2}+2 x^{2} \ln \left (5\right )+{\mathrm e}^{2}+x^{3}}{x}} x^{4}+x^{6}\)	\(64\)
parts	\({\mathrm e}^{\frac {2 x \ln \left (5\right )^{2}+4 x^{2} \ln \left (5\right )+2 \,{\mathrm e}^{2}+2 x^{3}}{x}} x^{2}+2 \,{\mathrm e}^{\frac {x \ln \left (5\right )^{2}+2 x^{2} \ln \left (5\right )+{\mathrm e}^{2}+x^{3}}{x}} x^{4}+x^{6}\)	\(64\)

[In]

int((4*x^2*ln(5)-2*exp(2)+4*x^3+2*x)*exp((x*ln(5)^2+2*x^2*ln(5)+exp(2)+x^3)/x)^2+(4*x^4*ln(5)-2*x^2*exp(2)+4*x
^5+8*x^3)*exp((x*ln(5)^2+2*x^2*ln(5)+exp(2)+x^3)/x)+6*x^5,x,method=_RETURNVERBOSE)

[Out]

(25^x)^2*exp(2*(x*ln(5)^2+x^3+exp(2))/x)*x^2+2*25^x*exp((x*ln(5)^2+x^3+exp(2))/x)*x^4+x^6

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.77 \[ \int \left (6 x^5+e^{\frac {2 \left (e^2+x^3+2 x^2 \log (5)+x \log ^2(5)\right )}{x}} \left (-2 e^2+2 x+4 x^3+4 x^2 \log (5)\right )+e^{\frac {e^2+x^3+2 x^2 \log (5)+x \log ^2(5)}{x}} \left (-2 e^2 x^2+8 x^3+4 x^5+4 x^4 \log (5)\right )\right ) \, dx=x^{6} + 2 \, x^{4} e^{\left (\frac {x^{3} + 2 \, x^{2} \log \left (5\right ) + x \log \left (5\right )^{2} + e^{2}}{x}\right )} + x^{2} e^{\left (\frac {2 \, {\left (x^{3} + 2 \, x^{2} \log \left (5\right ) + x \log \left (5\right )^{2} + e^{2}\right )}}{x}\right )} \]

[In]

integrate((4*x^2*log(5)-2*exp(2)+4*x^3+2*x)*exp((x*log(5)^2+2*x^2*log(5)+exp(2)+x^3)/x)^2+(4*x^4*log(5)-2*x^2*
exp(2)+4*x^5+8*x^3)*exp((x*log(5)^2+2*x^2*log(5)+exp(2)+x^3)/x)+6*x^5,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (27) = 54\).

Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.89 \[ \int \left (6 x^5+e^{\frac {2 \left (e^2+x^3+2 x^2 \log (5)+x \log ^2(5)\right )}{x}} \left (-2 e^2+2 x+4 x^3+4 x^2 \log (5)\right )+e^{\frac {e^2+x^3+2 x^2 \log (5)+x \log ^2(5)}{x}} \left (-2 e^2 x^2+8 x^3+4 x^5+4 x^4 \log (5)\right )\right ) \, dx=x^{6} + 2 x^{4} e^{\frac {x^{3} + 2 x^{2} \log {\left (5 \right )} + x \log {\left (5 \right )}^{2} + e^{2}}{x}} + x^{2} e^{\frac {2 \left (x^{3} + 2 x^{2} \log {\left (5 \right )} + x \log {\left (5 \right )}^{2} + e^{2}\right )}{x}} \]

[In]

integrate((4*x**2*ln(5)-2*exp(2)+4*x**3+2*x)*exp((x*ln(5)**2+2*x**2*ln(5)+exp(2)+x**3)/x)**2+(4*x**4*ln(5)-2*x
**2*exp(2)+4*x**5+8*x**3)*exp((x*ln(5)**2+2*x**2*ln(5)+exp(2)+x**3)/x)+6*x**5,x)

[Out]

x**6 + 2*x**4*exp((x**3 + 2*x**2*log(5) + x*log(5)**2 + exp(2))/x) + x**2*exp(2*(x**3 + 2*x**2*log(5) + x*log(
5)**2 + exp(2))/x)

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66 \[ \int \left (6 x^5+e^{\frac {2 \left (e^2+x^3+2 x^2 \log (5)+x \log ^2(5)\right )}{x}} \left (-2 e^2+2 x+4 x^3+4 x^2 \log (5)\right )+e^{\frac {e^2+x^3+2 x^2 \log (5)+x \log ^2(5)}{x}} \left (-2 e^2 x^2+8 x^3+4 x^5+4 x^4 \log (5)\right )\right ) \, dx=x^{6} + 2 \, x^{4} e^{\left (x^{2} + 2 \, x \log \left (5\right ) + \log \left (5\right )^{2} + \frac {e^{2}}{x}\right )} + x^{2} e^{\left (2 \, x^{2} + 4 \, x \log \left (5\right ) + 2 \, \log \left (5\right )^{2} + \frac {2 \, e^{2}}{x}\right )} \]

[In]

integrate((4*x^2*log(5)-2*exp(2)+4*x^3+2*x)*exp((x*log(5)^2+2*x^2*log(5)+exp(2)+x^3)/x)^2+(4*x^4*log(5)-2*x^2*
exp(2)+4*x^5+8*x^3)*exp((x*log(5)^2+2*x^2*log(5)+exp(2)+x^3)/x)+6*x^5,x, algorithm="maxima")

[Out]

x^6 + 2*x^4*e^(x^2 + 2*x*log(5) + log(5)^2 + e^2/x) + x^2*e^(2*x^2 + 4*x*log(5) + 2*log(5)^2 + 2*e^2/x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.77 \[ \int \left (6 x^5+e^{\frac {2 \left (e^2+x^3+2 x^2 \log (5)+x \log ^2(5)\right )}{x}} \left (-2 e^2+2 x+4 x^3+4 x^2 \log (5)\right )+e^{\frac {e^2+x^3+2 x^2 \log (5)+x \log ^2(5)}{x}} \left (-2 e^2 x^2+8 x^3+4 x^5+4 x^4 \log (5)\right )\right ) \, dx=x^{6} + 2 \, x^{4} e^{\left (\frac {x^{3} + 2 \, x^{2} \log \left (5\right ) + x \log \left (5\right )^{2} + e^{2}}{x}\right )} + x^{2} e^{\left (\frac {2 \, {\left (x^{3} + 2 \, x^{2} \log \left (5\right ) + x \log \left (5\right )^{2} + e^{2}\right )}}{x}\right )} \]

[In]

integrate((4*x^2*log(5)-2*exp(2)+4*x^3+2*x)*exp((x*log(5)^2+2*x^2*log(5)+exp(2)+x^3)/x)^2+(4*x^4*log(5)-2*x^2*
exp(2)+4*x^5+8*x^3)*exp((x*log(5)^2+2*x^2*log(5)+exp(2)+x^3)/x)+6*x^5,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \left (6 x^5+e^{\frac {2 \left (e^2+x^3+2 x^2 \log (5)+x \log ^2(5)\right )}{x}} \left (-2 e^2+2 x+4 x^3+4 x^2 \log (5)\right )+e^{\frac {e^2+x^3+2 x^2 \log (5)+x \log ^2(5)}{x}} \left (-2 e^2 x^2+8 x^3+4 x^5+4 x^4 \log (5)\right )\right ) \, dx=x^2\,{\left (5^{2\,x}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^2}{x}+{\ln \left (5\right )}^2+x^2}+x^2\right )}^2 \]

[In]

int(exp((2*(exp(2) + x*log(5)^2 + 2*x^2*log(5) + x^3))/x)*(2*x - 2*exp(2) + 4*x^2*log(5) + 4*x^3) + 6*x^5 + ex
p((exp(2) + x*log(5)^2 + 2*x^2*log(5) + x^3)/x)*(4*x^4*log(5) - 2*x^2*exp(2) + 8*x^3 + 4*x^5),x)

[Out]

x^2*(5^(2*x)*exp(exp(2)/x + log(5)^2 + x^2) + x^2)^2