Integrand size = 182, antiderivative size = 28 \begin {dmath*} \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=x \left (2+e^{3 e^{\left (25^{\frac {1}{2+\frac {x}{3+x}}}+x\right )^2}}+x\right ) \end {dmath*}
Time = 0.50 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \begin {dmath*} \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=x \left (2+e^{3 e^{\left (5^{\frac {2 (3+x)}{3 (2+x)}}+x\right )^2}}+x\right ) \end {dmath*}
Integrate[(8 + 16*x + 10*x^2 + 2*x^3 + E^(3*E^(25^((2*(3 + x))/(6 + 3*x)) + 2*25^((3 + x)/(6 + 3*x))*x + x^2))*(4 + 4*x + x^2 + E^(25^((2*(3 + x))/( 6 + 3*x)) + 2*25^((3 + x)/(6 + 3*x))*x + x^2)*(24*x^2 + 24*x^3 + 6*x^4 - 2 *25^((2*(3 + x))/(6 + 3*x))*x*Log[25] + 25^((3 + x)/(6 + 3*x))*(24*x + 24* x^2 + 6*x^3 - 2*x^2*Log[25]))))/(4 + 4*x + x^2),x]
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.
\(\displaystyle \int \frac {\exp \left (3 \exp \left (x^2+2\ 25^{\frac {x+3}{3 x+6}} x+25^{\frac {2 (x+3)}{3 x+6}}\right )\right ) \left (\exp \left (x^2+2\ 25^{\frac {x+3}{3 x+6}} x+25^{\frac {2 (x+3)}{3 x+6}}\right ) \left (6 x^4+24 x^3+24 x^2+25^{\frac {x+3}{3 x+6}} \left (6 x^3+24 x^2-2 x^2 \log (25)+24 x\right )-2\ 25^{\frac {2 (x+3)}{3 x+6}} x \log (25)\right )+x^2+4 x+4\right )+2 x^3+10 x^2+16 x+8}{x^2+4 x+4} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {\exp \left (3 \exp \left (x^2+2\ 25^{\frac {x+3}{3 x+6}} x+25^{\frac {2 (x+3)}{3 x+6}}\right )\right ) \left (\exp \left (x^2+2\ 25^{\frac {x+3}{3 x+6}} x+25^{\frac {2 (x+3)}{3 x+6}}\right ) \left (6 x^4+24 x^3+24 x^2+25^{\frac {x+3}{3 x+6}} \left (6 x^3+24 x^2-2 x^2 \log (25)+24 x\right )-2\ 25^{\frac {2 (x+3)}{3 x+6}} x \log (25)\right )+x^2+4 x+4\right )+2 x^3+10 x^2+16 x+8}{(x+2)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \left (x+5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}}\right ) x \exp \left (\left (x+5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}}\right )^2+3 e^{\left (x+5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}}\right )^2}\right ) \left (3 x^2+12 x-5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}} \log (25)+12\right )}{(x+2)^2}+\frac {2 x^3}{(x+2)^2}+\frac {e^{3 e^{\left (x+25^{\frac {x}{3 x+6}+\frac {3}{3 x+6}}\right )^2}} x^2}{(x+2)^2}+\frac {10 x^2}{(x+2)^2}+\frac {4 e^{3 e^{\left (x+25^{\frac {x}{3 x+6}+\frac {3}{3 x+6}}\right )^2}} x}{(x+2)^2}+\frac {16 x}{(x+2)^2}+\frac {4 e^{3 e^{\left (x+25^{\frac {x}{3 x+6}+\frac {3}{3 x+6}}\right )^2}}}{(x+2)^2}+\frac {8}{(x+2)^2}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {2 \left (x+5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}}\right ) x \exp \left (\left (x+5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}}\right )^2+3 e^{\left (x+5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}}\right )^2}\right ) \left (3 x^2+12 x-5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}} \log (25)+12\right )}{(x+2)^2}+\frac {2 x^3}{(x+2)^2}+\frac {e^{3 e^{\left (x+25^{\frac {x}{3 x+6}+\frac {3}{3 x+6}}\right )^2}} x^2}{(x+2)^2}+\frac {10 x^2}{(x+2)^2}+\frac {4 e^{3 e^{\left (x+25^{\frac {x}{3 x+6}+\frac {3}{3 x+6}}\right )^2}} x}{(x+2)^2}+\frac {16 x}{(x+2)^2}+\frac {4 e^{3 e^{\left (x+25^{\frac {x}{3 x+6}+\frac {3}{3 x+6}}\right )^2}}}{(x+2)^2}+\frac {8}{(x+2)^2}\right )dx\) |
Int[(8 + 16*x + 10*x^2 + 2*x^3 + E^(3*E^(25^((2*(3 + x))/(6 + 3*x)) + 2*25 ^((3 + x)/(6 + 3*x))*x + x^2))*(4 + 4*x + x^2 + E^(25^((2*(3 + x))/(6 + 3* x)) + 2*25^((3 + x)/(6 + 3*x))*x + x^2)*(24*x^2 + 24*x^3 + 6*x^4 - 2*25^(( 2*(3 + x))/(6 + 3*x))*x*Log[25] + 25^((3 + x)/(6 + 3*x))*(24*x + 24*x^2 + 6*x^3 - 2*x^2*Log[25]))))/(4 + 4*x + x^2),x]
3.12.6.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 17.63 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07
method | result | size |
risch | \(x^{2}+x \,{\mathrm e}^{3 \,{\mathrm e}^{\left (5^{\frac {2+\frac {2 x}{3}}{2+x}}+x \right )^{2}}}+2 x\) | \(30\) |
parallelrisch | \(x^{2}+{\mathrm e}^{3 \,{\mathrm e}^{{\mathrm e}^{\frac {4 \left (3+x \right ) \ln \left (5\right )}{3 \left (2+x \right )}}+2 x \,{\mathrm e}^{\frac {2 \left (3+x \right ) \ln \left (5\right )}{3 \left (2+x \right )}}+x^{2}}} x +2 x -40\) | \(50\) |
int((((-4*x*ln(5)*exp(2*(3+x)*ln(5)/(6+3*x))^2+(-4*x^2*ln(5)+6*x^3+24*x^2+ 24*x)*exp(2*(3+x)*ln(5)/(6+3*x))+6*x^4+24*x^3+24*x^2)*exp(exp(2*(3+x)*ln(5 )/(6+3*x))^2+2*x*exp(2*(3+x)*ln(5)/(6+3*x))+x^2)+x^2+4*x+4)*exp(3*exp(exp( 2*(3+x)*ln(5)/(6+3*x))^2+2*x*exp(2*(3+x)*ln(5)/(6+3*x))+x^2))+2*x^3+10*x^2 +16*x+8)/(x^2+4*x+4),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \begin {dmath*} \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=x^{2} + x e^{\left (3 \, e^{\left (2 \cdot 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x + x^{2} + 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}}\right )}\right )} + 2 \, x \end {dmath*}
integrate((((-4*x*log(5)*exp(2*(3+x)*log(5)/(6+3*x))^2+(-4*x^2*log(5)+6*x^ 3+24*x^2+24*x)*exp(2*(3+x)*log(5)/(6+3*x))+6*x^4+24*x^3+24*x^2)*exp(exp(2* (3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2)+x^2+4*x+4)*ex p(3*exp(exp(2*(3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2) )+2*x^3+10*x^2+16*x+8)/(x^2+4*x+4),x, algorithm=\
Timed out. \begin {dmath*} \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=\text {Timed out} \end {dmath*}
integrate((((-4*x*ln(5)*exp(2*(3+x)*ln(5)/(6+3*x))**2+(-4*x**2*ln(5)+6*x** 3+24*x**2+24*x)*exp(2*(3+x)*ln(5)/(6+3*x))+6*x**4+24*x**3+24*x**2)*exp(exp (2*(3+x)*ln(5)/(6+3*x))**2+2*x*exp(2*(3+x)*ln(5)/(6+3*x))+x**2)+x**2+4*x+4 )*exp(3*exp(exp(2*(3+x)*ln(5)/(6+3*x))**2+2*x*exp(2*(3+x)*ln(5)/(6+3*x))+x **2))+2*x**3+10*x**2+16*x+8)/(x**2+4*x+4),x)
\begin {dmath*} \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=\int { \frac {2 \, x^{3} + 10 \, x^{2} + {\left (x^{2} + 2 \, {\left (3 \, x^{4} + 12 \, x^{3} - 2 \cdot 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x \log \left (5\right ) + {\left (3 \, x^{3} - 2 \, x^{2} \log \left (5\right ) + 12 \, x^{2} + 12 \, x\right )} 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} + 12 \, x^{2}\right )} e^{\left (2 \cdot 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x + x^{2} + 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}}\right )} + 4 \, x + 4\right )} e^{\left (3 \, e^{\left (2 \cdot 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x + x^{2} + 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}}\right )}\right )} + 16 \, x + 8}{x^{2} + 4 \, x + 4} \,d x } \end {dmath*}
integrate((((-4*x*log(5)*exp(2*(3+x)*log(5)/(6+3*x))^2+(-4*x^2*log(5)+6*x^ 3+24*x^2+24*x)*exp(2*(3+x)*log(5)/(6+3*x))+6*x^4+24*x^3+24*x^2)*exp(exp(2* (3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2)+x^2+4*x+4)*ex p(3*exp(exp(2*(3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2) )+2*x^3+10*x^2+16*x+8)/(x^2+4*x+4),x, algorithm=\
x^2 + x*e^(3*e^(2*5^(2/3)*5^(2/3/(x + 2))*x + x^2 + 5*5^(1/3)*5^(4/3/(x + 2)))) + 2*x - integrate(0, x)
\begin {dmath*} \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=\int { \frac {2 \, x^{3} + 10 \, x^{2} + {\left (x^{2} + 2 \, {\left (3 \, x^{4} + 12 \, x^{3} - 2 \cdot 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x \log \left (5\right ) + {\left (3 \, x^{3} - 2 \, x^{2} \log \left (5\right ) + 12 \, x^{2} + 12 \, x\right )} 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} + 12 \, x^{2}\right )} e^{\left (2 \cdot 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x + x^{2} + 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}}\right )} + 4 \, x + 4\right )} e^{\left (3 \, e^{\left (2 \cdot 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x + x^{2} + 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}}\right )}\right )} + 16 \, x + 8}{x^{2} + 4 \, x + 4} \,d x } \end {dmath*}
integrate((((-4*x*log(5)*exp(2*(3+x)*log(5)/(6+3*x))^2+(-4*x^2*log(5)+6*x^ 3+24*x^2+24*x)*exp(2*(3+x)*log(5)/(6+3*x))+6*x^4+24*x^3+24*x^2)*exp(exp(2* (3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2)+x^2+4*x+4)*ex p(3*exp(exp(2*(3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2) )+2*x^3+10*x^2+16*x+8)/(x^2+4*x+4),x, algorithm=\
integrate((2*x^3 + 10*x^2 + (x^2 + 2*(3*x^4 + 12*x^3 - 2*5^(4/3*(x + 3)/(x + 2))*x*log(5) + (3*x^3 - 2*x^2*log(5) + 12*x^2 + 12*x)*5^(2/3*(x + 3)/(x + 2)) + 12*x^2)*e^(2*5^(2/3*(x + 3)/(x + 2))*x + x^2 + 5^(4/3*(x + 3)/(x + 2))) + 4*x + 4)*e^(3*e^(2*5^(2/3*(x + 3)/(x + 2))*x + x^2 + 5^(4/3*(x + 3)/(x + 2)))) + 16*x + 8)/(x^2 + 4*x + 4), x)
Time = 15.64 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \begin {dmath*} \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=x\,\left (x+{\mathrm {e}}^{3\,{\mathrm {e}}^{2\,5^{\frac {2\,\left (x+3\right )}{3\,\left (x+2\right )}}\,x}\,{\mathrm {e}}^{5^{\frac {4\,\left (x+3\right )}{3\,\left (x+2\right )}}}\,{\mathrm {e}}^{x^2}}+2\right ) \end {dmath*}
int((16*x + exp(3*exp(exp((4*log(5)*(x + 3))/(3*x + 6)) + 2*x*exp((2*log(5 )*(x + 3))/(3*x + 6)) + x^2))*(4*x + exp(exp((4*log(5)*(x + 3))/(3*x + 6)) + 2*x*exp((2*log(5)*(x + 3))/(3*x + 6)) + x^2)*(exp((2*log(5)*(x + 3))/(3 *x + 6))*(24*x - 4*x^2*log(5) + 24*x^2 + 6*x^3) + 24*x^2 + 24*x^3 + 6*x^4 - 4*x*exp((4*log(5)*(x + 3))/(3*x + 6))*log(5)) + x^2 + 4) + 10*x^2 + 2*x^ 3 + 8)/(4*x + x^2 + 4),x)