Integrand size = 65, antiderivative size = 16 \begin {dmath*} \int \frac {e^8 \left (12288 e x^2+12800 x^3\right )+e^8 \left (1536 e x^2+1568 x^3\right ) \log (e+x)+e^8 \left (48 e x^2+48 x^3\right ) \log ^2(e+x)}{e+x} \, dx=16 e^8 x^3 (16+\log (e+x))^2 \end {dmath*}
Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \begin {dmath*} \int \frac {e^8 \left (12288 e x^2+12800 x^3\right )+e^8 \left (1536 e x^2+1568 x^3\right ) \log (e+x)+e^8 \left (48 e x^2+48 x^3\right ) \log ^2(e+x)}{e+x} \, dx=16 e^8 \left (256 x^3+32 x^3 \log (e+x)+x^3 \log ^2(e+x)\right ) \end {dmath*}
Integrate[(E^8*(12288*E*x^2 + 12800*x^3) + E^8*(1536*E*x^2 + 1568*x^3)*Log [E + x] + E^8*(48*E*x^2 + 48*x^3)*Log[E + x]^2)/(E + x),x]
Leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(16)=32\).
Time = 0.74 (sec) , antiderivative size = 127, normalized size of antiderivative = 7.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {7239, 27, 7293, 2009}
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.
\(\displaystyle \int \frac {e^8 \left (12800 x^3+12288 e x^2\right )+e^8 \left (48 x^3+48 e x^2\right ) \log ^2(x+e)+e^8 \left (1568 x^3+1536 e x^2\right ) \log (x+e)}{x+e} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {16 e^8 x^2 (\log (x+e)+16) (50 x+3 (x+e) \log (x+e)+48 e)}{x+e}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 16 e^8 \int \frac {x^2 (\log (x+e)+16) (50 x+3 (x+e) \log (x+e)+48 e)}{x+e}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 16 e^8 \int \left (3 \log ^2(x+e) x^2+\frac {32 (25 x+24 e) x^2}{x+e}+\frac {2 (49 x+48 e) \log (x+e) x^2}{x+e}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 16 e^8 \left (\frac {2302 x^3}{9}+x^3 \log ^2(x+e)+\frac {98}{3} x^3 \log (x+e)+\frac {5 e x^2}{6}-e x^2 \log (x+e)+\frac {7 e^2 x}{3}+\frac {2}{9} (x+e)^3-\frac {3}{2} e (x+e)^2-\frac {2}{3} (x+e)^3 \log (x+e)+3 e (x+e)^2 \log (x+e)-4 e^2 (x+e) \log (x+e)+\frac {5}{3} e^3 \log (x+e)\right )\) |
Int[(E^8*(12288*E*x^2 + 12800*x^3) + E^8*(1536*E*x^2 + 1568*x^3)*Log[E + x ] + E^8*(48*E*x^2 + 48*x^3)*Log[E + x]^2)/(E + x),x]
16*E^8*((7*E^2*x)/3 + (5*E*x^2)/6 + (2302*x^3)/9 - (3*E*(E + x)^2)/2 + (2* (E + x)^3)/9 + (5*E^3*Log[E + x])/3 - E*x^2*Log[E + x] + (98*x^3*Log[E + x ])/3 - 4*E^2*(E + x)*Log[E + x] + 3*E*(E + x)^2*Log[E + x] - (2*(E + x)^3* Log[E + x])/3 + x^3*Log[E + x]^2)
3.9.2.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.92 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.19
method | result | size |
risch | \(16 x^{3} {\mathrm e}^{8} \ln \left (x +{\mathrm e}\right )^{2}+512 x^{3} {\mathrm e}^{8} \ln \left (x +{\mathrm e}\right )+4096 x^{3} {\mathrm e}^{8}\) | \(35\) |
norman | \(16 x^{3} {\mathrm e}^{8} \ln \left (x +{\mathrm e}\right )^{2}+512 x^{3} {\mathrm e}^{8} \ln \left (x +{\mathrm e}\right )+4096 x^{3} {\mathrm e}^{8}\) | \(41\) |
parallelrisch | \(16 x^{3} {\mathrm e}^{8} \ln \left (x +{\mathrm e}\right )^{2}+512 x^{3} {\mathrm e}^{8} \ln \left (x +{\mathrm e}\right )+4096 x^{3} {\mathrm e}^{8}\) | \(41\) |
parts | \(48 \,{\mathrm e}^{8} \left (\frac {\ln \left (x +{\mathrm e}\right )^{2} \left (x +{\mathrm e}\right )^{3}}{3}-\frac {2 \ln \left (x +{\mathrm e}\right ) \left (x +{\mathrm e}\right )^{3}}{9}+\frac {2 \left (x +{\mathrm e}\right )^{3}}{27}-2 \,{\mathrm e} \left (\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )^{2}}{2}-\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )}{2}+\frac {\left (x +{\mathrm e}\right )^{2}}{4}\right )+{\mathrm e}^{2} \left (\left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )^{2}-2 \left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )+2 x +2 \,{\mathrm e}\right )\right )+512 \,{\mathrm e}^{8} \left (\frac {25 x^{3}}{3}-\frac {x^{2} {\mathrm e}}{2}+{\mathrm e}^{2} x -{\mathrm e}^{3} \ln \left (x +{\mathrm e}\right )\right )+32 \,{\mathrm e}^{8} \left (-99 \,{\mathrm e} \left (\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )}{2}-\frac {\left (x +{\mathrm e}\right )^{2}}{4}\right )+\frac {49 \ln \left (x +{\mathrm e}\right ) \left (x +{\mathrm e}\right )^{3}}{3}-\frac {49 \left (x +{\mathrm e}\right )^{3}}{9}+51 \,{\mathrm e}^{2} \left (\left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )-x -{\mathrm e}\right )-\frac {{\mathrm e}^{3} \ln \left (x +{\mathrm e}\right )^{2}}{2}\right )\) | \(247\) |
derivativedivides | \(48 \,{\mathrm e}^{8} {\mathrm e}^{2} \left (\left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )^{2}-2 \left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )+2 x +2 \,{\mathrm e}\right )-96 \,{\mathrm e} \,{\mathrm e}^{8} \left (\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )^{2}}{2}-\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )}{2}+\frac {\left (x +{\mathrm e}\right )^{2}}{4}\right )+48 \,{\mathrm e}^{8} \left (\frac {\ln \left (x +{\mathrm e}\right )^{2} \left (x +{\mathrm e}\right )^{3}}{3}-\frac {2 \ln \left (x +{\mathrm e}\right ) \left (x +{\mathrm e}\right )^{3}}{9}+\frac {2 \left (x +{\mathrm e}\right )^{3}}{27}\right )-16 \,{\mathrm e}^{8} {\mathrm e}^{3} \ln \left (x +{\mathrm e}\right )^{2}+1632 \,{\mathrm e}^{8} {\mathrm e}^{2} \left (\left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )-x -{\mathrm e}\right )-3168 \,{\mathrm e} \,{\mathrm e}^{8} \left (\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )}{2}-\frac {\left (x +{\mathrm e}\right )^{2}}{4}\right )+1568 \,{\mathrm e}^{8} \left (\frac {\ln \left (x +{\mathrm e}\right ) \left (x +{\mathrm e}\right )^{3}}{3}-\frac {\left (x +{\mathrm e}\right )^{3}}{9}\right )-512 \,{\mathrm e}^{8} {\mathrm e}^{3} \ln \left (x +{\mathrm e}\right )+13824 \,{\mathrm e}^{8} {\mathrm e}^{2} \left (x +{\mathrm e}\right )-13056 \,{\mathrm e} \,{\mathrm e}^{8} \left (x +{\mathrm e}\right )^{2}+\frac {12800 \,{\mathrm e}^{8} \left (x +{\mathrm e}\right )^{3}}{3}\) | \(289\) |
default | \(48 \,{\mathrm e}^{8} {\mathrm e}^{2} \left (\left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )^{2}-2 \left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )+2 x +2 \,{\mathrm e}\right )-96 \,{\mathrm e} \,{\mathrm e}^{8} \left (\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )^{2}}{2}-\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )}{2}+\frac {\left (x +{\mathrm e}\right )^{2}}{4}\right )+48 \,{\mathrm e}^{8} \left (\frac {\ln \left (x +{\mathrm e}\right )^{2} \left (x +{\mathrm e}\right )^{3}}{3}-\frac {2 \ln \left (x +{\mathrm e}\right ) \left (x +{\mathrm e}\right )^{3}}{9}+\frac {2 \left (x +{\mathrm e}\right )^{3}}{27}\right )-16 \,{\mathrm e}^{8} {\mathrm e}^{3} \ln \left (x +{\mathrm e}\right )^{2}+1632 \,{\mathrm e}^{8} {\mathrm e}^{2} \left (\left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )-x -{\mathrm e}\right )-3168 \,{\mathrm e} \,{\mathrm e}^{8} \left (\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )}{2}-\frac {\left (x +{\mathrm e}\right )^{2}}{4}\right )+1568 \,{\mathrm e}^{8} \left (\frac {\ln \left (x +{\mathrm e}\right ) \left (x +{\mathrm e}\right )^{3}}{3}-\frac {\left (x +{\mathrm e}\right )^{3}}{9}\right )-512 \,{\mathrm e}^{8} {\mathrm e}^{3} \ln \left (x +{\mathrm e}\right )+13824 \,{\mathrm e}^{8} {\mathrm e}^{2} \left (x +{\mathrm e}\right )-13056 \,{\mathrm e} \,{\mathrm e}^{8} \left (x +{\mathrm e}\right )^{2}+\frac {12800 \,{\mathrm e}^{8} \left (x +{\mathrm e}\right )^{3}}{3}\) | \(289\) |
int(((48*x^2*exp(1)+48*x^3)*exp(4)^2*ln(x+exp(1))^2+(1536*x^2*exp(1)+1568* x^3)*exp(4)^2*ln(x+exp(1))+(12288*x^2*exp(1)+12800*x^3)*exp(4)^2)/(x+exp(1 )),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (16) = 32\).
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.12 \begin {dmath*} \int \frac {e^8 \left (12288 e x^2+12800 x^3\right )+e^8 \left (1536 e x^2+1568 x^3\right ) \log (e+x)+e^8 \left (48 e x^2+48 x^3\right ) \log ^2(e+x)}{e+x} \, dx=16 \, x^{3} e^{8} \log \left (x + e\right )^{2} + 512 \, x^{3} e^{8} \log \left (x + e\right ) + 4096 \, x^{3} e^{8} \end {dmath*}
integrate(((48*x^2*exp(1)+48*x^3)*exp(4)^2*log(x+exp(1))^2+(1536*x^2*exp(1 )+1568*x^3)*exp(4)^2*log(x+exp(1))+(12288*x^2*exp(1)+12800*x^3)*exp(4)^2)/ (x+exp(1)),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.44 \begin {dmath*} \int \frac {e^8 \left (12288 e x^2+12800 x^3\right )+e^8 \left (1536 e x^2+1568 x^3\right ) \log (e+x)+e^8 \left (48 e x^2+48 x^3\right ) \log ^2(e+x)}{e+x} \, dx=16 x^{3} e^{8} \log {\left (x + e \right )}^{2} + 512 x^{3} e^{8} \log {\left (x + e \right )} + 4096 x^{3} e^{8} \end {dmath*}
integrate(((48*x**2*exp(1)+48*x**3)*exp(4)**2*ln(x+exp(1))**2+(1536*x**2*e xp(1)+1568*x**3)*exp(4)**2*ln(x+exp(1))+(12288*x**2*exp(1)+12800*x**3)*exp (4)**2)/(x+exp(1)),x)
Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (16) = 32\).
Time = 0.21 (sec) , antiderivative size = 367, normalized size of antiderivative = 22.94 \begin {dmath*} \int \frac {e^8 \left (12288 e x^2+12800 x^3\right )+e^8 \left (1536 e x^2+1568 x^3\right ) \log (e+x)+e^8 \left (48 e x^2+48 x^3\right ) \log ^2(e+x)}{e+x} \, dx=768 \, {\left (x^{2} - 2 \, x e + 2 \, e^{2} \log \left (x + e\right )\right )} e^{9} \log \left (x + e\right ) + \frac {784}{3} \, {\left (2 \, x^{3} - 3 \, x^{2} e + 6 \, x e^{2} - 6 \, e^{3} \log \left (x + e\right )\right )} e^{8} \log \left (x + e\right ) + 4 \, {\left (4 \, e^{2} \log \left (x + e\right )^{3} + 3 \, {\left (2 \, \log \left (x + e\right )^{2} - 2 \, \log \left (x + e\right ) + 1\right )} {\left (x + e\right )}^{2} - 24 \, {\left (e \log \left (x + e\right )^{2} - 2 \, e \log \left (x + e\right ) + 2 \, e\right )} {\left (x + e\right )}\right )} e^{9} - 384 \, {\left (2 \, e^{2} \log \left (x + e\right )^{2} + x^{2} - 6 \, x e + 6 \, e^{2} \log \left (x + e\right )\right )} e^{9} + 6144 \, {\left (x^{2} - 2 \, x e + 2 \, e^{2} \log \left (x + e\right )\right )} e^{9} + \frac {4}{9} \, {\left (4 \, {\left (9 \, \log \left (x + e\right )^{2} - 6 \, \log \left (x + e\right ) + 2\right )} {\left (x + e\right )}^{3} - 36 \, e^{3} \log \left (x + e\right )^{3} - 81 \, {\left (2 \, e \log \left (x + e\right )^{2} - 2 \, e \log \left (x + e\right ) + e\right )} {\left (x + e\right )}^{2} + 324 \, {\left (e^{2} \log \left (x + e\right )^{2} - 2 \, e^{2} \log \left (x + e\right ) + 2 \, e^{2}\right )} {\left (x + e\right )}\right )} e^{8} - \frac {392}{9} \, {\left (4 \, x^{3} - 15 \, x^{2} e - 18 \, e^{3} \log \left (x + e\right )^{2} + 66 \, x e^{2} - 66 \, e^{3} \log \left (x + e\right )\right )} e^{8} + \frac {6400}{3} \, {\left (2 \, x^{3} - 3 \, x^{2} e + 6 \, x e^{2} - 6 \, e^{3} \log \left (x + e\right )\right )} e^{8} \end {dmath*}
integrate(((48*x^2*exp(1)+48*x^3)*exp(4)^2*log(x+exp(1))^2+(1536*x^2*exp(1 )+1568*x^3)*exp(4)^2*log(x+exp(1))+(12288*x^2*exp(1)+12800*x^3)*exp(4)^2)/ (x+exp(1)),x, algorithm=\
768*(x^2 - 2*x*e + 2*e^2*log(x + e))*e^9*log(x + e) + 784/3*(2*x^3 - 3*x^2 *e + 6*x*e^2 - 6*e^3*log(x + e))*e^8*log(x + e) + 4*(4*e^2*log(x + e)^3 + 3*(2*log(x + e)^2 - 2*log(x + e) + 1)*(x + e)^2 - 24*(e*log(x + e)^2 - 2*e *log(x + e) + 2*e)*(x + e))*e^9 - 384*(2*e^2*log(x + e)^2 + x^2 - 6*x*e + 6*e^2*log(x + e))*e^9 + 6144*(x^2 - 2*x*e + 2*e^2*log(x + e))*e^9 + 4/9*(4 *(9*log(x + e)^2 - 6*log(x + e) + 2)*(x + e)^3 - 36*e^3*log(x + e)^3 - 81* (2*e*log(x + e)^2 - 2*e*log(x + e) + e)*(x + e)^2 + 324*(e^2*log(x + e)^2 - 2*e^2*log(x + e) + 2*e^2)*(x + e))*e^8 - 392/9*(4*x^3 - 15*x^2*e - 18*e^ 3*log(x + e)^2 + 66*x*e^2 - 66*e^3*log(x + e))*e^8 + 6400/3*(2*x^3 - 3*x^2 *e + 6*x*e^2 - 6*e^3*log(x + e))*e^8
Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (16) = 32\).
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.12 \begin {dmath*} \int \frac {e^8 \left (12288 e x^2+12800 x^3\right )+e^8 \left (1536 e x^2+1568 x^3\right ) \log (e+x)+e^8 \left (48 e x^2+48 x^3\right ) \log ^2(e+x)}{e+x} \, dx=16 \, x^{3} e^{8} \log \left (x + e\right )^{2} + 512 \, x^{3} e^{8} \log \left (x + e\right ) + 4096 \, x^{3} e^{8} \end {dmath*}
integrate(((48*x^2*exp(1)+48*x^3)*exp(4)^2*log(x+exp(1))^2+(1536*x^2*exp(1 )+1568*x^3)*exp(4)^2*log(x+exp(1))+(12288*x^2*exp(1)+12800*x^3)*exp(4)^2)/ (x+exp(1)),x, algorithm=\
Time = 0.36 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {e^8 \left (12288 e x^2+12800 x^3\right )+e^8 \left (1536 e x^2+1568 x^3\right ) \log (e+x)+e^8 \left (48 e x^2+48 x^3\right ) \log ^2(e+x)}{e+x} \, dx=16\,x^3\,{\mathrm {e}}^8\,{\left (\ln \left (x+\mathrm {e}\right )+16\right )}^2 \end {dmath*}