Integrand size = 76, antiderivative size = 18 \[ \int \frac {18 x-24 x^2+8 x^3+e^{\frac {x}{-3+2 x+\log (25)}} (-3+\log (25))+\left (-12 x+8 x^2\right ) \log (25)+2 x \log ^2(25)}{9-12 x+4 x^2+(-6+4 x) \log (25)+\log ^2(25)} \, dx=4+e^{\frac {x}{-3+2 x+\log (25)}}+x^2 \]
Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).
Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {18 x-24 x^2+8 x^3+e^{\frac {x}{-3+2 x+\log (25)}} (-3+\log (25))+\left (-12 x+8 x^2\right ) \log (25)+2 x \log ^2(25)}{9-12 x+4 x^2+(-6+4 x) \log (25)+\log ^2(25)} \, dx=5^{\frac {1}{3-2 x-\log (25)}} e^{\frac {1}{2}+\frac {3}{2 (-3+2 x+\log (25))}}+x^2 \]
Integrate[(18*x - 24*x^2 + 8*x^3 + E^(x/(-3 + 2*x + Log[25]))*(-3 + Log[25 ]) + (-12*x + 8*x^2)*Log[25] + 2*x*Log[25]^2)/(9 - 12*x + 4*x^2 + (-6 + 4* x)*Log[25] + Log[25]^2),x]
Time = 0.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6, 2007, 7239, 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.
| \(\displaystyle \int \frac {8 x^3-24 x^2+\left (8 x^2-12 x\right ) \log (25)+18 x+2 x \log ^2(25)+(\log (25)-3) e^{\frac {x}{2 x-3+\log (25)}}}{4 x^2-12 x+(4 x-6) \log (25)+9+\log ^2(25)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {8 x^3-24 x^2+\left (8 x^2-12 x\right ) \log (25)+x \left (18+2 \log ^2(25)\right )+(\log (25)-3) e^{\frac {x}{2 x-3+\log (25)}}}{4 x^2-12 x+(4 x-6) \log (25)+9+\log ^2(25)}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {8 x^3-24 x^2+\left (8 x^2-12 x\right ) \log (25)+x \left (18+2 \log ^2(25)\right )+(\log (25)-3) e^{\frac {x}{2 x-3+\log (25)}}}{(2 x-3+\log (25))^2}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \left (2 x+\frac {(\log (25)-3) e^{\frac {x}{2 x-3+\log (25)}}}{(2 x-3+\log (25))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^2+e^{\frac {1}{2}-\frac {3-\log (25)}{2 (-2 x+3-\log (25))}}\) |
Int[(18*x - 24*x^2 + 8*x^3 + E^(x/(-3 + 2*x + Log[25]))*(-3 + Log[25]) + ( -12*x + 8*x^2)*Log[25] + 2*x*Log[25]^2)/(9 - 12*x + 4*x^2 + (-6 + 4*x)*Log [25] + Log[25]^2),x]
3.13.80.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 4.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(x^{2}+{\mathrm e}^{\frac {x}{2 \ln \left (5\right )+2 x -3}}\) | \(19\) |
| parallelrisch | \(-6 \ln \left (5\right )^{2}+x^{2}+18 \ln \left (5\right )+{\mathrm e}^{\frac {x}{2 \ln \left (5\right )+2 x -3}}-\frac {27}{2}\) | \(30\) |
| parts | \(-\frac {\left (2 \ln \left (5\right )-3\right ) \left (-4 \ln \left (5\right )+6\right ) {\mathrm e}^{\frac {1}{2}+\frac {-\ln \left (5\right )+\frac {3}{2}}{2 \ln \left (5\right )+2 x -3}}}{2 \left (4 \ln \left (5\right )^{2}-12 \ln \left (5\right )+9\right )}+x^{2}\) | \(54\) |
| norman | \(\frac {\left (2 \ln \left (5\right )-3\right ) x^{2}+\left (2 \ln \left (5\right )-3\right ) {\mathrm e}^{\frac {x}{2 \ln \left (5\right )+2 x -3}}+2 x^{3}+2 \,{\mathrm e}^{\frac {x}{2 \ln \left (5\right )+2 x -3}} x}{2 \ln \left (5\right )+2 x -3}\) | \(67\) |
| derivativedivides | \(-\frac {\left (-4 \ln \left (5\right )+6\right ) \left (-\frac {27 \left (2 \ln \left (5\right )+2 x -3\right )^{2}}{4 \left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )^{2}}-\frac {27 \left (2 \ln \left (5\right )+2 x -3\right )}{\left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )}-\frac {3 \,{\mathrm e}^{\frac {1}{2}+\frac {-\ln \left (5\right )+\frac {3}{2}}{2 \ln \left (5\right )+2 x -3}}}{4 \ln \left (5\right )^{2}-12 \ln \left (5\right )+9}+\frac {27 \ln \left (5\right ) \left (2 \ln \left (5\right )+2 x -3\right )^{2}}{2 \left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )^{2}}-\frac {9 \ln \left (5\right )^{2} \left (2 \ln \left (5\right )+2 x -3\right )^{2}}{\left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )^{2}}+\frac {2 \ln \left (5\right )^{3} \left (2 \ln \left (5\right )+2 x -3\right )^{2}}{\left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )^{2}}+\frac {54 \ln \left (5\right ) \left (2 \ln \left (5\right )+2 x -3\right )}{\left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )}-\frac {36 \ln \left (5\right )^{2} \left (2 \ln \left (5\right )+2 x -3\right )}{\left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )}+\frac {8 \ln \left (5\right )^{3} \left (2 \ln \left (5\right )+2 x -3\right )}{\left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )}+\frac {2 \ln \left (5\right ) {\mathrm e}^{\frac {1}{2}+\frac {-\ln \left (5\right )+\frac {3}{2}}{2 \ln \left (5\right )+2 x -3}}}{4 \ln \left (5\right )^{2}-12 \ln \left (5\right )+9}\right )}{2}\) | \(378\) |
| default | \(-\frac {\left (-4 \ln \left (5\right )+6\right ) \left (-\frac {27 \left (2 \ln \left (5\right )+2 x -3\right )^{2}}{4 \left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )^{2}}-\frac {27 \left (2 \ln \left (5\right )+2 x -3\right )}{\left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )}-\frac {3 \,{\mathrm e}^{\frac {1}{2}+\frac {-\ln \left (5\right )+\frac {3}{2}}{2 \ln \left (5\right )+2 x -3}}}{4 \ln \left (5\right )^{2}-12 \ln \left (5\right )+9}+\frac {27 \ln \left (5\right ) \left (2 \ln \left (5\right )+2 x -3\right )^{2}}{2 \left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )^{2}}-\frac {9 \ln \left (5\right )^{2} \left (2 \ln \left (5\right )+2 x -3\right )^{2}}{\left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )^{2}}+\frac {2 \ln \left (5\right )^{3} \left (2 \ln \left (5\right )+2 x -3\right )^{2}}{\left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )^{2}}+\frac {54 \ln \left (5\right ) \left (2 \ln \left (5\right )+2 x -3\right )}{\left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )}-\frac {36 \ln \left (5\right )^{2} \left (2 \ln \left (5\right )+2 x -3\right )}{\left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )}+\frac {8 \ln \left (5\right )^{3} \left (2 \ln \left (5\right )+2 x -3\right )}{\left (16 \ln \left (5\right )^{2}-48 \ln \left (5\right )+36\right ) \left (-\ln \left (5\right )+\frac {3}{2}\right )}+\frac {2 \ln \left (5\right ) {\mathrm e}^{\frac {1}{2}+\frac {-\ln \left (5\right )+\frac {3}{2}}{2 \ln \left (5\right )+2 x -3}}}{4 \ln \left (5\right )^{2}-12 \ln \left (5\right )+9}\right )}{2}\) | \(378\) |
int(((2*ln(5)-3)*exp(x/(2*ln(5)+2*x-3))+8*x*ln(5)^2+2*(8*x^2-12*x)*ln(5)+8 *x^3-24*x^2+18*x)/(4*ln(5)^2+2*(4*x-6)*ln(5)+4*x^2-12*x+9),x,method=_RETUR NVERBOSE)
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {18 x-24 x^2+8 x^3+e^{\frac {x}{-3+2 x+\log (25)}} (-3+\log (25))+\left (-12 x+8 x^2\right ) \log (25)+2 x \log ^2(25)}{9-12 x+4 x^2+(-6+4 x) \log (25)+\log ^2(25)} \, dx=x^{2} + e^{\left (\frac {x}{2 \, x + 2 \, \log \left (5\right ) - 3}\right )} \]
integrate(((2*log(5)-3)*exp(x/(2*log(5)+2*x-3))+8*x*log(5)^2+2*(8*x^2-12*x )*log(5)+8*x^3-24*x^2+18*x)/(4*log(5)^2+2*(4*x-6)*log(5)+4*x^2-12*x+9),x, algorithm=\
Time = 0.14 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {18 x-24 x^2+8 x^3+e^{\frac {x}{-3+2 x+\log (25)}} (-3+\log (25))+\left (-12 x+8 x^2\right ) \log (25)+2 x \log ^2(25)}{9-12 x+4 x^2+(-6+4 x) \log (25)+\log ^2(25)} \, dx=x^{2} + e^{\frac {x}{2 x - 3 + 2 \log {\left (5 \right )}}} \]
integrate(((2*ln(5)-3)*exp(x/(2*ln(5)+2*x-3))+8*x*ln(5)**2+2*(8*x**2-12*x) *ln(5)+8*x**3-24*x**2+18*x)/(4*ln(5)**2+2*(4*x-6)*ln(5)+4*x**2-12*x+9),x)
Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 347, normalized size of antiderivative = 19.28 \[ \int \frac {18 x-24 x^2+8 x^3+e^{\frac {x}{-3+2 x+\log (25)}} (-3+\log (25))+\left (-12 x+8 x^2\right ) \log (25)+2 x \log ^2(25)}{9-12 x+4 x^2+(-6+4 x) \log (25)+\log ^2(25)} \, dx=2 \, {\left (\frac {2 \, \log \left (5\right ) - 3}{2 \, x + 2 \, \log \left (5\right ) - 3} + \log \left (2 \, x + 2 \, \log \left (5\right ) - 3\right )\right )} \log \left (5\right )^{2} + x^{2} - 2 \, x {\left (2 \, \log \left (5\right ) - 3\right )} - 2 \, {\left (2 \, {\left (2 \, \log \left (5\right ) - 3\right )} \log \left (2 \, x + 2 \, \log \left (5\right ) - 3\right ) - 2 \, x + \frac {4 \, \log \left (5\right )^{2} - 12 \, \log \left (5\right ) + 9}{2 \, x + 2 \, \log \left (5\right ) - 3}\right )} \log \left (5\right ) - 6 \, {\left (\frac {2 \, \log \left (5\right ) - 3}{2 \, x + 2 \, \log \left (5\right ) - 3} + \log \left (2 \, x + 2 \, \log \left (5\right ) - 3\right )\right )} \log \left (5\right ) + \frac {3}{2} \, {\left (4 \, \log \left (5\right )^{2} - 12 \, \log \left (5\right ) + 9\right )} \log \left (2 \, x + 2 \, \log \left (5\right ) - 3\right ) + 6 \, {\left (2 \, \log \left (5\right ) - 3\right )} \log \left (2 \, x + 2 \, \log \left (5\right ) - 3\right ) - 6 \, x + \frac {2 \, e^{\left (-\frac {\log \left (5\right )}{2 \, x + 2 \, \log \left (5\right ) - 3} + \frac {3}{2 \, {\left (2 \, x + 2 \, \log \left (5\right ) - 3\right )}} + \frac {1}{2}\right )} \log \left (5\right )}{2 \, \log \left (5\right ) - 3} + \frac {8 \, \log \left (5\right )^{3} - 36 \, \log \left (5\right )^{2} + 54 \, \log \left (5\right ) - 27}{2 \, {\left (2 \, x + 2 \, \log \left (5\right ) - 3\right )}} + \frac {3 \, {\left (4 \, \log \left (5\right )^{2} - 12 \, \log \left (5\right ) + 9\right )}}{2 \, x + 2 \, \log \left (5\right ) - 3} + \frac {9 \, {\left (2 \, \log \left (5\right ) - 3\right )}}{2 \, {\left (2 \, x + 2 \, \log \left (5\right ) - 3\right )}} - \frac {3 \, e^{\left (-\frac {\log \left (5\right )}{2 \, x + 2 \, \log \left (5\right ) - 3} + \frac {3}{2 \, {\left (2 \, x + 2 \, \log \left (5\right ) - 3\right )}} + \frac {1}{2}\right )}}{2 \, \log \left (5\right ) - 3} + \frac {9}{2} \, \log \left (2 \, x + 2 \, \log \left (5\right ) - 3\right ) \]
integrate(((2*log(5)-3)*exp(x/(2*log(5)+2*x-3))+8*x*log(5)^2+2*(8*x^2-12*x )*log(5)+8*x^3-24*x^2+18*x)/(4*log(5)^2+2*(4*x-6)*log(5)+4*x^2-12*x+9),x, algorithm=\
2*((2*log(5) - 3)/(2*x + 2*log(5) - 3) + log(2*x + 2*log(5) - 3))*log(5)^2 + x^2 - 2*x*(2*log(5) - 3) - 2*(2*(2*log(5) - 3)*log(2*x + 2*log(5) - 3) - 2*x + (4*log(5)^2 - 12*log(5) + 9)/(2*x + 2*log(5) - 3))*log(5) - 6*((2* log(5) - 3)/(2*x + 2*log(5) - 3) + log(2*x + 2*log(5) - 3))*log(5) + 3/2*( 4*log(5)^2 - 12*log(5) + 9)*log(2*x + 2*log(5) - 3) + 6*(2*log(5) - 3)*log (2*x + 2*log(5) - 3) - 6*x + 2*e^(-log(5)/(2*x + 2*log(5) - 3) + 3/2/(2*x + 2*log(5) - 3) + 1/2)*log(5)/(2*log(5) - 3) + 1/2*(8*log(5)^3 - 36*log(5) ^2 + 54*log(5) - 27)/(2*x + 2*log(5) - 3) + 3*(4*log(5)^2 - 12*log(5) + 9) /(2*x + 2*log(5) - 3) + 9/2*(2*log(5) - 3)/(2*x + 2*log(5) - 3) - 3*e^(-lo g(5)/(2*x + 2*log(5) - 3) + 3/2/(2*x + 2*log(5) - 3) + 1/2)/(2*log(5) - 3) + 9/2*log(2*x + 2*log(5) - 3)
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (19) = 38\).
Time = 0.38 (sec) , antiderivative size = 282, normalized size of antiderivative = 15.67 \[ \int \frac {18 x-24 x^2+8 x^3+e^{\frac {x}{-3+2 x+\log (25)}} (-3+\log (25))+\left (-12 x+8 x^2\right ) \log (25)+2 x \log ^2(25)}{9-12 x+4 x^2+(-6+4 x) \log (25)+\log ^2(25)} \, dx=-\frac {\frac {32 \, x \log \left (5\right )^{3}}{2 \, x + 2 \, \log \left (5\right ) - 3} - 8 \, \log \left (5\right )^{3} - \frac {32 \, x e^{\left (\frac {x}{2 \, x + 2 \, \log \left (5\right ) - 3}\right )} \log \left (5\right )}{2 \, x + 2 \, \log \left (5\right ) - 3} + \frac {32 \, x^{2} e^{\left (\frac {x}{2 \, x + 2 \, \log \left (5\right ) - 3}\right )} \log \left (5\right )}{{\left (2 \, x + 2 \, \log \left (5\right ) - 3\right )}^{2}} + 8 \, e^{\left (\frac {x}{2 \, x + 2 \, \log \left (5\right ) - 3}\right )} \log \left (5\right ) - \frac {144 \, x \log \left (5\right )^{2}}{2 \, x + 2 \, \log \left (5\right ) - 3} + 36 \, \log \left (5\right )^{2} + \frac {48 \, x e^{\left (\frac {x}{2 \, x + 2 \, \log \left (5\right ) - 3}\right )}}{2 \, x + 2 \, \log \left (5\right ) - 3} - \frac {48 \, x^{2} e^{\left (\frac {x}{2 \, x + 2 \, \log \left (5\right ) - 3}\right )}}{{\left (2 \, x + 2 \, \log \left (5\right ) - 3\right )}^{2}} + \frac {216 \, x \log \left (5\right )}{2 \, x + 2 \, \log \left (5\right ) - 3} - \frac {108 \, x}{2 \, x + 2 \, \log \left (5\right ) - 3} - 12 \, e^{\left (\frac {x}{2 \, x + 2 \, \log \left (5\right ) - 3}\right )} - 54 \, \log \left (5\right ) + 27}{4 \, {\left (\frac {4 \, x}{2 \, x + 2 \, \log \left (5\right ) - 3} - \frac {4 \, x^{2}}{{\left (2 \, x + 2 \, \log \left (5\right ) - 3\right )}^{2}} - 1\right )} {\left (2 \, \log \left (5\right ) - 3\right )}} \]
integrate(((2*log(5)-3)*exp(x/(2*log(5)+2*x-3))+8*x*log(5)^2+2*(8*x^2-12*x )*log(5)+8*x^3-24*x^2+18*x)/(4*log(5)^2+2*(4*x-6)*log(5)+4*x^2-12*x+9),x, algorithm=\
-1/4*(32*x*log(5)^3/(2*x + 2*log(5) - 3) - 8*log(5)^3 - 32*x*e^(x/(2*x + 2 *log(5) - 3))*log(5)/(2*x + 2*log(5) - 3) + 32*x^2*e^(x/(2*x + 2*log(5) - 3))*log(5)/(2*x + 2*log(5) - 3)^2 + 8*e^(x/(2*x + 2*log(5) - 3))*log(5) - 144*x*log(5)^2/(2*x + 2*log(5) - 3) + 36*log(5)^2 + 48*x*e^(x/(2*x + 2*log (5) - 3))/(2*x + 2*log(5) - 3) - 48*x^2*e^(x/(2*x + 2*log(5) - 3))/(2*x + 2*log(5) - 3)^2 + 216*x*log(5)/(2*x + 2*log(5) - 3) - 108*x/(2*x + 2*log(5 ) - 3) - 12*e^(x/(2*x + 2*log(5) - 3)) - 54*log(5) + 27)/((4*x/(2*x + 2*lo g(5) - 3) - 4*x^2/(2*x + 2*log(5) - 3)^2 - 1)*(2*log(5) - 3))
Timed out. \[ \int \frac {18 x-24 x^2+8 x^3+e^{\frac {x}{-3+2 x+\log (25)}} (-3+\log (25))+\left (-12 x+8 x^2\right ) \log (25)+2 x \log ^2(25)}{9-12 x+4 x^2+(-6+4 x) \log (25)+\log ^2(25)} \, dx=\int \frac {18\,x+{\mathrm {e}}^{\frac {x}{2\,x+2\,\ln \left (5\right )-3}}\,\left (2\,\ln \left (5\right )-3\right )-2\,\ln \left (5\right )\,\left (12\,x-8\,x^2\right )+8\,x\,{\ln \left (5\right )}^2-24\,x^2+8\,x^3}{2\,\ln \left (5\right )\,\left (4\,x-6\right )-12\,x+4\,{\ln \left (5\right )}^2+4\,x^2+9} \,d x \]
int((18*x + exp(x/(2*x + 2*log(5) - 3))*(2*log(5) - 3) - 2*log(5)*(12*x - 8*x^2) + 8*x*log(5)^2 - 24*x^2 + 8*x^3)/(2*log(5)*(4*x - 6) - 12*x + 4*log (5)^2 + 4*x^2 + 9),x)