Integrand size = 202, antiderivative size = 36 \[ \int \frac {e^{-\frac {-50 x+25 x^2}{-56 x+20 x^2+4 x^3+(7+x) \log (5)}} \left (3136 x^2-2240 x^3-48 x^4+160 x^5+16 x^6+\left (-784 x+168 x^2+96 x^3+8 x^4\right ) \log (5)+\left (49+14 x+x^2\right ) \log ^2(5)+\left (400 x^3-400 x^4+100 x^5+\left (350 x-350 x^2-25 x^3\right ) \log (5)\right ) \log (x)\right )}{3136 x^3-2240 x^4-48 x^5+160 x^6+16 x^7+\left (-784 x^2+168 x^3+96 x^4+8 x^5\right ) \log (5)+\left (49 x+14 x^2+x^3\right ) \log ^2(5)} \, dx=e^{-\frac {5 x}{4 \left (1+\frac {2+x}{5}\right ) \left (x+\frac {\log (5)}{4 (-2+x)}\right )}} \log (x) \]
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int \frac {e^{-\frac {-50 x+25 x^2}{-56 x+20 x^2+4 x^3+(7+x) \log (5)}} \left (3136 x^2-2240 x^3-48 x^4+160 x^5+16 x^6+\left (-784 x+168 x^2+96 x^3+8 x^4\right ) \log (5)+\left (49+14 x+x^2\right ) \log ^2(5)+\left (400 x^3-400 x^4+100 x^5+\left (350 x-350 x^2-25 x^3\right ) \log (5)\right ) \log (x)\right )}{3136 x^3-2240 x^4-48 x^5+160 x^6+16 x^7+\left (-784 x^2+168 x^3+96 x^4+8 x^5\right ) \log (5)+\left (49 x+14 x^2+x^3\right ) \log ^2(5)} \, dx=5^{-\frac {25 (-9+x)}{(252+\log (5)) \left (-8 x+4 x^2+\log (5)\right )}} e^{-\frac {1575}{(7+x) (252+\log (5))}} \log (x) \]
Integrate[(3136*x^2 - 2240*x^3 - 48*x^4 + 160*x^5 + 16*x^6 + (-784*x + 168 *x^2 + 96*x^3 + 8*x^4)*Log[5] + (49 + 14*x + x^2)*Log[5]^2 + (400*x^3 - 40 0*x^4 + 100*x^5 + (350*x - 350*x^2 - 25*x^3)*Log[5])*Log[x])/(E^((-50*x + 25*x^2)/(-56*x + 20*x^2 + 4*x^3 + (7 + x)*Log[5]))*(3136*x^3 - 2240*x^4 - 48*x^5 + 160*x^6 + 16*x^7 + (-784*x^2 + 168*x^3 + 96*x^4 + 8*x^5)*Log[5] + (49*x + 14*x^2 + x^3)*Log[5]^2)),x]
Log[x]/(5^((25*(-9 + x))/((252 + Log[5])*(-8*x + 4*x^2 + Log[5])))*E^(1575 /((7 + x)*(252 + Log[5]))))
Leaf count is larger than twice the leaf count of optimal. \(221\) vs. \(2(36)=72\).
Time = 1.02 (sec) , antiderivative size = 221, normalized size of antiderivative = 6.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {2026, 2726}
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.
| \(\displaystyle \int \frac {\left (16 x^6+160 x^5-48 x^4-2240 x^3+3136 x^2+\left (x^2+14 x+49\right ) \log ^2(5)+\left (8 x^4+96 x^3+168 x^2-784 x\right ) \log (5)+\left (100 x^5-400 x^4+400 x^3+\left (-25 x^3-350 x^2+350 x\right ) \log (5)\right ) \log (x)\right ) \exp \left (-\frac {25 x^2-50 x}{4 x^3+20 x^2-56 x+(x+7) \log (5)}\right )}{16 x^7+160 x^6-48 x^5-2240 x^4+3136 x^3+\left (x^3+14 x^2+49 x\right ) \log ^2(5)+\left (8 x^5+96 x^4+168 x^3-784 x^2\right ) \log (5)} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (16 x^6+160 x^5-48 x^4-2240 x^3+3136 x^2+\left (x^2+14 x+49\right ) \log ^2(5)+\left (8 x^4+96 x^3+168 x^2-784 x\right ) \log (5)+\left (100 x^5-400 x^4+400 x^3+\left (-25 x^3-350 x^2+350 x\right ) \log (5)\right ) \log (x)\right ) \exp \left (-\frac {25 x^2-50 x}{4 x^3+20 x^2-56 x+(x+7) \log (5)}\right )}{x \left (16 x^6+160 x^5-8 x^4 (6-\log (5))-32 x^3 (70-\log (125))+x^2 \left (3136+\log ^2(5)+168 \log (5)\right )-14 x (56-\log (5)) \log (5)+49 \log ^2(5)\right )}dx\) |
| \(\Big \downarrow \) 2726 |
| \(\displaystyle \frac {\left (4 x^5-16 x^4+16 x^3+\left (-x^3-14 x^2+14 x\right ) \log (5)\right ) \log (x) \exp \left (-\frac {25 \left (2 x-x^2\right )}{-4 x^3-20 x^2+56 x-(x+7) \log (5)}\right )}{x \left (\frac {\left (2 x-x^2\right ) \left (-12 x^2-40 x+56-\log (5)\right )}{\left (-4 x^3-20 x^2+56 x-(x+7) \log (5)\right )^2}-\frac {2 (1-x)}{-4 x^3-20 x^2+56 x-(x+7) \log (5)}\right ) \left (16 x^6+160 x^5-8 x^4 (6-\log (5))-32 x^3 (70-\log (125))+x^2 \left (3136+\log ^2(5)+168 \log (5)\right )-14 x (56-\log (5)) \log (5)+49 \log ^2(5)\right )}\) |
Int[(3136*x^2 - 2240*x^3 - 48*x^4 + 160*x^5 + 16*x^6 + (-784*x + 168*x^2 + 96*x^3 + 8*x^4)*Log[5] + (49 + 14*x + x^2)*Log[5]^2 + (400*x^3 - 400*x^4 + 100*x^5 + (350*x - 350*x^2 - 25*x^3)*Log[5])*Log[x])/(E^((-50*x + 25*x^2 )/(-56*x + 20*x^2 + 4*x^3 + (7 + x)*Log[5]))*(3136*x^3 - 2240*x^4 - 48*x^5 + 160*x^6 + 16*x^7 + (-784*x^2 + 168*x^3 + 96*x^4 + 8*x^5)*Log[5] + (49*x + 14*x^2 + x^3)*Log[5]^2)),x]
((16*x^3 - 16*x^4 + 4*x^5 + (14*x - 14*x^2 - x^3)*Log[5])*Log[x])/(E^((25* (2*x - x^2))/(56*x - 20*x^2 - 4*x^3 - (7 + x)*Log[5]))*x*(((2*x - x^2)*(56 - 40*x - 12*x^2 - Log[5]))/(56*x - 20*x^2 - 4*x^3 - (7 + x)*Log[5])^2 - ( 2*(1 - x))/(56*x - 20*x^2 - 4*x^3 - (7 + x)*Log[5]))*(160*x^5 + 16*x^6 - 8 *x^4*(6 - Log[5]) - 14*x*(56 - Log[5])*Log[5] + 49*Log[5]^2 + x^2*(3136 + 168*Log[5] + Log[5]^2) - 32*x^3*(70 - Log[125])))
3.4.69.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
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]
Time = 35.93 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81
| method | result | size |
| risch | \({\mathrm e}^{-\frac {25 \left (-2+x \right ) x}{\left (x +7\right ) \left (4 x^{2}+\ln \left (5\right )-8 x \right )}} \ln \left (x \right )\) | \(29\) |
| parallelrisch | \(\frac {\left (-134456 \ln \left (5\right )^{4} x \ln \left (x \right )+9604 \ln \left (5\right )^{4} x^{3} \ln \left (x \right )+2401 \ln \left (5\right )^{5} x \ln \left (x \right )+48020 \ln \left (5\right )^{4} \ln \left (x \right ) x^{2}+16807 \ln \left (5\right )^{5} \ln \left (x \right )\right ) {\mathrm e}^{-\frac {25 x^{2}-50 x}{x \ln \left (5\right )+\ln \left (78125\right )+4 x^{3}+20 x^{2}-56 x}}}{2401 \left (4 x^{3}+x \ln \left (5\right )+20 x^{2}+7 \ln \left (5\right )-56 x \right ) \ln \left (5\right )^{4}}\) | \(117\) |
int((((-25*x^3-350*x^2+350*x)*ln(5)+100*x^5-400*x^4+400*x^3)*ln(x)+(x^2+14 *x+49)*ln(5)^2+(8*x^4+96*x^3+168*x^2-784*x)*ln(5)+16*x^6+160*x^5-48*x^4-22 40*x^3+3136*x^2)/((x^3+14*x^2+49*x)*ln(5)^2+(8*x^5+96*x^4+168*x^3-784*x^2) *ln(5)+16*x^7+160*x^6-48*x^5-2240*x^4+3136*x^3)/exp((25*x^2-50*x)/((x+7)*l n(5)+4*x^3+20*x^2-56*x)),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-\frac {-50 x+25 x^2}{-56 x+20 x^2+4 x^3+(7+x) \log (5)}} \left (3136 x^2-2240 x^3-48 x^4+160 x^5+16 x^6+\left (-784 x+168 x^2+96 x^3+8 x^4\right ) \log (5)+\left (49+14 x+x^2\right ) \log ^2(5)+\left (400 x^3-400 x^4+100 x^5+\left (350 x-350 x^2-25 x^3\right ) \log (5)\right ) \log (x)\right )}{3136 x^3-2240 x^4-48 x^5+160 x^6+16 x^7+\left (-784 x^2+168 x^3+96 x^4+8 x^5\right ) \log (5)+\left (49 x+14 x^2+x^3\right ) \log ^2(5)} \, dx=e^{\left (-\frac {25 \, {\left (x^{2} - 2 \, x\right )}}{4 \, x^{3} + 20 \, x^{2} + {\left (x + 7\right )} \log \left (5\right ) - 56 \, x}\right )} \log \left (x\right ) \]
integrate((((-25*x^3-350*x^2+350*x)*log(5)+100*x^5-400*x^4+400*x^3)*log(x) +(x^2+14*x+49)*log(5)^2+(8*x^4+96*x^3+168*x^2-784*x)*log(5)+16*x^6+160*x^5 -48*x^4-2240*x^3+3136*x^2)/((x^3+14*x^2+49*x)*log(5)^2+(8*x^5+96*x^4+168*x ^3-784*x^2)*log(5)+16*x^7+160*x^6-48*x^5-2240*x^4+3136*x^3)/exp((25*x^2-50 *x)/((x+7)*log(5)+4*x^3+20*x^2-56*x)),x, algorithm=\
Timed out. \[ \int \frac {e^{-\frac {-50 x+25 x^2}{-56 x+20 x^2+4 x^3+(7+x) \log (5)}} \left (3136 x^2-2240 x^3-48 x^4+160 x^5+16 x^6+\left (-784 x+168 x^2+96 x^3+8 x^4\right ) \log (5)+\left (49+14 x+x^2\right ) \log ^2(5)+\left (400 x^3-400 x^4+100 x^5+\left (350 x-350 x^2-25 x^3\right ) \log (5)\right ) \log (x)\right )}{3136 x^3-2240 x^4-48 x^5+160 x^6+16 x^7+\left (-784 x^2+168 x^3+96 x^4+8 x^5\right ) \log (5)+\left (49 x+14 x^2+x^3\right ) \log ^2(5)} \, dx=\text {Timed out} \]
integrate((((-25*x**3-350*x**2+350*x)*ln(5)+100*x**5-400*x**4+400*x**3)*ln (x)+(x**2+14*x+49)*ln(5)**2+(8*x**4+96*x**3+168*x**2-784*x)*ln(5)+16*x**6+ 160*x**5-48*x**4-2240*x**3+3136*x**2)/((x**3+14*x**2+49*x)*ln(5)**2+(8*x** 5+96*x**4+168*x**3-784*x**2)*ln(5)+16*x**7+160*x**6-48*x**5-2240*x**4+3136 *x**3)/exp((25*x**2-50*x)/((x+7)*ln(5)+4*x**3+20*x**2-56*x)),x)
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).
Time = 0.58 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.33 \[ \int \frac {e^{-\frac {-50 x+25 x^2}{-56 x+20 x^2+4 x^3+(7+x) \log (5)}} \left (3136 x^2-2240 x^3-48 x^4+160 x^5+16 x^6+\left (-784 x+168 x^2+96 x^3+8 x^4\right ) \log (5)+\left (49+14 x+x^2\right ) \log ^2(5)+\left (400 x^3-400 x^4+100 x^5+\left (350 x-350 x^2-25 x^3\right ) \log (5)\right ) \log (x)\right )}{3136 x^3-2240 x^4-48 x^5+160 x^6+16 x^7+\left (-784 x^2+168 x^3+96 x^4+8 x^5\right ) \log (5)+\left (49 x+14 x^2+x^3\right ) \log ^2(5)} \, dx=e^{\left (-\frac {25 \, x \log \left (5\right )}{4 \, x^{2} {\left (\log \left (5\right ) + 252\right )} - 8 \, x {\left (\log \left (5\right ) + 252\right )} + \log \left (5\right )^{2} + 252 \, \log \left (5\right )} + \frac {225 \, \log \left (5\right )}{4 \, x^{2} {\left (\log \left (5\right ) + 252\right )} - 8 \, x {\left (\log \left (5\right ) + 252\right )} + \log \left (5\right )^{2} + 252 \, \log \left (5\right )} - \frac {1575}{x {\left (\log \left (5\right ) + 252\right )} + 7 \, \log \left (5\right ) + 1764}\right )} \log \left (x\right ) \]
integrate((((-25*x^3-350*x^2+350*x)*log(5)+100*x^5-400*x^4+400*x^3)*log(x) +(x^2+14*x+49)*log(5)^2+(8*x^4+96*x^3+168*x^2-784*x)*log(5)+16*x^6+160*x^5 -48*x^4-2240*x^3+3136*x^2)/((x^3+14*x^2+49*x)*log(5)^2+(8*x^5+96*x^4+168*x ^3-784*x^2)*log(5)+16*x^7+160*x^6-48*x^5-2240*x^4+3136*x^3)/exp((25*x^2-50 *x)/((x+7)*log(5)+4*x^3+20*x^2-56*x)),x, algorithm=\
e^(-25*x*log(5)/(4*x^2*(log(5) + 252) - 8*x*(log(5) + 252) + log(5)^2 + 25 2*log(5)) + 225*log(5)/(4*x^2*(log(5) + 252) - 8*x*(log(5) + 252) + log(5) ^2 + 252*log(5)) - 1575/(x*(log(5) + 252) + 7*log(5) + 1764))*log(x)
Time = 0.56 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {-50 x+25 x^2}{-56 x+20 x^2+4 x^3+(7+x) \log (5)}} \left (3136 x^2-2240 x^3-48 x^4+160 x^5+16 x^6+\left (-784 x+168 x^2+96 x^3+8 x^4\right ) \log (5)+\left (49+14 x+x^2\right ) \log ^2(5)+\left (400 x^3-400 x^4+100 x^5+\left (350 x-350 x^2-25 x^3\right ) \log (5)\right ) \log (x)\right )}{3136 x^3-2240 x^4-48 x^5+160 x^6+16 x^7+\left (-784 x^2+168 x^3+96 x^4+8 x^5\right ) \log (5)+\left (49 x+14 x^2+x^3\right ) \log ^2(5)} \, dx=e^{\left (-\frac {25 \, {\left (x^{2} - 2 \, x\right )}}{4 \, x^{3} + 20 \, x^{2} + x \log \left (5\right ) - 56 \, x + 7 \, \log \left (5\right )}\right )} \log \left (x\right ) \]
integrate((((-25*x^3-350*x^2+350*x)*log(5)+100*x^5-400*x^4+400*x^3)*log(x) +(x^2+14*x+49)*log(5)^2+(8*x^4+96*x^3+168*x^2-784*x)*log(5)+16*x^6+160*x^5 -48*x^4-2240*x^3+3136*x^2)/((x^3+14*x^2+49*x)*log(5)^2+(8*x^5+96*x^4+168*x ^3-784*x^2)*log(5)+16*x^7+160*x^6-48*x^5-2240*x^4+3136*x^3)/exp((25*x^2-50 *x)/((x+7)*log(5)+4*x^3+20*x^2-56*x)),x, algorithm=\
Timed out. \[ \int \frac {e^{-\frac {-50 x+25 x^2}{-56 x+20 x^2+4 x^3+(7+x) \log (5)}} \left (3136 x^2-2240 x^3-48 x^4+160 x^5+16 x^6+\left (-784 x+168 x^2+96 x^3+8 x^4\right ) \log (5)+\left (49+14 x+x^2\right ) \log ^2(5)+\left (400 x^3-400 x^4+100 x^5+\left (350 x-350 x^2-25 x^3\right ) \log (5)\right ) \log (x)\right )}{3136 x^3-2240 x^4-48 x^5+160 x^6+16 x^7+\left (-784 x^2+168 x^3+96 x^4+8 x^5\right ) \log (5)+\left (49 x+14 x^2+x^3\right ) \log ^2(5)} \, dx=\int \frac {{\mathrm {e}}^{\frac {50\,x-25\,x^2}{\ln \left (5\right )\,\left (x+7\right )-56\,x+20\,x^2+4\,x^3}}\,\left ({\ln \left (5\right )}^2\,\left (x^2+14\,x+49\right )+\ln \left (5\right )\,\left (8\,x^4+96\,x^3+168\,x^2-784\,x\right )-\ln \left (x\right )\,\left (\ln \left (5\right )\,\left (25\,x^3+350\,x^2-350\,x\right )-400\,x^3+400\,x^4-100\,x^5\right )+3136\,x^2-2240\,x^3-48\,x^4+160\,x^5+16\,x^6\right )}{\ln \left (5\right )\,\left (8\,x^5+96\,x^4+168\,x^3-784\,x^2\right )+{\ln \left (5\right )}^2\,\left (x^3+14\,x^2+49\,x\right )+3136\,x^3-2240\,x^4-48\,x^5+160\,x^6+16\,x^7} \,d x \]
int((exp((50*x - 25*x^2)/(log(5)*(x + 7) - 56*x + 20*x^2 + 4*x^3))*(log(5) ^2*(14*x + x^2 + 49) + log(5)*(168*x^2 - 784*x + 96*x^3 + 8*x^4) - log(x)* (log(5)*(350*x^2 - 350*x + 25*x^3) - 400*x^3 + 400*x^4 - 100*x^5) + 3136*x ^2 - 2240*x^3 - 48*x^4 + 160*x^5 + 16*x^6))/(log(5)*(168*x^3 - 784*x^2 + 9 6*x^4 + 8*x^5) + log(5)^2*(49*x + 14*x^2 + x^3) + 3136*x^3 - 2240*x^4 - 48 *x^5 + 160*x^6 + 16*x^7),x)
int((exp((50*x - 25*x^2)/(log(5)*(x + 7) - 56*x + 20*x^2 + 4*x^3))*(log(5) ^2*(14*x + x^2 + 49) + log(5)*(168*x^2 - 784*x + 96*x^3 + 8*x^4) - log(x)* (log(5)*(350*x^2 - 350*x + 25*x^3) - 400*x^3 + 400*x^4 - 100*x^5) + 3136*x ^2 - 2240*x^3 - 48*x^4 + 160*x^5 + 16*x^6))/(log(5)*(168*x^3 - 784*x^2 + 9 6*x^4 + 8*x^5) + log(5)^2*(49*x + 14*x^2 + x^3) + 3136*x^3 - 2240*x^4 - 48 *x^5 + 160*x^6 + 16*x^7), x)