Integrand size = 127, antiderivative size = 20 \[ \int \frac {\left (2000000 x-800000 x^2+80000 x^3+\left (-800000 x+160000 x^2\right ) \log (x)+80000 x \log ^2(x)\right ) \log \left (e^5+5 x\right )+\left (600000 x-720000 x^2+120000 x^3+e^5 \left (120000-144000 x+24000 x^2\right )+\left (-320000 x+160000 x^2+e^5 (-64000+32000 x)\right ) \log (x)+\left (8000 e^5+40000 x\right ) \log ^2(x)\right ) \log ^2\left (e^5+5 x\right )}{e^5+5 x} \, dx=8000 x (-5+x+\log (x))^2 \log ^2\left (e^5+5 x\right ) \]
Time = 0.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2000000 x-800000 x^2+80000 x^3+\left (-800000 x+160000 x^2\right ) \log (x)+80000 x \log ^2(x)\right ) \log \left (e^5+5 x\right )+\left (600000 x-720000 x^2+120000 x^3+e^5 \left (120000-144000 x+24000 x^2\right )+\left (-320000 x+160000 x^2+e^5 (-64000+32000 x)\right ) \log (x)+\left (8000 e^5+40000 x\right ) \log ^2(x)\right ) \log ^2\left (e^5+5 x\right )}{e^5+5 x} \, dx=8000 x (-5+x+\log (x))^2 \log ^2\left (e^5+5 x\right ) \]
Integrate[((2000000*x - 800000*x^2 + 80000*x^3 + (-800000*x + 160000*x^2)* Log[x] + 80000*x*Log[x]^2)*Log[E^5 + 5*x] + (600000*x - 720000*x^2 + 12000 0*x^3 + E^5*(120000 - 144000*x + 24000*x^2) + (-320000*x + 160000*x^2 + E^ 5*(-64000 + 32000*x))*Log[x] + (8000*E^5 + 40000*x)*Log[x]^2)*Log[E^5 + 5* x]^2)/(E^5 + 5*x),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.
| \(\displaystyle \int \frac {\left (120000 x^3-720000 x^2+e^5 \left (24000 x^2-144000 x+120000\right )+\left (160000 x^2-320000 x+e^5 (32000 x-64000)\right ) \log (x)+600000 x+\left (40000 x+8000 e^5\right ) \log ^2(x)\right ) \log ^2\left (5 x+e^5\right )+\left (80000 x^3-800000 x^2+\left (160000 x^2-800000 x\right ) \log (x)+2000000 x+80000 x \log ^2(x)\right ) \log \left (5 x+e^5\right )}{5 x+e^5} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {8000 (-x-\log (x)+5) \log \left (5 x+e^5\right ) \left (-10 (x-5) x-3 (x-1) \left (5 x+e^5\right ) \log \left (5 x+e^5\right )-\log (x) \left (10 x+\left (5 x+e^5\right ) \log \left (5 x+e^5\right )\right )\right )}{5 x+e^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8000 \int \frac {(-x-\log (x)+5) \log \left (5 x+e^5\right ) \left (10 (5-x) x+3 (1-x) \left (5 x+e^5\right ) \log \left (5 x+e^5\right )-\log (x) \left (10 x+\left (5 x+e^5\right ) \log \left (5 x+e^5\right )\right )\right )}{5 x+e^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 8000 \int \left (\frac {10 x \log \left (5 x+e^5\right ) (x+\log (x)-5)^2}{5 x+e^5}+\left (3 x^2+4 \log (x) x-18 x+\log ^2(x)-8 \log (x)+15\right ) \log ^2\left (5 x+e^5\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 8000 \left (\log ^2\left (5 x+e^5\right ) x^3+\frac {2}{3} \log \left (5 x+e^5\right ) x^3-\frac {2 x^3}{9}-\log (x) x^2+2 \log (x) \log \left (5 x+e^5\right ) x^2-11 \log \left (5 x+e^5\right ) x^2+\frac {e^5 x^2}{15}+\frac {11 x^2}{2}-2 \log ^2(x) x-\frac {2}{5} e^5 \log (x) x+12 \log (x) x+2 \log ^2(x) \log \left (5 x+e^5\right ) x-4 \log (x) \log \left (5 x+e^5\right ) x+4 \log \left (5 x+e^5\right ) x+\frac {4 e^{10} x}{75}+\frac {11 e^5 x}{5}-24 x+\frac {2 \left (5 x+e^5\right )^3}{1125}-\frac {1}{125} e^5 \left (5 x+e^5\right )^2-\frac {11}{50} \left (5 x+e^5\right )^2-\frac {2}{5} \left (5 x+e^5\right )^2 \log ^2\left (5 x+e^5\right )+\frac {4}{5} e^5 \left (5 x+e^5\right ) \log ^2\left (5 x+e^5\right )+\frac {23}{5} \left (5 x+e^5\right ) \log ^2\left (5 x+e^5\right )+\frac {2}{25} \left (5 x+e^5\right )^2 \log (x) \log ^2\left (5 x+e^5\right )-\frac {4}{25} e^5 \left (5 x+e^5\right ) \log (x) \log ^2\left (5 x+e^5\right )-\frac {8}{5} \left (5 x+e^5\right ) \log (x) \log ^2\left (5 x+e^5\right )+\frac {2}{25} e^{10} \log (x) \log ^2\left (5 x+e^5\right )+2 e^5 \log (x) \log ^2\left (5 x+e^5\right )-\frac {2}{5} e^5 \log \left (-\frac {5 x}{e^5}\right ) \log ^2\left (5 x+e^5\right )-\frac {2}{5} e^{10} \log ^2\left (5 x+e^5\right )-5 e^5 \log ^2\left (5 x+e^5\right )+\frac {1}{25} \left (5 x+e^5\right )^2 \log (x)-\frac {1}{25} e^{10} \log (x)+4 e^5 \log (x)-\frac {2}{375} \left (5 x+e^5\right )^3 \log \left (5 x+e^5\right )+\frac {2}{125} e^5 \left (5 x+e^5\right )^2 \log \left (5 x+e^5\right )+\frac {11}{25} \left (5 x+e^5\right )^2 \log \left (5 x+e^5\right )-\frac {2}{125} e^{10} \left (5 x+e^5\right ) \log \left (5 x+e^5\right )-\frac {22}{25} e^5 \left (5 x+e^5\right ) \log \left (5 x+e^5\right )+\frac {8}{5} \left (5 x+e^5\right ) \log \left (5 x+e^5\right )-\frac {2}{25} \left (5 x+e^5\right )^2 \log (x) \log \left (5 x+e^5\right )+\frac {4}{25} e^5 \left (5 x+e^5\right ) \log (x) \log \left (5 x+e^5\right )-\frac {4}{5} \left (5 x+e^5\right ) \log (x) \log \left (5 x+e^5\right )-\frac {2}{25} e^{10} \log (x) \log \left (5 x+e^5\right )+\frac {2}{375} e^{15} \log \left (5 x+e^5\right )+\frac {11}{25} e^{10} \log \left (5 x+e^5\right )+\frac {4}{5} e^5 \log \left (5 x+e^5\right )+\frac {2}{5} e^5 \log ^2(x) \log \left (\frac {5 x}{e^5}+1\right )-\frac {4}{5} e^5 \log (x) \log \left (\frac {5 x}{e^5}+1\right )+\frac {4}{5} e^5 \log (x) \operatorname {PolyLog}\left (2,-\frac {5 x}{e^5}\right )-\frac {8}{5} e^5 \operatorname {PolyLog}\left (2,-\frac {5 x}{e^5}\right )-\frac {4}{5} e^5 \log \left (5 x+e^5\right ) \operatorname {PolyLog}\left (2,\frac {5 x}{e^5}+1\right )-\frac {4}{5} e^5 \operatorname {PolyLog}\left (3,-\frac {5 x}{e^5}\right )+\frac {4}{5} e^5 \operatorname {PolyLog}\left (3,\frac {5 x}{e^5}+1\right )-2 e^5 \int \frac {\log ^2(x) \log \left (5 x+e^5\right )}{5 x+e^5}dx+\int \log ^2(x) \log ^2\left (5 x+e^5\right )dx\right )\) |
Int[((2000000*x - 800000*x^2 + 80000*x^3 + (-800000*x + 160000*x^2)*Log[x] + 80000*x*Log[x]^2)*Log[E^5 + 5*x] + (600000*x - 720000*x^2 + 120000*x^3 + E^5*(120000 - 144000*x + 24000*x^2) + (-320000*x + 160000*x^2 + E^5*(-64 000 + 32000*x))*Log[x] + (8000*E^5 + 40000*x)*Log[x]^2)*Log[E^5 + 5*x]^2)/ (E^5 + 5*x),x]
3.4.55.3.1 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(43\) vs. \(2(19)=38\).
Time = 0.63 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.20
| method | result | size |
| risch | \(\left (8000 x^{3}+16000 x^{2} \ln \left (x \right )+8000 x \ln \left (x \right )^{2}-80000 x^{2}-80000 x \ln \left (x \right )+200000 x \right ) \ln \left ({\mathrm e}^{5}+5 x \right )^{2}\) | \(44\) |
| parallelrisch | \(-80000 \ln \left ({\mathrm e}^{5}+5 x \right )^{2} x^{2}+200000 \ln \left ({\mathrm e}^{5}+5 x \right )^{2} x +8000 \ln \left ({\mathrm e}^{5}+5 x \right )^{2} x^{3}+16000 \ln \left (x \right ) \ln \left ({\mathrm e}^{5}+5 x \right )^{2} x^{2}+8000 \ln \left ({\mathrm e}^{5}+5 x \right )^{2} \ln \left (x \right )^{2} x -80000 \ln \left ({\mathrm e}^{5}+5 x \right )^{2} x \ln \left (x \right )\) | \(88\) |
int((((8000*exp(5)+40000*x)*ln(x)^2+((32000*x-64000)*exp(5)+160000*x^2-320 000*x)*ln(x)+(24000*x^2-144000*x+120000)*exp(5)+120000*x^3-720000*x^2+6000 00*x)*ln(exp(5)+5*x)^2+(80000*x*ln(x)^2+(160000*x^2-800000*x)*ln(x)+80000* x^3-800000*x^2+2000000*x)*ln(exp(5)+5*x))/(exp(5)+5*x),x,method=_RETURNVER BOSE)
Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.00 \[ \int \frac {\left (2000000 x-800000 x^2+80000 x^3+\left (-800000 x+160000 x^2\right ) \log (x)+80000 x \log ^2(x)\right ) \log \left (e^5+5 x\right )+\left (600000 x-720000 x^2+120000 x^3+e^5 \left (120000-144000 x+24000 x^2\right )+\left (-320000 x+160000 x^2+e^5 (-64000+32000 x)\right ) \log (x)+\left (8000 e^5+40000 x\right ) \log ^2(x)\right ) \log ^2\left (e^5+5 x\right )}{e^5+5 x} \, dx=8000 \, {\left (x^{3} + x \log \left (x\right )^{2} - 10 \, x^{2} + 2 \, {\left (x^{2} - 5 \, x\right )} \log \left (x\right ) + 25 \, x\right )} \log \left (5 \, x + e^{5}\right )^{2} \]
integrate((((8000*exp(5)+40000*x)*log(x)^2+((32000*x-64000)*exp(5)+160000* x^2-320000*x)*log(x)+(24000*x^2-144000*x+120000)*exp(5)+120000*x^3-720000* x^2+600000*x)*log(exp(5)+5*x)^2+(80000*x*log(x)^2+(160000*x^2-800000*x)*lo g(x)+80000*x^3-800000*x^2+2000000*x)*log(exp(5)+5*x))/(exp(5)+5*x),x, algo rithm=\
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (20) = 40\).
Time = 2.63 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.30 \[ \int \frac {\left (2000000 x-800000 x^2+80000 x^3+\left (-800000 x+160000 x^2\right ) \log (x)+80000 x \log ^2(x)\right ) \log \left (e^5+5 x\right )+\left (600000 x-720000 x^2+120000 x^3+e^5 \left (120000-144000 x+24000 x^2\right )+\left (-320000 x+160000 x^2+e^5 (-64000+32000 x)\right ) \log (x)+\left (8000 e^5+40000 x\right ) \log ^2(x)\right ) \log ^2\left (e^5+5 x\right )}{e^5+5 x} \, dx=\left (8000 x^{3} + 16000 x^{2} \log {\left (x \right )} - 80000 x^{2} + 8000 x \log {\left (x \right )}^{2} - 80000 x \log {\left (x \right )} + 200000 x\right ) \log {\left (5 x + e^{5} \right )}^{2} \]
integrate((((8000*exp(5)+40000*x)*ln(x)**2+((32000*x-64000)*exp(5)+160000* x**2-320000*x)*ln(x)+(24000*x**2-144000*x+120000)*exp(5)+120000*x**3-72000 0*x**2+600000*x)*ln(exp(5)+5*x)**2+(80000*x*ln(x)**2+(160000*x**2-800000*x )*ln(x)+80000*x**3-800000*x**2+2000000*x)*ln(exp(5)+5*x))/(exp(5)+5*x),x)
(8000*x**3 + 16000*x**2*log(x) - 80000*x**2 + 8000*x*log(x)**2 - 80000*x*l og(x) + 200000*x)*log(5*x + exp(5))**2
Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (19) = 38\).
Time = 0.23 (sec) , antiderivative size = 312, normalized size of antiderivative = 15.60 \[ \int \frac {\left (2000000 x-800000 x^2+80000 x^3+\left (-800000 x+160000 x^2\right ) \log (x)+80000 x \log ^2(x)\right ) \log \left (e^5+5 x\right )+\left (600000 x-720000 x^2+120000 x^3+e^5 \left (120000-144000 x+24000 x^2\right )+\left (-320000 x+160000 x^2+e^5 (-64000+32000 x)\right ) \log (x)+\left (8000 e^5+40000 x\right ) \log ^2(x)\right ) \log ^2\left (e^5+5 x\right )}{e^5+5 x} \, dx=-\frac {4000}{3} \, x^{2} {\left (e^{5} + 30\right )} + \frac {4000}{3} \, x^{2} e^{5} + 64 \, {\left (125 \, x^{3} + 125 \, x \log \left (x\right )^{2} - 1250 \, x^{2} + 250 \, {\left (x^{2} - 5 \, x\right )} \log \left (x\right ) + 3125 \, x + e^{15} + 50 \, e^{10} + 625 \, e^{5}\right )} \log \left (5 \, x + e^{5}\right )^{2} + 64 \, e^{15} \log \left (5 \, x + e^{5}\right )^{2} + 3200 \, e^{10} \log \left (5 \, x + e^{5}\right )^{2} + 40000 \, e^{5} \log \left (5 \, x + e^{5}\right )^{2} + 40000 \, x^{2} + \frac {320}{3} \, x {\left (11 \, e^{10} + 450 \, e^{5} + 3750\right )} - \frac {3520}{3} \, x e^{10} - 48000 \, x e^{5} - \frac {64}{3} \, {\left (250 \, x^{3} - 75 \, x^{2} {\left (e^{5} + 50\right )} + 30 \, x {\left (e^{10} + 50 \, e^{5} + 625\right )} + 11 \, e^{15} + 450 \, e^{10} + 3750 \, e^{5}\right )} \log \left (5 \, x + e^{5}\right ) + \frac {64}{3} \, {\left (250 \, x^{3} - 75 \, x^{2} e^{5} + 30 \, x e^{10} - 6 \, e^{15} \log \left (5 \, x + e^{5}\right )\right )} \log \left (5 \, x + e^{5}\right ) - 3200 \, {\left (25 \, x^{2} - 10 \, x e^{5} + 2 \, e^{10} \log \left (5 \, x + e^{5}\right )\right )} \log \left (5 \, x + e^{5}\right ) - 80000 \, {\left (e^{5} \log \left (5 \, x + e^{5}\right ) - 5 \, x\right )} \log \left (5 \, x + e^{5}\right ) + \frac {704}{3} \, e^{15} \log \left (5 \, x + e^{5}\right ) + 9600 \, e^{10} \log \left (5 \, x + e^{5}\right ) + 80000 \, e^{5} \log \left (5 \, x + e^{5}\right ) - 400000 \, x \]
integrate((((8000*exp(5)+40000*x)*log(x)^2+((32000*x-64000)*exp(5)+160000* x^2-320000*x)*log(x)+(24000*x^2-144000*x+120000)*exp(5)+120000*x^3-720000* x^2+600000*x)*log(exp(5)+5*x)^2+(80000*x*log(x)^2+(160000*x^2-800000*x)*lo g(x)+80000*x^3-800000*x^2+2000000*x)*log(exp(5)+5*x))/(exp(5)+5*x),x, algo rithm=\
-4000/3*x^2*(e^5 + 30) + 4000/3*x^2*e^5 + 64*(125*x^3 + 125*x*log(x)^2 - 1 250*x^2 + 250*(x^2 - 5*x)*log(x) + 3125*x + e^15 + 50*e^10 + 625*e^5)*log( 5*x + e^5)^2 + 64*e^15*log(5*x + e^5)^2 + 3200*e^10*log(5*x + e^5)^2 + 400 00*e^5*log(5*x + e^5)^2 + 40000*x^2 + 320/3*x*(11*e^10 + 450*e^5 + 3750) - 3520/3*x*e^10 - 48000*x*e^5 - 64/3*(250*x^3 - 75*x^2*(e^5 + 50) + 30*x*(e ^10 + 50*e^5 + 625) + 11*e^15 + 450*e^10 + 3750*e^5)*log(5*x + e^5) + 64/3 *(250*x^3 - 75*x^2*e^5 + 30*x*e^10 - 6*e^15*log(5*x + e^5))*log(5*x + e^5) - 3200*(25*x^2 - 10*x*e^5 + 2*e^10*log(5*x + e^5))*log(5*x + e^5) - 80000 *(e^5*log(5*x + e^5) - 5*x)*log(5*x + e^5) + 704/3*e^15*log(5*x + e^5) + 9 600*e^10*log(5*x + e^5) + 80000*e^5*log(5*x + e^5) - 400000*x
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (19) = 38\).
Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.35 \[ \int \frac {\left (2000000 x-800000 x^2+80000 x^3+\left (-800000 x+160000 x^2\right ) \log (x)+80000 x \log ^2(x)\right ) \log \left (e^5+5 x\right )+\left (600000 x-720000 x^2+120000 x^3+e^5 \left (120000-144000 x+24000 x^2\right )+\left (-320000 x+160000 x^2+e^5 (-64000+32000 x)\right ) \log (x)+\left (8000 e^5+40000 x\right ) \log ^2(x)\right ) \log ^2\left (e^5+5 x\right )}{e^5+5 x} \, dx=8000 \, x^{3} \log \left (5 \, x + e^{5}\right )^{2} + 16000 \, x^{2} \log \left (5 \, x + e^{5}\right )^{2} \log \left (x\right ) + 8000 \, x \log \left (5 \, x + e^{5}\right )^{2} \log \left (x\right )^{2} - 80000 \, x^{2} \log \left (5 \, x + e^{5}\right )^{2} - 80000 \, x \log \left (5 \, x + e^{5}\right )^{2} \log \left (x\right ) + 200000 \, x \log \left (5 \, x + e^{5}\right )^{2} \]
integrate((((8000*exp(5)+40000*x)*log(x)^2+((32000*x-64000)*exp(5)+160000* x^2-320000*x)*log(x)+(24000*x^2-144000*x+120000)*exp(5)+120000*x^3-720000* x^2+600000*x)*log(exp(5)+5*x)^2+(80000*x*log(x)^2+(160000*x^2-800000*x)*lo g(x)+80000*x^3-800000*x^2+2000000*x)*log(exp(5)+5*x))/(exp(5)+5*x),x, algo rithm=\
8000*x^3*log(5*x + e^5)^2 + 16000*x^2*log(5*x + e^5)^2*log(x) + 8000*x*log (5*x + e^5)^2*log(x)^2 - 80000*x^2*log(5*x + e^5)^2 - 80000*x*log(5*x + e^ 5)^2*log(x) + 200000*x*log(5*x + e^5)^2
Time = 8.32 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.20 \[ \int \frac {\left (2000000 x-800000 x^2+80000 x^3+\left (-800000 x+160000 x^2\right ) \log (x)+80000 x \log ^2(x)\right ) \log \left (e^5+5 x\right )+\left (600000 x-720000 x^2+120000 x^3+e^5 \left (120000-144000 x+24000 x^2\right )+\left (-320000 x+160000 x^2+e^5 (-64000+32000 x)\right ) \log (x)+\left (8000 e^5+40000 x\right ) \log ^2(x)\right ) \log ^2\left (e^5+5 x\right )}{e^5+5 x} \, dx={\ln \left (5\,x+{\mathrm {e}}^5\right )}^2\,\left (200000\,x+8000\,x\,{\ln \left (x\right )}^2-\ln \left (x\right )\,\left (80000\,x-16000\,x^2\right )-80000\,x^2+8000\,x^3\right ) \]
int((log(5*x + exp(5))*(2000000*x + 80000*x*log(x)^2 - log(x)*(800000*x - 160000*x^2) - 800000*x^2 + 80000*x^3) + log(5*x + exp(5))^2*(600000*x + ex p(5)*(24000*x^2 - 144000*x + 120000) + log(x)^2*(40000*x + 8000*exp(5)) + log(x)*(160000*x^2 - 320000*x + exp(5)*(32000*x - 64000)) - 720000*x^2 + 1 20000*x^3))/(5*x + exp(5)),x)