Integrand size = 86, antiderivative size = 28 \[ \int \frac {5160 x-7380 x^2+2200 x^3+100 x^4+e^{2 x} (20000+800 x)+e^x \left (10320+400 x-5000 x^2-200 x^3\right )+\left (2064-1920 x-80 x^2+e^x (4000+160 x)\right ) \log \left (625+50 x+x^2\right )}{625+25 x} \, dx=16 \left (-e^x+\frac {1}{4} (-2+x) x-\frac {1}{5} \log \left ((25+x)^2\right )\right )^2 \]
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {5160 x-7380 x^2+2200 x^3+100 x^4+e^{2 x} (20000+800 x)+e^x \left (10320+400 x-5000 x^2-200 x^3\right )+\left (2064-1920 x-80 x^2+e^x (4000+160 x)\right ) \log \left (625+50 x+x^2\right )}{625+25 x} \, dx=\frac {1}{25} \left (20 e^x-5 (-2+x) x+4 \log \left ((25+x)^2\right )\right )^2 \]
Integrate[(5160*x - 7380*x^2 + 2200*x^3 + 100*x^4 + E^(2*x)*(20000 + 800*x ) + E^x*(10320 + 400*x - 5000*x^2 - 200*x^3) + (2064 - 1920*x - 80*x^2 + E ^x*(4000 + 160*x))*Log[625 + 50*x + x^2])/(625 + 25*x),x]
Time = 0.52 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {7239, 27, 7237}
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.
\(\displaystyle \int \frac {100 x^4+2200 x^3-7380 x^2+\left (-80 x^2-1920 x+e^x (160 x+4000)+2064\right ) \log \left (x^2+50 x+625\right )+e^x \left (-200 x^3-5000 x^2+400 x+10320\right )+5160 x+e^{2 x} (800 x+20000)}{25 x+625} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {4 \left (-5 x^2-120 x+10 e^x (x+25)+129\right ) \left (-5 (x-2) x+20 e^x+4 \log \left ((x+25)^2\right )\right )}{25 (x+25)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4}{25} \int \frac {\left (-5 x^2-120 x+10 e^x (x+25)+129\right ) \left (5 (2-x) x+20 e^x+4 \log \left ((x+25)^2\right )\right )}{x+25}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {1}{25} \left (5 (2-x) x+20 e^x+4 \log \left ((x+25)^2\right )\right )^2\) |
Int[(5160*x - 7380*x^2 + 2200*x^3 + 100*x^4 + E^(2*x)*(20000 + 800*x) + E^ x*(10320 + 400*x - 5000*x^2 - 200*x^3) + (2064 - 1920*x - 80*x^2 + E^x*(40 00 + 160*x))*Log[625 + 50*x + x^2])/(625 + 25*x),x]
3.12.27.3.1 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_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
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. \(101\) vs. \(2(39)=78\).
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.64
method | result | size |
parallelrisch | \(x^{4}-4 x^{3}-8 \,{\mathrm e}^{x} x^{2}-\frac {8 \ln \left (x^{2}+50 x +625\right ) x^{2}}{5}+4 x^{2}+16 \,{\mathrm e}^{x} x +\frac {16 \ln \left (x^{2}+50 x +625\right ) x}{5}+16 \,{\mathrm e}^{2 x}+\frac {32 \,{\mathrm e}^{x} \ln \left (x^{2}+50 x +625\right )}{5}+\frac {16 \ln \left (x^{2}+50 x +625\right )^{2}}{25}-1680 \ln \left (x +25\right )+840 \ln \left (x^{2}+50 x +625\right )\) | \(102\) |
default | \(16 \,{\mathrm e}^{x} x +\frac {32 \left (\ln \left (\left (x +25\right )^{2}\right )-2 \ln \left (x +25\right )\right ) {\mathrm e}^{x}}{5}-8 \,{\mathrm e}^{x} x^{2}+\frac {64 \ln \left (x +25\right ) {\mathrm e}^{x}}{5}+16 \,{\mathrm e}^{2 x}+x^{4}-4 x^{3}+4 x^{2}-\frac {8 \ln \left (x^{2}+50 x +625\right ) x^{2}}{5}+\frac {16 \ln \left (x^{2}+50 x +625\right ) x}{5}+\frac {64 \ln \left (x +25\right ) \ln \left (x^{2}+50 x +625\right )}{25}-\frac {64 \ln \left (x +25\right )^{2}}{25}\) | \(107\) |
parts | \(16 \,{\mathrm e}^{x} x +\frac {32 \left (\ln \left (\left (x +25\right )^{2}\right )-2 \ln \left (x +25\right )\right ) {\mathrm e}^{x}}{5}-8 \,{\mathrm e}^{x} x^{2}+\frac {64 \ln \left (x +25\right ) {\mathrm e}^{x}}{5}+16 \,{\mathrm e}^{2 x}+x^{4}-4 x^{3}+4 x^{2}-\frac {8 \ln \left (x^{2}+50 x +625\right ) x^{2}}{5}+\frac {16 \ln \left (x^{2}+50 x +625\right ) x}{5}+\frac {64 \ln \left (x +25\right ) \ln \left (x^{2}+50 x +625\right )}{25}-\frac {64 \ln \left (x +25\right )^{2}}{25}\) | \(107\) |
risch | \(-8 \,{\mathrm e}^{x} x^{2}+16 \,{\mathrm e}^{x} x +\frac {64 \ln \left (x +25\right )^{2}}{25}+16 \,{\mathrm e}^{2 x}+x^{4}-4 x^{3}+4 x^{2}-\frac {8 i \pi x \operatorname {csgn}\left (i \left (x +25\right )\right )^{2} \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )}{5}+\frac {16 i \pi x \,\operatorname {csgn}\left (i \left (x +25\right )\right ) \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{2}}{5}+\frac {4 i \pi \,x^{2} \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{3}}{5}-\frac {8 i \pi x \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{3}}{5}-\frac {32 i \pi \ln \left (x +25\right ) \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{3}}{25}-\frac {16 i \pi \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{3} {\mathrm e}^{x}}{5}+\frac {4 i \pi \,x^{2} \operatorname {csgn}\left (i \left (x +25\right )\right )^{2} \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )}{5}-\frac {8 i \pi \,x^{2} \operatorname {csgn}\left (i \left (x +25\right )\right ) \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{2}}{5}+\left (-\frac {16 x^{2}}{5}+\frac {32 x}{5}+\frac {64 \,{\mathrm e}^{x}}{5}\right ) \ln \left (x +25\right )-\frac {4 \pi ^{2} \operatorname {csgn}\left (i \left (x +25\right )\right )^{4} \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{2}}{25}+\frac {16 \pi ^{2} \operatorname {csgn}\left (i \left (x +25\right )\right )^{3} \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{3}}{25}-\frac {24 \pi ^{2} \operatorname {csgn}\left (i \left (x +25\right )\right )^{2} \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{4}}{25}+\frac {16 \pi ^{2} \operatorname {csgn}\left (i \left (x +25\right )\right ) \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{5}}{25}+\frac {64 i \pi \ln \left (x +25\right ) \operatorname {csgn}\left (i \left (x +25\right )\right ) \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{2}}{25}-\frac {16 i \pi \operatorname {csgn}\left (i \left (x +25\right )\right )^{2} \operatorname {csgn}\left (i \left (x +25\right )^{2}\right ) {\mathrm e}^{x}}{5}+\frac {32 i \pi \,\operatorname {csgn}\left (i \left (x +25\right )\right ) \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{2} {\mathrm e}^{x}}{5}-\frac {32 i \pi \ln \left (x +25\right ) \operatorname {csgn}\left (i \left (x +25\right )\right )^{2} \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )}{25}-\frac {4 \pi ^{2} \operatorname {csgn}\left (i \left (x +25\right )^{2}\right )^{6}}{25}\) | \(439\) |
int((((160*x+4000)*exp(x)-80*x^2-1920*x+2064)*ln(x^2+50*x+625)+(800*x+2000 0)*exp(x)^2+(-200*x^3-5000*x^2+400*x+10320)*exp(x)+100*x^4+2200*x^3-7380*x ^2+5160*x)/(25*x+625),x,method=_RETURNVERBOSE)
x^4-4*x^3-8*exp(x)*x^2-8/5*ln(x^2+50*x+625)*x^2+4*x^2+16*exp(x)*x+16/5*ln( x^2+50*x+625)*x+16*exp(x)^2+32/5*exp(x)*ln(x^2+50*x+625)+16/25*ln(x^2+50*x +625)^2-1680*ln(x+25)+840*ln(x^2+50*x+625)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int \frac {5160 x-7380 x^2+2200 x^3+100 x^4+e^{2 x} (20000+800 x)+e^x \left (10320+400 x-5000 x^2-200 x^3\right )+\left (2064-1920 x-80 x^2+e^x (4000+160 x)\right ) \log \left (625+50 x+x^2\right )}{625+25 x} \, dx=x^{4} - 4 \, x^{3} + 4 \, x^{2} - 8 \, {\left (x^{2} - 2 \, x\right )} e^{x} - \frac {8}{5} \, {\left (x^{2} - 2 \, x - 4 \, e^{x}\right )} \log \left (x^{2} + 50 \, x + 625\right ) + \frac {16}{25} \, \log \left (x^{2} + 50 \, x + 625\right )^{2} + 16 \, e^{\left (2 \, x\right )} \]
integrate((((160*x+4000)*exp(x)-80*x^2-1920*x+2064)*log(x^2+50*x+625)+(800 *x+20000)*exp(x)^2+(-200*x^3-5000*x^2+400*x+10320)*exp(x)+100*x^4+2200*x^3 -7380*x^2+5160*x)/(25*x+625),x, algorithm=\
x^4 - 4*x^3 + 4*x^2 - 8*(x^2 - 2*x)*e^x - 8/5*(x^2 - 2*x - 4*e^x)*log(x^2 + 50*x + 625) + 16/25*log(x^2 + 50*x + 625)^2 + 16*e^(2*x)
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.93 \[ \int \frac {5160 x-7380 x^2+2200 x^3+100 x^4+e^{2 x} (20000+800 x)+e^x \left (10320+400 x-5000 x^2-200 x^3\right )+\left (2064-1920 x-80 x^2+e^x (4000+160 x)\right ) \log \left (625+50 x+x^2\right )}{625+25 x} \, dx=x^{4} - 4 x^{3} + 4 x^{2} + \left (- \frac {8 x^{2}}{5} + \frac {16 x}{5}\right ) \log {\left (x^{2} + 50 x + 625 \right )} + \frac {\left (- 40 x^{2} + 80 x + 32 \log {\left (x^{2} + 50 x + 625 \right )}\right ) e^{x}}{5} + 16 e^{2 x} + \frac {16 \log {\left (x^{2} + 50 x + 625 \right )}^{2}}{25} \]
integrate((((160*x+4000)*exp(x)-80*x**2-1920*x+2064)*ln(x**2+50*x+625)+(80 0*x+20000)*exp(x)**2+(-200*x**3-5000*x**2+400*x+10320)*exp(x)+100*x**4+220 0*x**3-7380*x**2+5160*x)/(25*x+625),x)
x**4 - 4*x**3 + 4*x**2 + (-8*x**2/5 + 16*x/5)*log(x**2 + 50*x + 625) + (-4 0*x**2 + 80*x + 32*log(x**2 + 50*x + 625))*exp(x)/5 + 16*exp(2*x) + 16*log (x**2 + 50*x + 625)**2/25
\[ \int \frac {5160 x-7380 x^2+2200 x^3+100 x^4+e^{2 x} (20000+800 x)+e^x \left (10320+400 x-5000 x^2-200 x^3\right )+\left (2064-1920 x-80 x^2+e^x (4000+160 x)\right ) \log \left (625+50 x+x^2\right )}{625+25 x} \, dx=\int { \frac {4 \, {\left (25 \, x^{4} + 550 \, x^{3} - 1845 \, x^{2} + 200 \, {\left (x + 25\right )} e^{\left (2 \, x\right )} - 10 \, {\left (5 \, x^{3} + 125 \, x^{2} - 10 \, x - 258\right )} e^{x} - 4 \, {\left (5 \, x^{2} - 10 \, {\left (x + 25\right )} e^{x} + 120 \, x - 129\right )} \log \left (x^{2} + 50 \, x + 625\right ) + 1290 \, x\right )}}{25 \, {\left (x + 25\right )}} \,d x } \]
integrate((((160*x+4000)*exp(x)-80*x^2-1920*x+2064)*log(x^2+50*x+625)+(800 *x+20000)*exp(x)^2+(-200*x^3-5000*x^2+400*x+10320)*exp(x)+100*x^4+2200*x^3 -7380*x^2+5160*x)/(25*x+625),x, algorithm=\
x^4 - 4*x^3 + 4*x^2 - 8*(x^2 - 2*x)*e^x - 2064/5*e^(-25)*exp_integral_e(1, -x - 25) - 16/5*(x^2 - 2*x - 4*e^x - 675)*log(x + 25) + 64/25*log(x + 25) ^2 + 16*e^(2*x) - 2064/5*integrate(e^x/(x + 25), x) - 2160*log(x + 25)
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {5160 x-7380 x^2+2200 x^3+100 x^4+e^{2 x} (20000+800 x)+e^x \left (10320+400 x-5000 x^2-200 x^3\right )+\left (2064-1920 x-80 x^2+e^x (4000+160 x)\right ) \log \left (625+50 x+x^2\right )}{625+25 x} \, dx=x^{4} - 4 \, x^{3} - 8 \, x^{2} e^{x} - \frac {8}{5} \, x^{2} \log \left (x^{2} + 50 \, x + 625\right ) + 4 \, x^{2} + 16 \, x e^{x} + \frac {16}{5} \, x \log \left (x^{2} + 50 \, x + 625\right ) + \frac {32}{5} \, e^{x} \log \left (x^{2} + 50 \, x + 625\right ) + \frac {16}{25} \, \log \left (x^{2} + 50 \, x + 625\right )^{2} + 16 \, e^{\left (2 \, x\right )} \]
integrate((((160*x+4000)*exp(x)-80*x^2-1920*x+2064)*log(x^2+50*x+625)+(800 *x+20000)*exp(x)^2+(-200*x^3-5000*x^2+400*x+10320)*exp(x)+100*x^4+2200*x^3 -7380*x^2+5160*x)/(25*x+625),x, algorithm=\
x^4 - 4*x^3 - 8*x^2*e^x - 8/5*x^2*log(x^2 + 50*x + 625) + 4*x^2 + 16*x*e^x + 16/5*x*log(x^2 + 50*x + 625) + 32/5*e^x*log(x^2 + 50*x + 625) + 16/25*l og(x^2 + 50*x + 625)^2 + 16*e^(2*x)
Time = 8.90 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.43 \[ \int \frac {5160 x-7380 x^2+2200 x^3+100 x^4+e^{2 x} (20000+800 x)+e^x \left (10320+400 x-5000 x^2-200 x^3\right )+\left (2064-1920 x-80 x^2+e^x (4000+160 x)\right ) \log \left (625+50 x+x^2\right )}{625+25 x} \, dx=16\,{\mathrm {e}}^{2\,x}+\ln \left (x^2+50\,x+625\right )\,\left (\frac {16\,x}{5}+\frac {32\,{\mathrm {e}}^x}{5}-\frac {8\,x^2}{5}\right )+{\mathrm {e}}^x\,\left (16\,x-8\,x^2\right )+4\,x^2-4\,x^3+x^4+\frac {16\,{\ln \left (x^2+50\,x+625\right )}^2}{25} \]
int((5160*x - log(50*x + x^2 + 625)*(1920*x - exp(x)*(160*x + 4000) + 80*x ^2 - 2064) + exp(2*x)*(800*x + 20000) - 7380*x^2 + 2200*x^3 + 100*x^4 + ex p(x)*(400*x - 5000*x^2 - 200*x^3 + 10320))/(25*x + 625),x)