Integrand size = 83, antiderivative size = 25 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=25 \left (4-e^{x+4 x^2}+\frac {25 x}{-25+x}\right )^2 \] Output:
5*(25*x/(x-25)+4-exp(4*x^2+x))*(125*x/(x-25)+20-5*exp(4*x^2+x))
Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(25)=50\).
Time = 9.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=\frac {25 \left (e^{2 x (1+4 x)} (-25+x)^2+625 (-825+58 x)-2 e^{x+4 x^2} \left (2500-825 x+29 x^2\right )\right )}{(-25+x)^2} \] Input:
Integrate[(3125000 - 906250*x + E^(x + 4*x^2)*(2343750 + 23875000*x - 9172 500*x^2 + 618550*x^3 - 11600*x^4) + E^(2*x + 8*x^2)*(-781250 - 6156250*x + 746250*x^2 - 29950*x^3 + 400*x^4))/(-15625 + 1875*x - 75*x^2 + x^3),x]
Output:
(25*(E^(2*x*(1 + 4*x))*(-25 + x)^2 + 625*(-825 + 58*x) - 2*E^(x + 4*x^2)*( 2500 - 825*x + 29*x^2)))/(-25 + x)^2
Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(25)=50\).
Time = 0.71 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.36, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2007, 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.
\(\displaystyle \int \frac {e^{4 x^2+x} \left (-11600 x^4+618550 x^3-9172500 x^2+23875000 x+2343750\right )+e^{8 x^2+2 x} \left (400 x^4-29950 x^3+746250 x^2-6156250 x-781250\right )-906250 x+3125000}{x^3-75 x^2+1875 x-15625} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^{4 x^2+x} \left (-11600 x^4+618550 x^3-9172500 x^2+23875000 x+2343750\right )+e^{8 x^2+2 x} \left (400 x^4-29950 x^3+746250 x^2-6156250 x-781250\right )-906250 x+3125000}{(x-25)^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {50 e^{4 x^2+x} \left (232 x^3-6571 x^2+19175 x+1875\right )}{(x-25)^2}+50 e^{2 x (4 x+1)} (8 x+1)-\frac {31250 (29 x-100)}{(x-25)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -1450 e^{4 x^2+x}+25 e^{8 x^2+2 x}+\frac {31250 e^{4 x^2+x}}{25-x}+\frac {25 (100-29 x)^2}{(25-x)^2}\) |
Input:
Int[(3125000 - 906250*x + E^(x + 4*x^2)*(2343750 + 23875000*x - 9172500*x^ 2 + 618550*x^3 - 11600*x^4) + E^(2*x + 8*x^2)*(-781250 - 6156250*x + 74625 0*x^2 - 29950*x^3 + 400*x^4))/(-15625 + 1875*x - 75*x^2 + x^3),x]
Output:
-1450*E^(x + 4*x^2) + 25*E^(2*x + 8*x^2) + (25*(100 - 29*x)^2)/(25 - x)^2 + (31250*E^(x + 4*x^2))/(25 - 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]
Time = 0.63 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96
method | result | size |
risch | \(\frac {906250 x -12890625}{x^{2}-50 x +625}+25 \,{\mathrm e}^{2 x \left (1+4 x \right )}-\frac {50 \left (29 x -100\right ) {\mathrm e}^{x \left (1+4 x \right )}}{x -25}\) | \(49\) |
parts | \(25 \,{\mathrm e}^{8 x^{2}+2 x}+\frac {9765625}{\left (x -25\right )^{2}}+\frac {906250}{x -25}+\frac {-1450 \,{\mathrm e}^{4 x^{2}+x} x +5000 \,{\mathrm e}^{4 x^{2}+x}}{x -25}\) | \(56\) |
norman | \(\frac {906250 x +15625 \,{\mathrm e}^{8 x^{2}+2 x}+41250 \,{\mathrm e}^{4 x^{2}+x} x -1450 \,{\mathrm e}^{4 x^{2}+x} x^{2}-1250 \,{\mathrm e}^{8 x^{2}+2 x} x +25 \,{\mathrm e}^{8 x^{2}+2 x} x^{2}-125000 \,{\mathrm e}^{4 x^{2}+x}-12890625}{\left (x -25\right )^{2}}\) | \(86\) |
parallelrisch | \(\frac {906250 x +15625 \,{\mathrm e}^{8 x^{2}+2 x}+41250 \,{\mathrm e}^{4 x^{2}+x} x -1450 \,{\mathrm e}^{4 x^{2}+x} x^{2}-1250 \,{\mathrm e}^{8 x^{2}+2 x} x +25 \,{\mathrm e}^{8 x^{2}+2 x} x^{2}-125000 \,{\mathrm e}^{4 x^{2}+x}-12890625}{x^{2}-50 x +625}\) | \(91\) |
orering | \(\text {Expression too large to display}\) | \(1243\) |
Input:
int(((400*x^4-29950*x^3+746250*x^2-6156250*x-781250)*exp(4*x^2+x)^2+(-1160 0*x^4+618550*x^3-9172500*x^2+23875000*x+2343750)*exp(4*x^2+x)-906250*x+312 5000)/(x^3-75*x^2+1875*x-15625),x,method=_RETURNVERBOSE)
Output:
(906250*x-12890625)/(x^2-50*x+625)+25*exp(2*x*(1+4*x))-50*(29*x-100)/(x-25 )*exp(x*(1+4*x))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=\frac {25 \, {\left ({\left (x^{2} - 50 \, x + 625\right )} e^{\left (8 \, x^{2} + 2 \, x\right )} - 2 \, {\left (29 \, x^{2} - 825 \, x + 2500\right )} e^{\left (4 \, x^{2} + x\right )} + 36250 \, x - 515625\right )}}{x^{2} - 50 \, x + 625} \] Input:
integrate(((400*x^4-29950*x^3+746250*x^2-6156250*x-781250)*exp(4*x^2+x)^2+ (-11600*x^4+618550*x^3-9172500*x^2+23875000*x+2343750)*exp(4*x^2+x)-906250 *x+3125000)/(x^3-75*x^2+1875*x-15625),x, algorithm="fricas")
Output:
25*((x^2 - 50*x + 625)*e^(8*x^2 + 2*x) - 2*(29*x^2 - 825*x + 2500)*e^(4*x^ 2 + x) + 36250*x - 515625)/(x^2 - 50*x + 625)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=- \frac {12890625 - 906250 x}{x^{2} - 50 x + 625} + \frac {\left (5000 - 1450 x\right ) e^{4 x^{2} + x} + \left (25 x - 625\right ) e^{8 x^{2} + 2 x}}{x - 25} \] Input:
integrate(((400*x**4-29950*x**3+746250*x**2-6156250*x-781250)*exp(4*x**2+x )**2+(-11600*x**4+618550*x**3-9172500*x**2+23875000*x+2343750)*exp(4*x**2+ x)-906250*x+3125000)/(x**3-75*x**2+1875*x-15625),x)
Output:
-(12890625 - 906250*x)/(x**2 - 50*x + 625) + ((5000 - 1450*x)*exp(4*x**2 + x) + (25*x - 625)*exp(8*x**2 + 2*x))/(x - 25)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=\frac {453125 \, {\left (2 \, x - 25\right )}}{x^{2} - 50 \, x + 625} + \frac {25 \, {\left ({\left (x - 25\right )} e^{\left (8 \, x^{2} + 2 \, x\right )} - 2 \, {\left (29 \, x - 100\right )} e^{\left (4 \, x^{2} + x\right )}\right )}}{x - 25} - \frac {1562500}{x^{2} - 50 \, x + 625} \] Input:
integrate(((400*x^4-29950*x^3+746250*x^2-6156250*x-781250)*exp(4*x^2+x)^2+ (-11600*x^4+618550*x^3-9172500*x^2+23875000*x+2343750)*exp(4*x^2+x)-906250 *x+3125000)/(x^3-75*x^2+1875*x-15625),x, algorithm="maxima")
Output:
453125*(2*x - 25)/(x^2 - 50*x + 625) + 25*((x - 25)*e^(8*x^2 + 2*x) - 2*(2 9*x - 100)*e^(4*x^2 + x))/(x - 25) - 1562500/(x^2 - 50*x + 625)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (24) = 48\).
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.60 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=\frac {25 \, {\left (x^{2} e^{\left (8 \, x^{2} + 2 \, x\right )} - 58 \, x^{2} e^{\left (4 \, x^{2} + x\right )} - 50 \, x e^{\left (8 \, x^{2} + 2 \, x\right )} + 1650 \, x e^{\left (4 \, x^{2} + x\right )} + 36250 \, x + 625 \, e^{\left (8 \, x^{2} + 2 \, x\right )} - 5000 \, e^{\left (4 \, x^{2} + x\right )} - 515625\right )}}{x^{2} - 50 \, x + 625} \] Input:
integrate(((400*x^4-29950*x^3+746250*x^2-6156250*x-781250)*exp(4*x^2+x)^2+ (-11600*x^4+618550*x^3-9172500*x^2+23875000*x+2343750)*exp(4*x^2+x)-906250 *x+3125000)/(x^3-75*x^2+1875*x-15625),x, algorithm="giac")
Output:
25*(x^2*e^(8*x^2 + 2*x) - 58*x^2*e^(4*x^2 + x) - 50*x*e^(8*x^2 + 2*x) + 16 50*x*e^(4*x^2 + x) + 36250*x + 625*e^(8*x^2 + 2*x) - 5000*e^(4*x^2 + x) - 515625)/(x^2 - 50*x + 625)
Time = 3.95 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=25\,{\mathrm {e}}^{8\,x^2+2\,x}-1450\,{\mathrm {e}}^{4\,x^2+x}-\frac {x\,\left (31250\,{\mathrm {e}}^{4\,x^2+x}-906250\right )-781250\,{\mathrm {e}}^{4\,x^2+x}+12890625}{{\left (x-25\right )}^2} \] Input:
int(-(906250*x + exp(2*x + 8*x^2)*(6156250*x - 746250*x^2 + 29950*x^3 - 40 0*x^4 + 781250) - exp(x + 4*x^2)*(23875000*x - 9172500*x^2 + 618550*x^3 - 11600*x^4 + 2343750) - 3125000)/(1875*x - 75*x^2 + x^3 - 15625),x)
Output:
25*exp(2*x + 8*x^2) - 1450*exp(x + 4*x^2) - (x*(31250*exp(x + 4*x^2) - 906 250) - 781250*exp(x + 4*x^2) + 12890625)/(x - 25)^2
Time = 0.17 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.92 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=\frac {25 e^{8 x^{2}+2 x} x^{2}-1250 e^{8 x^{2}+2 x} x +15625 e^{8 x^{2}+2 x}-1450 e^{4 x^{2}+x} x^{2}+41250 e^{4 x^{2}+x} x -125000 e^{4 x^{2}+x}+18125 x^{2}-1562500}{x^{2}-50 x +625} \] Input:
int(((400*x^4-29950*x^3+746250*x^2-6156250*x-781250)*exp(4*x^2+x)^2+(-1160 0*x^4+618550*x^3-9172500*x^2+23875000*x+2343750)*exp(4*x^2+x)-906250*x+312 5000)/(x^3-75*x^2+1875*x-15625),x)
Output:
(25*(e**(8*x**2 + 2*x)*x**2 - 50*e**(8*x**2 + 2*x)*x + 625*e**(8*x**2 + 2* x) - 58*e**(4*x**2 + x)*x**2 + 1650*e**(4*x**2 + x)*x - 5000*e**(4*x**2 + x) + 725*x**2 - 62500))/(x**2 - 50*x + 625)