Integrand size = 139, antiderivative size = 21 \[ \int \frac {618-164770 x-24360 x^2-33206 x^3-4820 x^4-240 x^5-4 x^6+e^{15} \left (20 x+4 x^3\right )+e^{10} \left (-1212 x-60 x^2-240 x^3-12 x^4\right )+e^5 \left (-30+24480 x+2418 x^2+4860 x^3+480 x^4+12 x^5\right )}{-8000+e^{15}+e^{10} (-60-3 x)-1200 x-60 x^2-x^3+e^5 \left (1200+120 x+3 x^2\right )} \, dx=3+\left (5+x^2+\frac {3}{20-e^5+x}\right )^2 \] Output:
3+(3/(20-exp(5)+x)+x^2+5)^2
Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(21)=42\).
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.95 \[ \int \frac {618-164770 x-24360 x^2-33206 x^3-4820 x^4-240 x^5-4 x^6+e^{15} \left (20 x+4 x^3\right )+e^{10} \left (-1212 x-60 x^2-240 x^3-12 x^4\right )+e^5 \left (-30+24480 x+2418 x^2+4860 x^3+480 x^4+12 x^5\right )}{-8000+e^{15}+e^{10} (-60-3 x)-1200 x-60 x^2-x^3+e^5 \left (1200+120 x+3 x^2\right )} \, dx=-163880+80 e^{15}-e^{20}+e^5 \left (32394+\frac {240}{-20+e^5-x}\right )+6 x+10 x^2+x^4+\frac {9}{\left (20-e^5+x\right )^2}+\frac {2430}{20-e^5+x}+e^{10} \left (-2410+\frac {6}{20-e^5+x}\right ) \] Input:
Integrate[(618 - 164770*x - 24360*x^2 - 33206*x^3 - 4820*x^4 - 240*x^5 - 4 *x^6 + E^15*(20*x + 4*x^3) + E^10*(-1212*x - 60*x^2 - 240*x^3 - 12*x^4) + E^5*(-30 + 24480*x + 2418*x^2 + 4860*x^3 + 480*x^4 + 12*x^5))/(-8000 + E^1 5 + E^10*(-60 - 3*x) - 1200*x - 60*x^2 - x^3 + E^5*(1200 + 120*x + 3*x^2)) ,x]
Output:
-163880 + 80*E^15 - E^20 + E^5*(32394 + 240/(-20 + E^5 - x)) + 6*x + 10*x^ 2 + x^4 + 9/(20 - E^5 + x)^2 + 2430/(20 - E^5 + x) + E^10*(-2410 + 6/(20 - E^5 + x))
Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(21)=42\).
Time = 0.37 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {2007, 2389, 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.
\(\displaystyle \int \frac {-4 x^6-240 x^5-4820 x^4-33206 x^3+e^{15} \left (4 x^3+20 x\right )-24360 x^2+e^{10} \left (-12 x^4-240 x^3-60 x^2-1212 x\right )+e^5 \left (12 x^5+480 x^4+4860 x^3+2418 x^2+24480 x-30\right )-164770 x+618}{-x^3-60 x^2+e^5 \left (3 x^2+120 x+1200\right )-1200 x+e^{10} (-3 x-60)+e^{15}-8000} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-4 x^6-240 x^5-4820 x^4-33206 x^3+e^{15} \left (4 x^3+20 x\right )-24360 x^2+e^{10} \left (-12 x^4-240 x^3-60 x^2-1212 x\right )+e^5 \left (12 x^5+480 x^4+4860 x^3+2418 x^2+24480 x-30\right )-164770 x+618}{\left (-x+e^5-20\right )^3}dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (4 x^3+20 x-\frac {6 \left (405-40 e^5+e^{10}\right )}{\left (-x+e^5-20\right )^2}+\frac {18}{\left (-x+e^5-20\right )^3}+6\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^4+10 x^2+6 x+\frac {6 \left (405-40 e^5+e^{10}\right )}{x-e^5+20}+\frac {9}{\left (x-e^5+20\right )^2}\) |
Input:
Int[(618 - 164770*x - 24360*x^2 - 33206*x^3 - 4820*x^4 - 240*x^5 - 4*x^6 + E^15*(20*x + 4*x^3) + E^10*(-1212*x - 60*x^2 - 240*x^3 - 12*x^4) + E^5*(- 30 + 24480*x + 2418*x^2 + 4860*x^3 + 480*x^4 + 12*x^5))/(-8000 + E^15 + E^ 10*(-60 - 3*x) - 1200*x - 60*x^2 - x^3 + E^5*(1200 + 120*x + 3*x^2)),x]
Output:
6*x + 10*x^2 + x^4 + 9/(20 - E^5 + x)^2 + (6*(405 - 40*E^5 + E^10))/(20 - E^5 + 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[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs. \(2(20)=40\).
Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.90
method | result | size |
risch | \(x^{4}+10 x^{2}+6 x +\frac {\left (6 \,{\mathrm e}^{10}-240 \,{\mathrm e}^{5}+2430\right ) x -6 \,{\mathrm e}^{15}+360 \,{\mathrm e}^{10}-7230 \,{\mathrm e}^{5}+48609}{{\mathrm e}^{10}-2 x \,{\mathrm e}^{5}+x^{2}-40 \,{\mathrm e}^{5}+40 x +400}\) | \(61\) |
norman | \(\frac {x^{6}+\left (-20 \,{\mathrm e}^{5}+406\right ) x^{3}+\left (-2 \,{\mathrm e}^{5}+40\right ) x^{5}+\left ({\mathrm e}^{10}-40 \,{\mathrm e}^{5}+410\right ) x^{4}+\left (20 \,{\mathrm e}^{15}-1212 \,{\mathrm e}^{10}+24480 \,{\mathrm e}^{5}-164770\right ) x -10 \,{\mathrm e}^{20}+806 \,{\mathrm e}^{15}-24360 \,{\mathrm e}^{10}+327170 \,{\mathrm e}^{5}-1647391}{\left ({\mathrm e}^{5}-x -20\right )^{2}}\) | \(92\) |
default | \(x^{4}+10 x^{2}+6 x +2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 \,{\mathrm e}^{5}+60\right ) \textit {\_Z}^{2}+\left (3 \,{\mathrm e}^{10}-120 \,{\mathrm e}^{5}+1200\right ) \textit {\_Z} -{\mathrm e}^{15}+8000+60 \,{\mathrm e}^{10}-1200 \,{\mathrm e}^{5}\right )}{\sum }\frac {\left (-8103-{\mathrm e}^{10} \textit {\_R} +40 \textit {\_R} \,{\mathrm e}^{5}-60 \,{\mathrm e}^{10}+{\mathrm e}^{15}+1205 \,{\mathrm e}^{5}-405 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{400+{\mathrm e}^{10}-2 \textit {\_R} \,{\mathrm e}^{5}+\textit {\_R}^{2}-40 \,{\mathrm e}^{5}+40 \textit {\_R}}\right )\) | \(111\) |
gosper | \(-\frac {-{\mathrm e}^{10} x^{4}+2 x^{5} {\mathrm e}^{5}-x^{6}+40 x^{4} {\mathrm e}^{5}-40 x^{5}+10 \,{\mathrm e}^{20}-20 x \,{\mathrm e}^{15}+20 x^{3} {\mathrm e}^{5}-410 x^{4}-806 \,{\mathrm e}^{15}+1212 \,{\mathrm e}^{10} x -406 x^{3}+24360 \,{\mathrm e}^{10}-24480 x \,{\mathrm e}^{5}-327170 \,{\mathrm e}^{5}+164770 x +1647391}{{\mathrm e}^{10}-2 x \,{\mathrm e}^{5}+x^{2}-40 \,{\mathrm e}^{5}+40 x +400}\) | \(122\) |
parallelrisch | \(-\frac {-{\mathrm e}^{10} x^{4}+2 x^{5} {\mathrm e}^{5}-x^{6}+40 x^{4} {\mathrm e}^{5}-40 x^{5}+10 \,{\mathrm e}^{20}-20 x \,{\mathrm e}^{15}+20 x^{3} {\mathrm e}^{5}-410 x^{4}-806 \,{\mathrm e}^{15}+1212 \,{\mathrm e}^{10} x -406 x^{3}+24360 \,{\mathrm e}^{10}-24480 x \,{\mathrm e}^{5}-327170 \,{\mathrm e}^{5}+164770 x +1647391}{{\mathrm e}^{10}-2 x \,{\mathrm e}^{5}+x^{2}-40 \,{\mathrm e}^{5}+40 x +400}\) | \(122\) |
Input:
int(((4*x^3+20*x)*exp(5)^3+(-12*x^4-240*x^3-60*x^2-1212*x)*exp(5)^2+(12*x^ 5+480*x^4+4860*x^3+2418*x^2+24480*x-30)*exp(5)-4*x^6-240*x^5-4820*x^4-3320 6*x^3-24360*x^2-164770*x+618)/(exp(5)^3+(-3*x-60)*exp(5)^2+(3*x^2+120*x+12 00)*exp(5)-x^3-60*x^2-1200*x-8000),x,method=_RETURNVERBOSE)
Output:
x^4+10*x^2+6*x+((6*exp(10)-240*exp(5)+2430)*x-6*exp(15)+360*exp(10)-7230*e xp(5)+48609)/(exp(10)-2*x*exp(5)+x^2-40*exp(5)+40*x+400)
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (20) = 40\).
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 4.52 \[ \int \frac {618-164770 x-24360 x^2-33206 x^3-4820 x^4-240 x^5-4 x^6+e^{15} \left (20 x+4 x^3\right )+e^{10} \left (-1212 x-60 x^2-240 x^3-12 x^4\right )+e^5 \left (-30+24480 x+2418 x^2+4860 x^3+480 x^4+12 x^5\right )}{-8000+e^{15}+e^{10} (-60-3 x)-1200 x-60 x^2-x^3+e^5 \left (1200+120 x+3 x^2\right )} \, dx=\frac {x^{6} + 40 \, x^{5} + 410 \, x^{4} + 406 \, x^{3} + 4240 \, x^{2} + {\left (x^{4} + 10 \, x^{2} + 12 \, x + 360\right )} e^{10} - 2 \, {\left (x^{5} + 20 \, x^{4} + 10 \, x^{3} + 206 \, x^{2} + 240 \, x + 3615\right )} e^{5} + 4830 \, x - 6 \, e^{15} + 48609}{x^{2} - 2 \, {\left (x + 20\right )} e^{5} + 40 \, x + e^{10} + 400} \] Input:
integrate(((4*x^3+20*x)*exp(5)^3+(-12*x^4-240*x^3-60*x^2-1212*x)*exp(5)^2+ (12*x^5+480*x^4+4860*x^3+2418*x^2+24480*x-30)*exp(5)-4*x^6-240*x^5-4820*x^ 4-33206*x^3-24360*x^2-164770*x+618)/(exp(5)^3+(-3*x-60)*exp(5)^2+(3*x^2+12 0*x+1200)*exp(5)-x^3-60*x^2-1200*x-8000),x, algorithm="fricas")
Output:
(x^6 + 40*x^5 + 410*x^4 + 406*x^3 + 4240*x^2 + (x^4 + 10*x^2 + 12*x + 360) *e^10 - 2*(x^5 + 20*x^4 + 10*x^3 + 206*x^2 + 240*x + 3615)*e^5 + 4830*x - 6*e^15 + 48609)/(x^2 - 2*(x + 20)*e^5 + 40*x + e^10 + 400)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (15) = 30\).
Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.00 \[ \int \frac {618-164770 x-24360 x^2-33206 x^3-4820 x^4-240 x^5-4 x^6+e^{15} \left (20 x+4 x^3\right )+e^{10} \left (-1212 x-60 x^2-240 x^3-12 x^4\right )+e^5 \left (-30+24480 x+2418 x^2+4860 x^3+480 x^4+12 x^5\right )}{-8000+e^{15}+e^{10} (-60-3 x)-1200 x-60 x^2-x^3+e^5 \left (1200+120 x+3 x^2\right )} \, dx=x^{4} + 10 x^{2} + 6 x + \frac {x \left (- 240 e^{5} + 2430 + 6 e^{10}\right ) - 6 e^{15} - 7230 e^{5} + 48609 + 360 e^{10}}{x^{2} + x \left (40 - 2 e^{5}\right ) - 40 e^{5} + 400 + e^{10}} \] Input:
integrate(((4*x**3+20*x)*exp(5)**3+(-12*x**4-240*x**3-60*x**2-1212*x)*exp( 5)**2+(12*x**5+480*x**4+4860*x**3+2418*x**2+24480*x-30)*exp(5)-4*x**6-240* x**5-4820*x**4-33206*x**3-24360*x**2-164770*x+618)/(exp(5)**3+(-3*x-60)*ex p(5)**2+(3*x**2+120*x+1200)*exp(5)-x**3-60*x**2-1200*x-8000),x)
Output:
x**4 + 10*x**2 + 6*x + (x*(-240*exp(5) + 2430 + 6*exp(10)) - 6*exp(15) - 7 230*exp(5) + 48609 + 360*exp(10))/(x**2 + x*(40 - 2*exp(5)) - 40*exp(5) + 400 + exp(10))
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (20) = 40\).
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.81 \[ \int \frac {618-164770 x-24360 x^2-33206 x^3-4820 x^4-240 x^5-4 x^6+e^{15} \left (20 x+4 x^3\right )+e^{10} \left (-1212 x-60 x^2-240 x^3-12 x^4\right )+e^5 \left (-30+24480 x+2418 x^2+4860 x^3+480 x^4+12 x^5\right )}{-8000+e^{15}+e^{10} (-60-3 x)-1200 x-60 x^2-x^3+e^5 \left (1200+120 x+3 x^2\right )} \, dx=x^{4} + 10 \, x^{2} + 6 \, x + \frac {3 \, {\left (2 \, x {\left (e^{10} - 40 \, e^{5} + 405\right )} - 2 \, e^{15} + 120 \, e^{10} - 2410 \, e^{5} + 16203\right )}}{x^{2} - 2 \, x {\left (e^{5} - 20\right )} + e^{10} - 40 \, e^{5} + 400} \] Input:
integrate(((4*x^3+20*x)*exp(5)^3+(-12*x^4-240*x^3-60*x^2-1212*x)*exp(5)^2+ (12*x^5+480*x^4+4860*x^3+2418*x^2+24480*x-30)*exp(5)-4*x^6-240*x^5-4820*x^ 4-33206*x^3-24360*x^2-164770*x+618)/(exp(5)^3+(-3*x-60)*exp(5)^2+(3*x^2+12 0*x+1200)*exp(5)-x^3-60*x^2-1200*x-8000),x, algorithm="maxima")
Output:
x^4 + 10*x^2 + 6*x + 3*(2*x*(e^10 - 40*e^5 + 405) - 2*e^15 + 120*e^10 - 24 10*e^5 + 16203)/(x^2 - 2*x*(e^5 - 20) + e^10 - 40*e^5 + 400)
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (20) = 40\).
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.38 \[ \int \frac {618-164770 x-24360 x^2-33206 x^3-4820 x^4-240 x^5-4 x^6+e^{15} \left (20 x+4 x^3\right )+e^{10} \left (-1212 x-60 x^2-240 x^3-12 x^4\right )+e^5 \left (-30+24480 x+2418 x^2+4860 x^3+480 x^4+12 x^5\right )}{-8000+e^{15}+e^{10} (-60-3 x)-1200 x-60 x^2-x^3+e^5 \left (1200+120 x+3 x^2\right )} \, dx=x^{4} + 10 \, x^{2} + 6 \, x + \frac {3 \, {\left (2 \, x e^{10} - 80 \, x e^{5} + 810 \, x - 2 \, e^{15} + 120 \, e^{10} - 2410 \, e^{5} + 16203\right )}}{{\left (x - e^{5} + 20\right )}^{2}} \] Input:
integrate(((4*x^3+20*x)*exp(5)^3+(-12*x^4-240*x^3-60*x^2-1212*x)*exp(5)^2+ (12*x^5+480*x^4+4860*x^3+2418*x^2+24480*x-30)*exp(5)-4*x^6-240*x^5-4820*x^ 4-33206*x^3-24360*x^2-164770*x+618)/(exp(5)^3+(-3*x-60)*exp(5)^2+(3*x^2+12 0*x+1200)*exp(5)-x^3-60*x^2-1200*x-8000),x, algorithm="giac")
Output:
x^4 + 10*x^2 + 6*x + 3*(2*x*e^10 - 80*x*e^5 + 810*x - 2*e^15 + 120*e^10 - 2410*e^5 + 16203)/(x - e^5 + 20)^2
Time = 7.58 (sec) , antiderivative size = 126, normalized size of antiderivative = 6.00 \[ \int \frac {618-164770 x-24360 x^2-33206 x^3-4820 x^4-240 x^5-4 x^6+e^{15} \left (20 x+4 x^3\right )+e^{10} \left (-1212 x-60 x^2-240 x^3-12 x^4\right )+e^5 \left (-30+24480 x+2418 x^2+4860 x^3+480 x^4+12 x^5\right )}{-8000+e^{15}+e^{10} (-60-3 x)-1200 x-60 x^2-x^3+e^5 \left (1200+120 x+3 x^2\right )} \, dx=\frac {360\,{\mathrm {e}}^{10}-7230\,{\mathrm {e}}^5-6\,{\mathrm {e}}^{15}+x\,\left (6\,{\mathrm {e}}^{10}-240\,{\mathrm {e}}^5+2430\right )+48609}{x^2+\left (40-2\,{\mathrm {e}}^5\right )\,x-40\,{\mathrm {e}}^5+{\mathrm {e}}^{10}+400}-x^2\,\left (240\,{\mathrm {e}}^5-6\,{\mathrm {e}}^{10}+6\,{\left ({\mathrm {e}}^5-20\right )}^2-2410\right )+x^4-x\,\left (4860\,{\mathrm {e}}^5-240\,{\mathrm {e}}^{10}+4\,{\mathrm {e}}^{15}-4\,{\left ({\mathrm {e}}^5-20\right )}^3+\left (3\,{\mathrm {e}}^5-60\right )\,\left (480\,{\mathrm {e}}^5-12\,{\mathrm {e}}^{10}+12\,{\left ({\mathrm {e}}^5-20\right )}^2-4820\right )-33206\right ) \] Input:
int((164770*x - exp(15)*(20*x + 4*x^3) + exp(10)*(1212*x + 60*x^2 + 240*x^ 3 + 12*x^4) - exp(5)*(24480*x + 2418*x^2 + 4860*x^3 + 480*x^4 + 12*x^5 - 3 0) + 24360*x^2 + 33206*x^3 + 4820*x^4 + 240*x^5 + 4*x^6 - 618)/(1200*x - e xp(15) - exp(5)*(120*x + 3*x^2 + 1200) + 60*x^2 + x^3 + exp(10)*(3*x + 60) + 8000),x)
Output:
(360*exp(10) - 7230*exp(5) - 6*exp(15) + x*(6*exp(10) - 240*exp(5) + 2430) + 48609)/(exp(10) - 40*exp(5) + x^2 - x*(2*exp(5) - 40) + 400) - x^2*(240 *exp(5) - 6*exp(10) + 6*(exp(5) - 20)^2 - 2410) + x^4 - x*(4860*exp(5) - 2 40*exp(10) + 4*exp(15) - 4*(exp(5) - 20)^3 + (3*exp(5) - 60)*(480*exp(5) - 12*exp(10) + 12*(exp(5) - 20)^2 - 4820) - 33206)
Time = 0.22 (sec) , antiderivative size = 168, normalized size of antiderivative = 8.00 \[ \int \frac {618-164770 x-24360 x^2-33206 x^3-4820 x^4-240 x^5-4 x^6+e^{15} \left (20 x+4 x^3\right )+e^{10} \left (-1212 x-60 x^2-240 x^3-12 x^4\right )+e^5 \left (-30+24480 x+2418 x^2+4860 x^3+480 x^4+12 x^5\right )}{-8000+e^{15}+e^{10} (-60-3 x)-1200 x-60 x^2-x^3+e^5 \left (1200+120 x+3 x^2\right )} \, dx=\frac {e^{15} x^{4}+10 e^{15} x^{2}-2 e^{10} x^{5}-60 e^{10} x^{4}-20 e^{10} x^{3}-606 e^{10} x^{2}+e^{5} x^{6}-15 e^{10}+80 e^{5} x^{5}+1210 e^{5} x^{4}+806 e^{5} x^{3}+12240 e^{5} x^{2}-20 x^{6}+609 e^{5}-800 x^{5}-8200 x^{4}-8120 x^{3}-82385 x^{2}-6180}{e^{15}-2 e^{10} x -60 e^{10}+e^{5} x^{2}+80 e^{5} x +1200 e^{5}-20 x^{2}-800 x -8000} \] Input:
int(((4*x^3+20*x)*exp(5)^3+(-12*x^4-240*x^3-60*x^2-1212*x)*exp(5)^2+(12*x^ 5+480*x^4+4860*x^3+2418*x^2+24480*x-30)*exp(5)-4*x^6-240*x^5-4820*x^4-3320 6*x^3-24360*x^2-164770*x+618)/(exp(5)^3+(-3*x-60)*exp(5)^2+(3*x^2+120*x+12 00)*exp(5)-x^3-60*x^2-1200*x-8000),x)
Output:
(e**15*x**4 + 10*e**15*x**2 - 2*e**10*x**5 - 60*e**10*x**4 - 20*e**10*x**3 - 606*e**10*x**2 - 15*e**10 + e**5*x**6 + 80*e**5*x**5 + 1210*e**5*x**4 + 806*e**5*x**3 + 12240*e**5*x**2 + 609*e**5 - 20*x**6 - 800*x**5 - 8200*x* *4 - 8120*x**3 - 82385*x**2 - 6180)/(e**15 - 2*e**10*x - 60*e**10 + e**5*x **2 + 80*e**5*x + 1200*e**5 - 20*x**2 - 800*x - 8000)