Integrand size = 195, antiderivative size = 22 \[ \int \frac {-1476225+25 e^{15}+4920714 x-6561024 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2186996 x+2187000 x^2-972000 x^3+162000 x^4\right )}{-1476225+25 e^{15}+4920750 x-6561000 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2187000 x+2187000 x^2-972000 x^3+162000 x^4\right )} \, dx=x+\frac {2 x^2}{25 \left (e^3-3 (3-2 x)\right )^4} \] Output:
x+2/25*x^2/(exp(3)+6*x-9)^4
Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(22)=44\).
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.73 \[ \int \frac {-1476225+25 e^{15}+4920714 x-6561024 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2186996 x+2187000 x^2-972000 x^3+162000 x^4\right )}{-1476225+25 e^{15}+4920750 x-6561000 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2187000 x+2187000 x^2-972000 x^3+162000 x^4\right )} \, dx=\frac {81-e^6+6 e^3 (3-2 x)+54 (-3+2 x)+\left (-9+e^3+6 x\right )^2+75 \left (-9+e^3+6 x\right )^5}{450 \left (-9+e^3+6 x\right )^4} \] Input:
Integrate[(-1476225 + 25*E^15 + 4920714*x - 6561024*x^2 + 4374000*x^3 - 14 58000*x^4 + 194400*x^5 + E^12*(-1125 + 750*x) + E^9*(20250 - 27000*x + 900 0*x^2) + E^6*(-182250 + 364500*x - 243000*x^2 + 54000*x^3) + E^3*(820125 - 2186996*x + 2187000*x^2 - 972000*x^3 + 162000*x^4))/(-1476225 + 25*E^15 + 4920750*x - 6561000*x^2 + 4374000*x^3 - 1458000*x^4 + 194400*x^5 + E^12*( -1125 + 750*x) + E^9*(20250 - 27000*x + 9000*x^2) + E^6*(-182250 + 364500* x - 243000*x^2 + 54000*x^3) + E^3*(820125 - 2187000*x + 2187000*x^2 - 9720 00*x^3 + 162000*x^4)),x]
Output:
(81 - E^6 + 6*E^3*(3 - 2*x) + 54*(-3 + 2*x) + (-9 + E^3 + 6*x)^2 + 75*(-9 + E^3 + 6*x)^5)/(450*(-9 + E^3 + 6*x)^4)
Leaf count is larger than twice the leaf count of optimal. \(66\) vs. \(2(22)=44\).
Time = 0.47 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, 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.
| \(\displaystyle \int \frac {194400 x^5-1458000 x^4+4374000 x^3-6561024 x^2+e^9 \left (9000 x^2-27000 x+20250\right )+e^6 \left (54000 x^3-243000 x^2+364500 x-182250\right )+e^3 \left (162000 x^4-972000 x^3+2187000 x^2-2186996 x+820125\right )+4920714 x+e^{12} (750 x-1125)+25 e^{15}-1476225}{194400 x^5-1458000 x^4+4374000 x^3-6561000 x^2+e^9 \left (9000 x^2-27000 x+20250\right )+e^6 \left (54000 x^3-243000 x^2+364500 x-182250\right )+e^3 \left (162000 x^4-972000 x^3+2187000 x^2-2187000 x+820125\right )+4920750 x+e^{12} (750 x-1125)+25 e^{15}-1476225} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {194400 x^5-1458000 x^4+4374000 x^3-6561024 x^2+e^9 \left (9000 x^2-27000 x+20250\right )+e^6 \left (54000 x^3-243000 x^2+364500 x-182250\right )+e^3 \left (162000 x^4-972000 x^3+2187000 x^2-2186996 x+820125\right )+4920714 x+e^{12} (750 x-1125)+25 e^{15}-1476225}{\left (6\ 5^{2/5} x+5^{2/5} \left (e^3-9\right )\right )^5}dx\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \int \left (-\frac {2}{75 \left (6 x+e^3-9\right )^3}+\frac {2 \left (e^3-9\right )}{25 \left (6 x+e^3-9\right )^4}-\frac {4 \left (e^3-9\right )^2}{75 \left (6 x+e^3-9\right )^5}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x+\frac {1}{450 \left (-6 x-e^3+9\right )^2}-\frac {9-e^3}{225 \left (-6 x-e^3+9\right )^3}+\frac {\left (9-e^3\right )^2}{450 \left (-6 x-e^3+9\right )^4}\) |
Input:
Int[(-1476225 + 25*E^15 + 4920714*x - 6561024*x^2 + 4374000*x^3 - 1458000* x^4 + 194400*x^5 + E^12*(-1125 + 750*x) + E^9*(20250 - 27000*x + 9000*x^2) + E^6*(-182250 + 364500*x - 243000*x^2 + 54000*x^3) + E^3*(820125 - 21869 96*x + 2187000*x^2 - 972000*x^3 + 162000*x^4))/(-1476225 + 25*E^15 + 49207 50*x - 6561000*x^2 + 4374000*x^3 - 1458000*x^4 + 194400*x^5 + E^12*(-1125 + 750*x) + E^9*(20250 - 27000*x + 9000*x^2) + E^6*(-182250 + 364500*x - 24 3000*x^2 + 54000*x^3) + E^3*(820125 - 2187000*x + 2187000*x^2 - 972000*x^3 + 162000*x^4)),x]
Output:
(9 - E^3)^2/(450*(9 - E^3 - 6*x)^4) - (9 - E^3)/(225*(9 - E^3 - 6*x)^3) + 1/(450*(9 - E^3 - 6*x)^2) + 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. \(79\) vs. \(2(16)=32\).
Time = 0.59 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.64
| method | result | size |
| risch | \(x +\frac {2 x^{2}}{25 \left ({\mathrm e}^{12}+24 x \,{\mathrm e}^{9}+216 x^{2} {\mathrm e}^{6}+864 x^{3} {\mathrm e}^{3}+1296 x^{4}-36 \,{\mathrm e}^{9}-648 x \,{\mathrm e}^{6}-3888 x^{2} {\mathrm e}^{3}-7776 x^{3}+486 \,{\mathrm e}^{6}+5832 x \,{\mathrm e}^{3}+17496 x^{2}-2916 \,{\mathrm e}^{3}-17496 x +6561\right )}\) | \(80\) |
| norman | \(\frac {\left (-360 \,{\mathrm e}^{6}+6480 \,{\mathrm e}^{3}-29160\right ) x^{3}+\left (-120 \,{\mathrm e}^{9}+3240 \,{\mathrm e}^{6}-29160 \,{\mathrm e}^{3}+\frac {2187002}{25}\right ) x^{2}+\left (-15 \,{\mathrm e}^{12}+540 \,{\mathrm e}^{9}-7290 \,{\mathrm e}^{6}+43740 \,{\mathrm e}^{3}-98415\right ) x +1296 x^{5}-\frac {2 \,{\mathrm e}^{15}}{3}+30 \,{\mathrm e}^{12}-540 \,{\mathrm e}^{9}+4860 \,{\mathrm e}^{6}-21870 \,{\mathrm e}^{3}+39366}{\left ({\mathrm e}^{3}+6 x -9\right )^{4}}\) | \(110\) |
| default | \(x +\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (7776 \textit {\_Z}^{5}+\left (6480 \,{\mathrm e}^{3}-58320\right ) \textit {\_Z}^{4}+\left (2160 \,{\mathrm e}^{6}-38880 \,{\mathrm e}^{3}+174960\right ) \textit {\_Z}^{3}+\left (360 \,{\mathrm e}^{9}-9720 \,{\mathrm e}^{6}+87480 \,{\mathrm e}^{3}-262440\right ) \textit {\_Z}^{2}+\left (30 \,{\mathrm e}^{12}-1080 \,{\mathrm e}^{9}+14580 \,{\mathrm e}^{6}-87480 \,{\mathrm e}^{3}+196830\right ) \textit {\_Z} +{\mathrm e}^{15}-45 \,{\mathrm e}^{12}+810 \,{\mathrm e}^{9}-7290 \,{\mathrm e}^{6}+32805 \,{\mathrm e}^{3}-59049\right )}{\sum }\frac {\left (-6 \textit {\_R}^{2}+\left ({\mathrm e}^{3}-9\right ) \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{6561+{\mathrm e}^{12}+24 \textit {\_R} \,{\mathrm e}^{9}+216 \textit {\_R}^{2} {\mathrm e}^{6}+864 \textit {\_R}^{3} {\mathrm e}^{3}+1296 \textit {\_R}^{4}-36 \,{\mathrm e}^{9}-648 \textit {\_R} \,{\mathrm e}^{6}-3888 \textit {\_R}^{2} {\mathrm e}^{3}-7776 \textit {\_R}^{3}+486 \,{\mathrm e}^{6}+5832 \textit {\_R} \,{\mathrm e}^{3}+17496 \textit {\_R}^{2}-2916 \,{\mathrm e}^{3}-17496 \textit {\_R}}\right )}{375}\) | \(187\) |
| gosper | \(-\frac {50 \,{\mathrm e}^{15}+1125 x \,{\mathrm e}^{12}+9000 x^{2} {\mathrm e}^{9}+27000 x^{3} {\mathrm e}^{6}-97200 x^{5}-2250 \,{\mathrm e}^{12}-40500 x \,{\mathrm e}^{9}-243000 x^{2} {\mathrm e}^{6}-486000 x^{3} {\mathrm e}^{3}+40500 \,{\mathrm e}^{9}+546750 x \,{\mathrm e}^{6}+2187000 x^{2} {\mathrm e}^{3}+2187000 x^{3}-364500 \,{\mathrm e}^{6}-3280500 x \,{\mathrm e}^{3}-6561006 x^{2}+1640250 \,{\mathrm e}^{3}+7381125 x -2952450}{75 \left ({\mathrm e}^{12}+24 x \,{\mathrm e}^{9}+216 x^{2} {\mathrm e}^{6}+864 x^{3} {\mathrm e}^{3}+1296 x^{4}-36 \,{\mathrm e}^{9}-648 x \,{\mathrm e}^{6}-3888 x^{2} {\mathrm e}^{3}-7776 x^{3}+486 \,{\mathrm e}^{6}+5832 x \,{\mathrm e}^{3}+17496 x^{2}-2916 \,{\mathrm e}^{3}-17496 x +6561\right )}\) | \(202\) |
| parallelrisch | \(-\frac {-1275458400+3188646000 x -104976000 x^{2} {\mathrm e}^{6}+11664000 x^{3} {\mathrm e}^{6}-209952000 x^{3} {\mathrm e}^{3}+3888000 x^{2} {\mathrm e}^{9}-1417176000 x \,{\mathrm e}^{3}+236196000 x \,{\mathrm e}^{6}+944784000 x^{2} {\mathrm e}^{3}-17496000 x \,{\mathrm e}^{9}+486000 x \,{\mathrm e}^{12}+21600 \,{\mathrm e}^{15}-972000 \,{\mathrm e}^{12}+17496000 \,{\mathrm e}^{9}-41990400 x^{5}-2834354592 x^{2}+944784000 x^{3}+708588000 \,{\mathrm e}^{3}-157464000 \,{\mathrm e}^{6}}{32400 \left ({\mathrm e}^{12}+24 x \,{\mathrm e}^{9}+216 x^{2} {\mathrm e}^{6}+864 x^{3} {\mathrm e}^{3}+1296 x^{4}-36 \,{\mathrm e}^{9}-648 x \,{\mathrm e}^{6}-3888 x^{2} {\mathrm e}^{3}-7776 x^{3}+486 \,{\mathrm e}^{6}+5832 x \,{\mathrm e}^{3}+17496 x^{2}-2916 \,{\mathrm e}^{3}-17496 x +6561\right )}\) | \(202\) |
Input:
int((25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp(3)^3+( 54000*x^3-243000*x^2+364500*x-182250)*exp(3)^2+(162000*x^4-972000*x^3+2187 000*x^2-2186996*x+820125)*exp(3)+194400*x^5-1458000*x^4+4374000*x^3-656102 4*x^2+4920714*x-1476225)/(25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-2700 0*x+20250)*exp(3)^3+(54000*x^3-243000*x^2+364500*x-182250)*exp(3)^2+(16200 0*x^4-972000*x^3+2187000*x^2-2187000*x+820125)*exp(3)+194400*x^5-1458000*x ^4+4374000*x^3-6561000*x^2+4920750*x-1476225),x,method=_RETURNVERBOSE)
Output:
x+2/25*x^2/(exp(12)+24*x*exp(9)+216*x^2*exp(6)+864*x^3*exp(3)+1296*x^4-36* exp(9)-648*x*exp(6)-3888*x^2*exp(3)-7776*x^3+486*exp(6)+5832*x*exp(3)+1749 6*x^2-2916*exp(3)-17496*x+6561)
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (16) = 32\).
Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 6.86 \[ \int \frac {-1476225+25 e^{15}+4920714 x-6561024 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2186996 x+2187000 x^2-972000 x^3+162000 x^4\right )}{-1476225+25 e^{15}+4920750 x-6561000 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2187000 x+2187000 x^2-972000 x^3+162000 x^4\right )} \, dx=\frac {32400 \, x^{5} - 194400 \, x^{4} + 437400 \, x^{3} - 437398 \, x^{2} + 25 \, x e^{12} + 300 \, {\left (2 \, x^{2} - 3 \, x\right )} e^{9} + 1350 \, {\left (4 \, x^{3} - 12 \, x^{2} + 9 \, x\right )} e^{6} + 2700 \, {\left (8 \, x^{4} - 36 \, x^{3} + 54 \, x^{2} - 27 \, x\right )} e^{3} + 164025 \, x}{25 \, {\left (1296 \, x^{4} - 7776 \, x^{3} + 17496 \, x^{2} + 12 \, {\left (2 \, x - 3\right )} e^{9} + 54 \, {\left (4 \, x^{2} - 12 \, x + 9\right )} e^{6} + 108 \, {\left (8 \, x^{3} - 36 \, x^{2} + 54 \, x - 27\right )} e^{3} - 17496 \, x + e^{12} + 6561\right )}} \] Input:
integrate((25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp( 3)^3+(54000*x^3-243000*x^2+364500*x-182250)*exp(3)^2+(162000*x^4-972000*x^ 3+2187000*x^2-2186996*x+820125)*exp(3)+194400*x^5-1458000*x^4+4374000*x^3- 6561024*x^2+4920714*x-1476225)/(25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^ 2-27000*x+20250)*exp(3)^3+(54000*x^3-243000*x^2+364500*x-182250)*exp(3)^2+ (162000*x^4-972000*x^3+2187000*x^2-2187000*x+820125)*exp(3)+194400*x^5-145 8000*x^4+4374000*x^3-6561000*x^2+4920750*x-1476225),x, algorithm="fricas")
Output:
1/25*(32400*x^5 - 194400*x^4 + 437400*x^3 - 437398*x^2 + 25*x*e^12 + 300*( 2*x^2 - 3*x)*e^9 + 1350*(4*x^3 - 12*x^2 + 9*x)*e^6 + 2700*(8*x^4 - 36*x^3 + 54*x^2 - 27*x)*e^3 + 164025*x)/(1296*x^4 - 7776*x^3 + 17496*x^2 + 12*(2* x - 3)*e^9 + 54*(4*x^2 - 12*x + 9)*e^6 + 108*(8*x^3 - 36*x^2 + 54*x - 27)* e^3 - 17496*x + e^12 + 6561)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (17) = 34\).
Time = 0.55 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.45 \[ \int \frac {-1476225+25 e^{15}+4920714 x-6561024 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2186996 x+2187000 x^2-972000 x^3+162000 x^4\right )}{-1476225+25 e^{15}+4920750 x-6561000 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2187000 x+2187000 x^2-972000 x^3+162000 x^4\right )} \, dx=\frac {2 x^{2}}{32400 x^{4} + x^{3} \left (-194400 + 21600 e^{3}\right ) + x^{2} \left (- 97200 e^{3} + 437400 + 5400 e^{6}\right ) + x \left (- 16200 e^{6} - 437400 + 145800 e^{3} + 600 e^{9}\right ) - 900 e^{9} - 72900 e^{3} + 164025 + 25 e^{12} + 12150 e^{6}} + x \] Input:
integrate((25*exp(3)**5+(750*x-1125)*exp(3)**4+(9000*x**2-27000*x+20250)*e xp(3)**3+(54000*x**3-243000*x**2+364500*x-182250)*exp(3)**2+(162000*x**4-9 72000*x**3+2187000*x**2-2186996*x+820125)*exp(3)+194400*x**5-1458000*x**4+ 4374000*x**3-6561024*x**2+4920714*x-1476225)/(25*exp(3)**5+(750*x-1125)*ex p(3)**4+(9000*x**2-27000*x+20250)*exp(3)**3+(54000*x**3-243000*x**2+364500 *x-182250)*exp(3)**2+(162000*x**4-972000*x**3+2187000*x**2-2187000*x+82012 5)*exp(3)+194400*x**5-1458000*x**4+4374000*x**3-6561000*x**2+4920750*x-147 6225),x)
Output:
2*x**2/(32400*x**4 + x**3*(-194400 + 21600*exp(3)) + x**2*(-97200*exp(3) + 437400 + 5400*exp(6)) + x*(-16200*exp(6) - 437400 + 145800*exp(3) + 600*e xp(9)) - 900*exp(9) - 72900*exp(3) + 164025 + 25*exp(12) + 12150*exp(6)) + x
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (16) = 32\).
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.05 \[ \int \frac {-1476225+25 e^{15}+4920714 x-6561024 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2186996 x+2187000 x^2-972000 x^3+162000 x^4\right )}{-1476225+25 e^{15}+4920750 x-6561000 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2187000 x+2187000 x^2-972000 x^3+162000 x^4\right )} \, dx=x + \frac {2 \, x^{2}}{25 \, {\left (1296 \, x^{4} + 864 \, x^{3} {\left (e^{3} - 9\right )} + 216 \, x^{2} {\left (e^{6} - 18 \, e^{3} + 81\right )} + 24 \, x {\left (e^{9} - 27 \, e^{6} + 243 \, e^{3} - 729\right )} + e^{12} - 36 \, e^{9} + 486 \, e^{6} - 2916 \, e^{3} + 6561\right )}} \] Input:
integrate((25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp( 3)^3+(54000*x^3-243000*x^2+364500*x-182250)*exp(3)^2+(162000*x^4-972000*x^ 3+2187000*x^2-2186996*x+820125)*exp(3)+194400*x^5-1458000*x^4+4374000*x^3- 6561024*x^2+4920714*x-1476225)/(25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^ 2-27000*x+20250)*exp(3)^3+(54000*x^3-243000*x^2+364500*x-182250)*exp(3)^2+ (162000*x^4-972000*x^3+2187000*x^2-2187000*x+820125)*exp(3)+194400*x^5-145 8000*x^4+4374000*x^3-6561000*x^2+4920750*x-1476225),x, algorithm="maxima")
Output:
x + 2/25*x^2/(1296*x^4 + 864*x^3*(e^3 - 9) + 216*x^2*(e^6 - 18*e^3 + 81) + 24*x*(e^9 - 27*e^6 + 243*e^3 - 729) + e^12 - 36*e^9 + 486*e^6 - 2916*e^3 + 6561)
Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {-1476225+25 e^{15}+4920714 x-6561024 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2186996 x+2187000 x^2-972000 x^3+162000 x^4\right )}{-1476225+25 e^{15}+4920750 x-6561000 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2187000 x+2187000 x^2-972000 x^3+162000 x^4\right )} \, dx=x + \frac {2 \, x^{2}}{25 \, {\left (6 \, x + e^{3} - 9\right )}^{4}} \] Input:
integrate((25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp( 3)^3+(54000*x^3-243000*x^2+364500*x-182250)*exp(3)^2+(162000*x^4-972000*x^ 3+2187000*x^2-2186996*x+820125)*exp(3)+194400*x^5-1458000*x^4+4374000*x^3- 6561024*x^2+4920714*x-1476225)/(25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^ 2-27000*x+20250)*exp(3)^3+(54000*x^3-243000*x^2+364500*x-182250)*exp(3)^2+ (162000*x^4-972000*x^3+2187000*x^2-2187000*x+820125)*exp(3)+194400*x^5-145 8000*x^4+4374000*x^3-6561000*x^2+4920750*x-1476225),x, algorithm="giac")
Output:
x + 2/25*x^2/(6*x + e^3 - 9)^4
Time = 3.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.82 \[ \int \frac {-1476225+25 e^{15}+4920714 x-6561024 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2186996 x+2187000 x^2-972000 x^3+162000 x^4\right )}{-1476225+25 e^{15}+4920750 x-6561000 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2187000 x+2187000 x^2-972000 x^3+162000 x^4\right )} \, dx=\frac {x\,\left (12150\,{\mathrm {e}}^6-72900\,{\mathrm {e}}^3-437398\,x-900\,{\mathrm {e}}^9+25\,{\mathrm {e}}^{12}+145800\,x\,{\mathrm {e}}^3-16200\,x\,{\mathrm {e}}^6+600\,x\,{\mathrm {e}}^9-97200\,x^2\,{\mathrm {e}}^3+21600\,x^3\,{\mathrm {e}}^3+5400\,x^2\,{\mathrm {e}}^6+437400\,x^2-194400\,x^3+32400\,x^4+164025\right )}{25\,{\left (6\,x+{\mathrm {e}}^3-9\right )}^4} \] Input:
int((4920714*x + 25*exp(15) + exp(9)*(9000*x^2 - 27000*x + 20250) + exp(6) *(364500*x - 243000*x^2 + 54000*x^3 - 182250) + exp(3)*(2187000*x^2 - 2186 996*x - 972000*x^3 + 162000*x^4 + 820125) - 6561024*x^2 + 4374000*x^3 - 14 58000*x^4 + 194400*x^5 + exp(12)*(750*x - 1125) - 1476225)/(4920750*x + 25 *exp(15) + exp(9)*(9000*x^2 - 27000*x + 20250) + exp(6)*(364500*x - 243000 *x^2 + 54000*x^3 - 182250) + exp(3)*(2187000*x^2 - 2187000*x - 972000*x^3 + 162000*x^4 + 820125) - 6561000*x^2 + 4374000*x^3 - 1458000*x^4 + 194400* x^5 + exp(12)*(750*x - 1125) - 1476225),x)
Output:
(x*(12150*exp(6) - 72900*exp(3) - 437398*x - 900*exp(9) + 25*exp(12) + 145 800*x*exp(3) - 16200*x*exp(6) + 600*x*exp(9) - 97200*x^2*exp(3) + 21600*x^ 3*exp(3) + 5400*x^2*exp(6) + 437400*x^2 - 194400*x^3 + 32400*x^4 + 164025) )/(25*(6*x + exp(3) - 9)^4)
Time = 0.17 (sec) , antiderivative size = 186, normalized size of antiderivative = 8.45 \[ \int \frac {-1476225+25 e^{15}+4920714 x-6561024 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2186996 x+2187000 x^2-972000 x^3+162000 x^4\right )}{-1476225+25 e^{15}+4920750 x-6561000 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 \left (20250-27000 x+9000 x^2\right )+e^6 \left (-182250+364500 x-243000 x^2+54000 x^3\right )+e^3 \left (820125-2187000 x+2187000 x^2-972000 x^3+162000 x^4\right )} \, dx=\frac {-25 e^{15}-500 e^{12} x +1125 e^{12}-3000 e^{9} x^{2}+18000 e^{9} x -20250 e^{9}+81000 e^{6} x^{2}-243000 e^{6} x +54000 e^{3} x^{4}+182250 e^{6}-729000 e^{3} x^{2}+129600 x^{5}+1458000 e^{3} x -486000 x^{4}-820125 e^{3}+2187008 x^{2}-3280500 x +1476225}{100 e^{12}+2400 e^{9} x -3600 e^{9}+21600 e^{6} x^{2}-64800 e^{6} x +48600 e^{6}+86400 e^{3} x^{3}-388800 e^{3} x^{2}+583200 e^{3} x +129600 x^{4}-291600 e^{3}-777600 x^{3}+1749600 x^{2}-1749600 x +656100} \] Input:
int((25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp(3)^3+( 54000*x^3-243000*x^2+364500*x-182250)*exp(3)^2+(162000*x^4-972000*x^3+2187 000*x^2-2186996*x+820125)*exp(3)+194400*x^5-1458000*x^4+4374000*x^3-656102 4*x^2+4920714*x-1476225)/(25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-2700 0*x+20250)*exp(3)^3+(54000*x^3-243000*x^2+364500*x-182250)*exp(3)^2+(16200 0*x^4-972000*x^3+2187000*x^2-2187000*x+820125)*exp(3)+194400*x^5-1458000*x ^4+4374000*x^3-6561000*x^2+4920750*x-1476225),x)
Output:
( - 25*e**15 - 500*e**12*x + 1125*e**12 - 3000*e**9*x**2 + 18000*e**9*x - 20250*e**9 + 81000*e**6*x**2 - 243000*e**6*x + 182250*e**6 + 54000*e**3*x* *4 - 729000*e**3*x**2 + 1458000*e**3*x - 820125*e**3 + 129600*x**5 - 48600 0*x**4 + 2187008*x**2 - 3280500*x + 1476225)/(100*(e**12 + 24*e**9*x - 36* e**9 + 216*e**6*x**2 - 648*e**6*x + 486*e**6 + 864*e**3*x**3 - 3888*e**3*x **2 + 5832*e**3*x - 2916*e**3 + 1296*x**4 - 7776*x**3 + 17496*x**2 - 17496 *x + 6561))