Integrand size = 120, antiderivative size = 24 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=e^x+x^2-\frac {75}{x \left (-7-x-x^2\right )^2} \] Output:
exp(x)+x^2-3/x/(-1/5*x^2-1/5*x-7/5)^2
Time = 2.96 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=e^x+\frac {-75+49 x^3+14 x^4+15 x^5+2 x^6+x^7}{x \left (7+x+x^2\right )^2} \] Input:
Integrate[(525 + 225*x + 375*x^2 + 686*x^3 + 294*x^4 + 336*x^5 + 86*x^6 + 48*x^7 + 6*x^8 + 2*x^9 + E^x*(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^ 6 + 3*x^7 + x^8))/(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3*x^7 + x^8),x]
Output:
E^x + (-75 + 49*x^3 + 14*x^4 + 15*x^5 + 2*x^6 + x^7)/(x*(7 + x + x^2)^2)
Leaf count is larger than twice the leaf count of optimal. \(354\) vs. \(2(24)=48\).
Time = 12.25 (sec) , antiderivative size = 354, normalized size of antiderivative = 14.75, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2026, 2463, 7239, 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.
| \(\displaystyle \int \frac {2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525}{x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525}{x^2 \left (x^6+3 x^5+24 x^4+43 x^3+168 x^2+147 x+343\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {4 i \left (2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525\right )}{729 \sqrt {3} x^2 \left (2 x+3 i \sqrt {3}+1\right )}+\frac {4 i \left (2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525\right )}{729 \sqrt {3} \left (-2 x+3 i \sqrt {3}-1\right ) x^2}-\frac {4 \left (2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525\right )}{243 \left (-2 x+3 i \sqrt {3}-1\right )^2 x^2}-\frac {8 i \left (2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525\right )}{81 \sqrt {3} \left (-2 x+3 i \sqrt {3}-1\right )^3 x^2}-\frac {4 \left (2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525\right )}{243 x^2 \left (2 x+3 i \sqrt {3}+1\right )^2}-\frac {8 i \left (2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525\right )}{81 \sqrt {3} x^2 \left (2 x+3 i \sqrt {3}+1\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 x^9+\left (e^x+6\right ) x^8+3 \left (e^x+16\right ) x^7+\left (24 e^x+86\right ) x^6+\left (43 e^x+336\right ) x^5+42 \left (4 e^x+7\right ) x^4+49 \left (3 e^x+14\right ) x^3+\left (343 e^x+375\right ) x^2+225 x+525}{x^2 \left (x^2+x+7\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {294 x^2}{\left (x^2+x+7\right )^3}+\frac {686 x}{\left (x^2+x+7\right )^3}+\frac {375}{\left (x^2+x+7\right )^3}+\frac {225}{\left (x^2+x+7\right )^3 x}+\frac {525}{\left (x^2+x+7\right )^3 x^2}+\frac {2 x^7}{\left (x^2+x+7\right )^3}+\frac {6 x^6}{\left (x^2+x+7\right )^3}+\frac {48 x^5}{\left (x^2+x+7\right )^3}+\frac {86 x^4}{\left (x^2+x+7\right )^3}+\frac {336 x^3}{\left (x^2+x+7\right )^3}+e^x\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {50 x^3}{243}-\frac {8 (23 x+259) x^2}{81 \left (x^2+x+7\right )}+\frac {818 x^2}{243}-\frac {434 (x+14) x}{243 \left (x^2+x+7\right )}-\frac {49 (x+14) x}{9 \left (x^2+x+7\right )^2}+\frac {25 (227-32 x)}{2646 \left (x^2+x+7\right )}+\frac {32 (2 x+1)}{243 \left (x^2+x+7\right )}+\frac {98 (5 x+7)}{81 \left (x^2+x+7\right )}+\frac {25 (13-x)}{42 \left (x^2+x+7\right )^2}-\frac {343 (x+14)}{27 \left (x^2+x+7\right )^2}+\frac {125 (2 x+1)}{18 \left (x^2+x+7\right )^2}+\frac {25 (281-41 x)}{1134 \left (x^2+x+7\right ) x}+\frac {25 (13-x)}{18 \left (x^2+x+7\right )^2 x}-\frac {(x+14) x^6}{27 \left (x^2+x+7\right )^2}-\frac {(x+14) x^5}{9 \left (x^2+x+7\right )^2}-\frac {(41 x+385) x^4}{243 \left (x^2+x+7\right )}-\frac {8 (x+14) x^4}{9 \left (x^2+x+7\right )^2}-\frac {2 (16 x+161) x^3}{81 \left (x^2+x+7\right )}-\frac {43 (x+14) x^3}{27 \left (x^2+x+7\right )^2}+\frac {56 (2 x+1) x^3}{9 \left (x^2+x+7\right )^2}+\frac {980 x}{243}+e^x-\frac {11050}{3969 x}\) |
Input:
Int[(525 + 225*x + 375*x^2 + 686*x^3 + 294*x^4 + 336*x^5 + 86*x^6 + 48*x^7 + 6*x^8 + 2*x^9 + E^x*(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3* x^7 + x^8))/(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3*x^7 + x^8), x]
Output:
E^x - 11050/(3969*x) + (980*x)/243 + (818*x^2)/243 + (50*x^3)/243 + (25*(1 3 - x))/(42*(7 + x + x^2)^2) + (25*(13 - x))/(18*x*(7 + x + x^2)^2) - (343 *(14 + x))/(27*(7 + x + x^2)^2) - (49*x*(14 + x))/(9*(7 + x + x^2)^2) - (4 3*x^3*(14 + x))/(27*(7 + x + x^2)^2) - (8*x^4*(14 + x))/(9*(7 + x + x^2)^2 ) - (x^5*(14 + x))/(9*(7 + x + x^2)^2) - (x^6*(14 + x))/(27*(7 + x + x^2)^ 2) + (125*(1 + 2*x))/(18*(7 + x + x^2)^2) + (56*x^3*(1 + 2*x))/(9*(7 + x + x^2)^2) + (25*(227 - 32*x))/(2646*(7 + x + x^2)) + (25*(281 - 41*x))/(113 4*x*(7 + x + x^2)) - (434*x*(14 + x))/(243*(7 + x + x^2)) + (32*(1 + 2*x)) /(243*(7 + x + x^2)) + (98*(7 + 5*x))/(81*(7 + x + x^2)) - (2*x^3*(161 + 1 6*x))/(81*(7 + x + x^2)) - (8*x^2*(259 + 23*x))/(81*(7 + x + x^2)) - (x^4* (385 + 41*x))/(243*(7 + x + x^2))
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 2.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(x^{2}-\frac {75}{x \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+{\mathrm e}^{x}\) | \(32\) |
| parts | \(x^{2}-\frac {75 \left (-x^{3}-2 x^{2}-15 x -14\right )}{49 \left (x^{2}+x +7\right )^{2}}-\frac {75}{49 x}+{\mathrm e}^{x}\) | \(37\) |
| norman | \(\frac {-75+x^{7}+x^{5} {\mathrm e}^{x}-735 x -210 x^{2}-176 x^{3}-16 x^{4}+2 x^{6}+49 \,{\mathrm e}^{x} x +14 \,{\mathrm e}^{x} x^{2}+15 \,{\mathrm e}^{x} x^{3}+2 \,{\mathrm e}^{x} x^{4}}{x \left (x^{2}+x +7\right )^{2}}\) | \(73\) |
| parallelrisch | \(\frac {-75+x^{7}+x^{5} {\mathrm e}^{x}-735 x -210 x^{2}-176 x^{3}-16 x^{4}+2 x^{6}+49 \,{\mathrm e}^{x} x +14 \,{\mathrm e}^{x} x^{2}+15 \,{\mathrm e}^{x} x^{3}+2 \,{\mathrm e}^{x} x^{4}}{x \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}\) | \(85\) |
| default | \(-\frac {6 \left (-\frac {1280}{243} x^{3}+\frac {899}{81} x^{2}-\frac {1232}{81} x +\frac {42385}{486}\right )}{\left (x^{2}+x +7\right )^{2}}-\frac {2 \left (\frac {5797}{243} x^{3}+\frac {11305}{162} x^{2}+\frac {14602}{81} x +\frac {138229}{486}\right )}{\left (x^{2}+x +7\right )^{2}}-\frac {75}{49 x}+x^{2}+{\mathrm e}^{x}-\frac {43 \,{\mathrm e}^{x} \left (68 x^{3}+255 x^{2}+483 x +490\right )}{486 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+\frac {\frac {5984}{81} x^{3}+\frac {10120}{27} x^{2}+\frac {17024}{27} x +\frac {134848}{81}}{\left (x^{2}+x +7\right )^{2}}+\frac {-\frac {12470}{243} x^{3}+\frac {731}{81} x^{2}-\frac {15050}{81} x +\frac {48461}{243}}{\left (x^{2}+x +7\right )^{2}}+\frac {-\frac {784}{81} x^{3}-\frac {4928}{27} x^{2}-\frac {3136}{27} x -\frac {52136}{81}}{\left (x^{2}+x +7\right )^{2}}+\frac {\frac {490}{81} x^{3}+\frac {245}{27} x^{2}-\frac {686}{27} x +\frac {4802}{81}}{\left (x^{2}+x +7\right )^{2}}+\frac {-\frac {343 x}{27}-\frac {4802}{27}}{\left (x^{2}+x +7\right )^{2}}+\frac {\frac {125}{18}+\frac {125 x}{9}}{\left (x^{2}+x +7\right )^{2}}+\frac {\frac {32}{243}+\frac {64 x}{243}}{x^{2}+x +7}-\frac {225 \left (\frac {112}{243} x^{3}-\frac {455}{162} x^{2}+\frac {70}{81} x -\frac {8428}{243}\right )}{343 \left (x^{2}+x +7\right )^{2}}+\frac {-\frac {4975}{3969} x^{3}-\frac {9025}{2646} x^{2}-\frac {17875}{1323} x -\frac {21625}{1134}}{\left (x^{2}+x +7\right )^{2}}+\frac {{\mathrm e}^{x} \left (1480 x^{3}-8301 x^{2}-1995 x -55174\right )}{486 x^{4}+972 x^{3}+7290 x^{2}+6804 x +23814}+\frac {28 \,{\mathrm e}^{x} \left (x^{3}-57 x^{2}-168 x -343\right )}{81 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+\frac {343 \,{\mathrm e}^{x} \left (4 x^{3}+15 x^{2}+57 x +86\right )}{486 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}-\frac {4 \,{\mathrm e}^{x} \left (173 x^{3}-546 x^{2}-147 x -3773\right )}{81 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+\frac {49 \,{\mathrm e}^{x} \left (7 x^{3}+6 x^{2}+39 x -133\right )}{162 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+\frac {{\mathrm e}^{x} \left (1009 x^{3}+3237 x^{2}+7392 x +11123\right )}{162 x^{4}+324 x^{3}+2430 x^{2}+2268 x +7938}\) | \(528\) |
| orering | \(\frac {\left (x^{13}+4 x^{12}+14 x^{11}+8 x^{10}-301 x^{9}-1144 x^{8}-5914 x^{7}-11023 x^{6}-31069 x^{5}-26315 x^{4}-46370 x^{3}+1425 x^{2}+4200 x +7350\right ) \left (\left (x^{8}+3 x^{7}+24 x^{6}+43 x^{5}+168 x^{4}+147 x^{3}+343 x^{2}\right ) {\mathrm e}^{x}+2 x^{9}+6 x^{8}+48 x^{7}+86 x^{6}+336 x^{5}+294 x^{4}+686 x^{3}+375 x^{2}+225 x +525\right )}{\left (2 x^{12}+6 x^{11}+60 x^{10}+108 x^{9}+582 x^{8}+474 x^{7}+2100 x^{6}-213 x^{5}+4908 x^{4}+1273 x^{3}+7200 x^{2}+7875 x +7350\right ) \left (x^{8}+3 x^{7}+24 x^{6}+43 x^{5}+168 x^{4}+147 x^{3}+343 x^{2}\right )}-\frac {\left (x^{10}+x^{9}-59 x^{7}-350 x^{6}-963 x^{5}-2975 x^{4}-3407 x^{3}-6624 x^{2}-750 x -525\right ) x \left (x^{2}+x +7\right ) \left (\frac {\left (8 x^{7}+21 x^{6}+144 x^{5}+215 x^{4}+672 x^{3}+441 x^{2}+686 x \right ) {\mathrm e}^{x}+\left (x^{8}+3 x^{7}+24 x^{6}+43 x^{5}+168 x^{4}+147 x^{3}+343 x^{2}\right ) {\mathrm e}^{x}+18 x^{8}+48 x^{7}+336 x^{6}+516 x^{5}+1680 x^{4}+1176 x^{3}+2058 x^{2}+750 x +225}{x^{8}+3 x^{7}+24 x^{6}+43 x^{5}+168 x^{4}+147 x^{3}+343 x^{2}}-\frac {\left (\left (x^{8}+3 x^{7}+24 x^{6}+43 x^{5}+168 x^{4}+147 x^{3}+343 x^{2}\right ) {\mathrm e}^{x}+2 x^{9}+6 x^{8}+48 x^{7}+86 x^{6}+336 x^{5}+294 x^{4}+686 x^{3}+375 x^{2}+225 x +525\right ) \left (8 x^{7}+21 x^{6}+144 x^{5}+215 x^{4}+672 x^{3}+441 x^{2}+686 x \right )}{\left (x^{8}+3 x^{7}+24 x^{6}+43 x^{5}+168 x^{4}+147 x^{3}+343 x^{2}\right )^{2}}\right )}{2 x^{12}+6 x^{11}+60 x^{10}+108 x^{9}+582 x^{8}+474 x^{7}+2100 x^{6}-213 x^{5}+4908 x^{4}+1273 x^{3}+7200 x^{2}+7875 x +7350}\) | \(664\) |
Input:
int(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+6*x^8+ 48*x^7+86*x^6+336*x^5+294*x^4+686*x^3+375*x^2+225*x+525)/(x^8+3*x^7+24*x^6 +43*x^5+168*x^4+147*x^3+343*x^2),x,method=_RETURNVERBOSE)
Output:
x^2-75/x/(x^4+2*x^3+15*x^2+14*x+49)+exp(x)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (19) = 38\).
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.12 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=\frac {x^{7} + 2 \, x^{6} + 15 \, x^{5} + 14 \, x^{4} + 49 \, x^{3} + {\left (x^{5} + 2 \, x^{4} + 15 \, x^{3} + 14 \, x^{2} + 49 \, x\right )} e^{x} - 75}{x^{5} + 2 \, x^{4} + 15 \, x^{3} + 14 \, x^{2} + 49 \, x} \] Input:
integrate(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+ 6*x^8+48*x^7+86*x^6+336*x^5+294*x^4+686*x^3+375*x^2+225*x+525)/(x^8+3*x^7+ 24*x^6+43*x^5+168*x^4+147*x^3+343*x^2),x, algorithm="fricas")
Output:
(x^7 + 2*x^6 + 15*x^5 + 14*x^4 + 49*x^3 + (x^5 + 2*x^4 + 15*x^3 + 14*x^2 + 49*x)*e^x - 75)/(x^5 + 2*x^4 + 15*x^3 + 14*x^2 + 49*x)
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=x^{2} + e^{x} - \frac {75}{x^{5} + 2 x^{4} + 15 x^{3} + 14 x^{2} + 49 x} \] Input:
integrate(((x**8+3*x**7+24*x**6+43*x**5+168*x**4+147*x**3+343*x**2)*exp(x) +2*x**9+6*x**8+48*x**7+86*x**6+336*x**5+294*x**4+686*x**3+375*x**2+225*x+5 25)/(x**8+3*x**7+24*x**6+43*x**5+168*x**4+147*x**3+343*x**2),x)
Output:
x**2 + exp(x) - 75/(x**5 + 2*x**4 + 15*x**3 + 14*x**2 + 49*x)
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (19) = 38\).
Time = 0.18 (sec) , antiderivative size = 385, normalized size of antiderivative = 16.04 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=x^{2} - \frac {25 \, {\left (884 \, x^{4} + 2055 \, x^{3} + 11580 \, x^{2} + 12859 \, x + 23814\right )}}{7938 \, {\left (x^{5} + 2 \, x^{4} + 15 \, x^{3} + 14 \, x^{2} + 49 \, x\right )}} - \frac {11594 \, x^{3} + 33915 \, x^{2} + 87612 \, x + 138229}{243 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + \frac {2560 \, x^{3} - 5394 \, x^{2} + 7392 \, x - 42385}{81 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + \frac {8 \, {\left (748 \, x^{3} + 3795 \, x^{2} + 6384 \, x + 16856\right )}}{81 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} - \frac {43 \, {\left (290 \, x^{3} - 51 \, x^{2} + 1050 \, x - 1127\right )}}{243 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} - \frac {25 \, {\left (32 \, x^{3} - 195 \, x^{2} + 60 \, x - 2408\right )}}{2646 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} - \frac {56 \, {\left (14 \, x^{3} + 264 \, x^{2} + 168 \, x + 931\right )}}{81 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + \frac {49 \, {\left (10 \, x^{3} + 15 \, x^{2} - 42 \, x + 98\right )}}{81 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + \frac {125 \, {\left (4 \, x^{3} + 6 \, x^{2} + 48 \, x + 23\right )}}{162 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} - \frac {343 \, {\left (2 \, x^{3} + 3 \, x^{2} + 24 \, x + 133\right )}}{243 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + e^{x} \] Input:
integrate(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+ 6*x^8+48*x^7+86*x^6+336*x^5+294*x^4+686*x^3+375*x^2+225*x+525)/(x^8+3*x^7+ 24*x^6+43*x^5+168*x^4+147*x^3+343*x^2),x, algorithm="maxima")
Output:
x^2 - 25/7938*(884*x^4 + 2055*x^3 + 11580*x^2 + 12859*x + 23814)/(x^5 + 2* x^4 + 15*x^3 + 14*x^2 + 49*x) - 1/243*(11594*x^3 + 33915*x^2 + 87612*x + 1 38229)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + 1/81*(2560*x^3 - 5394*x^2 + 73 92*x - 42385)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + 8/81*(748*x^3 + 3795*x^ 2 + 6384*x + 16856)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) - 43/243*(290*x^3 - 51*x^2 + 1050*x - 1127)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) - 25/2646*(32* x^3 - 195*x^2 + 60*x - 2408)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) - 56/81*(1 4*x^3 + 264*x^2 + 168*x + 931)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + 49/81* (10*x^3 + 15*x^2 - 42*x + 98)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + 125/162 *(4*x^3 + 6*x^2 + 48*x + 23)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) - 343/243* (2*x^3 + 3*x^2 + 24*x + 133)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + e^x
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.42 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=\frac {x^{7} + 2 \, x^{6} + x^{5} e^{x} + 15 \, x^{5} + 2 \, x^{4} e^{x} + 14 \, x^{4} + 15 \, x^{3} e^{x} + 49 \, x^{3} + 14 \, x^{2} e^{x} + 49 \, x e^{x} - 75}{x^{5} + 2 \, x^{4} + 15 \, x^{3} + 14 \, x^{2} + 49 \, x} \] Input:
integrate(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+ 6*x^8+48*x^7+86*x^6+336*x^5+294*x^4+686*x^3+375*x^2+225*x+525)/(x^8+3*x^7+ 24*x^6+43*x^5+168*x^4+147*x^3+343*x^2),x, algorithm="giac")
Output:
(x^7 + 2*x^6 + x^5*e^x + 15*x^5 + 2*x^4*e^x + 14*x^4 + 15*x^3*e^x + 49*x^3 + 14*x^2*e^x + 49*x*e^x - 75)/(x^5 + 2*x^4 + 15*x^3 + 14*x^2 + 49*x)
Time = 1.49 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx={\mathrm {e}}^x-\frac {75}{x^5+2\,x^4+15\,x^3+14\,x^2+49\,x}+x^2 \] Input:
int((225*x + exp(x)*(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3*x^7 + x^8) + 375*x^2 + 686*x^3 + 294*x^4 + 336*x^5 + 86*x^6 + 48*x^7 + 6*x^8 + 2*x^9 + 525)/(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3*x^7 + x^ 8),x)
Output:
exp(x) - 75/(49*x + 14*x^2 + 15*x^3 + 2*x^4 + x^5) + x^2
Time = 0.23 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.71 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=\frac {e^{x} x^{5}+2 e^{x} x^{4}+15 e^{x} x^{3}+14 e^{x} x^{2}+49 e^{x} x +x^{7}+2 x^{6}+8 x^{5}-56 x^{3}-98 x^{2}-343 x -75}{x \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )} \] Input:
int(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+6*x^8+ 48*x^7+86*x^6+336*x^5+294*x^4+686*x^3+375*x^2+225*x+525)/(x^8+3*x^7+24*x^6 +43*x^5+168*x^4+147*x^3+343*x^2),x)
Output:
(e**x*x**5 + 2*e**x*x**4 + 15*e**x*x**3 + 14*e**x*x**2 + 49*e**x*x + x**7 + 2*x**6 + 8*x**5 - 56*x**3 - 98*x**2 - 343*x - 75)/(x*(x**4 + 2*x**3 + 15 *x**2 + 14*x + 49))