Integrand size = 134, antiderivative size = 16 \[ \int \frac {e^5 \left (-65537-262144 x-344064 x^2-229376 x^3-89600 x^4-21504 x^5-3136 x^6-256 x^7-9 x^8\right )}{25-655370 x+4293787649 x^2+17178984448 x^3+32211910656 x^4+37580899328 x^5+30534533120 x^6+18320722560 x^7+8396997184 x^8+2998927414 x^9+843448322 x^{10}+187432960 x^{11}+32800768 x^{12}+4472832 x^{13}+465920 x^{14}+35840 x^{15}+1920 x^{16}+64 x^{17}+x^{18}} \, dx=\frac {e^5}{-5+x+x (4+x)^8} \] Output:
exp(5)/(x*(4+x)^8+x-5)
Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(16)=32\).
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 3.06 \[ \int \frac {e^5 \left (-65537-262144 x-344064 x^2-229376 x^3-89600 x^4-21504 x^5-3136 x^6-256 x^7-9 x^8\right )}{25-655370 x+4293787649 x^2+17178984448 x^3+32211910656 x^4+37580899328 x^5+30534533120 x^6+18320722560 x^7+8396997184 x^8+2998927414 x^9+843448322 x^{10}+187432960 x^{11}+32800768 x^{12}+4472832 x^{13}+465920 x^{14}+35840 x^{15}+1920 x^{16}+64 x^{17}+x^{18}} \, dx=\frac {e^5}{-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9} \] Input:
Integrate[(E^5*(-65537 - 262144*x - 344064*x^2 - 229376*x^3 - 89600*x^4 - 21504*x^5 - 3136*x^6 - 256*x^7 - 9*x^8))/(25 - 655370*x + 4293787649*x^2 + 17178984448*x^3 + 32211910656*x^4 + 37580899328*x^5 + 30534533120*x^6 + 1 8320722560*x^7 + 8396997184*x^8 + 2998927414*x^9 + 843448322*x^10 + 187432 960*x^11 + 32800768*x^12 + 4472832*x^13 + 465920*x^14 + 35840*x^15 + 1920* x^16 + 64*x^17 + x^18),x]
Output:
E^5/(-5 + 65537*x + 131072*x^2 + 114688*x^3 + 57344*x^4 + 17920*x^5 + 3584 *x^6 + 448*x^7 + 32*x^8 + x^9)
Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(16)=32\).
Time = 3.89 (sec) , antiderivative size = 52, normalized size of antiderivative = 3.25, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {27, 25, 2462, 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.
| \(\displaystyle \int \frac {e^5 \left (-9 x^8-256 x^7-3136 x^6-21504 x^5-89600 x^4-229376 x^3-344064 x^2-262144 x-65537\right )}{x^{18}+64 x^{17}+1920 x^{16}+35840 x^{15}+465920 x^{14}+4472832 x^{13}+32800768 x^{12}+187432960 x^{11}+843448322 x^{10}+2998927414 x^9+8396997184 x^8+18320722560 x^7+30534533120 x^6+37580899328 x^5+32211910656 x^4+17178984448 x^3+4293787649 x^2-655370 x+25} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e^5 \int -\frac {9 x^8+256 x^7+3136 x^6+21504 x^5+89600 x^4+229376 x^3+344064 x^2+262144 x+65537}{x^{18}+64 x^{17}+1920 x^{16}+35840 x^{15}+465920 x^{14}+4472832 x^{13}+32800768 x^{12}+187432960 x^{11}+843448322 x^{10}+2998927414 x^9+8396997184 x^8+18320722560 x^7+30534533120 x^6+37580899328 x^5+32211910656 x^4+17178984448 x^3+4293787649 x^2-655370 x+25}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -e^5 \int \frac {9 x^8+256 x^7+3136 x^6+21504 x^5+89600 x^4+229376 x^3+344064 x^2+262144 x+65537}{x^{18}+64 x^{17}+1920 x^{16}+35840 x^{15}+465920 x^{14}+4472832 x^{13}+32800768 x^{12}+187432960 x^{11}+843448322 x^{10}+2998927414 x^9+8396997184 x^8+18320722560 x^7+30534533120 x^6+37580899328 x^5+32211910656 x^4+17178984448 x^3+4293787649 x^2-655370 x+25}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle -e^5 \int \left (\frac {9 x^8}{\left (x^9+32 x^8+448 x^7+3584 x^6+17920 x^5+57344 x^4+114688 x^3+131072 x^2+65537 x-5\right )^2}+\frac {256 x^7}{\left (x^9+32 x^8+448 x^7+3584 x^6+17920 x^5+57344 x^4+114688 x^3+131072 x^2+65537 x-5\right )^2}+\frac {3136 x^6}{\left (x^9+32 x^8+448 x^7+3584 x^6+17920 x^5+57344 x^4+114688 x^3+131072 x^2+65537 x-5\right )^2}+\frac {21504 x^5}{\left (x^9+32 x^8+448 x^7+3584 x^6+17920 x^5+57344 x^4+114688 x^3+131072 x^2+65537 x-5\right )^2}+\frac {89600 x^4}{\left (x^9+32 x^8+448 x^7+3584 x^6+17920 x^5+57344 x^4+114688 x^3+131072 x^2+65537 x-5\right )^2}+\frac {229376 x^3}{\left (x^9+32 x^8+448 x^7+3584 x^6+17920 x^5+57344 x^4+114688 x^3+131072 x^2+65537 x-5\right )^2}+\frac {344064 x^2}{\left (x^9+32 x^8+448 x^7+3584 x^6+17920 x^5+57344 x^4+114688 x^3+131072 x^2+65537 x-5\right )^2}+\frac {262144 x}{\left (x^9+32 x^8+448 x^7+3584 x^6+17920 x^5+57344 x^4+114688 x^3+131072 x^2+65537 x-5\right )^2}+\frac {65537}{\left (x^9+32 x^8+448 x^7+3584 x^6+17920 x^5+57344 x^4+114688 x^3+131072 x^2+65537 x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {e^5}{-x^9-32 x^8-448 x^7-3584 x^6-17920 x^5-57344 x^4-114688 x^3-131072 x^2-65537 x+5}\) |
Input:
Int[(E^5*(-65537 - 262144*x - 344064*x^2 - 229376*x^3 - 89600*x^4 - 21504* x^5 - 3136*x^6 - 256*x^7 - 9*x^8))/(25 - 655370*x + 4293787649*x^2 + 17178 984448*x^3 + 32211910656*x^4 + 37580899328*x^5 + 30534533120*x^6 + 1832072 2560*x^7 + 8396997184*x^8 + 2998927414*x^9 + 843448322*x^10 + 187432960*x^ 11 + 32800768*x^12 + 4472832*x^13 + 465920*x^14 + 35840*x^15 + 1920*x^16 + 64*x^17 + x^18),x]
Output:
-(E^5/(5 - 65537*x - 131072*x^2 - 114688*x^3 - 57344*x^4 - 17920*x^5 - 358 4*x^6 - 448*x^7 - 32*x^8 - x^9))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(15)=30\).
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 3.06
| method | result | size |
| gosper | \(\frac {{\mathrm e}^{5}}{x^{9}+32 x^{8}+448 x^{7}+3584 x^{6}+17920 x^{5}+57344 x^{4}+114688 x^{3}+131072 x^{2}+65537 x -5}\) | \(49\) |
| default | \(\frac {{\mathrm e}^{5}}{x^{9}+32 x^{8}+448 x^{7}+3584 x^{6}+17920 x^{5}+57344 x^{4}+114688 x^{3}+131072 x^{2}+65537 x -5}\) | \(49\) |
| norman | \(\frac {{\mathrm e}^{5}}{x^{9}+32 x^{8}+448 x^{7}+3584 x^{6}+17920 x^{5}+57344 x^{4}+114688 x^{3}+131072 x^{2}+65537 x -5}\) | \(49\) |
| risch | \(\frac {{\mathrm e}^{5}}{x^{9}+32 x^{8}+448 x^{7}+3584 x^{6}+17920 x^{5}+57344 x^{4}+114688 x^{3}+131072 x^{2}+65537 x -5}\) | \(49\) |
| parallelrisch | \(\frac {{\mathrm e}^{5}}{x^{9}+32 x^{8}+448 x^{7}+3584 x^{6}+17920 x^{5}+57344 x^{4}+114688 x^{3}+131072 x^{2}+65537 x -5}\) | \(49\) |
| orering | \(-\frac {\left (x^{9}+32 x^{8}+448 x^{7}+3584 x^{6}+17920 x^{5}+57344 x^{4}+114688 x^{3}+131072 x^{2}+65537 x -5\right ) \left (-9 x^{8}-256 x^{7}-3136 x^{6}-21504 x^{5}-89600 x^{4}-229376 x^{3}-344064 x^{2}-262144 x -65537\right ) {\mathrm e}^{5}}{\left (9 x^{8}+256 x^{7}+3136 x^{6}+21504 x^{5}+89600 x^{4}+229376 x^{3}+344064 x^{2}+262144 x +65537\right ) \left (x^{18}+64 x^{17}+1920 x^{16}+35840 x^{15}+465920 x^{14}+4472832 x^{13}+32800768 x^{12}+187432960 x^{11}+843448322 x^{10}+2998927414 x^{9}+8396997184 x^{8}+18320722560 x^{7}+30534533120 x^{6}+37580899328 x^{5}+32211910656 x^{4}+17178984448 x^{3}+4293787649 x^{2}-655370 x +25\right )}\) | \(220\) |
Input:
int((-9*x^8-256*x^7-3136*x^6-21504*x^5-89600*x^4-229376*x^3-344064*x^2-262 144*x-65537)*exp(5)/(x^18+64*x^17+1920*x^16+35840*x^15+465920*x^14+4472832 *x^13+32800768*x^12+187432960*x^11+843448322*x^10+2998927414*x^9+839699718 4*x^8+18320722560*x^7+30534533120*x^6+37580899328*x^5+32211910656*x^4+1717 8984448*x^3+4293787649*x^2-655370*x+25),x,method=_RETURNVERBOSE)
Output:
exp(5)/(x^9+32*x^8+448*x^7+3584*x^6+17920*x^5+57344*x^4+114688*x^3+131072* x^2+65537*x-5)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (15) = 30\).
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 3.00 \[ \int \frac {e^5 \left (-65537-262144 x-344064 x^2-229376 x^3-89600 x^4-21504 x^5-3136 x^6-256 x^7-9 x^8\right )}{25-655370 x+4293787649 x^2+17178984448 x^3+32211910656 x^4+37580899328 x^5+30534533120 x^6+18320722560 x^7+8396997184 x^8+2998927414 x^9+843448322 x^{10}+187432960 x^{11}+32800768 x^{12}+4472832 x^{13}+465920 x^{14}+35840 x^{15}+1920 x^{16}+64 x^{17}+x^{18}} \, dx=\frac {e^{5}}{x^{9} + 32 \, x^{8} + 448 \, x^{7} + 3584 \, x^{6} + 17920 \, x^{5} + 57344 \, x^{4} + 114688 \, x^{3} + 131072 \, x^{2} + 65537 \, x - 5} \] Input:
integrate((-9*x^8-256*x^7-3136*x^6-21504*x^5-89600*x^4-229376*x^3-344064*x ^2-262144*x-65537)*exp(5)/(x^18+64*x^17+1920*x^16+35840*x^15+465920*x^14+4 472832*x^13+32800768*x^12+187432960*x^11+843448322*x^10+2998927414*x^9+839 6997184*x^8+18320722560*x^7+30534533120*x^6+37580899328*x^5+32211910656*x^ 4+17178984448*x^3+4293787649*x^2-655370*x+25),x, algorithm="fricas")
Output:
e^5/(x^9 + 32*x^8 + 448*x^7 + 3584*x^6 + 17920*x^5 + 57344*x^4 + 114688*x^ 3 + 131072*x^2 + 65537*x - 5)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (12) = 24\).
Time = 0.96 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.88 \[ \int \frac {e^5 \left (-65537-262144 x-344064 x^2-229376 x^3-89600 x^4-21504 x^5-3136 x^6-256 x^7-9 x^8\right )}{25-655370 x+4293787649 x^2+17178984448 x^3+32211910656 x^4+37580899328 x^5+30534533120 x^6+18320722560 x^7+8396997184 x^8+2998927414 x^9+843448322 x^{10}+187432960 x^{11}+32800768 x^{12}+4472832 x^{13}+465920 x^{14}+35840 x^{15}+1920 x^{16}+64 x^{17}+x^{18}} \, dx=\frac {e^{5}}{x^{9} + 32 x^{8} + 448 x^{7} + 3584 x^{6} + 17920 x^{5} + 57344 x^{4} + 114688 x^{3} + 131072 x^{2} + 65537 x - 5} \] Input:
integrate((-9*x**8-256*x**7-3136*x**6-21504*x**5-89600*x**4-229376*x**3-34 4064*x**2-262144*x-65537)*exp(5)/(x**18+64*x**17+1920*x**16+35840*x**15+46 5920*x**14+4472832*x**13+32800768*x**12+187432960*x**11+843448322*x**10+29 98927414*x**9+8396997184*x**8+18320722560*x**7+30534533120*x**6+3758089932 8*x**5+32211910656*x**4+17178984448*x**3+4293787649*x**2-655370*x+25),x)
Output:
exp(5)/(x**9 + 32*x**8 + 448*x**7 + 3584*x**6 + 17920*x**5 + 57344*x**4 + 114688*x**3 + 131072*x**2 + 65537*x - 5)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (15) = 30\).
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 3.00 \[ \int \frac {e^5 \left (-65537-262144 x-344064 x^2-229376 x^3-89600 x^4-21504 x^5-3136 x^6-256 x^7-9 x^8\right )}{25-655370 x+4293787649 x^2+17178984448 x^3+32211910656 x^4+37580899328 x^5+30534533120 x^6+18320722560 x^7+8396997184 x^8+2998927414 x^9+843448322 x^{10}+187432960 x^{11}+32800768 x^{12}+4472832 x^{13}+465920 x^{14}+35840 x^{15}+1920 x^{16}+64 x^{17}+x^{18}} \, dx=\frac {e^{5}}{x^{9} + 32 \, x^{8} + 448 \, x^{7} + 3584 \, x^{6} + 17920 \, x^{5} + 57344 \, x^{4} + 114688 \, x^{3} + 131072 \, x^{2} + 65537 \, x - 5} \] Input:
integrate((-9*x^8-256*x^7-3136*x^6-21504*x^5-89600*x^4-229376*x^3-344064*x ^2-262144*x-65537)*exp(5)/(x^18+64*x^17+1920*x^16+35840*x^15+465920*x^14+4 472832*x^13+32800768*x^12+187432960*x^11+843448322*x^10+2998927414*x^9+839 6997184*x^8+18320722560*x^7+30534533120*x^6+37580899328*x^5+32211910656*x^ 4+17178984448*x^3+4293787649*x^2-655370*x+25),x, algorithm="maxima")
Output:
e^5/(x^9 + 32*x^8 + 448*x^7 + 3584*x^6 + 17920*x^5 + 57344*x^4 + 114688*x^ 3 + 131072*x^2 + 65537*x - 5)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (15) = 30\).
Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 3.00 \[ \int \frac {e^5 \left (-65537-262144 x-344064 x^2-229376 x^3-89600 x^4-21504 x^5-3136 x^6-256 x^7-9 x^8\right )}{25-655370 x+4293787649 x^2+17178984448 x^3+32211910656 x^4+37580899328 x^5+30534533120 x^6+18320722560 x^7+8396997184 x^8+2998927414 x^9+843448322 x^{10}+187432960 x^{11}+32800768 x^{12}+4472832 x^{13}+465920 x^{14}+35840 x^{15}+1920 x^{16}+64 x^{17}+x^{18}} \, dx=\frac {e^{5}}{x^{9} + 32 \, x^{8} + 448 \, x^{7} + 3584 \, x^{6} + 17920 \, x^{5} + 57344 \, x^{4} + 114688 \, x^{3} + 131072 \, x^{2} + 65537 \, x - 5} \] Input:
integrate((-9*x^8-256*x^7-3136*x^6-21504*x^5-89600*x^4-229376*x^3-344064*x ^2-262144*x-65537)*exp(5)/(x^18+64*x^17+1920*x^16+35840*x^15+465920*x^14+4 472832*x^13+32800768*x^12+187432960*x^11+843448322*x^10+2998927414*x^9+839 6997184*x^8+18320722560*x^7+30534533120*x^6+37580899328*x^5+32211910656*x^ 4+17178984448*x^3+4293787649*x^2-655370*x+25),x, algorithm="giac")
Output:
e^5/(x^9 + 32*x^8 + 448*x^7 + 3584*x^6 + 17920*x^5 + 57344*x^4 + 114688*x^ 3 + 131072*x^2 + 65537*x - 5)
Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 3.00 \[ \int \frac {e^5 \left (-65537-262144 x-344064 x^2-229376 x^3-89600 x^4-21504 x^5-3136 x^6-256 x^7-9 x^8\right )}{25-655370 x+4293787649 x^2+17178984448 x^3+32211910656 x^4+37580899328 x^5+30534533120 x^6+18320722560 x^7+8396997184 x^8+2998927414 x^9+843448322 x^{10}+187432960 x^{11}+32800768 x^{12}+4472832 x^{13}+465920 x^{14}+35840 x^{15}+1920 x^{16}+64 x^{17}+x^{18}} \, dx=\frac {{\mathrm {e}}^5}{x^9+32\,x^8+448\,x^7+3584\,x^6+17920\,x^5+57344\,x^4+114688\,x^3+131072\,x^2+65537\,x-5} \] Input:
int(-(exp(5)*(262144*x + 344064*x^2 + 229376*x^3 + 89600*x^4 + 21504*x^5 + 3136*x^6 + 256*x^7 + 9*x^8 + 65537))/(4293787649*x^2 - 655370*x + 1717898 4448*x^3 + 32211910656*x^4 + 37580899328*x^5 + 30534533120*x^6 + 183207225 60*x^7 + 8396997184*x^8 + 2998927414*x^9 + 843448322*x^10 + 187432960*x^11 + 32800768*x^12 + 4472832*x^13 + 465920*x^14 + 35840*x^15 + 1920*x^16 + 6 4*x^17 + x^18 + 25),x)
Output:
exp(5)/(65537*x + 131072*x^2 + 114688*x^3 + 57344*x^4 + 17920*x^5 + 3584*x ^6 + 448*x^7 + 32*x^8 + x^9 - 5)
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 3.06 \[ \int \frac {e^5 \left (-65537-262144 x-344064 x^2-229376 x^3-89600 x^4-21504 x^5-3136 x^6-256 x^7-9 x^8\right )}{25-655370 x+4293787649 x^2+17178984448 x^3+32211910656 x^4+37580899328 x^5+30534533120 x^6+18320722560 x^7+8396997184 x^8+2998927414 x^9+843448322 x^{10}+187432960 x^{11}+32800768 x^{12}+4472832 x^{13}+465920 x^{14}+35840 x^{15}+1920 x^{16}+64 x^{17}+x^{18}} \, dx=\frac {e^{5}}{x^{9}+32 x^{8}+448 x^{7}+3584 x^{6}+17920 x^{5}+57344 x^{4}+114688 x^{3}+131072 x^{2}+65537 x -5} \] Input:
int((-9*x^8-256*x^7-3136*x^6-21504*x^5-89600*x^4-229376*x^3-344064*x^2-262 144*x-65537)*exp(5)/(x^18+64*x^17+1920*x^16+35840*x^15+465920*x^14+4472832 *x^13+32800768*x^12+187432960*x^11+843448322*x^10+2998927414*x^9+839699718 4*x^8+18320722560*x^7+30534533120*x^6+37580899328*x^5+32211910656*x^4+1717 8984448*x^3+4293787649*x^2-655370*x+25),x)
Output:
e**5/(x**9 + 32*x**8 + 448*x**7 + 3584*x**6 + 17920*x**5 + 57344*x**4 + 11 4688*x**3 + 131072*x**2 + 65537*x - 5)