Integrand size = 152, antiderivative size = 33 \[ \int \frac {40200250 x-120300 x^2-31823280 x^3-32496 x^4+576 x^5-192 x^6+128 x^7+e^{2 e^3+2 x} \left (250-300 x+720 x^2-496 x^3+576 x^4-192 x^5+128 x^6\right )+e^{e^3+x} \left (-100250-59950 x+79580 x^2-208224 x^3+127920 x^4-64384 x^5+64 x^6-128 x^7\right )}{125-150 x+360 x^2-248 x^3+288 x^4-96 x^5+64 x^6} \, dx=\left (e^{e^3+x}-x-\frac {400}{\frac {1}{5} \left (-2+\frac {5}{x}-x\right )+x}\right )^2 \] Output:
(exp(exp(3)+x)-400/(4/5*x-2/5+1/x)-x)^2
Leaf count is larger than twice the leaf count of optimal. \(92\) vs. \(2(33)=66\).
Time = 5.76 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.79 \[ \int \frac {40200250 x-120300 x^2-31823280 x^3-32496 x^4+576 x^5-192 x^6+128 x^7+e^{2 e^3+2 x} \left (250-300 x+720 x^2-496 x^3+576 x^4-192 x^5+128 x^6\right )+e^{e^3+x} \left (-100250-59950 x+79580 x^2-208224 x^3+127920 x^4-64384 x^5+64 x^6-128 x^7\right )}{125-150 x+360 x^2-248 x^3+288 x^4-96 x^5+64 x^6} \, dx=-2 \left (-\frac {1}{2} e^{2 \left (e^3+x\right )}-\frac {x^2}{2}-\frac {500000 (-5+2 x)}{\left (5-2 x+4 x^2\right )^2}-\frac {500 (995+2 x)}{5-2 x+4 x^2}+\frac {e^{e^3+x} x \left (2005-2 x+4 x^2\right )}{5-2 x+4 x^2}\right ) \] Input:
Integrate[(40200250*x - 120300*x^2 - 31823280*x^3 - 32496*x^4 + 576*x^5 - 192*x^6 + 128*x^7 + E^(2*E^3 + 2*x)*(250 - 300*x + 720*x^2 - 496*x^3 + 576 *x^4 - 192*x^5 + 128*x^6) + E^(E^3 + x)*(-100250 - 59950*x + 79580*x^2 - 2 08224*x^3 + 127920*x^4 - 64384*x^5 + 64*x^6 - 128*x^7))/(125 - 150*x + 360 *x^2 - 248*x^3 + 288*x^4 - 96*x^5 + 64*x^6),x]
Output:
-2*(-1/2*E^(2*(E^3 + x)) - x^2/2 - (500000*(-5 + 2*x))/(5 - 2*x + 4*x^2)^2 - (500*(995 + 2*x))/(5 - 2*x + 4*x^2) + (E^(E^3 + x)*x*(2005 - 2*x + 4*x^ 2))/(5 - 2*x + 4*x^2))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 13.68 (sec) , antiderivative size = 587, normalized size of antiderivative = 17.79, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2463, 7239, 27, 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.
| \(\displaystyle \int \frac {128 x^7-192 x^6+576 x^5-32496 x^4-31823280 x^3-120300 x^2+e^{2 x+2 e^3} \left (128 x^6-192 x^5+576 x^4-496 x^3+720 x^2-300 x+250\right )+e^{x+e^3} \left (-128 x^7+64 x^6-64384 x^5+127920 x^4-208224 x^3+79580 x^2-59950 x-100250\right )+40200250 x}{64 x^6-96 x^5+288 x^4-248 x^3+360 x^2-150 x+125} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {24 i \left (128 x^7-192 x^6+576 x^5-32496 x^4-31823280 x^3-120300 x^2+40200250 x+e^{2 x+2 e^3} \left (128 x^6-192 x^5+576 x^4-496 x^3+720 x^2-300 x+250\right )+e^{x+e^3} \left (-128 x^7+64 x^6-64384 x^5+127920 x^4-208224 x^3+79580 x^2-59950 x-100250\right )\right )}{361 \sqrt {19} \left (-8 x+2 i \sqrt {19}+2\right )}+\frac {24 i \left (128 x^7-192 x^6+576 x^5-32496 x^4-31823280 x^3-120300 x^2+40200250 x+e^{2 x+2 e^3} \left (128 x^6-192 x^5+576 x^4-496 x^3+720 x^2-300 x+250\right )+e^{x+e^3} \left (-128 x^7+64 x^6-64384 x^5+127920 x^4-208224 x^3+79580 x^2-59950 x-100250\right )\right )}{361 \sqrt {19} \left (8 x+2 i \sqrt {19}-2\right )}-\frac {48 \left (128 x^7-192 x^6+576 x^5-32496 x^4-31823280 x^3-120300 x^2+40200250 x+e^{2 x+2 e^3} \left (128 x^6-192 x^5+576 x^4-496 x^3+720 x^2-300 x+250\right )+e^{x+e^3} \left (-128 x^7+64 x^6-64384 x^5+127920 x^4-208224 x^3+79580 x^2-59950 x-100250\right )\right )}{361 \left (-8 x+2 i \sqrt {19}+2\right )^2}-\frac {48 \left (128 x^7-192 x^6+576 x^5-32496 x^4-31823280 x^3-120300 x^2+40200250 x+e^{2 x+2 e^3} \left (128 x^6-192 x^5+576 x^4-496 x^3+720 x^2-300 x+250\right )+e^{x+e^3} \left (-128 x^7+64 x^6-64384 x^5+127920 x^4-208224 x^3+79580 x^2-59950 x-100250\right )\right )}{361 \left (8 x+2 i \sqrt {19}-2\right )^2}-\frac {64 i \left (128 x^7-192 x^6+576 x^5-32496 x^4-31823280 x^3-120300 x^2+40200250 x+e^{2 x+2 e^3} \left (128 x^6-192 x^5+576 x^4-496 x^3+720 x^2-300 x+250\right )+e^{x+e^3} \left (-128 x^7+64 x^6-64384 x^5+127920 x^4-208224 x^3+79580 x^2-59950 x-100250\right )\right )}{19 \sqrt {19} \left (-8 x+2 i \sqrt {19}+2\right )^3}-\frac {64 i \left (128 x^7-192 x^6+576 x^5-32496 x^4-31823280 x^3-120300 x^2+40200250 x+e^{2 x+2 e^3} \left (128 x^6-192 x^5+576 x^4-496 x^3+720 x^2-300 x+250\right )+e^{x+e^3} \left (-128 x^7+64 x^6-64384 x^5+127920 x^4-208224 x^3+79580 x^2-59950 x-100250\right )\right )}{19 \sqrt {19} \left (8 x+2 i \sqrt {19}-2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (e^{2 \left (x+e^3\right )} \left (4 x^2-2 x+5\right )^3-x \left (-64 x^6+96 x^5-288 x^4+16248 x^3+15911640 x^2+60150 x-20100125\right )-e^{x+e^3} \left (64 x^7-32 x^6+32192 x^5-63960 x^4+104112 x^3-39790 x^2+29975 x+50125\right )\right )}{\left (4 x^2-2 x+5\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {e^{2 \left (x+e^3\right )} \left (4 x^2-2 x+5\right )^3+x \left (64 x^6-96 x^5+288 x^4-16248 x^3-15911640 x^2-60150 x+20100125\right )-e^{x+e^3} \left (64 x^7-32 x^6+32192 x^5-63960 x^4+104112 x^3-39790 x^2+29975 x+50125\right )}{\left (4 x^2-2 x+5\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {x \left (4 x^2-2 x+2005\right ) \left (16 x^4-16 x^3-7956 x^2-20 x+10025\right )}{\left (4 x^2-2 x+5\right )^3}+e^{2 x+2 e^3}-\frac {e^{x+e^3} \left (16 x^5+8028 x^3-11976 x^2+10005 x+10025\right )}{\left (4 x^2-2 x+5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {250}{19} \left (19+3 i \sqrt {19}\right ) e^{\frac {1}{4}+\frac {i \sqrt {19}}{4}+e^3} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-i \sqrt {19}-1\right )\right )-\frac {250}{19} \left (1+i \sqrt {19}\right ) e^{\frac {1}{4}+\frac {i \sqrt {19}}{4}+e^3} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-i \sqrt {19}-1\right )\right )+\frac {1000 i e^{\frac {1}{4}+\frac {i \sqrt {19}}{4}+e^3} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-i \sqrt {19}-1\right )\right )}{\sqrt {19}}+\frac {5000}{19} e^{\frac {1}{4}+\frac {i \sqrt {19}}{4}+e^3} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-i \sqrt {19}-1\right )\right )-\frac {250}{19} \left (19-3 i \sqrt {19}\right ) e^{\frac {1}{4}-\frac {i \sqrt {19}}{4}+e^3} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+i \sqrt {19}-1\right )\right )-\frac {250}{19} \left (1-i \sqrt {19}\right ) e^{\frac {1}{4}-\frac {i \sqrt {19}}{4}+e^3} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+i \sqrt {19}-1\right )\right )-\frac {1000 i e^{\frac {1}{4}-\frac {i \sqrt {19}}{4}+e^3} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+i \sqrt {19}-1\right )\right )}{\sqrt {19}}+\frac {5000}{19} e^{\frac {1}{4}-\frac {i \sqrt {19}}{4}+e^3} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+i \sqrt {19}-1\right )\right )+\frac {x^2 \left (4 x^2-2 x+2005\right )^2}{2 \left (4 x^2-2 x+5\right )^2}+\frac {1}{2} e^{2 x+2 e^3}-e^{x+e^3} x-\frac {1000 \left (1-i \sqrt {19}\right ) e^{x+e^3}}{19 \left (-4 x-i \sqrt {19}+1\right )}+\frac {20000 e^{x+e^3}}{19 \left (-4 x-i \sqrt {19}+1\right )}-\frac {1000 \left (1+i \sqrt {19}\right ) e^{x+e^3}}{19 \left (-4 x+i \sqrt {19}+1\right )}+\frac {20000 e^{x+e^3}}{19 \left (-4 x+i \sqrt {19}+1\right )}\right )\) |
Input:
Int[(40200250*x - 120300*x^2 - 31823280*x^3 - 32496*x^4 + 576*x^5 - 192*x^ 6 + 128*x^7 + E^(2*E^3 + 2*x)*(250 - 300*x + 720*x^2 - 496*x^3 + 576*x^4 - 192*x^5 + 128*x^6) + E^(E^3 + x)*(-100250 - 59950*x + 79580*x^2 - 208224* x^3 + 127920*x^4 - 64384*x^5 + 64*x^6 - 128*x^7))/(125 - 150*x + 360*x^2 - 248*x^3 + 288*x^4 - 96*x^5 + 64*x^6),x]
Output:
2*(E^(2*E^3 + 2*x)/2 + (20000*E^(E^3 + x))/(19*(1 - I*Sqrt[19] - 4*x)) - ( 1000*(1 - I*Sqrt[19])*E^(E^3 + x))/(19*(1 - I*Sqrt[19] - 4*x)) + (20000*E^ (E^3 + x))/(19*(1 + I*Sqrt[19] - 4*x)) - (1000*(1 + I*Sqrt[19])*E^(E^3 + x ))/(19*(1 + I*Sqrt[19] - 4*x)) - E^(E^3 + x)*x + (x^2*(2005 - 2*x + 4*x^2) ^2)/(2*(5 - 2*x + 4*x^2)^2) + (5000*E^(1/4 + (I/4)*Sqrt[19] + E^3)*ExpInte gralEi[(-1 - I*Sqrt[19] + 4*x)/4])/19 + ((1000*I)*E^(1/4 + (I/4)*Sqrt[19] + E^3)*ExpIntegralEi[(-1 - I*Sqrt[19] + 4*x)/4])/Sqrt[19] - (250*(1 + I*Sq rt[19])*E^(1/4 + (I/4)*Sqrt[19] + E^3)*ExpIntegralEi[(-1 - I*Sqrt[19] + 4* x)/4])/19 - (250*(19 + (3*I)*Sqrt[19])*E^(1/4 + (I/4)*Sqrt[19] + E^3)*ExpI ntegralEi[(-1 - I*Sqrt[19] + 4*x)/4])/19 + (5000*E^(1/4 - (I/4)*Sqrt[19] + E^3)*ExpIntegralEi[(-1 + I*Sqrt[19] + 4*x)/4])/19 - ((1000*I)*E^(1/4 - (I /4)*Sqrt[19] + E^3)*ExpIntegralEi[(-1 + I*Sqrt[19] + 4*x)/4])/Sqrt[19] - ( 250*(1 - I*Sqrt[19])*E^(1/4 - (I/4)*Sqrt[19] + E^3)*ExpIntegralEi[(-1 + I* Sqrt[19] + 4*x)/4])/19 - (250*(19 - (3*I)*Sqrt[19])*E^(1/4 - (I/4)*Sqrt[19 ] + E^3)*ExpIntegralEi[(-1 + I*Sqrt[19] + 4*x)/4])/19)
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] && 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(23)=46\).
Time = 3.43 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42
| method | result | size |
| risch | \(x^{2}+\frac {500 x^{3}+248500 x^{2}+1250 x -\frac {3125}{2}}{x^{4}-x^{3}+\frac {11}{4} x^{2}-\frac {5}{4} x +\frac {25}{16}}+{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x}-\frac {2 x \left (4 x^{2}-2 x +2005\right ) {\mathrm e}^{{\mathrm e}^{3}+x}}{4 x^{2}-2 x +5}\) | \(80\) |
| norman | \(\frac {8024 x^{3}+20055 x +3975904 x^{2}-16 x^{5}+16 x^{6}+25 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x}-20050 \,{\mathrm e}^{{\mathrm e}^{3}+x} x +8040 \,{\mathrm e}^{{\mathrm e}^{3}+x} x^{2}-16088 \,{\mathrm e}^{{\mathrm e}^{3}+x} x^{3}+32 \,{\mathrm e}^{{\mathrm e}^{3}+x} x^{4}-32 \,{\mathrm e}^{{\mathrm e}^{3}+x} x^{5}-20 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x} x +44 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x} x^{2}-16 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x} x^{3}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x} x^{4}-\frac {100275}{4}}{\left (4 x^{2}-2 x +5\right )^{2}}\) | \(142\) |
| parallelrisch | \(\frac {256 x^{6}-512 \,{\mathrm e}^{{\mathrm e}^{3}+x} x^{5}+256 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x} x^{4}-256 x^{5}+512 \,{\mathrm e}^{{\mathrm e}^{3}+x} x^{4}-401100-256 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x} x^{3}-257408 \,{\mathrm e}^{{\mathrm e}^{3}+x} x^{3}+704 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x} x^{2}+128384 x^{3}+128640 \,{\mathrm e}^{{\mathrm e}^{3}+x} x^{2}-320 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x} x +63614464 x^{2}-320800 \,{\mathrm e}^{{\mathrm e}^{3}+x} x +400 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x}+320880 x}{256 x^{4}-256 x^{3}+704 x^{2}-320 x +400}\) | \(153\) |
| orering | \(\text {Expression too large to display}\) | \(1940\) |
| parts | \(\text {Expression too large to display}\) | \(14490\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(94595\) |
| default | \(\text {Expression too large to display}\) | \(94595\) |
Input:
int(((128*x^6-192*x^5+576*x^4-496*x^3+720*x^2-300*x+250)*exp(exp(3)+x)^2+( -128*x^7+64*x^6-64384*x^5+127920*x^4-208224*x^3+79580*x^2-59950*x-100250)* exp(exp(3)+x)+128*x^7-192*x^6+576*x^5-32496*x^4-31823280*x^3-120300*x^2+40 200250*x)/(64*x^6-96*x^5+288*x^4-248*x^3+360*x^2-150*x+125),x,method=_RETU RNVERBOSE)
Output:
x^2+(500*x^3+248500*x^2+1250*x-3125/2)/(x^4-x^3+11/4*x^2-5/4*x+25/16)+exp( 2*exp(3)+2*x)-2*x*(4*x^2-2*x+2005)/(4*x^2-2*x+5)*exp(exp(3)+x)
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (25) = 50\).
Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.45 \[ \int \frac {40200250 x-120300 x^2-31823280 x^3-32496 x^4+576 x^5-192 x^6+128 x^7+e^{2 e^3+2 x} \left (250-300 x+720 x^2-496 x^3+576 x^4-192 x^5+128 x^6\right )+e^{e^3+x} \left (-100250-59950 x+79580 x^2-208224 x^3+127920 x^4-64384 x^5+64 x^6-128 x^7\right )}{125-150 x+360 x^2-248 x^3+288 x^4-96 x^5+64 x^6} \, dx=\frac {16 \, x^{6} - 16 \, x^{5} + 44 \, x^{4} + 7980 \, x^{3} + 3976025 \, x^{2} + {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )} e^{\left (2 \, x + 2 \, e^{3}\right )} - 2 \, {\left (16 \, x^{5} - 16 \, x^{4} + 8044 \, x^{3} - 4020 \, x^{2} + 10025 \, x\right )} e^{\left (x + e^{3}\right )} + 20000 \, x - 25000}{16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25} \] Input:
integrate(((128*x^6-192*x^5+576*x^4-496*x^3+720*x^2-300*x+250)*exp(exp(3)+ x)^2+(-128*x^7+64*x^6-64384*x^5+127920*x^4-208224*x^3+79580*x^2-59950*x-10 0250)*exp(exp(3)+x)+128*x^7-192*x^6+576*x^5-32496*x^4-31823280*x^3-120300* x^2+40200250*x)/(64*x^6-96*x^5+288*x^4-248*x^3+360*x^2-150*x+125),x, algor ithm="fricas")
Output:
(16*x^6 - 16*x^5 + 44*x^4 + 7980*x^3 + 3976025*x^2 + (16*x^4 - 16*x^3 + 44 *x^2 - 20*x + 25)*e^(2*x + 2*e^3) - 2*(16*x^5 - 16*x^4 + 8044*x^3 - 4020*x ^2 + 10025*x)*e^(x + e^3) + 20000*x - 25000)/(16*x^4 - 16*x^3 + 44*x^2 - 2 0*x + 25)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (22) = 44\).
Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.67 \[ \int \frac {40200250 x-120300 x^2-31823280 x^3-32496 x^4+576 x^5-192 x^6+128 x^7+e^{2 e^3+2 x} \left (250-300 x+720 x^2-496 x^3+576 x^4-192 x^5+128 x^6\right )+e^{e^3+x} \left (-100250-59950 x+79580 x^2-208224 x^3+127920 x^4-64384 x^5+64 x^6-128 x^7\right )}{125-150 x+360 x^2-248 x^3+288 x^4-96 x^5+64 x^6} \, dx=x^{2} + \frac {\left (4 x^{2} - 2 x + 5\right ) e^{2 x + 2 e^{3}} + \left (- 8 x^{3} + 4 x^{2} - 4010 x\right ) e^{x + e^{3}}}{4 x^{2} - 2 x + 5} + \frac {8000 x^{3} + 3976000 x^{2} + 20000 x - 25000}{16 x^{4} - 16 x^{3} + 44 x^{2} - 20 x + 25} \] Input:
integrate(((128*x**6-192*x**5+576*x**4-496*x**3+720*x**2-300*x+250)*exp(ex p(3)+x)**2+(-128*x**7+64*x**6-64384*x**5+127920*x**4-208224*x**3+79580*x** 2-59950*x-100250)*exp(exp(3)+x)+128*x**7-192*x**6+576*x**5-32496*x**4-3182 3280*x**3-120300*x**2+40200250*x)/(64*x**6-96*x**5+288*x**4-248*x**3+360*x **2-150*x+125),x)
Output:
x**2 + ((4*x**2 - 2*x + 5)*exp(2*x + 2*exp(3)) + (-8*x**3 + 4*x**2 - 4010* x)*exp(x + exp(3)))/(4*x**2 - 2*x + 5) + (8000*x**3 + 3976000*x**2 + 20000 *x - 25000)/(16*x**4 - 16*x**3 + 44*x**2 - 20*x + 25)
Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (25) = 50\).
Time = 0.21 (sec) , antiderivative size = 349, normalized size of antiderivative = 10.58 \[ \int \frac {40200250 x-120300 x^2-31823280 x^3-32496 x^4+576 x^5-192 x^6+128 x^7+e^{2 e^3+2 x} \left (250-300 x+720 x^2-496 x^3+576 x^4-192 x^5+128 x^6\right )+e^{e^3+x} \left (-100250-59950 x+79580 x^2-208224 x^3+127920 x^4-64384 x^5+64 x^6-128 x^7\right )}{125-150 x+360 x^2-248 x^3+288 x^4-96 x^5+64 x^6} \, dx=x^{2} + \frac {92336 \, x^{3} - 98132 \, x^{2} + 120160 \, x - 59825}{1444 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} - \frac {3 \, {\left (17472 \, x^{3} + 24440 \, x^{2} + 3200 \, x + 31875\right )}}{1444 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} - \frac {9 \, {\left (4416 \, x^{3} - 8366 \, x^{2} + 6680 \, x - 6225\right )}}{361 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} + \frac {2031 \, {\left (3376 \, x^{3} + 356 \, x^{2} + 2020 \, x + 1225\right )}}{1444 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} - \frac {1988955 \, {\left (240 \, x^{3} - 1624 \, x^{2} + 520 \, x - 1025\right )}}{722 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} - \frac {30075 \, {\left (88 \, x^{3} - 66 \, x^{2} - 50 \, x - 75\right )}}{722 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} + \frac {20100125 \, {\left (48 \, x^{3} - 36 \, x^{2} + 104 \, x - 205\right )}}{1444 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} + \frac {{\left (4 \, x^{2} e^{\left (2 \, e^{3}\right )} - 2 \, x e^{\left (2 \, e^{3}\right )} + 5 \, e^{\left (2 \, e^{3}\right )}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (4 \, x^{3} e^{\left (e^{3}\right )} - 2 \, x^{2} e^{\left (e^{3}\right )} + 2005 \, x e^{\left (e^{3}\right )}\right )} e^{x}}{4 \, x^{2} - 2 \, x + 5} \] Input:
integrate(((128*x^6-192*x^5+576*x^4-496*x^3+720*x^2-300*x+250)*exp(exp(3)+ x)^2+(-128*x^7+64*x^6-64384*x^5+127920*x^4-208224*x^3+79580*x^2-59950*x-10 0250)*exp(exp(3)+x)+128*x^7-192*x^6+576*x^5-32496*x^4-31823280*x^3-120300* x^2+40200250*x)/(64*x^6-96*x^5+288*x^4-248*x^3+360*x^2-150*x+125),x, algor ithm="maxima")
Output:
x^2 + 1/1444*(92336*x^3 - 98132*x^2 + 120160*x - 59825)/(16*x^4 - 16*x^3 + 44*x^2 - 20*x + 25) - 3/1444*(17472*x^3 + 24440*x^2 + 3200*x + 31875)/(16 *x^4 - 16*x^3 + 44*x^2 - 20*x + 25) - 9/361*(4416*x^3 - 8366*x^2 + 6680*x - 6225)/(16*x^4 - 16*x^3 + 44*x^2 - 20*x + 25) + 2031/1444*(3376*x^3 + 356 *x^2 + 2020*x + 1225)/(16*x^4 - 16*x^3 + 44*x^2 - 20*x + 25) - 1988955/722 *(240*x^3 - 1624*x^2 + 520*x - 1025)/(16*x^4 - 16*x^3 + 44*x^2 - 20*x + 25 ) - 30075/722*(88*x^3 - 66*x^2 - 50*x - 75)/(16*x^4 - 16*x^3 + 44*x^2 - 20 *x + 25) + 20100125/1444*(48*x^3 - 36*x^2 + 104*x - 205)/(16*x^4 - 16*x^3 + 44*x^2 - 20*x + 25) + ((4*x^2*e^(2*e^3) - 2*x*e^(2*e^3) + 5*e^(2*e^3))*e ^(2*x) - 2*(4*x^3*e^(e^3) - 2*x^2*e^(e^3) + 2005*x*e^(e^3))*e^x)/(4*x^2 - 2*x + 5)
Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (25) = 50\).
Time = 0.35 (sec) , antiderivative size = 753, normalized size of antiderivative = 22.82 \[ \int \frac {40200250 x-120300 x^2-31823280 x^3-32496 x^4+576 x^5-192 x^6+128 x^7+e^{2 e^3+2 x} \left (250-300 x+720 x^2-496 x^3+576 x^4-192 x^5+128 x^6\right )+e^{e^3+x} \left (-100250-59950 x+79580 x^2-208224 x^3+127920 x^4-64384 x^5+64 x^6-128 x^7\right )}{125-150 x+360 x^2-248 x^3+288 x^4-96 x^5+64 x^6} \, dx=\text {Too large to display} \] Input:
integrate(((128*x^6-192*x^5+576*x^4-496*x^3+720*x^2-300*x+250)*exp(exp(3)+ x)^2+(-128*x^7+64*x^6-64384*x^5+127920*x^4-208224*x^3+79580*x^2-59950*x-10 0250)*exp(exp(3)+x)+128*x^7-192*x^6+576*x^5-32496*x^4-31823280*x^3-120300* x^2+40200250*x)/(64*x^6-96*x^5+288*x^4-248*x^3+360*x^2-150*x+125),x, algor ithm="giac")
Output:
(16*(x + e^3)^6 - 96*(x + e^3)^5*e^3 - 32*(x + e^3)^5*e^(x + e^3) - 16*(x + e^3)^5 + 224*(x + e^3)^4*e^6 + 80*(x + e^3)^4*e^3 + 16*(x + e^3)^4*e^(2* x + 2*e^3) + 160*(x + e^3)^4*e^(x + e^3 + 3) + 32*(x + e^3)^4*e^(x + e^3) + 44*(x + e^3)^4 - 256*(x + e^3)^3*e^9 - 144*(x + e^3)^3*e^6 - 176*(x + e^ 3)^3*e^3 - 64*(x + e^3)^3*e^(2*x + 2*e^3 + 3) - 16*(x + e^3)^3*e^(2*x + 2* e^3) - 320*(x + e^3)^3*e^(x + e^3 + 6) - 128*(x + e^3)^3*e^(x + e^3 + 3) - 32088*(x + e^3)^3*e^(x + e^3) + 15980*(x + e^3)^3 + 144*(x + e^3)^2*e^12 + 112*(x + e^3)^2*e^9 + 220*(x + e^3)^2*e^6 - 47940*(x + e^3)^2*e^3 + 96*( x + e^3)^2*e^(2*x + 2*e^3 + 6) + 48*(x + e^3)^2*e^(2*x + 2*e^3 + 3) + 44*( x + e^3)^2*e^(2*x + 2*e^3) + 320*(x + e^3)^2*e^(x + e^3 + 9) + 192*(x + e^ 3)^2*e^(x + e^3 + 6) + 96264*(x + e^3)^2*e^(x + e^3 + 3) + 16040*(x + e^3) ^2*e^(x + e^3) + 7952025*(x + e^3)^2 - 32*(x + e^3)*e^15 - 32*(x + e^3)*e^ 12 - 88*(x + e^3)*e^9 + 47960*(x + e^3)*e^6 - 15904050*(x + e^3)*e^3 - 64* (x + e^3)*e^(2*x + 2*e^3 + 9) - 48*(x + e^3)*e^(2*x + 2*e^3 + 6) - 88*(x + e^3)*e^(2*x + 2*e^3 + 3) - 20*(x + e^3)*e^(2*x + 2*e^3) - 160*(x + e^3)*e ^(x + e^3 + 12) - 128*(x + e^3)*e^(x + e^3 + 9) - 96264*(x + e^3)*e^(x + e ^3 + 6) - 32080*(x + e^3)*e^(x + e^3 + 3) - 40050*(x + e^3)*e^(x + e^3) + 40000*x - 16000*e^9 + 7952000*e^6 + 16*e^(2*x + 2*e^3 + 12) + 16*e^(2*x + 2*e^3 + 9) + 44*e^(2*x + 2*e^3 + 6) + 20*e^(2*x + 2*e^3 + 3) + 25*e^(2*x + 2*e^3) + 32*e^(x + e^3 + 15) + 32*e^(x + e^3 + 12) + 32088*e^(x + e^3 ...
Time = 2.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42 \[ \int \frac {40200250 x-120300 x^2-31823280 x^3-32496 x^4+576 x^5-192 x^6+128 x^7+e^{2 e^3+2 x} \left (250-300 x+720 x^2-496 x^3+576 x^4-192 x^5+128 x^6\right )+e^{e^3+x} \left (-100250-59950 x+79580 x^2-208224 x^3+127920 x^4-64384 x^5+64 x^6-128 x^7\right )}{125-150 x+360 x^2-248 x^3+288 x^4-96 x^5+64 x^6} \, dx={\mathrm {e}}^{2\,x+2\,{\mathrm {e}}^3}+x^2+\frac {500\,x^3+248500\,x^2+1250\,x-\frac {3125}{2}}{x^4-x^3+\frac {11\,x^2}{4}-\frac {5\,x}{4}+\frac {25}{16}}-\frac {{\mathrm {e}}^{x+{\mathrm {e}}^3}\,\left (2\,x^3-x^2+\frac {2005\,x}{2}\right )}{x^2-\frac {x}{2}+\frac {5}{4}} \] Input:
int(-(exp(x + exp(3))*(59950*x - 79580*x^2 + 208224*x^3 - 127920*x^4 + 643 84*x^5 - 64*x^6 + 128*x^7 + 100250) - exp(2*x + 2*exp(3))*(720*x^2 - 300*x - 496*x^3 + 576*x^4 - 192*x^5 + 128*x^6 + 250) - 40200250*x + 120300*x^2 + 31823280*x^3 + 32496*x^4 - 576*x^5 + 192*x^6 - 128*x^7)/(360*x^2 - 150*x - 248*x^3 + 288*x^4 - 96*x^5 + 64*x^6 + 125),x)
Output:
exp(2*x + 2*exp(3)) + x^2 + (1250*x + 248500*x^2 + 500*x^3 - 3125/2)/((11* x^2)/4 - (5*x)/4 - x^3 + x^4 + 25/16) - (exp(x + exp(3))*((2005*x)/2 - x^2 + 2*x^3))/(x^2 - x/2 + 5/4)
Time = 0.18 (sec) , antiderivative size = 181, normalized size of antiderivative = 5.48 \[ \int \frac {40200250 x-120300 x^2-31823280 x^3-32496 x^4+576 x^5-192 x^6+128 x^7+e^{2 e^3+2 x} \left (250-300 x+720 x^2-496 x^3+576 x^4-192 x^5+128 x^6\right )+e^{e^3+x} \left (-100250-59950 x+79580 x^2-208224 x^3+127920 x^4-64384 x^5+64 x^6-128 x^7\right )}{125-150 x+360 x^2-248 x^3+288 x^4-96 x^5+64 x^6} \, dx=\frac {64 e^{2 e^{3}+2 x} x^{4}-64 e^{2 e^{3}+2 x} x^{3}+176 e^{2 e^{3}+2 x} x^{2}-80 e^{2 e^{3}+2 x} x +100 e^{2 e^{3}+2 x}-128 e^{e^{3}+x} x^{5}+128 e^{e^{3}+x} x^{4}-64352 e^{e^{3}+x} x^{3}+32160 e^{e^{3}+x} x^{2}-80200 e^{e^{3}+x} x +64 x^{6}-64 x^{5}+32096 x^{4}+15991880 x^{2}+40100 x -50125}{64 x^{4}-64 x^{3}+176 x^{2}-80 x +100} \] Input:
int(((128*x^6-192*x^5+576*x^4-496*x^3+720*x^2-300*x+250)*exp(exp(3)+x)^2+( -128*x^7+64*x^6-64384*x^5+127920*x^4-208224*x^3+79580*x^2-59950*x-100250)* exp(exp(3)+x)+128*x^7-192*x^6+576*x^5-32496*x^4-31823280*x^3-120300*x^2+40 200250*x)/(64*x^6-96*x^5+288*x^4-248*x^3+360*x^2-150*x+125),x)
Output:
(64*e**(2*e**3 + 2*x)*x**4 - 64*e**(2*e**3 + 2*x)*x**3 + 176*e**(2*e**3 + 2*x)*x**2 - 80*e**(2*e**3 + 2*x)*x + 100*e**(2*e**3 + 2*x) - 128*e**(e**3 + x)*x**5 + 128*e**(e**3 + x)*x**4 - 64352*e**(e**3 + x)*x**3 + 32160*e**( e**3 + x)*x**2 - 80200*e**(e**3 + x)*x + 64*x**6 - 64*x**5 + 32096*x**4 + 15991880*x**2 + 40100*x - 50125)/(4*(16*x**4 - 16*x**3 + 44*x**2 - 20*x + 25))