Integrand size = 67, antiderivative size = 29 \[ \int \frac {-50-350 x-1600 x^3-2000 x^4+2400 x^5+1600 x^6+e^{2 x} (-18+18 x)+e^x \left (-60-180 x+210 x^2-240 x^3-120 x^4\right )}{9 x^3} \, dx=\left (\frac {e^x}{x}-\frac {5 (-1+x) \left (-3+(2+2 x)^2\right )}{3 x}\right )^2 \] Output:
(exp(x)/x-5/3*(-1+x)*((2+2*x)^2-3)/x)^2
Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(29)=58\).
Time = 3.58 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.14 \[ \int \frac {-50-350 x-1600 x^3-2000 x^4+2400 x^5+1600 x^6+e^{2 x} (-18+18 x)+e^x \left (-60-180 x+210 x^2-240 x^3-120 x^4\right )}{9 x^3} \, dx=\frac {1}{9} \left (e^x \left (-120+\frac {30}{x^2}+\frac {210}{x}-120 x\right )+\frac {25}{x^2}+\frac {9 e^{2 x}}{x^2}+\frac {350}{x}-1600 x-1000 x^2+800 x^3+400 x^4\right ) \] Input:
Integrate[(-50 - 350*x - 1600*x^3 - 2000*x^4 + 2400*x^5 + 1600*x^6 + E^(2* x)*(-18 + 18*x) + E^x*(-60 - 180*x + 210*x^2 - 240*x^3 - 120*x^4))/(9*x^3) ,x]
Output:
(E^x*(-120 + 30/x^2 + 210/x - 120*x) + 25/x^2 + (9*E^(2*x))/x^2 + 350/x - 1600*x - 1000*x^2 + 800*x^3 + 400*x^4)/9
Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(29)=58\).
Time = 0.40 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.55, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {27, 27, 2010, 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.
\(\displaystyle \int \frac {1600 x^6+2400 x^5-2000 x^4-1600 x^3+e^x \left (-120 x^4-240 x^3+210 x^2-180 x-60\right )-350 x+e^{2 x} (18 x-18)-50}{9 x^3} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int -\frac {2 \left (-800 x^6-1200 x^5+1000 x^4+800 x^3+175 x+9 e^{2 x} (1-x)+15 e^x \left (4 x^4+8 x^3-7 x^2+6 x+2\right )+25\right )}{x^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2}{9} \int \frac {-800 x^6-1200 x^5+1000 x^4+800 x^3+175 x+9 e^{2 x} (1-x)+15 e^x \left (4 x^4+8 x^3-7 x^2+6 x+2\right )+25}{x^3}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle -\frac {2}{9} \int \left (-\frac {9 e^{2 x} (x-1)}{x^3}+\frac {15 e^x \left (4 x^4+8 x^3-7 x^2+6 x+2\right )}{x^3}-\frac {25 \left (32 x^6+48 x^5-40 x^4-32 x^3-7 x-1\right )}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2}{9} \left (-200 x^4-400 x^3+500 x^2-\frac {15 e^x}{x^2}-\frac {9 e^{2 x}}{2 x^2}-\frac {25}{2 x^2}+60 e^x x+800 x+60 e^x-\frac {105 e^x}{x}-\frac {175}{x}\right )\) |
Input:
Int[(-50 - 350*x - 1600*x^3 - 2000*x^4 + 2400*x^5 + 1600*x^6 + E^(2*x)*(-1 8 + 18*x) + E^x*(-60 - 180*x + 210*x^2 - 240*x^3 - 120*x^4))/(9*x^3),x]
Output:
(-2*(60*E^x - 25/(2*x^2) - (15*E^x)/x^2 - (9*E^(2*x))/(2*x^2) - 175/x - (1 05*E^x)/x + 800*x + 60*E^x*x + 500*x^2 - 400*x^3 - 200*x^4))/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_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(26)=52\).
Time = 1.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97
method | result | size |
norman | \(\frac {\frac {25}{9}+{\mathrm e}^{2 x}+\frac {350 x}{9}-\frac {1600 x^{3}}{9}-\frac {1000 x^{4}}{9}+\frac {800 x^{5}}{9}+\frac {400 x^{6}}{9}+\frac {70 \,{\mathrm e}^{x} x}{3}-\frac {40 \,{\mathrm e}^{x} x^{2}}{3}-\frac {40 \,{\mathrm e}^{x} x^{3}}{3}+\frac {10 \,{\mathrm e}^{x}}{3}}{x^{2}}\) | \(57\) |
risch | \(\frac {400 x^{4}}{9}+\frac {800 x^{3}}{9}-\frac {1000 x^{2}}{9}-\frac {1600 x}{9}+\frac {350 x +25}{9 x^{2}}+\frac {{\mathrm e}^{2 x}}{x^{2}}-\frac {10 \left (4 x^{3}+4 x^{2}-7 x -1\right ) {\mathrm e}^{x}}{3 x^{2}}\) | \(60\) |
parallelrisch | \(\frac {400 x^{6}+800 x^{5}-120 \,{\mathrm e}^{x} x^{3}-1000 x^{4}-120 \,{\mathrm e}^{x} x^{2}-1600 x^{3}+210 \,{\mathrm e}^{x} x +9 \,{\mathrm e}^{2 x}+30 \,{\mathrm e}^{x}+350 x +25}{9 x^{2}}\) | \(60\) |
default | \(-\frac {1000 x^{2}}{9}-\frac {1600 x}{9}+\frac {25}{9 x^{2}}+\frac {350}{9 x}+\frac {800 x^{3}}{9}+\frac {400 x^{4}}{9}+\frac {10 \,{\mathrm e}^{x}}{3 x^{2}}+\frac {70 \,{\mathrm e}^{x}}{3 x}+\frac {{\mathrm e}^{2 x}}{x^{2}}-\frac {40 \,{\mathrm e}^{x} x}{3}-\frac {40 \,{\mathrm e}^{x}}{3}\) | \(61\) |
parts | \(-\frac {1000 x^{2}}{9}-\frac {1600 x}{9}+\frac {25}{9 x^{2}}+\frac {350}{9 x}+\frac {800 x^{3}}{9}+\frac {400 x^{4}}{9}+\frac {10 \,{\mathrm e}^{x}}{3 x^{2}}+\frac {70 \,{\mathrm e}^{x}}{3 x}+\frac {{\mathrm e}^{2 x}}{x^{2}}-\frac {40 \,{\mathrm e}^{x} x}{3}-\frac {40 \,{\mathrm e}^{x}}{3}\) | \(61\) |
orering | \(\frac {\left (32 x^{11}-1016 x^{9}+2336 x^{8}+2844 x^{7}-11852 x^{6}+2271 x^{5}+26265 x^{4}-33885 x^{3}+16938 x^{2}-10449 x +3321\right ) \left (\left (18 x -18\right ) {\mathrm e}^{2 x}+\left (-120 x^{4}-240 x^{3}+210 x^{2}-180 x -60\right ) {\mathrm e}^{x}+1600 x^{6}+2400 x^{5}-2000 x^{4}-1600 x^{3}-350 x -50\right )}{9 \left (128 x^{10}-640 x^{9}+64 x^{8}+4784 x^{7}-8728 x^{6}-2216 x^{5}+14760 x^{4}-8691 x^{3}+4356 x^{2}+387 x -486\right ) x^{3}}-\frac {3 x \left (32 x^{10}-160 x^{9}-104 x^{8}+1336 x^{7}-4536 x^{5}+1215 x^{4}+10728 x^{3}-15699 x^{2}+7452 x -2214\right ) \left (\frac {18 \,{\mathrm e}^{2 x}+2 \left (18 x -18\right ) {\mathrm e}^{2 x}+\left (-480 x^{3}-720 x^{2}+420 x -180\right ) {\mathrm e}^{x}+\left (-120 x^{4}-240 x^{3}+210 x^{2}-180 x -60\right ) {\mathrm e}^{x}+9600 x^{5}+12000 x^{4}-8000 x^{3}-4800 x^{2}-350}{9 x^{3}}-\frac {\left (18 x -18\right ) {\mathrm e}^{2 x}+\left (-120 x^{4}-240 x^{3}+210 x^{2}-180 x -60\right ) {\mathrm e}^{x}+1600 x^{6}+2400 x^{5}-2000 x^{4}-1600 x^{3}-350 x -50}{3 x^{4}}\right )}{2 \left (128 x^{10}-640 x^{9}+64 x^{8}+4784 x^{7}-8728 x^{6}-2216 x^{5}+14760 x^{4}-8691 x^{3}+4356 x^{2}+387 x -486\right )}+\frac {\left (32 x^{9}-192 x^{8}+24 x^{7}+1328 x^{6}-4536 x^{4}-1809 x^{3}+8919 x^{2}-3807 x +1107\right ) x^{2} \left (\frac {72 \,{\mathrm e}^{2 x}+4 \left (18 x -18\right ) {\mathrm e}^{2 x}+\left (-1440 x^{2}-1440 x +420\right ) {\mathrm e}^{x}+2 \left (-480 x^{3}-720 x^{2}+420 x -180\right ) {\mathrm e}^{x}+\left (-120 x^{4}-240 x^{3}+210 x^{2}-180 x -60\right ) {\mathrm e}^{x}+48000 x^{4}+48000 x^{3}-24000 x^{2}-9600 x}{9 x^{3}}-\frac {2 \left (18 \,{\mathrm e}^{2 x}+2 \left (18 x -18\right ) {\mathrm e}^{2 x}+\left (-480 x^{3}-720 x^{2}+420 x -180\right ) {\mathrm e}^{x}+\left (-120 x^{4}-240 x^{3}+210 x^{2}-180 x -60\right ) {\mathrm e}^{x}+9600 x^{5}+12000 x^{4}-8000 x^{3}-4800 x^{2}-350\right )}{3 x^{4}}+\frac {\frac {4 \left (18 x -18\right ) {\mathrm e}^{2 x}}{3}+\frac {4 \left (-120 x^{4}-240 x^{3}+210 x^{2}-180 x -60\right ) {\mathrm e}^{x}}{3}+\frac {6400 x^{6}}{3}+3200 x^{5}-\frac {8000 x^{4}}{3}-\frac {6400 x^{3}}{3}-\frac {1400 x}{3}-\frac {200}{3}}{x^{5}}\right )}{2 \left (4 x^{3}-8 x^{2}-15 x +6\right ) \left (32 x^{7}-96 x^{6}-56 x^{5}+676 x^{4}-896 x^{3}+273 x^{2}-138 x -81\right )}\) | \(760\) |
Input:
int(1/9*((18*x-18)*exp(x)^2+(-120*x^4-240*x^3+210*x^2-180*x-60)*exp(x)+160 0*x^6+2400*x^5-2000*x^4-1600*x^3-350*x-50)/x^3,x,method=_RETURNVERBOSE)
Output:
(25/9+exp(x)^2+350/9*x-1600/9*x^3-1000/9*x^4+800/9*x^5+400/9*x^6+70/3*exp( x)*x-40/3*exp(x)*x^2-40/3*exp(x)*x^3+10/3*exp(x))/x^2
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90 \[ \int \frac {-50-350 x-1600 x^3-2000 x^4+2400 x^5+1600 x^6+e^{2 x} (-18+18 x)+e^x \left (-60-180 x+210 x^2-240 x^3-120 x^4\right )}{9 x^3} \, dx=\frac {400 \, x^{6} + 800 \, x^{5} - 1000 \, x^{4} - 1600 \, x^{3} - 30 \, {\left (4 \, x^{3} + 4 \, x^{2} - 7 \, x - 1\right )} e^{x} + 350 \, x + 9 \, e^{\left (2 \, x\right )} + 25}{9 \, x^{2}} \] Input:
integrate(1/9*((18*x-18)*exp(x)^2+(-120*x^4-240*x^3+210*x^2-180*x-60)*exp( x)+1600*x^6+2400*x^5-2000*x^4-1600*x^3-350*x-50)/x^3,x, algorithm="fricas" )
Output:
1/9*(400*x^6 + 800*x^5 - 1000*x^4 - 1600*x^3 - 30*(4*x^3 + 4*x^2 - 7*x - 1 )*e^x + 350*x + 9*e^(2*x) + 25)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (24) = 48\).
Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.52 \[ \int \frac {-50-350 x-1600 x^3-2000 x^4+2400 x^5+1600 x^6+e^{2 x} (-18+18 x)+e^x \left (-60-180 x+210 x^2-240 x^3-120 x^4\right )}{9 x^3} \, dx=\frac {400 x^{4}}{9} + \frac {800 x^{3}}{9} - \frac {1000 x^{2}}{9} - \frac {1600 x}{9} + \frac {350 x + 25}{9 x^{2}} + \frac {3 x^{2} e^{2 x} + \left (- 40 x^{5} - 40 x^{4} + 70 x^{3} + 10 x^{2}\right ) e^{x}}{3 x^{4}} \] Input:
integrate(1/9*((18*x-18)*exp(x)**2+(-120*x**4-240*x**3+210*x**2-180*x-60)* exp(x)+1600*x**6+2400*x**5-2000*x**4-1600*x**3-350*x-50)/x**3,x)
Output:
400*x**4/9 + 800*x**3/9 - 1000*x**2/9 - 1600*x/9 + (350*x + 25)/(9*x**2) + (3*x**2*exp(2*x) + (-40*x**5 - 40*x**4 + 70*x**3 + 10*x**2)*exp(x))/(3*x* *4)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.48 \[ \int \frac {-50-350 x-1600 x^3-2000 x^4+2400 x^5+1600 x^6+e^{2 x} (-18+18 x)+e^x \left (-60-180 x+210 x^2-240 x^3-120 x^4\right )}{9 x^3} \, dx=\frac {400}{9} \, x^{4} + \frac {800}{9} \, x^{3} - \frac {1000}{9} \, x^{2} - \frac {40}{3} \, {\left (x - 1\right )} e^{x} - \frac {1600}{9} \, x + \frac {350}{9 \, x} + \frac {25}{9 \, x^{2}} + \frac {70}{3} \, {\rm Ei}\left (x\right ) - \frac {80}{3} \, e^{x} - 20 \, \Gamma \left (-1, -x\right ) + 4 \, \Gamma \left (-1, -2 \, x\right ) + \frac {20}{3} \, \Gamma \left (-2, -x\right ) + 8 \, \Gamma \left (-2, -2 \, x\right ) \] Input:
integrate(1/9*((18*x-18)*exp(x)^2+(-120*x^4-240*x^3+210*x^2-180*x-60)*exp( x)+1600*x^6+2400*x^5-2000*x^4-1600*x^3-350*x-50)/x^3,x, algorithm="maxima" )
Output:
400/9*x^4 + 800/9*x^3 - 1000/9*x^2 - 40/3*(x - 1)*e^x - 1600/9*x + 350/9/x + 25/9/x^2 + 70/3*Ei(x) - 80/3*e^x - 20*gamma(-1, -x) + 4*gamma(-1, -2*x) + 20/3*gamma(-2, -x) + 8*gamma(-2, -2*x)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (29) = 58\).
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {-50-350 x-1600 x^3-2000 x^4+2400 x^5+1600 x^6+e^{2 x} (-18+18 x)+e^x \left (-60-180 x+210 x^2-240 x^3-120 x^4\right )}{9 x^3} \, dx=\frac {400 \, x^{6} + 800 \, x^{5} - 1000 \, x^{4} - 120 \, x^{3} e^{x} - 1600 \, x^{3} - 120 \, x^{2} e^{x} + 210 \, x e^{x} + 350 \, x + 9 \, e^{\left (2 \, x\right )} + 30 \, e^{x} + 25}{9 \, x^{2}} \] Input:
integrate(1/9*((18*x-18)*exp(x)^2+(-120*x^4-240*x^3+210*x^2-180*x-60)*exp( x)+1600*x^6+2400*x^5-2000*x^4-1600*x^3-350*x-50)/x^3,x, algorithm="giac")
Output:
1/9*(400*x^6 + 800*x^5 - 1000*x^4 - 120*x^3*e^x - 1600*x^3 - 120*x^2*e^x + 210*x*e^x + 350*x + 9*e^(2*x) + 30*e^x + 25)/x^2
Time = 4.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {-50-350 x-1600 x^3-2000 x^4+2400 x^5+1600 x^6+e^{2 x} (-18+18 x)+e^x \left (-60-180 x+210 x^2-240 x^3-120 x^4\right )}{9 x^3} \, dx=\frac {{\mathrm {e}}^{2\,x}+\frac {10\,{\mathrm {e}}^x}{3}+x\,\left (\frac {70\,{\mathrm {e}}^x}{3}+\frac {350}{9}\right )+\frac {25}{9}}{x^2}-x\,\left (\frac {40\,{\mathrm {e}}^x}{3}+\frac {1600}{9}\right )-\frac {40\,{\mathrm {e}}^x}{3}-\frac {1000\,x^2}{9}+\frac {800\,x^3}{9}+\frac {400\,x^4}{9} \] Input:
int(-((350*x)/9 + (exp(x)*(180*x - 210*x^2 + 240*x^3 + 120*x^4 + 60))/9 - (exp(2*x)*(18*x - 18))/9 + (1600*x^3)/9 + (2000*x^4)/9 - (800*x^5)/3 - (16 00*x^6)/9 + 50/9)/x^3,x)
Output:
(exp(2*x) + (10*exp(x))/3 + x*((70*exp(x))/3 + 350/9) + 25/9)/x^2 - x*((40 *exp(x))/3 + 1600/9) - (40*exp(x))/3 - (1000*x^2)/9 + (800*x^3)/9 + (400*x ^4)/9
Time = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21 \[ \int \frac {-50-350 x-1600 x^3-2000 x^4+2400 x^5+1600 x^6+e^{2 x} (-18+18 x)+e^x \left (-60-180 x+210 x^2-240 x^3-120 x^4\right )}{9 x^3} \, dx=\frac {9 e^{2 x}-120 e^{x} x^{3}-120 e^{x} x^{2}+210 e^{x} x +30 e^{x}+400 x^{6}+800 x^{5}-1000 x^{4}-1600 x^{3}+350 x +25}{9 x^{2}} \] Input:
int(1/9*((18*x-18)*exp(x)^2+(-120*x^4-240*x^3+210*x^2-180*x-60)*exp(x)+160 0*x^6+2400*x^5-2000*x^4-1600*x^3-350*x-50)/x^3,x)
Output:
(9*e**(2*x) - 120*e**x*x**3 - 120*e**x*x**2 + 210*e**x*x + 30*e**x + 400*x **6 + 800*x**5 - 1000*x**4 - 1600*x**3 + 350*x + 25)/(9*x**2)