Integrand size = 77, antiderivative size = 28 \[ \int \frac {e^{2 x} \left (843780+226968 x-30576 x^2-5880 x^3+588 x^4\right )+e^{4 x} \left (7031500+4302200 x+198520 x^2-257736 x^3-17616 x^4+7080 x^5+136 x^6-88 x^7+4 x^8\right )}{2401} \, dx=\left (3+e^{2 x} \left (5+x+\frac {1}{7} \left (-x-x^2\right )\right )^2\right )^2 \] Output:
(exp(x)^2*(5-1/7*x^2+6/7*x)^2+3)^2
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^{2 x} \left (843780+226968 x-30576 x^2-5880 x^3+588 x^4\right )+e^{4 x} \left (7031500+4302200 x+198520 x^2-257736 x^3-17616 x^4+7080 x^5+136 x^6-88 x^7+4 x^8\right )}{2401} \, dx=\frac {\left (147+e^{2 x} \left (-35-6 x+x^2\right )^2\right )^2}{2401} \] Input:
Integrate[(E^(2*x)*(843780 + 226968*x - 30576*x^2 - 5880*x^3 + 588*x^4) + E^(4*x)*(7031500 + 4302200*x + 198520*x^2 - 257736*x^3 - 17616*x^4 + 7080* x^5 + 136*x^6 - 88*x^7 + 4*x^8))/2401,x]
Output:
(147 + E^(2*x)*(-35 - 6*x + x^2)^2)^2/2401
Leaf count is larger than twice the leaf count of optimal. \(134\) vs. \(2(28)=56\).
Time = 0.78 (sec) , antiderivative size = 134, normalized size of antiderivative = 4.79, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {27, 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.
\(\displaystyle \int \frac {e^{2 x} \left (588 x^4-5880 x^3-30576 x^2+226968 x+843780\right )+e^{4 x} \left (4 x^8-88 x^7+136 x^6+7080 x^5-17616 x^4-257736 x^3+198520 x^2+4302200 x+7031500\right )}{2401} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \left (588 e^{2 x} \left (x^4-10 x^3-52 x^2+386 x+1435\right )+4 e^{4 x} \left (x^8-22 x^7+34 x^6+1770 x^5-4404 x^4-64434 x^3+49630 x^2+1075550 x+1757875\right )\right )dx}{2401}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^{4 x} x^8-24 e^{4 x} x^7+76 e^{4 x} x^6+1656 e^{4 x} x^5+294 e^{2 x} x^4-6474 e^{4 x} x^4-3528 e^{2 x} x^3-57960 e^{4 x} x^3-9996 e^{2 x} x^2+93100 e^{4 x} x^2+123480 e^{2 x} x+1029000 e^{4 x} x+360150 e^{2 x}+1500625 e^{4 x}}{2401}\) |
Input:
Int[(E^(2*x)*(843780 + 226968*x - 30576*x^2 - 5880*x^3 + 588*x^4) + E^(4*x )*(7031500 + 4302200*x + 198520*x^2 - 257736*x^3 - 17616*x^4 + 7080*x^5 + 136*x^6 - 88*x^7 + 4*x^8))/2401,x]
Output:
(360150*E^(2*x) + 1500625*E^(4*x) + 123480*E^(2*x)*x + 1029000*E^(4*x)*x - 9996*E^(2*x)*x^2 + 93100*E^(4*x)*x^2 - 3528*E^(2*x)*x^3 - 57960*E^(4*x)*x ^3 + 294*E^(2*x)*x^4 - 6474*E^(4*x)*x^4 + 1656*E^(4*x)*x^5 + 76*E^(4*x)*x^ 6 - 24*E^(4*x)*x^7 + E^(4*x)*x^8)/2401
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(21)=42\).
Time = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.57
method | result | size |
risch | \(\frac {\left (x^{8}-24 x^{7}+76 x^{6}+1656 x^{5}-6474 x^{4}-57960 x^{3}+93100 x^{2}+1029000 x +1500625\right ) {\mathrm e}^{4 x}}{2401}+\frac {\left (294 x^{4}-3528 x^{3}-9996 x^{2}+123480 x +360150\right ) {\mathrm e}^{2 x}}{2401}\) | \(72\) |
default | \(\frac {{\mathrm e}^{4 x} x^{8}}{2401}-\frac {24 \,{\mathrm e}^{4 x} x^{7}}{2401}+\frac {76 \,{\mathrm e}^{4 x} x^{6}}{2401}+\frac {1656 \,{\mathrm e}^{4 x} x^{5}}{2401}-\frac {6474 x^{4} {\mathrm e}^{4 x}}{2401}-\frac {8280 x^{3} {\mathrm e}^{4 x}}{343}+\frac {1900 x^{2} {\mathrm e}^{4 x}}{49}+\frac {3000 x \,{\mathrm e}^{4 x}}{7}+625 \,{\mathrm e}^{4 x}+\frac {6 \,{\mathrm e}^{2 x} x^{4}}{49}-\frac {72 x^{3} {\mathrm e}^{2 x}}{49}-\frac {204 \,{\mathrm e}^{2 x} x^{2}}{49}+\frac {360 x \,{\mathrm e}^{2 x}}{7}+150 \,{\mathrm e}^{2 x}\) | \(118\) |
parallelrisch | \(\frac {{\mathrm e}^{4 x} x^{8}}{2401}-\frac {24 \,{\mathrm e}^{4 x} x^{7}}{2401}+\frac {76 \,{\mathrm e}^{4 x} x^{6}}{2401}+\frac {1656 \,{\mathrm e}^{4 x} x^{5}}{2401}-\frac {6474 x^{4} {\mathrm e}^{4 x}}{2401}-\frac {8280 x^{3} {\mathrm e}^{4 x}}{343}+\frac {1900 x^{2} {\mathrm e}^{4 x}}{49}+\frac {3000 x \,{\mathrm e}^{4 x}}{7}+625 \,{\mathrm e}^{4 x}+\frac {6 \,{\mathrm e}^{2 x} x^{4}}{49}-\frac {72 x^{3} {\mathrm e}^{2 x}}{49}-\frac {204 \,{\mathrm e}^{2 x} x^{2}}{49}+\frac {360 x \,{\mathrm e}^{2 x}}{7}+150 \,{\mathrm e}^{2 x}\) | \(118\) |
parts | \(\frac {{\mathrm e}^{4 x} x^{8}}{2401}-\frac {24 \,{\mathrm e}^{4 x} x^{7}}{2401}+\frac {76 \,{\mathrm e}^{4 x} x^{6}}{2401}+\frac {1656 \,{\mathrm e}^{4 x} x^{5}}{2401}-\frac {6474 x^{4} {\mathrm e}^{4 x}}{2401}-\frac {8280 x^{3} {\mathrm e}^{4 x}}{343}+\frac {1900 x^{2} {\mathrm e}^{4 x}}{49}+\frac {3000 x \,{\mathrm e}^{4 x}}{7}+625 \,{\mathrm e}^{4 x}+\frac {6 \,{\mathrm e}^{2 x} x^{4}}{49}-\frac {72 x^{3} {\mathrm e}^{2 x}}{49}-\frac {204 \,{\mathrm e}^{2 x} x^{2}}{49}+\frac {360 x \,{\mathrm e}^{2 x}}{7}+150 \,{\mathrm e}^{2 x}\) | \(118\) |
orering | \(\frac {\left (3 x^{6}-42 x^{5}-160 x^{4}+3024 x^{3}+6015 x^{2}-64590 x -174650\right ) \left (\frac {\left (4 x^{8}-88 x^{7}+136 x^{6}+7080 x^{5}-17616 x^{4}-257736 x^{3}+198520 x^{2}+4302200 x +7031500\right ) {\mathrm e}^{4 x}}{2401}+\frac {\left (588 x^{4}-5880 x^{3}-30576 x^{2}+226968 x +843780\right ) {\mathrm e}^{2 x}}{2401}\right )}{4 \left (x^{2}-4 x -41\right )^{3}}-\frac {\left (x^{4}-12 x^{3}-34 x^{2}+420 x +1225\right ) \left (\frac {\left (32 x^{7}-616 x^{6}+816 x^{5}+35400 x^{4}-70464 x^{3}-773208 x^{2}+397040 x +4302200\right ) {\mathrm e}^{4 x}}{2401}+\frac {4 \left (4 x^{8}-88 x^{7}+136 x^{6}+7080 x^{5}-17616 x^{4}-257736 x^{3}+198520 x^{2}+4302200 x +7031500\right ) {\mathrm e}^{4 x}}{2401}+\frac {\left (2352 x^{3}-17640 x^{2}-61152 x +226968\right ) {\mathrm e}^{2 x}}{2401}+\frac {2 \left (588 x^{4}-5880 x^{3}-30576 x^{2}+226968 x +843780\right ) {\mathrm e}^{2 x}}{2401}\right )}{8 \left (x^{2}-4 x -41\right )^{2}}\) | \(282\) |
meijerg | \(-775+\frac {5125 \,{\mathrm e}^{4 x}}{7}+\frac {\left (589824 x^{8}-1179648 x^{7}+2064384 x^{6}-3096576 x^{5}+3870720 x^{4}-3870720 x^{3}+2903040 x^{2}-1451520 x +362880\right ) {\mathrm e}^{4 x}}{1416167424}+\frac {11 \left (-131072 x^{7}+229376 x^{6}-344064 x^{5}+430080 x^{4}-430080 x^{3}+322560 x^{2}-161280 x +40320\right ) {\mathrm e}^{4 x}}{157351936}+\frac {17 \left (28672 x^{6}-43008 x^{5}+53760 x^{4}-53760 x^{3}+40320 x^{2}-20160 x +5040\right ) {\mathrm e}^{4 x}}{34420736}-\frac {295 \left (-6144 x^{5}+7680 x^{4}-7680 x^{3}+5760 x^{2}-2880 x +720\right ) {\mathrm e}^{4 x}}{2458624}-\frac {1101 \left (1280 x^{4}-1280 x^{3}+960 x^{2}-480 x +120\right ) {\mathrm e}^{4 x}}{768320}+\frac {32217 \left (-256 x^{3}+192 x^{2}-96 x +24\right ) {\mathrm e}^{4 x}}{307328}+\frac {3545 \left (48 x^{2}-24 x +6\right ) {\mathrm e}^{4 x}}{8232}-\frac {10975 \left (-8 x +2\right ) {\mathrm e}^{4 x}}{196}+\frac {1230 \,{\mathrm e}^{2 x}}{7}+\frac {3 \left (80 x^{4}-160 x^{3}+240 x^{2}-240 x +120\right ) {\mathrm e}^{2 x}}{1960}+\frac {15 \left (-32 x^{3}+48 x^{2}-48 x +24\right ) {\mathrm e}^{2 x}}{392}-\frac {26 \left (12 x^{2}-12 x +6\right ) {\mathrm e}^{2 x}}{49}-\frac {579 \left (-4 x +2\right ) {\mathrm e}^{2 x}}{49}\) | \(317\) |
Input:
int(1/2401*(4*x^8-88*x^7+136*x^6+7080*x^5-17616*x^4-257736*x^3+198520*x^2+ 4302200*x+7031500)*exp(x)^4+1/2401*(588*x^4-5880*x^3-30576*x^2+226968*x+84 3780)*exp(x)^2,x,method=_RETURNVERBOSE)
Output:
1/2401*(x^8-24*x^7+76*x^6+1656*x^5-6474*x^4-57960*x^3+93100*x^2+1029000*x+ 1500625)*exp(4*x)+1/2401*(294*x^4-3528*x^3-9996*x^2+123480*x+360150)*exp(2 *x)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.46 \[ \int \frac {e^{2 x} \left (843780+226968 x-30576 x^2-5880 x^3+588 x^4\right )+e^{4 x} \left (7031500+4302200 x+198520 x^2-257736 x^3-17616 x^4+7080 x^5+136 x^6-88 x^7+4 x^8\right )}{2401} \, dx=\frac {1}{2401} \, {\left (x^{8} - 24 \, x^{7} + 76 \, x^{6} + 1656 \, x^{5} - 6474 \, x^{4} - 57960 \, x^{3} + 93100 \, x^{2} + 1029000 \, x + 1500625\right )} e^{\left (4 \, x\right )} + \frac {6}{49} \, {\left (x^{4} - 12 \, x^{3} - 34 \, x^{2} + 420 \, x + 1225\right )} e^{\left (2 \, x\right )} \] Input:
integrate(1/2401*(4*x^8-88*x^7+136*x^6+7080*x^5-17616*x^4-257736*x^3+19852 0*x^2+4302200*x+7031500)*exp(x)^4+1/2401*(588*x^4-5880*x^3-30576*x^2+22696 8*x+843780)*exp(x)^2,x, algorithm="fricas")
Output:
1/2401*(x^8 - 24*x^7 + 76*x^6 + 1656*x^5 - 6474*x^4 - 57960*x^3 + 93100*x^ 2 + 1029000*x + 1500625)*e^(4*x) + 6/49*(x^4 - 12*x^3 - 34*x^2 + 420*x + 1 225)*e^(2*x)
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61 \[ \int \frac {e^{2 x} \left (843780+226968 x-30576 x^2-5880 x^3+588 x^4\right )+e^{4 x} \left (7031500+4302200 x+198520 x^2-257736 x^3-17616 x^4+7080 x^5+136 x^6-88 x^7+4 x^8\right )}{2401} \, dx=\frac {\left (14406 x^{4} - 172872 x^{3} - 489804 x^{2} + 6050520 x + 17647350\right ) e^{2 x}}{117649} + \frac {\left (49 x^{8} - 1176 x^{7} + 3724 x^{6} + 81144 x^{5} - 317226 x^{4} - 2840040 x^{3} + 4561900 x^{2} + 50421000 x + 73530625\right ) e^{4 x}}{117649} \] Input:
integrate(1/2401*(4*x**8-88*x**7+136*x**6+7080*x**5-17616*x**4-257736*x**3 +198520*x**2+4302200*x+7031500)*exp(x)**4+1/2401*(588*x**4-5880*x**3-30576 *x**2+226968*x+843780)*exp(x)**2,x)
Output:
(14406*x**4 - 172872*x**3 - 489804*x**2 + 6050520*x + 17647350)*exp(2*x)/1 17649 + (49*x**8 - 1176*x**7 + 3724*x**6 + 81144*x**5 - 317226*x**4 - 2840 040*x**3 + 4561900*x**2 + 50421000*x + 73530625)*exp(4*x)/117649
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (21) = 42\).
Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 4.46 \[ \int \frac {e^{2 x} \left (843780+226968 x-30576 x^2-5880 x^3+588 x^4\right )+e^{4 x} \left (7031500+4302200 x+198520 x^2-257736 x^3-17616 x^4+7080 x^5+136 x^6-88 x^7+4 x^8\right )}{2401} \, dx=\frac {1}{2401} \, {\left (x^{8} - 24 \, x^{7} + 76 \, x^{6} + 1656 \, x^{5} - 6474 \, x^{4} - 57960 \, x^{3} + 93100 \, x^{2} + 1029000 \, x + 1500625\right )} e^{\left (4 \, x\right )} + \frac {3}{49} \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x\right )} - \frac {15}{49} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} - \frac {156}{49} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {1158}{49} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + \frac {1230}{7} \, e^{\left (2 \, x\right )} \] Input:
integrate(1/2401*(4*x^8-88*x^7+136*x^6+7080*x^5-17616*x^4-257736*x^3+19852 0*x^2+4302200*x+7031500)*exp(x)^4+1/2401*(588*x^4-5880*x^3-30576*x^2+22696 8*x+843780)*exp(x)^2,x, algorithm="maxima")
Output:
1/2401*(x^8 - 24*x^7 + 76*x^6 + 1656*x^5 - 6474*x^4 - 57960*x^3 + 93100*x^ 2 + 1029000*x + 1500625)*e^(4*x) + 3/49*(2*x^4 - 4*x^3 + 6*x^2 - 6*x + 3)* e^(2*x) - 15/49*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) - 156/49*(2*x^2 - 2*x + 1)*e^(2*x) + 1158/49*(2*x - 1)*e^(2*x) + 1230/7*e^(2*x)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (21) = 42\).
Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.46 \[ \int \frac {e^{2 x} \left (843780+226968 x-30576 x^2-5880 x^3+588 x^4\right )+e^{4 x} \left (7031500+4302200 x+198520 x^2-257736 x^3-17616 x^4+7080 x^5+136 x^6-88 x^7+4 x^8\right )}{2401} \, dx=\frac {1}{2401} \, {\left (x^{8} - 24 \, x^{7} + 76 \, x^{6} + 1656 \, x^{5} - 6474 \, x^{4} - 57960 \, x^{3} + 93100 \, x^{2} + 1029000 \, x + 1500625\right )} e^{\left (4 \, x\right )} + \frac {6}{49} \, {\left (x^{4} - 12 \, x^{3} - 34 \, x^{2} + 420 \, x + 1225\right )} e^{\left (2 \, x\right )} \] Input:
integrate(1/2401*(4*x^8-88*x^7+136*x^6+7080*x^5-17616*x^4-257736*x^3+19852 0*x^2+4302200*x+7031500)*exp(x)^4+1/2401*(588*x^4-5880*x^3-30576*x^2+22696 8*x+843780)*exp(x)^2,x, algorithm="giac")
Output:
1/2401*(x^8 - 24*x^7 + 76*x^6 + 1656*x^5 - 6474*x^4 - 57960*x^3 + 93100*x^ 2 + 1029000*x + 1500625)*e^(4*x) + 6/49*(x^4 - 12*x^3 - 34*x^2 + 420*x + 1 225)*e^(2*x)
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {e^{2 x} \left (843780+226968 x-30576 x^2-5880 x^3+588 x^4\right )+e^{4 x} \left (7031500+4302200 x+198520 x^2-257736 x^3-17616 x^4+7080 x^5+136 x^6-88 x^7+4 x^8\right )}{2401} \, dx=\frac {{\mathrm {e}}^{2\,x}\,{\left (-x^2+6\,x+35\right )}^2\,\left (1225\,{\mathrm {e}}^{2\,x}+420\,x\,{\mathrm {e}}^{2\,x}-34\,x^2\,{\mathrm {e}}^{2\,x}-12\,x^3\,{\mathrm {e}}^{2\,x}+x^4\,{\mathrm {e}}^{2\,x}+294\right )}{2401} \] Input:
int((exp(4*x)*(4302200*x + 198520*x^2 - 257736*x^3 - 17616*x^4 + 7080*x^5 + 136*x^6 - 88*x^7 + 4*x^8 + 7031500))/2401 + (exp(2*x)*(226968*x - 30576* x^2 - 5880*x^3 + 588*x^4 + 843780))/2401,x)
Output:
(exp(2*x)*(6*x - x^2 + 35)^2*(1225*exp(2*x) + 420*x*exp(2*x) - 34*x^2*exp( 2*x) - 12*x^3*exp(2*x) + x^4*exp(2*x) + 294))/2401
Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.96 \[ \int \frac {e^{2 x} \left (843780+226968 x-30576 x^2-5880 x^3+588 x^4\right )+e^{4 x} \left (7031500+4302200 x+198520 x^2-257736 x^3-17616 x^4+7080 x^5+136 x^6-88 x^7+4 x^8\right )}{2401} \, dx=\frac {e^{2 x} \left (e^{2 x} x^{8}-24 e^{2 x} x^{7}+76 e^{2 x} x^{6}+1656 e^{2 x} x^{5}-6474 e^{2 x} x^{4}-57960 e^{2 x} x^{3}+93100 e^{2 x} x^{2}+1029000 e^{2 x} x +1500625 e^{2 x}+294 x^{4}-3528 x^{3}-9996 x^{2}+123480 x +360150\right )}{2401} \] Input:
int(1/2401*(4*x^8-88*x^7+136*x^6+7080*x^5-17616*x^4-257736*x^3+198520*x^2+ 4302200*x+7031500)*exp(x)^4+1/2401*(588*x^4-5880*x^3-30576*x^2+226968*x+84 3780)*exp(x)^2,x)
Output:
(e**(2*x)*(e**(2*x)*x**8 - 24*e**(2*x)*x**7 + 76*e**(2*x)*x**6 + 1656*e**( 2*x)*x**5 - 6474*e**(2*x)*x**4 - 57960*e**(2*x)*x**3 + 93100*e**(2*x)*x**2 + 1029000*e**(2*x)*x + 1500625*e**(2*x) + 294*x**4 - 3528*x**3 - 9996*x** 2 + 123480*x + 360150))/2401