Integrand size = 124, antiderivative size = 21 \[ \int \frac {-e^{7+8 x}+e^{7+4 x} \left (-3000+2300 x-660 x^2+84 x^3-4 x^4\right )}{390625-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{8 x} \left (4+4 x+x^2\right )+e^{4 x} \left (-2500+750 x+400 x^2-220 x^3+36 x^4-2 x^5\right )} \, dx=\frac {e^7}{2-e^{-4 x} (-5+x)^4+x} \] Output:
1/(2+x-(-5+x)^4/exp(x)^4)*exp(7)
Time = 15.93 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {-e^{7+8 x}+e^{7+4 x} \left (-3000+2300 x-660 x^2+84 x^3-4 x^4\right )}{390625-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{8 x} \left (4+4 x+x^2\right )+e^{4 x} \left (-2500+750 x+400 x^2-220 x^3+36 x^4-2 x^5\right )} \, dx=\frac {e^{7+4 x}}{-(-5+x)^4+e^{4 x} (2+x)} \] Input:
Integrate[(-E^(7 + 8*x) + E^(7 + 4*x)*(-3000 + 2300*x - 660*x^2 + 84*x^3 - 4*x^4))/(390625 - 625000*x + 437500*x^2 - 175000*x^3 + 43750*x^4 - 7000*x ^5 + 700*x^6 - 40*x^7 + x^8 + E^(8*x)*(4 + 4*x + x^2) + E^(4*x)*(-2500 + 7 50*x + 400*x^2 - 220*x^3 + 36*x^4 - 2*x^5)),x]
Output:
E^(7 + 4*x)/(-(-5 + x)^4 + E^(4*x)*(2 + x))
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.
\(\displaystyle \int \frac {e^{4 x+7} \left (-4 x^4+84 x^3-660 x^2+2300 x-3000\right )-e^{8 x+7}}{x^8-40 x^7+700 x^6-7000 x^5+43750 x^4-175000 x^3+437500 x^2+e^{8 x} \left (x^2+4 x+4\right )+e^{4 x} \left (-2 x^5+36 x^4-220 x^3+400 x^2+750 x-2500\right )-625000 x+390625} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{4 x+7} \left (-4 (x-6) (x-5)^3-e^{4 x}\right )}{\left ((x-5)^4-e^{4 x} (x+2)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{4 x+7}}{(x+2) \left (x^4-20 x^3+150 x^2-e^{4 x} x-500 x-2 e^{4 x}+625\right )}-\frac {e^{4 x+7} (x-5)^3 \left (4 x^2-15 x-53\right )}{(x+2) \left (x^4-20 x^3+150 x^2-e^{4 x} x-500 x-2 e^{4 x}+625\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int \frac {e^{4 x+7} x^4}{\left ((x-5)^4-e^{4 x} (x+2)\right )^2}dx+83 \int \frac {e^{4 x+7} x^3}{\left ((x-5)^4-e^{4 x} (x+2)\right )^2}dx-638 \int \frac {e^{4 x+7} x^2}{\left ((x-5)^4-e^{4 x} (x+2)\right )^2}dx-2112 \int \frac {e^{4 x+7}}{\left ((x-5)^4-e^{4 x} (x+2)\right )^2}dx+2106 \int \frac {e^{4 x+7} x}{\left ((x-5)^4-e^{4 x} (x+2)\right )^2}dx-2401 \int \frac {e^{4 x+7}}{(x+2) \left ((x-5)^4-e^{4 x} (x+2)\right )^2}dx+\int \frac {e^{4 x+7}}{(x+2) \left ((x-5)^4-e^{4 x} (x+2)\right )}dx\) |
Input:
Int[(-E^(7 + 8*x) + E^(7 + 4*x)*(-3000 + 2300*x - 660*x^2 + 84*x^3 - 4*x^4 ))/(390625 - 625000*x + 437500*x^2 - 175000*x^3 + 43750*x^4 - 7000*x^5 + 7 00*x^6 - 40*x^7 + x^8 + E^(8*x)*(4 + 4*x + x^2) + E^(4*x)*(-2500 + 750*x + 400*x^2 - 220*x^3 + 36*x^4 - 2*x^5)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs. \(2(19)=38\).
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00
method | result | size |
risch | \(\frac {{\mathrm e}^{7+4 x}}{-x^{4}+20 x^{3}+x \,{\mathrm e}^{4 x}-150 x^{2}+2 \,{\mathrm e}^{4 x}+500 x -625}\) | \(42\) |
parallelrisch | \(-\frac {{\mathrm e}^{7} {\mathrm e}^{4 x}}{-x \,{\mathrm e}^{4 x}+x^{4}-2 \,{\mathrm e}^{4 x}-20 x^{3}+150 x^{2}-500 x +625}\) | \(42\) |
Input:
int((-exp(7)*exp(x)^8+(-4*x^4+84*x^3-660*x^2+2300*x-3000)*exp(7)*exp(x)^4) /((x^2+4*x+4)*exp(x)^8+(-2*x^5+36*x^4-220*x^3+400*x^2+750*x-2500)*exp(x)^4 +x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-625000*x+3906 25),x,method=_RETURNVERBOSE)
Output:
1/(-x^4+20*x^3+x*exp(4*x)-150*x^2+2*exp(4*x)+500*x-625)*exp(7+4*x)
Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (21) = 42\).
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.05 \[ \int \frac {-e^{7+8 x}+e^{7+4 x} \left (-3000+2300 x-660 x^2+84 x^3-4 x^4\right )}{390625-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{8 x} \left (4+4 x+x^2\right )+e^{4 x} \left (-2500+750 x+400 x^2-220 x^3+36 x^4-2 x^5\right )} \, dx=-\frac {e^{\left (4 \, x + 14\right )}}{{\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )} e^{7} - {\left (x + 2\right )} e^{\left (4 \, x + 7\right )}} \] Input:
integrate((-exp(7)*exp(x)^8+(-4*x^4+84*x^3-660*x^2+2300*x-3000)*exp(7)*exp (x)^4)/((x^2+4*x+4)*exp(x)^8+(-2*x^5+36*x^4-220*x^3+400*x^2+750*x-2500)*ex p(x)^4+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-625000* x+390625),x, algorithm="fricas")
Output:
-e^(4*x + 14)/((x^4 - 20*x^3 + 150*x^2 - 500*x + 625)*e^7 - (x + 2)*e^(4*x + 7))
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (15) = 30\).
Time = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.71 \[ \int \frac {-e^{7+8 x}+e^{7+4 x} \left (-3000+2300 x-660 x^2+84 x^3-4 x^4\right )}{390625-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{8 x} \left (4+4 x+x^2\right )+e^{4 x} \left (-2500+750 x+400 x^2-220 x^3+36 x^4-2 x^5\right )} \, dx=\frac {x^{4} e^{7} - 20 x^{3} e^{7} + 150 x^{2} e^{7} - 500 x e^{7} + 625 e^{7}}{- x^{5} + 18 x^{4} - 110 x^{3} + 200 x^{2} + 375 x + \left (x^{2} + 4 x + 4\right ) e^{4 x} - 1250} + \frac {e^{7}}{x + 2} \] Input:
integrate((-exp(7)*exp(x)**8+(-4*x**4+84*x**3-660*x**2+2300*x-3000)*exp(7) *exp(x)**4)/((x**2+4*x+4)*exp(x)**8+(-2*x**5+36*x**4-220*x**3+400*x**2+750 *x-2500)*exp(x)**4+x**8-40*x**7+700*x**6-7000*x**5+43750*x**4-175000*x**3+ 437500*x**2-625000*x+390625),x)
Output:
(x**4*exp(7) - 20*x**3*exp(7) + 150*x**2*exp(7) - 500*x*exp(7) + 625*exp(7 ))/(-x**5 + 18*x**4 - 110*x**3 + 200*x**2 + 375*x + (x**2 + 4*x + 4)*exp(4 *x) - 1250) + exp(7)/(x + 2)
Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {-e^{7+8 x}+e^{7+4 x} \left (-3000+2300 x-660 x^2+84 x^3-4 x^4\right )}{390625-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{8 x} \left (4+4 x+x^2\right )+e^{4 x} \left (-2500+750 x+400 x^2-220 x^3+36 x^4-2 x^5\right )} \, dx=-\frac {e^{\left (4 \, x + 7\right )}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - {\left (x + 2\right )} e^{\left (4 \, x\right )} - 500 \, x + 625} \] Input:
integrate((-exp(7)*exp(x)^8+(-4*x^4+84*x^3-660*x^2+2300*x-3000)*exp(7)*exp (x)^4)/((x^2+4*x+4)*exp(x)^8+(-2*x^5+36*x^4-220*x^3+400*x^2+750*x-2500)*ex p(x)^4+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-625000* x+390625),x, algorithm="maxima")
Output:
-e^(4*x + 7)/(x^4 - 20*x^3 + 150*x^2 - (x + 2)*e^(4*x) - 500*x + 625)
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (21) = 42\).
Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.67 \[ \int \frac {-e^{7+8 x}+e^{7+4 x} \left (-3000+2300 x-660 x^2+84 x^3-4 x^4\right )}{390625-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{8 x} \left (4+4 x+x^2\right )+e^{4 x} \left (-2500+750 x+400 x^2-220 x^3+36 x^4-2 x^5\right )} \, dx=-\frac {256 \, e^{\left (4 \, x + 14\right )}}{{\left (4 \, x + 7\right )}^{4} e^{7} - 108 \, {\left (4 \, x + 7\right )}^{3} e^{7} + 4374 \, {\left (4 \, x + 7\right )}^{2} e^{7} - 78732 \, {\left (4 \, x + 7\right )} e^{7} - 64 \, {\left (4 \, x + 7\right )} e^{\left (4 \, x + 7\right )} + 531441 \, e^{7} - 64 \, e^{\left (4 \, x + 7\right )}} \] Input:
integrate((-exp(7)*exp(x)^8+(-4*x^4+84*x^3-660*x^2+2300*x-3000)*exp(7)*exp (x)^4)/((x^2+4*x+4)*exp(x)^8+(-2*x^5+36*x^4-220*x^3+400*x^2+750*x-2500)*ex p(x)^4+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-625000* x+390625),x, algorithm="giac")
Output:
-256*e^(4*x + 14)/((4*x + 7)^4*e^7 - 108*(4*x + 7)^3*e^7 + 4374*(4*x + 7)^ 2*e^7 - 78732*(4*x + 7)*e^7 - 64*(4*x + 7)*e^(4*x + 7) + 531441*e^7 - 64*e ^(4*x + 7))
Timed out. \[ \int \frac {-e^{7+8 x}+e^{7+4 x} \left (-3000+2300 x-660 x^2+84 x^3-4 x^4\right )}{390625-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{8 x} \left (4+4 x+x^2\right )+e^{4 x} \left (-2500+750 x+400 x^2-220 x^3+36 x^4-2 x^5\right )} \, dx=\int -\frac {{\mathrm {e}}^{8\,x}\,{\mathrm {e}}^7+{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^7\,\left (4\,x^4-84\,x^3+660\,x^2-2300\,x+3000\right )}{{\mathrm {e}}^{8\,x}\,\left (x^2+4\,x+4\right )-625000\,x+{\mathrm {e}}^{4\,x}\,\left (-2\,x^5+36\,x^4-220\,x^3+400\,x^2+750\,x-2500\right )+437500\,x^2-175000\,x^3+43750\,x^4-7000\,x^5+700\,x^6-40\,x^7+x^8+390625} \,d x \] Input:
int(-(exp(8*x)*exp(7) + exp(4*x)*exp(7)*(660*x^2 - 2300*x - 84*x^3 + 4*x^4 + 3000))/(exp(8*x)*(4*x + x^2 + 4) - 625000*x + exp(4*x)*(750*x + 400*x^2 - 220*x^3 + 36*x^4 - 2*x^5 - 2500) + 437500*x^2 - 175000*x^3 + 43750*x^4 - 7000*x^5 + 700*x^6 - 40*x^7 + x^8 + 390625),x)
Output:
int(-(exp(8*x)*exp(7) + exp(4*x)*exp(7)*(660*x^2 - 2300*x - 84*x^3 + 4*x^4 + 3000))/(exp(8*x)*(4*x + x^2 + 4) - 625000*x + exp(4*x)*(750*x + 400*x^2 - 220*x^3 + 36*x^4 - 2*x^5 - 2500) + 437500*x^2 - 175000*x^3 + 43750*x^4 - 7000*x^5 + 700*x^6 - 40*x^7 + x^8 + 390625), x)
Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.14 \[ \int \frac {-e^{7+8 x}+e^{7+4 x} \left (-3000+2300 x-660 x^2+84 x^3-4 x^4\right )}{390625-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{8 x} \left (4+4 x+x^2\right )+e^{4 x} \left (-2500+750 x+400 x^2-220 x^3+36 x^4-2 x^5\right )} \, dx=\frac {e^{4 x} e^{7}}{e^{4 x} x +2 e^{4 x}-x^{4}+20 x^{3}-150 x^{2}+500 x -625} \] Input:
int((-exp(7)*exp(x)^8+(-4*x^4+84*x^3-660*x^2+2300*x-3000)*exp(7)*exp(x)^4) /((x^2+4*x+4)*exp(x)^8+(-2*x^5+36*x^4-220*x^3+400*x^2+750*x-2500)*exp(x)^4 +x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-625000*x+3906 25),x)
Output:
(e**(4*x)*e**7)/(e**(4*x)*x + 2*e**(4*x) - x**4 + 20*x**3 - 150*x**2 + 500 *x - 625)