Integrand size = 90, antiderivative size = 33 \[ \int \frac {131250 x^6-46875 x^8+e^{-5+x} \left (13125 x^6-3750 x^7\right )}{432000+432 e^{-15+3 x}-1080000 x^2+900000 x^4-250000 x^6+e^{-10+2 x} \left (12960-10800 x^2\right )+e^{-5+x} \left (129600-216000 x^2+90000 x^4\right )} \, dx=4+\frac {x^5}{16 \left (\frac {3 \left (2+\frac {e^{-5+x}}{5}\right )}{5 x}-x\right )^2} \] Output:
1/16*x^5/(3/5*(2+1/5*exp(-5+x))/x-x)^2+4
Time = 2.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {131250 x^6-46875 x^8+e^{-5+x} \left (13125 x^6-3750 x^7\right )}{432000+432 e^{-15+3 x}-1080000 x^2+900000 x^4-250000 x^6+e^{-10+2 x} \left (12960-10800 x^2\right )+e^{-5+x} \left (129600-216000 x^2+90000 x^4\right )} \, dx=\frac {625 e^{10} x^7}{16 \left (30 e^5+3 e^x-25 e^5 x^2\right )^2} \] Input:
Integrate[(131250*x^6 - 46875*x^8 + E^(-5 + x)*(13125*x^6 - 3750*x^7))/(43 2000 + 432*E^(-15 + 3*x) - 1080000*x^2 + 900000*x^4 - 250000*x^6 + E^(-10 + 2*x)*(12960 - 10800*x^2) + E^(-5 + x)*(129600 - 216000*x^2 + 90000*x^4)) ,x]
Output:
(625*E^10*x^7)/(16*(30*E^5 + 3*E^x - 25*E^5*x^2)^2)
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.
\(\displaystyle \int \frac {-46875 x^8+131250 x^6+e^{x-5} \left (13125 x^6-3750 x^7\right )}{-250000 x^6+900000 x^4-1080000 x^2+e^{2 x-10} \left (12960-10800 x^2\right )+e^{x-5} \left (90000 x^4-216000 x^2+129600\right )+432 e^{3 x-15}+432000} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {1875 e^{10} x^6 \left (-5 e^5 \left (5 x^2-14\right )-e^x (2 x-7)\right )}{16 \left (3 e^x-5 e^5 \left (5 x^2-6\right )\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1875}{16} e^{10} \int \frac {x^6 \left (e^x (7-2 x)+5 e^5 \left (14-5 x^2\right )\right )}{\left (5 e^5 \left (6-5 x^2\right )+3 e^x\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1875}{16} e^{10} \int \left (\frac {10 e^5 x^7 \left (5 x^2-10 x-6\right )}{3 \left (25 e^5 x^2-3 e^x-30 e^5\right )^3}-\frac {x^6 (2 x-7)}{3 \left (25 e^5 x^2-3 e^x-30 e^5\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1875}{16} e^{10} \left (\frac {50}{3} e^5 \int \frac {x^9}{\left (25 e^5 x^2-3 e^x-30 e^5\right )^3}dx-\frac {100}{3} e^5 \int \frac {x^8}{\left (25 e^5 x^2-3 e^x-30 e^5\right )^3}dx-20 e^5 \int \frac {x^7}{\left (25 e^5 x^2-3 e^x-30 e^5\right )^3}dx-\frac {2}{3} \int \frac {x^7}{\left (25 e^5 x^2-3 e^x-30 e^5\right )^2}dx+\frac {7}{3} \int \frac {x^6}{\left (25 e^5 x^2-3 e^x-30 e^5\right )^2}dx\right )\) |
Input:
Int[(131250*x^6 - 46875*x^8 + E^(-5 + x)*(13125*x^6 - 3750*x^7))/(432000 + 432*E^(-15 + 3*x) - 1080000*x^2 + 900000*x^4 - 250000*x^6 + E^(-10 + 2*x) *(12960 - 10800*x^2) + E^(-5 + x)*(129600 - 216000*x^2 + 90000*x^4)),x]
Output:
$Aborted
Time = 0.62 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.64
method | result | size |
risch | \(\frac {625 x^{7}}{16 \left (25 x^{2}-3 \,{\mathrm e}^{-5+x}-30\right )^{2}}\) | \(21\) |
parallelrisch | \(\frac {625 x^{7}}{16 \left (625 x^{4}-150 \,{\mathrm e}^{-5+x} x^{2}-1500 x^{2}+9 \,{\mathrm e}^{2 x -10}+180 \,{\mathrm e}^{-5+x}+900\right )}\) | \(43\) |
Input:
int(((-3750*x^7+13125*x^6)*exp(-5+x)-46875*x^8+131250*x^6)/(432*exp(-5+x)^ 3+(-10800*x^2+12960)*exp(-5+x)^2+(90000*x^4-216000*x^2+129600)*exp(-5+x)-2 50000*x^6+900000*x^4-1080000*x^2+432000),x,method=_RETURNVERBOSE)
Output:
625/16*x^7/(25*x^2-3*exp(-5+x)-30)^2
Time = 0.11 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {131250 x^6-46875 x^8+e^{-5+x} \left (13125 x^6-3750 x^7\right )}{432000+432 e^{-15+3 x}-1080000 x^2+900000 x^4-250000 x^6+e^{-10+2 x} \left (12960-10800 x^2\right )+e^{-5+x} \left (129600-216000 x^2+90000 x^4\right )} \, dx=\frac {625 \, x^{7}}{16 \, {\left (625 \, x^{4} - 1500 \, x^{2} - 30 \, {\left (5 \, x^{2} - 6\right )} e^{\left (x - 5\right )} + 9 \, e^{\left (2 \, x - 10\right )} + 900\right )}} \] Input:
integrate(((-3750*x^7+13125*x^6)*exp(-5+x)-46875*x^8+131250*x^6)/(432*exp( -5+x)^3+(-10800*x^2+12960)*exp(-5+x)^2+(90000*x^4-216000*x^2+129600)*exp(- 5+x)-250000*x^6+900000*x^4-1080000*x^2+432000),x, algorithm="fricas")
Output:
625/16*x^7/(625*x^4 - 1500*x^2 - 30*(5*x^2 - 6)*e^(x - 5) + 9*e^(2*x - 10) + 900)
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {131250 x^6-46875 x^8+e^{-5+x} \left (13125 x^6-3750 x^7\right )}{432000+432 e^{-15+3 x}-1080000 x^2+900000 x^4-250000 x^6+e^{-10+2 x} \left (12960-10800 x^2\right )+e^{-5+x} \left (129600-216000 x^2+90000 x^4\right )} \, dx=\frac {625 x^{7}}{10000 x^{4} - 24000 x^{2} + \left (2880 - 2400 x^{2}\right ) e^{x - 5} + 144 e^{2 x - 10} + 14400} \] Input:
integrate(((-3750*x**7+13125*x**6)*exp(-5+x)-46875*x**8+131250*x**6)/(432* exp(-5+x)**3+(-10800*x**2+12960)*exp(-5+x)**2+(90000*x**4-216000*x**2+1296 00)*exp(-5+x)-250000*x**6+900000*x**4-1080000*x**2+432000),x)
Output:
625*x**7/(10000*x**4 - 24000*x**2 + (2880 - 2400*x**2)*exp(x - 5) + 144*ex p(2*x - 10) + 14400)
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {131250 x^6-46875 x^8+e^{-5+x} \left (13125 x^6-3750 x^7\right )}{432000+432 e^{-15+3 x}-1080000 x^2+900000 x^4-250000 x^6+e^{-10+2 x} \left (12960-10800 x^2\right )+e^{-5+x} \left (129600-216000 x^2+90000 x^4\right )} \, dx=\frac {625 \, x^{7} e^{10}}{16 \, {\left (625 \, x^{4} e^{10} - 1500 \, x^{2} e^{10} - 30 \, {\left (5 \, x^{2} e^{5} - 6 \, e^{5}\right )} e^{x} + 900 \, e^{10} + 9 \, e^{\left (2 \, x\right )}\right )}} \] Input:
integrate(((-3750*x^7+13125*x^6)*exp(-5+x)-46875*x^8+131250*x^6)/(432*exp( -5+x)^3+(-10800*x^2+12960)*exp(-5+x)^2+(90000*x^4-216000*x^2+129600)*exp(- 5+x)-250000*x^6+900000*x^4-1080000*x^2+432000),x, algorithm="maxima")
Output:
625/16*x^7*e^10/(625*x^4*e^10 - 1500*x^2*e^10 - 30*(5*x^2*e^5 - 6*e^5)*e^x + 900*e^10 + 9*e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {131250 x^6-46875 x^8+e^{-5+x} \left (13125 x^6-3750 x^7\right )}{432000+432 e^{-15+3 x}-1080000 x^2+900000 x^4-250000 x^6+e^{-10+2 x} \left (12960-10800 x^2\right )+e^{-5+x} \left (129600-216000 x^2+90000 x^4\right )} \, dx=\frac {625 \, x^{7} e^{10}}{16 \, {\left (625 \, x^{4} e^{10} - 1500 \, x^{2} e^{10} - 150 \, x^{2} e^{\left (x + 5\right )} + 900 \, e^{10} + 9 \, e^{\left (2 \, x\right )} + 180 \, e^{\left (x + 5\right )}\right )}} \] Input:
integrate(((-3750*x^7+13125*x^6)*exp(-5+x)-46875*x^8+131250*x^6)/(432*exp( -5+x)^3+(-10800*x^2+12960)*exp(-5+x)^2+(90000*x^4-216000*x^2+129600)*exp(- 5+x)-250000*x^6+900000*x^4-1080000*x^2+432000),x, algorithm="giac")
Output:
625/16*x^7*e^10/(625*x^4*e^10 - 1500*x^2*e^10 - 150*x^2*e^(x + 5) + 900*e^ 10 + 9*e^(2*x) + 180*e^(x + 5))
Timed out. \[ \int \frac {131250 x^6-46875 x^8+e^{-5+x} \left (13125 x^6-3750 x^7\right )}{432000+432 e^{-15+3 x}-1080000 x^2+900000 x^4-250000 x^6+e^{-10+2 x} \left (12960-10800 x^2\right )+e^{-5+x} \left (129600-216000 x^2+90000 x^4\right )} \, dx=\int \frac {{\mathrm {e}}^{x-5}\,\left (13125\,x^6-3750\,x^7\right )+131250\,x^6-46875\,x^8}{432\,{\mathrm {e}}^{3\,x-15}+{\mathrm {e}}^{x-5}\,\left (90000\,x^4-216000\,x^2+129600\right )-{\mathrm {e}}^{2\,x-10}\,\left (10800\,x^2-12960\right )-1080000\,x^2+900000\,x^4-250000\,x^6+432000} \,d x \] Input:
int((exp(x - 5)*(13125*x^6 - 3750*x^7) + 131250*x^6 - 46875*x^8)/(432*exp( 3*x - 15) + exp(x - 5)*(90000*x^4 - 216000*x^2 + 129600) - exp(2*x - 10)*( 10800*x^2 - 12960) - 1080000*x^2 + 900000*x^4 - 250000*x^6 + 432000),x)
Output:
int((exp(x - 5)*(13125*x^6 - 3750*x^7) + 131250*x^6 - 46875*x^8)/(432*exp( 3*x - 15) + exp(x - 5)*(90000*x^4 - 216000*x^2 + 129600) - exp(2*x - 10)*( 10800*x^2 - 12960) - 1080000*x^2 + 900000*x^4 - 250000*x^6 + 432000), x)
Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \frac {131250 x^6-46875 x^8+e^{-5+x} \left (13125 x^6-3750 x^7\right )}{432000+432 e^{-15+3 x}-1080000 x^2+900000 x^4-250000 x^6+e^{-10+2 x} \left (12960-10800 x^2\right )+e^{-5+x} \left (129600-216000 x^2+90000 x^4\right )} \, dx=\frac {625 e^{10} x^{7}}{144 e^{2 x}-2400 e^{x} e^{5} x^{2}+2880 e^{x} e^{5}+10000 e^{10} x^{4}-24000 e^{10} x^{2}+14400 e^{10}} \] Input:
int(((-3750*x^7+13125*x^6)*exp(-5+x)-46875*x^8+131250*x^6)/(432*exp(-5+x)^ 3+(-10800*x^2+12960)*exp(-5+x)^2+(90000*x^4-216000*x^2+129600)*exp(-5+x)-2 50000*x^6+900000*x^4-1080000*x^2+432000),x)
Output:
(625*e**10*x**7)/(16*(9*e**(2*x) - 150*e**x*e**5*x**2 + 180*e**x*e**5 + 62 5*e**10*x**4 - 1500*e**10*x**2 + 900*e**10))