Integrand size = 154, antiderivative size = 37 \[ \int \frac {40 x-28 x^2-42 x^3+12 x^4+18 x^5+3 x^6+e^{2 x} \left (10 x^3+3 x^4\right )+e^x \left (-40 x^2+8 x^3+27 x^4+7 x^5\right )}{100+20 x-179 x^2-78 x^3+75 x^4+54 x^5+9 x^6+e^{2 x} \left (25 x^2+30 x^3+9 x^4\right )+e^x \left (-100 x-70 x^2+84 x^3+84 x^4+18 x^5\right )} \, dx=\frac {x}{3-\frac {1}{-2-e^x+\frac {4}{x}+\frac {-2+x}{x}-x}+\frac {5}{x}} \]
Time = 7.58 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54 \[ \int \frac {40 x-28 x^2-42 x^3+12 x^4+18 x^5+3 x^6+e^{2 x} \left (10 x^3+3 x^4\right )+e^x \left (-40 x^2+8 x^3+27 x^4+7 x^5\right )}{100+20 x-179 x^2-78 x^3+75 x^4+54 x^5+9 x^6+e^{2 x} \left (25 x^2+30 x^3+9 x^4\right )+e^x \left (-100 x-70 x^2+84 x^3+84 x^4+18 x^5\right )} \, dx=\frac {x}{3}+\frac {25}{9 (5+3 x)}-\frac {x^4}{(5+3 x) \left (-10+\left (-1+5 e^x\right ) x+3 \left (3+e^x\right ) x^2+3 x^3\right )} \]
Integrate[(40*x - 28*x^2 - 42*x^3 + 12*x^4 + 18*x^5 + 3*x^6 + E^(2*x)*(10* x^3 + 3*x^4) + E^x*(-40*x^2 + 8*x^3 + 27*x^4 + 7*x^5))/(100 + 20*x - 179*x ^2 - 78*x^3 + 75*x^4 + 54*x^5 + 9*x^6 + E^(2*x)*(25*x^2 + 30*x^3 + 9*x^4) + E^x*(-100*x - 70*x^2 + 84*x^3 + 84*x^4 + 18*x^5)),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.
\(\displaystyle \int \frac {3 x^6+18 x^5+12 x^4-42 x^3-28 x^2+e^{2 x} \left (3 x^4+10 x^3\right )+e^x \left (7 x^5+27 x^4+8 x^3-40 x^2\right )+40 x}{9 x^6+54 x^5+75 x^4-78 x^3-179 x^2+e^{2 x} \left (9 x^4+30 x^3+25 x^2\right )+e^x \left (18 x^5+84 x^4+84 x^3-70 x^2-100 x\right )+20 x+100} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {3 x^6+18 x^5+12 x^4-42 x^3-28 x^2+e^{2 x} \left (3 x^4+10 x^3\right )+e^x \left (7 x^5+27 x^4+8 x^3-40 x^2\right )+40 x}{\left (-3 x^3-3 e^x x^2-9 x^2-5 e^x x+x+10\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\left (3 x^2+2 x-15\right ) x^3}{(3 x+5)^2 \left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )}-\frac {\left (9 x^5+33 x^4+12 x^3-83 x^2-110 x-50\right ) x^3}{(3 x+5)^2 \left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )^2}+\frac {(3 x+10) x}{(3 x+5)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4375}{729} \int \frac {1}{\left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )^2}dx+\frac {500}{243} \int \frac {x}{\left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )^2}dx-\frac {25}{81} \int \frac {x^2}{\left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )^2}dx+\frac {44}{27} \int \frac {x^3}{\left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )^2}dx-\frac {15625}{243} \int \frac {1}{(3 x+5)^2 \left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )^2}dx+\frac {31250}{729} \int \frac {1}{(3 x+5) \left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )^2}dx+\frac {100}{81} \int \frac {1}{3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10}dx+\frac {10}{27} \int \frac {x}{3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10}dx-\frac {8}{9} \int \frac {x^2}{3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10}dx+\frac {1}{3} \int \frac {x^3}{3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10}dx+\frac {1250}{27} \int \frac {1}{(3 x+5)^2 \left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )}dx-\frac {1250}{81} \int \frac {1}{(3 x+5) \left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )}dx-\int \frac {x^6}{\left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )^2}dx-\frac {1}{3} \int \frac {x^5}{\left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )^2}dx+\frac {23}{9} \int \frac {x^4}{\left (3 x^3+3 e^x x^2+9 x^2+5 e^x x-x-10\right )^2}dx+\frac {(3 x+10)^2}{9 (3 x+5)}\) |
Int[(40*x - 28*x^2 - 42*x^3 + 12*x^4 + 18*x^5 + 3*x^6 + E^(2*x)*(10*x^3 + 3*x^4) + E^x*(-40*x^2 + 8*x^3 + 27*x^4 + 7*x^5))/(100 + 20*x - 179*x^2 - 7 8*x^3 + 75*x^4 + 54*x^5 + 9*x^6 + E^(2*x)*(25*x^2 + 30*x^3 + 9*x^4) + E^x* (-100*x - 70*x^2 + 84*x^3 + 84*x^4 + 18*x^5)),x]
3.24.83.3.1 Defintions of rubi rules used
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43
method | result | size |
risch | \(\frac {x}{3}+\frac {25}{27 \left (x +\frac {5}{3}\right )}-\frac {x^{4}}{\left (3 x +5\right ) \left (3 \,{\mathrm e}^{x} x^{2}+3 x^{3}+5 \,{\mathrm e}^{x} x +9 x^{2}-x -10\right )}\) | \(53\) |
parallelrisch | \(\frac {3 x^{4}+3 \,{\mathrm e}^{x} x^{3}+3 x^{3}-6 x^{2}}{9 \,{\mathrm e}^{x} x^{2}+9 x^{3}+15 \,{\mathrm e}^{x} x +27 x^{2}-3 x -30}\) | \(55\) |
norman | \(\frac {x^{4}-5 x^{2}+\frac {x}{3}+{\mathrm e}^{x} x^{3}-{\mathrm e}^{x} x^{2}-\frac {5 \,{\mathrm e}^{x} x}{3}+\frac {10}{3}}{3 \,{\mathrm e}^{x} x^{2}+3 x^{3}+5 \,{\mathrm e}^{x} x +9 x^{2}-x -10}\) | \(62\) |
int(((3*x^4+10*x^3)*exp(x)^2+(7*x^5+27*x^4+8*x^3-40*x^2)*exp(x)+3*x^6+18*x ^5+12*x^4-42*x^3-28*x^2+40*x)/((9*x^4+30*x^3+25*x^2)*exp(x)^2+(18*x^5+84*x ^4+84*x^3-70*x^2-100*x)*exp(x)+9*x^6+54*x^5+75*x^4-78*x^3-179*x^2+20*x+100 ),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (31) = 62\).
Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.84 \[ \int \frac {40 x-28 x^2-42 x^3+12 x^4+18 x^5+3 x^6+e^{2 x} \left (10 x^3+3 x^4\right )+e^x \left (-40 x^2+8 x^3+27 x^4+7 x^5\right )}{100+20 x-179 x^2-78 x^3+75 x^4+54 x^5+9 x^6+e^{2 x} \left (25 x^2+30 x^3+9 x^4\right )+e^x \left (-100 x-70 x^2+84 x^3+84 x^4+18 x^5\right )} \, dx=\frac {9 \, x^{4} + 24 \, x^{3} + 27 \, x^{2} + {\left (9 \, x^{3} + 15 \, x^{2} + 25 \, x\right )} e^{x} - 5 \, x - 50}{9 \, {\left (3 \, x^{3} + 9 \, x^{2} + {\left (3 \, x^{2} + 5 \, x\right )} e^{x} - x - 10\right )}} \]
integrate(((3*x^4+10*x^3)*exp(x)^2+(7*x^5+27*x^4+8*x^3-40*x^2)*exp(x)+3*x^ 6+18*x^5+12*x^4-42*x^3-28*x^2+40*x)/((9*x^4+30*x^3+25*x^2)*exp(x)^2+(18*x^ 5+84*x^4+84*x^3-70*x^2-100*x)*exp(x)+9*x^6+54*x^5+75*x^4-78*x^3-179*x^2+20 *x+100),x, algorithm=\
1/9*(9*x^4 + 24*x^3 + 27*x^2 + (9*x^3 + 15*x^2 + 25*x)*e^x - 5*x - 50)/(3* x^3 + 9*x^2 + (3*x^2 + 5*x)*e^x - x - 10)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.32 \[ \int \frac {40 x-28 x^2-42 x^3+12 x^4+18 x^5+3 x^6+e^{2 x} \left (10 x^3+3 x^4\right )+e^x \left (-40 x^2+8 x^3+27 x^4+7 x^5\right )}{100+20 x-179 x^2-78 x^3+75 x^4+54 x^5+9 x^6+e^{2 x} \left (25 x^2+30 x^3+9 x^4\right )+e^x \left (-100 x-70 x^2+84 x^3+84 x^4+18 x^5\right )} \, dx=- \frac {x^{4}}{9 x^{4} + 42 x^{3} + 42 x^{2} - 35 x + \left (9 x^{3} + 30 x^{2} + 25 x\right ) e^{x} - 50} + \frac {x}{3} + \frac {25}{27 x + 45} \]
integrate(((3*x**4+10*x**3)*exp(x)**2+(7*x**5+27*x**4+8*x**3-40*x**2)*exp( x)+3*x**6+18*x**5+12*x**4-42*x**3-28*x**2+40*x)/((9*x**4+30*x**3+25*x**2)* exp(x)**2+(18*x**5+84*x**4+84*x**3-70*x**2-100*x)*exp(x)+9*x**6+54*x**5+75 *x**4-78*x**3-179*x**2+20*x+100),x)
-x**4/(9*x**4 + 42*x**3 + 42*x**2 - 35*x + (9*x**3 + 30*x**2 + 25*x)*exp(x ) - 50) + x/3 + 25/(27*x + 45)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (31) = 62\).
Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.84 \[ \int \frac {40 x-28 x^2-42 x^3+12 x^4+18 x^5+3 x^6+e^{2 x} \left (10 x^3+3 x^4\right )+e^x \left (-40 x^2+8 x^3+27 x^4+7 x^5\right )}{100+20 x-179 x^2-78 x^3+75 x^4+54 x^5+9 x^6+e^{2 x} \left (25 x^2+30 x^3+9 x^4\right )+e^x \left (-100 x-70 x^2+84 x^3+84 x^4+18 x^5\right )} \, dx=\frac {9 \, x^{4} + 24 \, x^{3} + 27 \, x^{2} + {\left (9 \, x^{3} + 15 \, x^{2} + 25 \, x\right )} e^{x} - 5 \, x - 50}{9 \, {\left (3 \, x^{3} + 9 \, x^{2} + {\left (3 \, x^{2} + 5 \, x\right )} e^{x} - x - 10\right )}} \]
integrate(((3*x^4+10*x^3)*exp(x)^2+(7*x^5+27*x^4+8*x^3-40*x^2)*exp(x)+3*x^ 6+18*x^5+12*x^4-42*x^3-28*x^2+40*x)/((9*x^4+30*x^3+25*x^2)*exp(x)^2+(18*x^ 5+84*x^4+84*x^3-70*x^2-100*x)*exp(x)+9*x^6+54*x^5+75*x^4-78*x^3-179*x^2+20 *x+100),x, algorithm=\
1/9*(9*x^4 + 24*x^3 + 27*x^2 + (9*x^3 + 15*x^2 + 25*x)*e^x - 5*x - 50)/(3* x^3 + 9*x^2 + (3*x^2 + 5*x)*e^x - x - 10)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (31) = 62\).
Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.89 \[ \int \frac {40 x-28 x^2-42 x^3+12 x^4+18 x^5+3 x^6+e^{2 x} \left (10 x^3+3 x^4\right )+e^x \left (-40 x^2+8 x^3+27 x^4+7 x^5\right )}{100+20 x-179 x^2-78 x^3+75 x^4+54 x^5+9 x^6+e^{2 x} \left (25 x^2+30 x^3+9 x^4\right )+e^x \left (-100 x-70 x^2+84 x^3+84 x^4+18 x^5\right )} \, dx=\frac {9 \, x^{4} + 9 \, x^{3} e^{x} + 24 \, x^{3} + 15 \, x^{2} e^{x} + 27 \, x^{2} + 25 \, x e^{x} - 5 \, x - 50}{9 \, {\left (3 \, x^{3} + 3 \, x^{2} e^{x} + 9 \, x^{2} + 5 \, x e^{x} - x - 10\right )}} \]
integrate(((3*x^4+10*x^3)*exp(x)^2+(7*x^5+27*x^4+8*x^3-40*x^2)*exp(x)+3*x^ 6+18*x^5+12*x^4-42*x^3-28*x^2+40*x)/((9*x^4+30*x^3+25*x^2)*exp(x)^2+(18*x^ 5+84*x^4+84*x^3-70*x^2-100*x)*exp(x)+9*x^6+54*x^5+75*x^4-78*x^3-179*x^2+20 *x+100),x, algorithm=\
1/9*(9*x^4 + 9*x^3*e^x + 24*x^3 + 15*x^2*e^x + 27*x^2 + 25*x*e^x - 5*x - 5 0)/(3*x^3 + 3*x^2*e^x + 9*x^2 + 5*x*e^x - x - 10)
Timed out. \[ \int \frac {40 x-28 x^2-42 x^3+12 x^4+18 x^5+3 x^6+e^{2 x} \left (10 x^3+3 x^4\right )+e^x \left (-40 x^2+8 x^3+27 x^4+7 x^5\right )}{100+20 x-179 x^2-78 x^3+75 x^4+54 x^5+9 x^6+e^{2 x} \left (25 x^2+30 x^3+9 x^4\right )+e^x \left (-100 x-70 x^2+84 x^3+84 x^4+18 x^5\right )} \, dx=\int \frac {40\,x+{\mathrm {e}}^{2\,x}\,\left (3\,x^4+10\,x^3\right )+{\mathrm {e}}^x\,\left (7\,x^5+27\,x^4+8\,x^3-40\,x^2\right )-28\,x^2-42\,x^3+12\,x^4+18\,x^5+3\,x^6}{20\,x+{\mathrm {e}}^x\,\left (18\,x^5+84\,x^4+84\,x^3-70\,x^2-100\,x\right )+{\mathrm {e}}^{2\,x}\,\left (9\,x^4+30\,x^3+25\,x^2\right )-179\,x^2-78\,x^3+75\,x^4+54\,x^5+9\,x^6+100} \,d x \]
int((40*x + exp(2*x)*(10*x^3 + 3*x^4) + exp(x)*(8*x^3 - 40*x^2 + 27*x^4 + 7*x^5) - 28*x^2 - 42*x^3 + 12*x^4 + 18*x^5 + 3*x^6)/(20*x + exp(x)*(84*x^3 - 70*x^2 - 100*x + 84*x^4 + 18*x^5) + exp(2*x)*(25*x^2 + 30*x^3 + 9*x^4) - 179*x^2 - 78*x^3 + 75*x^4 + 54*x^5 + 9*x^6 + 100),x)