Integrand size = 81, antiderivative size = 24 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {x^2}{\frac {2}{5}+\frac {3125 \left (-2+3 x^4\right )^5}{e}} \] Output:
x^2/(exp(5*ln(15*x^4-10)-1)+2/5)
Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {10 e x^2}{4 e+31250 \left (-2+3 x^4\right )^5} \] Input:
Integrate[(-40*x + 60*x^5 + ((-10 + 15*x^4)^5*(-100*x - 1350*x^5))/E)/(-8 + 12*x^4 + ((-10 + 15*x^4)^5*(-40 + 60*x^4))/E + ((-10 + 15*x^4)^10*(-50 + 75*x^4))/E^2),x]
Output:
(10*E*x^2)/(4*E + 31250*(-2 + 3*x^4)^5)
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.
\(\displaystyle \int \frac {60 x^5+\frac {\left (15 x^4-10\right )^5 \left (-1350 x^5-100 x\right )}{e}-40 x}{\frac {\left (75 x^4-50\right ) \left (15 x^4-10\right )^{10}}{e^2}+\frac {\left (60 x^4-40\right ) \left (15 x^4-10\right )^5}{e}+12 x^4-8} \, dx\) |
\(\Big \downarrow \) 2457 |
\(\displaystyle \int \frac {-341718750 e x^{21}+885937500 e x^{17}-843750000 e x^{13}+337500000 e x^9-37500000 e x^5+\left (20 e^2-5000000 e\right ) x}{14416259765625 x^{40}-96108398437500 x^{36}+288325195312500 x^{32}-512578125000000 x^{28}+598007812500000 x^{24}+(15187500 e-478406250000000) x^{20}+(265781250000000-50625000 e) x^{16}+(67500000 e-101250000000000) x^{12}+(25312500000000-45000000 e) x^8+(15000000 e-3750000000000) x^4+4 e^2-2000000 e+250000000000}dx\) |
\(\Big \downarrow \) 2460 |
\(\displaystyle \int \left (\frac {90 e x}{-3796875 x^{20}+12656250 x^{16}-16875000 x^{12}+11250000 x^8-3750000 x^4+500000 \left (1-\frac {e}{250000}\right )}+\frac {200 e \left (-1265625 x^{16}+3375000 x^{12}-3375000 x^8+1500000 x^4-250000 \left (1-\frac {e}{250000}\right )\right ) x}{\left (-3796875 x^{20}+12656250 x^{16}-16875000 x^{12}+11250000 x^8-3750000 x^4+500000 \left (1-\frac {e}{250000}\right )\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -100 (250000-e) e \text {Subst}\left (\int \frac {1}{\left (15625 \left (3 x^2-2\right )^5+2 e\right )^2}dx,x,x^2\right )+150000000 e \text {Subst}\left (\int \frac {x^2}{\left (15625 \left (3 x^2-2\right )^5+2 e\right )^2}dx,x,x^2\right )-126562500 e \text {Subst}\left (\int \frac {x^8}{\left (15625 \left (3 x^2-2\right )^5+2 e\right )^2}dx,x,x^2\right )+337500000 e \text {Subst}\left (\int \frac {x^6}{\left (15625 \left (3 x^2-2\right )^5+2 e\right )^2}dx,x,x^2\right )-337500000 e \text {Subst}\left (\int \frac {x^4}{\left (15625 \left (3 x^2-2\right )^5+2 e\right )^2}dx,x,x^2\right )+\frac {3 \sqrt {3 \left (50+5^{4/5} \sqrt [5]{-2 e}\right )} e \text {arctanh}\left (\frac {5 x^2}{\sqrt {\frac {1}{3} \left (50+5^{4/5} \sqrt [5]{-2 e}\right )}}\right )}{10 \left (5 \sqrt [5]{5} (-2 e)^{4/5}-e\right )}+\frac {3 \sqrt {\frac {3}{5\ 2^{4/5} \sqrt [5]{\frac {5}{e}}-1}} \sqrt [10]{e} \text {arctanh}\left (\frac {5^{3/5} \sqrt {\frac {3}{5\ 2^{4/5} \sqrt [5]{\frac {5}{e}}-1}} x^2}{\sqrt [10]{2 e}}\right )}{2^{9/10} 5^{3/5}}-\frac {3 e \sqrt {3 \left (50-(-5)^{4/5} \sqrt [5]{2 e}\right )} \text {arctanh}\left (\frac {5 x^2}{\sqrt {\frac {1}{3} \left (50-(-5)^{4/5} \sqrt [5]{2 e}\right )}}\right )}{10 \left (e+5 \sqrt [5]{-5} (2 e)^{4/5}\right )}-\frac {3 e \sqrt {3 \left (50-(-1)^{2/5} 5^{4/5} \sqrt [5]{2 e}\right )} \text {arctanh}\left (\frac {5 x^2}{\sqrt {\frac {1}{3} \left (50-(-1)^{2/5} 5^{4/5} \sqrt [5]{2 e}\right )}}\right )}{10 \left (e+5 (-1)^{3/5} \sqrt [5]{5} (2 e)^{4/5}\right )}-\frac {3 e \sqrt {3 \left (50+(-1)^{3/5} 5^{4/5} \sqrt [5]{2 e}\right )} \text {arctanh}\left (\frac {5 x^2}{\sqrt {\frac {1}{3} \left (50+(-1)^{3/5} 5^{4/5} \sqrt [5]{2 e}\right )}}\right )}{10 \left (e-5 (-1)^{2/5} \sqrt [5]{5} (2 e)^{4/5}\right )}\) |
Input:
Int[(-40*x + 60*x^5 + ((-10 + 15*x^4)^5*(-100*x - 1350*x^5))/E)/(-8 + 12*x ^4 + ((-10 + 15*x^4)^5*(-40 + 60*x^4))/E + ((-10 + 15*x^4)^10*(-50 + 75*x^ 4))/E^2),x]
Output:
$Aborted
Time = 1.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\frac {5 x^{2}}{2+5 \,{\mathrm e}^{5 \ln \left (15 x^{4}-10\right )-1}}\) | \(25\) |
risch | \(\frac {x^{2}}{759375 \,{\mathrm e}^{-1} x^{20}-2531250 \,{\mathrm e}^{-1} x^{16}+3375000 \,{\mathrm e}^{-1} x^{12}-2250000 \,{\mathrm e}^{-1} x^{8}+750000 \,{\mathrm e}^{-1} x^{4}-100000 \,{\mathrm e}^{-1}+\frac {2}{5}}\) | \(48\) |
Input:
int(((-1350*x^5-100*x)*exp(5*ln(15*x^4-10)-1)+60*x^5-40*x)/((75*x^4-50)*ex p(5*ln(15*x^4-10)-1)^2+(60*x^4-40)*exp(5*ln(15*x^4-10)-1)+12*x^4-8),x,meth od=_RETURNVERBOSE)
Output:
5*x^2/(2+5*exp(5*ln(15*x^4-10)-1))
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {5 \, x^{2} e}{3796875 \, x^{20} - 12656250 \, x^{16} + 16875000 \, x^{12} - 11250000 \, x^{8} + 3750000 \, x^{4} + 2 \, e - 500000} \] Input:
integrate(((-1350*x^5-100*x)*exp(5*log(15*x^4-10)-1)+60*x^5-40*x)/((75*x^4 -50)*exp(5*log(15*x^4-10)-1)^2+(60*x^4-40)*exp(5*log(15*x^4-10)-1)+12*x^4- 8),x, algorithm="fricas")
Output:
5*x^2*e/(3796875*x^20 - 12656250*x^16 + 16875000*x^12 - 11250000*x^8 + 375 0000*x^4 + 2*e - 500000)
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 3.83 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {5 e x^{2}}{3796875 x^{20} - 12656250 x^{16} + 16875000 x^{12} - 11250000 x^{8} + 3750000 x^{4} - 500000 + 2 e} \] Input:
integrate(((-1350*x**5-100*x)*exp(5*ln(15*x**4-10)-1)+60*x**5-40*x)/((75*x **4-50)*exp(5*ln(15*x**4-10)-1)**2+(60*x**4-40)*exp(5*ln(15*x**4-10)-1)+12 *x**4-8),x)
Output:
5*E*x**2/(3796875*x**20 - 12656250*x**16 + 16875000*x**12 - 11250000*x**8 + 3750000*x**4 - 500000 + 2*E)
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {5 \, x^{2} e}{3796875 \, x^{20} - 12656250 \, x^{16} + 16875000 \, x^{12} - 11250000 \, x^{8} + 3750000 \, x^{4} + 2 \, e - 500000} \] Input:
integrate(((-1350*x^5-100*x)*exp(5*log(15*x^4-10)-1)+60*x^5-40*x)/((75*x^4 -50)*exp(5*log(15*x^4-10)-1)^2+(60*x^4-40)*exp(5*log(15*x^4-10)-1)+12*x^4- 8),x, algorithm="maxima")
Output:
5*x^2*e/(3796875*x^20 - 12656250*x^16 + 16875000*x^12 - 11250000*x^8 + 375 0000*x^4 + 2*e - 500000)
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {5 \, x^{2} e}{3796875 \, x^{20} - 12656250 \, x^{16} + 16875000 \, x^{12} - 11250000 \, x^{8} + 3750000 \, x^{4} + 2 \, e - 500000} \] Input:
integrate(((-1350*x^5-100*x)*exp(5*log(15*x^4-10)-1)+60*x^5-40*x)/((75*x^4 -50)*exp(5*log(15*x^4-10)-1)^2+(60*x^4-40)*exp(5*log(15*x^4-10)-1)+12*x^4- 8),x, algorithm="giac")
Output:
5*x^2*e/(3796875*x^20 - 12656250*x^16 + 16875000*x^12 - 11250000*x^8 + 375 0000*x^4 + 2*e - 500000)
Time = 7.62 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {5\,x^2\,\mathrm {e}}{3796875\,x^{20}-12656250\,x^{16}+16875000\,x^{12}-11250000\,x^8+3750000\,x^4+2\,\mathrm {e}-500000} \] Input:
int(-(40*x - 60*x^5 + exp(5*log(15*x^4 - 10) - 1)*(100*x + 1350*x^5))/(exp (5*log(15*x^4 - 10) - 1)*(60*x^4 - 40) + exp(10*log(15*x^4 - 10) - 2)*(75* x^4 - 50) + 12*x^4 - 8),x)
Output:
(5*x^2*exp(1))/(2*exp(1) + 3750000*x^4 - 11250000*x^8 + 16875000*x^12 - 12 656250*x^16 + 3796875*x^20 - 500000)
Time = 0.16 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {5 e \,x^{2}}{3796875 x^{20}-12656250 x^{16}+16875000 x^{12}-11250000 x^{8}+3750000 x^{4}+2 e -500000} \] Input:
int(((-1350*x^5-100*x)*exp(5*log(15*x^4-10)-1)+60*x^5-40*x)/((75*x^4-50)*e xp(5*log(15*x^4-10)-1)^2+(60*x^4-40)*exp(5*log(15*x^4-10)-1)+12*x^4-8),x)
Output:
(5*e*x**2)/(2*e + 3796875*x**20 - 12656250*x**16 + 16875000*x**12 - 112500 00*x**8 + 3750000*x**4 - 500000)