Integrand size = 73, antiderivative size = 19 \[ \int \frac {1}{125} \left (-21349+20690 x+9450 x^2-13000 x^3+3125 x^4+\left (34020-5400 x-25500 x^2+10000 x^3\right ) \log (3)+\left (-14850-12000 x+11250 x^2\right ) \log ^2(3)+(500+5000 x) \log ^3(3)+625 \log ^4(3)\right ) \, dx=5 \left (-3+(2+x) \left (x+\left (-\frac {9}{5}+x+\log (3)\right )^4\right )\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(105\) vs. \(2(19)=38\).
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 5.53 \[ \int \frac {1}{125} \left (-21349+20690 x+9450 x^2-13000 x^3+3125 x^4+\left (34020-5400 x-25500 x^2+10000 x^3\right ) \log (3)+\left (-14850-12000 x+11250 x^2\right ) \log ^2(3)+(500+5000 x) \log ^3(3)+625 \log ^4(3)\right ) \, dx=-\frac {21349 x}{125}+\frac {2069 x^2}{25}+5 x^5+\frac {6804}{25} x \log (3)-\frac {108}{5} x^2 \log (3)-\frac {594}{5} x \log ^2(3)-48 x^2 \log ^2(3)+4 x \log ^3(3)+20 x^2 \log ^3(3)+5 x \log ^4(3)+2 x^4 (-13+10 \log (3))+\frac {2}{5} x^3 \left (63-170 \log (3)+75 \log ^2(3)\right ) \]
Integrate[(-21349 + 20690*x + 9450*x^2 - 13000*x^3 + 3125*x^4 + (34020 - 5 400*x - 25500*x^2 + 10000*x^3)*Log[3] + (-14850 - 12000*x + 11250*x^2)*Log [3]^2 + (500 + 5000*x)*Log[3]^3 + 625*Log[3]^4)/125,x]
(-21349*x)/125 + (2069*x^2)/25 + 5*x^5 + (6804*x*Log[3])/25 - (108*x^2*Log [3])/5 - (594*x*Log[3]^2)/5 - 48*x^2*Log[3]^2 + 4*x*Log[3]^3 + 20*x^2*Log[ 3]^3 + 5*x*Log[3]^4 + 2*x^4*(-13 + 10*Log[3]) + (2*x^3*(63 - 170*Log[3] + 75*Log[3]^2))/5
Leaf count is larger than twice the leaf count of optimal. \(100\) vs. \(2(19)=38\).
Time = 0.24 (sec) , antiderivative size = 100, normalized size of antiderivative = 5.26, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, 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.
\(\displaystyle \int \frac {1}{125} \left (3125 x^4-13000 x^3+9450 x^2+\left (11250 x^2-12000 x-14850\right ) \log ^2(3)+\left (10000 x^3-25500 x^2-5400 x+34020\right ) \log (3)+20690 x+(5000 x+500) \log ^3(3)-21349+625 \log ^4(3)\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{125} \int \left (3125 x^4-13000 x^3+9450 x^2+20690 x+625 \log ^4(3)+500 (10 x+1) \log ^3(3)-150 \left (-75 x^2+80 x+99\right ) \log ^2(3)+20 \left (500 x^3-1275 x^2-270 x+1701\right ) \log (3)-21349\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{125} \left (625 x^5-3250 x^4+2500 x^4 \log (3)+3150 x^3+3750 x^3 \log ^2(3)-8500 x^3 \log (3)+10345 x^2-6000 x^2 \log ^2(3)-2700 x^2 \log (3)-x \left (21349-625 \log ^4(3)\right )+25 (10 x+1)^2 \log ^3(3)-14850 x \log ^2(3)+34020 x \log (3)\right )\) |
Int[(-21349 + 20690*x + 9450*x^2 - 13000*x^3 + 3125*x^4 + (34020 - 5400*x - 25500*x^2 + 10000*x^3)*Log[3] + (-14850 - 12000*x + 11250*x^2)*Log[3]^2 + (500 + 5000*x)*Log[3]^3 + 625*Log[3]^4)/125,x]
(10345*x^2 + 3150*x^3 - 3250*x^4 + 625*x^5 + 34020*x*Log[3] - 2700*x^2*Log [3] - 8500*x^3*Log[3] + 2500*x^4*Log[3] - 14850*x*Log[3]^2 - 6000*x^2*Log[ 3]^2 + 3750*x^3*Log[3]^2 + 25*(1 + 10*x)^2*Log[3]^3 - x*(21349 - 625*Log[3 ]^4))/125
3.17.91.3.1 Defintions of rubi rules used
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. \(80\) vs. \(2(16)=32\).
Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 4.26
method | result | size |
norman | \(\left (20 \ln \left (3\right )-26\right ) x^{4}+\left (30 \ln \left (3\right )^{2}-68 \ln \left (3\right )+\frac {126}{5}\right ) x^{3}+\left (20 \ln \left (3\right )^{3}-48 \ln \left (3\right )^{2}-\frac {108 \ln \left (3\right )}{5}+\frac {2069}{25}\right ) x^{2}+\left (5 \ln \left (3\right )^{4}+4 \ln \left (3\right )^{3}-\frac {594 \ln \left (3\right )^{2}}{5}+\frac {6804 \ln \left (3\right )}{25}-\frac {21349}{125}\right ) x +5 x^{5}\) | \(81\) |
gosper | \(\frac {x \left (625 \ln \left (3\right )^{4}+2500 x \ln \left (3\right )^{3}+3750 x^{2} \ln \left (3\right )^{2}+2500 x^{3} \ln \left (3\right )+625 x^{4}+500 \ln \left (3\right )^{3}-6000 x \ln \left (3\right )^{2}-8500 x^{2} \ln \left (3\right )-3250 x^{3}-14850 \ln \left (3\right )^{2}-2700 x \ln \left (3\right )+3150 x^{2}+34020 \ln \left (3\right )+10345 x -21349\right )}{125}\) | \(88\) |
default | \(5 x \ln \left (3\right )^{4}+20 x^{2} \ln \left (3\right )^{3}+30 x^{3} \ln \left (3\right )^{2}+20 x^{4} \ln \left (3\right )+5 x^{5}+4 x \ln \left (3\right )^{3}-48 x^{2} \ln \left (3\right )^{2}-68 x^{3} \ln \left (3\right )-26 x^{4}-\frac {594 x \ln \left (3\right )^{2}}{5}-\frac {108 x^{2} \ln \left (3\right )}{5}+\frac {126 x^{3}}{5}+\frac {6804 x \ln \left (3\right )}{25}+\frac {2069 x^{2}}{25}-\frac {21349 x}{125}\) | \(99\) |
risch | \(5 x \ln \left (3\right )^{4}+20 x^{2} \ln \left (3\right )^{3}+30 x^{3} \ln \left (3\right )^{2}+20 x^{4} \ln \left (3\right )+5 x^{5}+4 x \ln \left (3\right )^{3}-48 x^{2} \ln \left (3\right )^{2}-68 x^{3} \ln \left (3\right )-26 x^{4}-\frac {594 x \ln \left (3\right )^{2}}{5}-\frac {108 x^{2} \ln \left (3\right )}{5}+\frac {126 x^{3}}{5}+\frac {6804 x \ln \left (3\right )}{25}+\frac {2069 x^{2}}{25}-\frac {21349 x}{125}\) | \(99\) |
parallelrisch | \(5 x^{5}+20 x^{4} \ln \left (3\right )-26 x^{4}+30 x^{3} \ln \left (3\right )^{2}-68 x^{3} \ln \left (3\right )+\frac {126 x^{3}}{5}+20 x^{2} \ln \left (3\right )^{3}-48 x^{2} \ln \left (3\right )^{2}-\frac {108 x^{2} \ln \left (3\right )}{5}+\frac {2069 x^{2}}{25}+4 x \ln \left (3\right )^{3}-\frac {594 x \ln \left (3\right )^{2}}{5}+\frac {6804 x \ln \left (3\right )}{25}+\left (5 \ln \left (3\right )^{4}-\frac {21349}{125}\right ) x\) | \(99\) |
parts | \(5 x \ln \left (3\right )^{4}+20 x^{2} \ln \left (3\right )^{3}+30 x^{3} \ln \left (3\right )^{2}+20 x^{4} \ln \left (3\right )+5 x^{5}+4 x \ln \left (3\right )^{3}-48 x^{2} \ln \left (3\right )^{2}-68 x^{3} \ln \left (3\right )-26 x^{4}-\frac {594 x \ln \left (3\right )^{2}}{5}-\frac {108 x^{2} \ln \left (3\right )}{5}+\frac {126 x^{3}}{5}+\frac {6804 x \ln \left (3\right )}{25}+\frac {2069 x^{2}}{25}-\frac {21349 x}{125}\) | \(99\) |
int(5*ln(3)^4+1/125*(5000*x+500)*ln(3)^3+1/125*(11250*x^2-12000*x-14850)*l n(3)^2+1/125*(10000*x^3-25500*x^2-5400*x+34020)*ln(3)+25*x^4-104*x^3+378/5 *x^2+4138/25*x-21349/125,x,method=_RETURNVERBOSE)
(20*ln(3)-26)*x^4+(30*ln(3)^2-68*ln(3)+126/5)*x^3+(20*ln(3)^3-48*ln(3)^2-1 08/5*ln(3)+2069/25)*x^2+(5*ln(3)^4+4*ln(3)^3-594/5*ln(3)^2+6804/25*ln(3)-2 1349/125)*x+5*x^5
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (22) = 44\).
Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.58 \[ \int \frac {1}{125} \left (-21349+20690 x+9450 x^2-13000 x^3+3125 x^4+\left (34020-5400 x-25500 x^2+10000 x^3\right ) \log (3)+\left (-14850-12000 x+11250 x^2\right ) \log ^2(3)+(500+5000 x) \log ^3(3)+625 \log ^4(3)\right ) \, dx=5 \, x^{5} + 5 \, x \log \left (3\right )^{4} - 26 \, x^{4} + 4 \, {\left (5 \, x^{2} + x\right )} \log \left (3\right )^{3} + \frac {126}{5} \, x^{3} + \frac {6}{5} \, {\left (25 \, x^{3} - 40 \, x^{2} - 99 \, x\right )} \log \left (3\right )^{2} + \frac {2069}{25} \, x^{2} + \frac {4}{25} \, {\left (125 \, x^{4} - 425 \, x^{3} - 135 \, x^{2} + 1701 \, x\right )} \log \left (3\right ) - \frac {21349}{125} \, x \]
integrate(5*log(3)^4+1/125*(5000*x+500)*log(3)^3+1/125*(11250*x^2-12000*x- 14850)*log(3)^2+1/125*(10000*x^3-25500*x^2-5400*x+34020)*log(3)+25*x^4-104 *x^3+378/5*x^2+4138/25*x-21349/125,x, algorithm=\
5*x^5 + 5*x*log(3)^4 - 26*x^4 + 4*(5*x^2 + x)*log(3)^3 + 126/5*x^3 + 6/5*( 25*x^3 - 40*x^2 - 99*x)*log(3)^2 + 2069/25*x^2 + 4/25*(125*x^4 - 425*x^3 - 135*x^2 + 1701*x)*log(3) - 21349/125*x
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (17) = 34\).
Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.95 \[ \int \frac {1}{125} \left (-21349+20690 x+9450 x^2-13000 x^3+3125 x^4+\left (34020-5400 x-25500 x^2+10000 x^3\right ) \log (3)+\left (-14850-12000 x+11250 x^2\right ) \log ^2(3)+(500+5000 x) \log ^3(3)+625 \log ^4(3)\right ) \, dx=5 x^{5} + x^{4} \left (-26 + 20 \log {\left (3 \right )}\right ) + x^{3} \left (- 68 \log {\left (3 \right )} + \frac {126}{5} + 30 \log {\left (3 \right )}^{2}\right ) + x^{2} \left (- 48 \log {\left (3 \right )}^{2} - \frac {108 \log {\left (3 \right )}}{5} + 20 \log {\left (3 \right )}^{3} + \frac {2069}{25}\right ) + x \left (- \frac {21349}{125} - \frac {594 \log {\left (3 \right )}^{2}}{5} + 4 \log {\left (3 \right )}^{3} + 5 \log {\left (3 \right )}^{4} + \frac {6804 \log {\left (3 \right )}}{25}\right ) \]
integrate(5*ln(3)**4+1/125*(5000*x+500)*ln(3)**3+1/125*(11250*x**2-12000*x -14850)*ln(3)**2+1/125*(10000*x**3-25500*x**2-5400*x+34020)*ln(3)+25*x**4- 104*x**3+378/5*x**2+4138/25*x-21349/125,x)
5*x**5 + x**4*(-26 + 20*log(3)) + x**3*(-68*log(3) + 126/5 + 30*log(3)**2) + x**2*(-48*log(3)**2 - 108*log(3)/5 + 20*log(3)**3 + 2069/25) + x*(-2134 9/125 - 594*log(3)**2/5 + 4*log(3)**3 + 5*log(3)**4 + 6804*log(3)/25)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (22) = 44\).
Time = 0.18 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.58 \[ \int \frac {1}{125} \left (-21349+20690 x+9450 x^2-13000 x^3+3125 x^4+\left (34020-5400 x-25500 x^2+10000 x^3\right ) \log (3)+\left (-14850-12000 x+11250 x^2\right ) \log ^2(3)+(500+5000 x) \log ^3(3)+625 \log ^4(3)\right ) \, dx=5 \, x^{5} + 5 \, x \log \left (3\right )^{4} - 26 \, x^{4} + 4 \, {\left (5 \, x^{2} + x\right )} \log \left (3\right )^{3} + \frac {126}{5} \, x^{3} + \frac {6}{5} \, {\left (25 \, x^{3} - 40 \, x^{2} - 99 \, x\right )} \log \left (3\right )^{2} + \frac {2069}{25} \, x^{2} + \frac {4}{25} \, {\left (125 \, x^{4} - 425 \, x^{3} - 135 \, x^{2} + 1701 \, x\right )} \log \left (3\right ) - \frac {21349}{125} \, x \]
integrate(5*log(3)^4+1/125*(5000*x+500)*log(3)^3+1/125*(11250*x^2-12000*x- 14850)*log(3)^2+1/125*(10000*x^3-25500*x^2-5400*x+34020)*log(3)+25*x^4-104 *x^3+378/5*x^2+4138/25*x-21349/125,x, algorithm=\
5*x^5 + 5*x*log(3)^4 - 26*x^4 + 4*(5*x^2 + x)*log(3)^3 + 126/5*x^3 + 6/5*( 25*x^3 - 40*x^2 - 99*x)*log(3)^2 + 2069/25*x^2 + 4/25*(125*x^4 - 425*x^3 - 135*x^2 + 1701*x)*log(3) - 21349/125*x
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.58 \[ \int \frac {1}{125} \left (-21349+20690 x+9450 x^2-13000 x^3+3125 x^4+\left (34020-5400 x-25500 x^2+10000 x^3\right ) \log (3)+\left (-14850-12000 x+11250 x^2\right ) \log ^2(3)+(500+5000 x) \log ^3(3)+625 \log ^4(3)\right ) \, dx=5 \, x^{5} + 5 \, x \log \left (3\right )^{4} - 26 \, x^{4} + 4 \, {\left (5 \, x^{2} + x\right )} \log \left (3\right )^{3} + \frac {126}{5} \, x^{3} + \frac {6}{5} \, {\left (25 \, x^{3} - 40 \, x^{2} - 99 \, x\right )} \log \left (3\right )^{2} + \frac {2069}{25} \, x^{2} + \frac {4}{25} \, {\left (125 \, x^{4} - 425 \, x^{3} - 135 \, x^{2} + 1701 \, x\right )} \log \left (3\right ) - \frac {21349}{125} \, x \]
integrate(5*log(3)^4+1/125*(5000*x+500)*log(3)^3+1/125*(11250*x^2-12000*x- 14850)*log(3)^2+1/125*(10000*x^3-25500*x^2-5400*x+34020)*log(3)+25*x^4-104 *x^3+378/5*x^2+4138/25*x-21349/125,x, algorithm=\
5*x^5 + 5*x*log(3)^4 - 26*x^4 + 4*(5*x^2 + x)*log(3)^3 + 126/5*x^3 + 6/5*( 25*x^3 - 40*x^2 - 99*x)*log(3)^2 + 2069/25*x^2 + 4/25*(125*x^4 - 425*x^3 - 135*x^2 + 1701*x)*log(3) - 21349/125*x
Time = 11.66 (sec) , antiderivative size = 81, normalized size of antiderivative = 4.26 \[ \int \frac {1}{125} \left (-21349+20690 x+9450 x^2-13000 x^3+3125 x^4+\left (34020-5400 x-25500 x^2+10000 x^3\right ) \log (3)+\left (-14850-12000 x+11250 x^2\right ) \log ^2(3)+(500+5000 x) \log ^3(3)+625 \log ^4(3)\right ) \, dx=5\,x^5+\left (20\,\ln \left (3\right )-26\right )\,x^4+\left (30\,{\ln \left (3\right )}^2-68\,\ln \left (3\right )+\frac {126}{5}\right )\,x^3+\left (20\,{\ln \left (3\right )}^3-48\,{\ln \left (3\right )}^2-\frac {108\,\ln \left (3\right )}{5}+\frac {2069}{25}\right )\,x^2+\left (\frac {6804\,\ln \left (3\right )}{25}-\frac {594\,{\ln \left (3\right )}^2}{5}+4\,{\ln \left (3\right )}^3+5\,{\ln \left (3\right )}^4-\frac {21349}{125}\right )\,x \]
int((4138*x)/25 + (log(3)^3*(5000*x + 500))/125 - (log(3)*(5400*x + 25500* x^2 - 10000*x^3 - 34020))/125 - (log(3)^2*(12000*x - 11250*x^2 + 14850))/1 25 + 5*log(3)^4 + (378*x^2)/5 - 104*x^3 + 25*x^4 - 21349/125,x)