Integrand size = 127, antiderivative size = 29 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=1-\left (e^3+\left (-x+\frac {4}{-5+3 x^3}\right )^2\right ) \log (3 x) \] Output:
1-ln(3*x)*(exp(3)+(4/(3*x^3-5)-x)^2)
Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=-e^3 \log (x)-\frac {\left (-4-5 x+3 x^4\right )^2 \log (3 x)}{\left (-5+3 x^3\right )^2} \] Input:
Integrate[(80 + 200*x + 125*x^2 - 48*x^3 - 240*x^4 - 225*x^5 + 72*x^7 + 13 5*x^8 - 27*x^11 + E^3*(125 - 225*x^3 + 135*x^6 - 27*x^9) + (200*x + 250*x^ 2 + 288*x^3 + 120*x^4 - 450*x^5 - 144*x^7 + 270*x^8 - 54*x^11)*Log[3*x])/( -125*x + 225*x^4 - 135*x^7 + 27*x^10),x]
Output:
-(E^3*Log[x]) - ((-4 - 5*x + 3*x^4)^2*Log[3*x])/(-5 + 3*x^3)^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.
| \(\displaystyle \int \frac {-27 x^{11}+135 x^8+72 x^7-225 x^5-240 x^4-48 x^3+125 x^2+e^3 \left (-27 x^9+135 x^6-225 x^3+125\right )+\left (-54 x^{11}+270 x^8-144 x^7-450 x^5+120 x^4+288 x^3+250 x^2+200 x\right ) \log (3 x)+200 x+80}{27 x^{10}-135 x^7+225 x^4-125 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-27 x^{11}+135 x^8+72 x^7-225 x^5-240 x^4-48 x^3+125 x^2+e^3 \left (-27 x^9+135 x^6-225 x^3+125\right )+\left (-54 x^{11}+270 x^8-144 x^7-450 x^5+120 x^4+288 x^3+250 x^2+200 x\right ) \log (3 x)+200 x+80}{x \left (27 x^9-135 x^6+225 x^3-125\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {2 \left (-2+\sqrt [3]{-1}\right ) \left (-27 x^{11}+135 x^8+72 x^7-225 x^5-240 x^4-48 x^3+125 x^2+200 x+e^3 \left (-27 x^9+135 x^6-225 x^3+125\right )+\left (-54 x^{11}+270 x^8-144 x^7-450 x^5+120 x^4+288 x^3+250 x^2+200 x\right ) \log (3 x)+80\right )}{45\ 3^{5/6} 5^{2/3} \left (-i+\sqrt {3}\right ) \left (3^{2/3} \sqrt [3]{5}-3 x\right ) x}+\frac {9 \sqrt [3]{\frac {3}{5}} \left (-2+\sqrt [3]{-1}\right ) \left (-27 x^{11}+135 x^8+72 x^7-225 x^5-240 x^4-48 x^3+125 x^2+200 x+e^3 \left (-27 x^9+135 x^6-225 x^3+125\right )+\left (-54 x^{11}+270 x^8-144 x^7-450 x^5+120 x^4+288 x^3+250 x^2+200 x\right ) \log (3 x)+80\right )}{25 \left (-1+\sqrt [3]{-1}\right )^4 \left (1+\sqrt [3]{-1}\right )^7 \left (3^{2/3} \sqrt [3]{5}-3 x\right )^2 x}-\frac {-27 x^{11}+135 x^8+72 x^7-225 x^5-240 x^4-48 x^3+125 x^2+200 x+e^3 \left (-27 x^9+135 x^6-225 x^3+125\right )+\left (-54 x^{11}+270 x^8-144 x^7-450 x^5+120 x^4+288 x^3+250 x^2+200 x\right ) \log (3 x)+80}{75 \left (3^{2/3} \sqrt [3]{5}-3 x\right )^3 x}+\frac {3 \sqrt [3]{-1} \left (\frac {3}{5}\right )^{2/3} \left (-27 x^{11}+135 x^8+72 x^7-225 x^5-240 x^4-48 x^3+125 x^2+200 x+e^3 \left (-27 x^9+135 x^6-225 x^3+125\right )+\left (-54 x^{11}+270 x^8-144 x^7-450 x^5+120 x^4+288 x^3+250 x^2+200 x\right ) \log (3 x)+80\right )}{5 \left (1+\sqrt [3]{-1}\right )^8 x \left (3 \sqrt [3]{-1} x+3^{2/3} \sqrt [3]{5}\right )}+\frac {2 (-1)^{5/6} \left (-27 x^{11}+135 x^8+72 x^7-225 x^5-240 x^4-48 x^3+125 x^2+200 x+e^3 \left (-27 x^9+135 x^6-225 x^3+125\right )+\left (-54 x^{11}+270 x^8-144 x^7-450 x^5+120 x^4+288 x^3+250 x^2+200 x\right ) \log (3 x)+80\right )}{135 x \left (-2 i \sqrt [3]{3} 5^{2/3} x+5 \sqrt {3}-5 i\right )}+\frac {9 \sqrt [6]{-1} 3^{5/6} \left (-27 x^{11}+135 x^8+72 x^7-225 x^5-240 x^4-48 x^3+125 x^2+200 x+e^3 \left (-27 x^9+135 x^6-225 x^3+125\right )+\left (-54 x^{11}+270 x^8-144 x^7-450 x^5+120 x^4+288 x^3+250 x^2+200 x\right ) \log (3 x)+80\right )}{25 \sqrt [3]{5} \left (1+\sqrt [3]{-1}\right )^7 x \left (3 \sqrt [3]{-1} x+3^{2/3} \sqrt [3]{5}\right )^2}-\frac {(-1)^{2/3} \left (-27 x^{11}+135 x^8+72 x^7-225 x^5-240 x^4-48 x^3+125 x^2+200 x+e^3 \left (-27 x^9+135 x^6-225 x^3+125\right )+\left (-54 x^{11}+270 x^8-144 x^7-450 x^5+120 x^4+288 x^3+250 x^2+200 x\right ) \log (3 x)+80\right )}{25\ 3^{2/3} \sqrt [3]{5} \left (-1+\sqrt [3]{-1}\right )^4 x \left (3 (-1)^{2/3} x-3^{2/3} \sqrt [3]{5}\right )^2}-\frac {-27 x^{11}+135 x^8+72 x^7-225 x^5-240 x^4-48 x^3+125 x^2+200 x+e^3 \left (-27 x^9+135 x^6-225 x^3+125\right )+\left (-54 x^{11}+270 x^8-144 x^7-450 x^5+120 x^4+288 x^3+250 x^2+200 x\right ) \log (3 x)+80}{75 x \left (3 \sqrt [3]{-1} x+3^{2/3} \sqrt [3]{5}\right )^3}-\frac {-27 x^{11}+135 x^8+72 x^7-225 x^5-240 x^4-48 x^3+125 x^2+200 x+e^3 \left (-27 x^9+135 x^6-225 x^3+125\right )+\left (-54 x^{11}+270 x^8-144 x^7-450 x^5+120 x^4+288 x^3+250 x^2+200 x\right ) \log (3 x)+80}{75 x \left (3^{2/3} \sqrt [3]{5}-3 (-1)^{2/3} x\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {81 i \left (1+i \sqrt {3}\right ) \left (\left (3 x^3-5\right ) \left (\left (-3 x^4+5 x+4\right )^2+e^3 \left (5-3 x^3\right )^2\right )+2 x \left (27 x^{10}-135 x^7+72 x^6+225 x^4-60 x^3-144 x^2-125 x-100\right ) \log (3 x)\right )}{\left (1-\sqrt [3]{-1}\right )^4 \left (1+\sqrt [3]{-1}\right )^8 \left (-\sqrt {3}+i\right ) x \left (5-3 x^3\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int -\frac {\left (5-3 x^3\right ) \left (\left (-3 x^4+5 x+4\right )^2+e^3 \left (5-3 x^3\right )^2\right )+2 x \left (-27 x^{10}+135 x^7-72 x^6-225 x^4+60 x^3+144 x^2+125 x+100\right ) \log (3 x)}{x \left (5-3 x^3\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (5-3 x^3\right ) \left (e^3 \left (5-3 x^3\right )^2+\left (-3 x^4+5 x+4\right )^2\right )+2 x \left (-27 x^{10}+135 x^7-72 x^6-225 x^4+60 x^3+144 x^2+125 x+100\right ) \log (3 x)}{x \left (5-3 x^3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\int \left (\frac {9 x^8+9 e^3 x^6-30 x^5-24 x^4-30 e^3 x^3+25 x^2+40 x+16 \left (1+\frac {25 e^3}{16}\right )}{x \left (5-3 x^3\right )^2}+\frac {2 \left (3 x^4-5 x-4\right ) \left (9 x^6-30 x^3+36 x^2+25\right ) \log (3 x)}{\left (3 x^3-5\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 \int \frac {\log (3 x)}{-3 x^2+2\ 3^{2/3} \sqrt [3]{5} x-\sqrt [3]{3} 5^{2/3}}dx}{3^{2/3} \sqrt [3]{5}}-\frac {8 \arctan \left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{5}}+\frac {1}{\sqrt {3}}\right )}{3^{5/6} 5^{2/3}}-\frac {48 i \sqrt [6]{3} \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {3}{5}} x\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {16 (-1)^{2/3} \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {3}{5}} x\right )}{3 \sqrt [3]{3} 5^{2/3}}+\frac {16 \sqrt [3]{-\frac {1}{3}} \operatorname {PolyLog}\left (2,(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}-\frac {32 \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}-\frac {16 x^3}{25 \left (5-3 x^3\right )}+\frac {16}{15 \left (5-3 x^3\right )}-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}-\frac {16}{75} \log \left (5-3 x^3\right )-x^2 \log (3 x)+\frac {4}{225} \left (12-5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (3^{2/3} x^2+\sqrt [3]{15} x+5^{2/3}\right )-\frac {8 \sqrt [3]{-1} x \log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{3} x+\sqrt [3]{-5}\right )}-\frac {8 x \log (3 x)}{15^{2/3} \left (\sqrt [3]{-1} 3^{2/3} x+\sqrt [3]{15}\right )}+\frac {16 \log (45) \log \left (3^{2/3} \sqrt [3]{5}-3 x\right )}{9 \sqrt [3]{3} 5^{2/3}}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)+\frac {16 \log (x)}{25}+\frac {8 \sqrt [3]{-1} 3^{2/3} \log \left (\sqrt [3]{-3} x+\sqrt [3]{5}\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^4}-\frac {48 i \sqrt [6]{3} \log (3 x) \log \left (\sqrt [3]{-\frac {3}{5}} x+1\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {16 (-1)^{2/3} \log (3 x) \log \left (\sqrt [3]{-\frac {3}{5}} x+1\right )}{3 \sqrt [3]{3} 5^{2/3}}+\frac {16 \sqrt [3]{-\frac {1}{3}} \log (3 x) \log \left (1-(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}-\frac {16 \log (45) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}{9 \sqrt [3]{3} 5^{2/3}}+\frac {8}{225} \left (6+5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )+\frac {8 \sqrt [3]{-\frac {1}{3}} \log \left (\sqrt [3]{3} x+\sqrt [3]{-5}\right )}{3\ 5^{2/3}}-\frac {32 \log (3 x) \log \left (1+\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}\) |
Input:
Int[(80 + 200*x + 125*x^2 - 48*x^3 - 240*x^4 - 225*x^5 + 72*x^7 + 135*x^8 - 27*x^11 + E^3*(125 - 225*x^3 + 135*x^6 - 27*x^9) + (200*x + 250*x^2 + 28 8*x^3 + 120*x^4 - 450*x^5 - 144*x^7 + 270*x^8 - 54*x^11)*Log[3*x])/(-125*x + 225*x^4 - 135*x^7 + 27*x^10),x]
Output:
$Aborted
Time = 4.97 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83
| method | result | size |
| risch | \(-\frac {\left (9 x^{8}-30 x^{5}-24 x^{4}+25 x^{2}+40 x +16\right ) \ln \left (3 x \right )}{9 x^{6}-30 x^{3}+25}-\ln \left (x \right ) {\mathrm e}^{3}\) | \(53\) |
| derivativedivides | \(-\frac {\left (9 \,{\mathrm e}^{3}+\frac {144}{25}\right ) \ln \left (3 x \right )}{9}-x^{2} \ln \left (3 x \right )+\frac {72 \ln \left (3 x \right ) x}{27 x^{3}-45}+\frac {432 \ln \left (3 x \right ) x^{3} \left (27 x^{3}-90\right )}{25 \left (27 x^{3}-45\right )^{2}}\) | \(64\) |
| default | \(-\frac {\left (9 \,{\mathrm e}^{3}+\frac {144}{25}\right ) \ln \left (3 x \right )}{9}-x^{2} \ln \left (3 x \right )+\frac {72 \ln \left (3 x \right ) x}{27 x^{3}-45}+\frac {432 \ln \left (3 x \right ) x^{3} \left (27 x^{3}-90\right )}{25 \left (27 x^{3}-45\right )^{2}}\) | \(64\) |
| norman | \(\frac {-16 \ln \left (3 x \right )-40 x \ln \left (3 x \right )-25 x^{2} \ln \left (3 x \right )+24 \ln \left (3 x \right ) x^{4}+30 \ln \left (3 x \right ) x^{5}-9 \ln \left (3 x \right ) x^{8}}{\left (3 x^{3}-5\right )^{2}}-\ln \left (x \right ) {\mathrm e}^{3}\) | \(68\) |
| parallelrisch | \(-\frac {81 \ln \left (3 x \right ) x^{8}+81 x^{6} \ln \left (x \right ) {\mathrm e}^{3}-270 \ln \left (3 x \right ) x^{5}-270 x^{3} \ln \left (x \right ) {\mathrm e}^{3}-216 \ln \left (3 x \right ) x^{4}+225 x^{2} \ln \left (3 x \right )+225 \ln \left (x \right ) {\mathrm e}^{3}+360 x \ln \left (3 x \right )+144 \ln \left (3 x \right )}{9 \left (9 x^{6}-30 x^{3}+25\right )}\) | \(91\) |
| parts | \(-\left (\frac {16}{25}+{\mathrm e}^{3}\right ) \ln \left (x \right )+\frac {16}{45 \left (x^{3}-\frac {5}{3}\right )}+\frac {8 \,5^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (x -\frac {5^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{45}-\frac {4 \,5^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (x^{2}+\frac {5^{\frac {1}{3}} 3^{\frac {2}{3}} x}{3}+\frac {5^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{45}-\frac {8 \,5^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,5^{\frac {2}{3}} 3^{\frac {1}{3}} x}{5}+1\right )}{3}\right )}{15}+\frac {16 \ln \left (3 x^{3}-5\right )}{75}-x^{2} \ln \left (3 x \right )-\frac {48}{5 \left (27 x^{3}-45\right )}-\frac {16 \ln \left (27 x^{3}-45\right )}{75}+\frac {432 \ln \left (3 x \right ) x^{3} \left (27 x^{3}-90\right )}{25 \left (27 x^{3}-45\right )^{2}}-\frac {8 \,45^{\frac {1}{3}} \ln \left (3 x -45^{\frac {1}{3}}\right )}{45}+\frac {4 \,45^{\frac {1}{3}} \ln \left (9 x^{2}+3 \,45^{\frac {1}{3}} x +45^{\frac {2}{3}}\right )}{45}+\frac {8 \,45^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,45^{\frac {2}{3}} x}{15}+1\right )}{3}\right )}{45}+\frac {72 \ln \left (3 x \right ) x}{27 x^{3}-45}\) | \(232\) |
Input:
int(((-54*x^11+270*x^8-144*x^7-450*x^5+120*x^4+288*x^3+250*x^2+200*x)*ln(3 *x)+(-27*x^9+135*x^6-225*x^3+125)*exp(3)-27*x^11+135*x^8+72*x^7-225*x^5-24 0*x^4-48*x^3+125*x^2+200*x+80)/(27*x^10-135*x^7+225*x^4-125*x),x,method=_R ETURNVERBOSE)
Output:
-(9*x^8-30*x^5-24*x^4+25*x^2+40*x+16)/(9*x^6-30*x^3+25)*ln(3*x)-ln(x)*exp( 3)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=-\frac {{\left (9 \, x^{8} - 30 \, x^{5} - 24 \, x^{4} + 25 \, x^{2} + {\left (9 \, x^{6} - 30 \, x^{3} + 25\right )} e^{3} + 40 \, x + 16\right )} \log \left (3 \, x\right )}{9 \, x^{6} - 30 \, x^{3} + 25} \] Input:
integrate(((-54*x^11+270*x^8-144*x^7-450*x^5+120*x^4+288*x^3+250*x^2+200*x )*log(3*x)+(-27*x^9+135*x^6-225*x^3+125)*exp(3)-27*x^11+135*x^8+72*x^7-225 *x^5-240*x^4-48*x^3+125*x^2+200*x+80)/(27*x^10-135*x^7+225*x^4-125*x),x, a lgorithm="fricas")
Output:
-(9*x^8 - 30*x^5 - 24*x^4 + 25*x^2 + (9*x^6 - 30*x^3 + 25)*e^3 + 40*x + 16 )*log(3*x)/(9*x^6 - 30*x^3 + 25)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=- e^{3} \log {\left (x \right )} + \frac {\left (- 9 x^{8} + 30 x^{5} + 24 x^{4} - 25 x^{2} - 40 x - 16\right ) \log {\left (3 x \right )}}{9 x^{6} - 30 x^{3} + 25} \] Input:
integrate(((-54*x**11+270*x**8-144*x**7-450*x**5+120*x**4+288*x**3+250*x** 2+200*x)*ln(3*x)+(-27*x**9+135*x**6-225*x**3+125)*exp(3)-27*x**11+135*x**8 +72*x**7-225*x**5-240*x**4-48*x**3+125*x**2+200*x+80)/(27*x**10-135*x**7+2 25*x**4-125*x),x)
Output:
-exp(3)*log(x) + (-9*x**8 + 30*x**5 + 24*x**4 - 25*x**2 - 40*x - 16)*log(3 *x)/(9*x**6 - 30*x**3 + 25)
Exception generated. \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(((-54*x^11+270*x^8-144*x^7-450*x^5+120*x^4+288*x^3+250*x^2+200*x )*log(3*x)+(-27*x^9+135*x^6-225*x^3+125)*exp(3)-27*x^11+135*x^8+72*x^7-225 *x^5-240*x^4-48*x^3+125*x^2+200*x+80)/(27*x^10-135*x^7+225*x^4-125*x),x, a lgorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (26) = 52\).
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.10 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=-\frac {9 \, x^{8} \log \left (3 \, x\right ) + 9 \, x^{6} e^{3} \log \left (x\right ) - 30 \, x^{5} \log \left (3 \, x\right ) - 24 \, x^{4} \log \left (3 \, x\right ) - 30 \, x^{3} e^{3} \log \left (x\right ) + 25 \, x^{2} \log \left (3 \, x\right ) + 40 \, x \log \left (3 \, x\right ) + 25 \, e^{3} \log \left (x\right ) + 16 \, \log \left (3 \, x\right )}{9 \, x^{6} - 30 \, x^{3} + 25} \] Input:
integrate(((-54*x^11+270*x^8-144*x^7-450*x^5+120*x^4+288*x^3+250*x^2+200*x )*log(3*x)+(-27*x^9+135*x^6-225*x^3+125)*exp(3)-27*x^11+135*x^8+72*x^7-225 *x^5-240*x^4-48*x^3+125*x^2+200*x+80)/(27*x^10-135*x^7+225*x^4-125*x),x, a lgorithm="giac")
Output:
-(9*x^8*log(3*x) + 9*x^6*e^3*log(x) - 30*x^5*log(3*x) - 24*x^4*log(3*x) - 30*x^3*e^3*log(x) + 25*x^2*log(3*x) + 40*x*log(3*x) + 25*e^3*log(x) + 16*l og(3*x))/(9*x^6 - 30*x^3 + 25)
Time = 3.45 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=-{\mathrm {e}}^3\,\ln \left (x\right )-\frac {\ln \left (3\,x\right )\,\left (x^8-\frac {10\,x^5}{3}-\frac {8\,x^4}{3}+\frac {25\,x^2}{9}+\frac {40\,x}{9}+\frac {16}{9}\right )}{x^6-\frac {10\,x^3}{3}+\frac {25}{9}} \] Input:
int(-(200*x + log(3*x)*(200*x + 250*x^2 + 288*x^3 + 120*x^4 - 450*x^5 - 14 4*x^7 + 270*x^8 - 54*x^11) - exp(3)*(225*x^3 - 135*x^6 + 27*x^9 - 125) + 1 25*x^2 - 48*x^3 - 240*x^4 - 225*x^5 + 72*x^7 + 135*x^8 - 27*x^11 + 80)/(12 5*x - 225*x^4 + 135*x^7 - 27*x^10),x)
Output:
- exp(3)*log(x) - (log(3*x)*((40*x)/9 + (25*x^2)/9 - (8*x^4)/3 - (10*x^5)/ 3 + x^8 + 16/9))/(x^6 - (10*x^3)/3 + 25/9)
Time = 0.18 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.59 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=\frac {-675 \,\mathrm {log}\left (3 x \right ) x^{8}+432 \,\mathrm {log}\left (3 x \right ) x^{6}+2250 \,\mathrm {log}\left (3 x \right ) x^{5}+1800 \,\mathrm {log}\left (3 x \right ) x^{4}-1440 x^{3} \mathrm {log}\left (3 x \right )-1875 \,\mathrm {log}\left (3 x \right ) x^{2}-3000 \,\mathrm {log}\left (3 x \right ) x -675 \,\mathrm {log}\left (x \right ) e^{3} x^{6}+2250 \,\mathrm {log}\left (x \right ) e^{3} x^{3}-1875 \,\mathrm {log}\left (x \right ) e^{3}-432 \,\mathrm {log}\left (x \right ) x^{6}+1440 \,\mathrm {log}\left (x \right ) x^{3}-1200 \,\mathrm {log}\left (x \right )+72 x^{6}-240 x^{3}+200}{675 x^{6}-2250 x^{3}+1875} \] Input:
int(((-54*x^11+270*x^8-144*x^7-450*x^5+120*x^4+288*x^3+250*x^2+200*x)*log( 3*x)+(-27*x^9+135*x^6-225*x^3+125)*exp(3)-27*x^11+135*x^8+72*x^7-225*x^5-2 40*x^4-48*x^3+125*x^2+200*x+80)/(27*x^10-135*x^7+225*x^4-125*x),x)
Output:
( - 675*log(3*x)*x**8 + 432*log(3*x)*x**6 + 2250*log(3*x)*x**5 + 1800*log( 3*x)*x**4 - 1440*log(3*x)*x**3 - 1875*log(3*x)*x**2 - 3000*log(3*x)*x - 67 5*log(x)*e**3*x**6 + 2250*log(x)*e**3*x**3 - 1875*log(x)*e**3 - 432*log(x) *x**6 + 1440*log(x)*x**3 - 1200*log(x) + 72*x**6 - 240*x**3 + 200)/(75*(9* x**6 - 30*x**3 + 25))