Integrand size = 38, antiderivative size = 353 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^2} \, dx=\frac {\left (700 d^6+270 d^5 e+555 d^4 e^2+148 d^3 e^3+444 d^2 e^4-130 d e^5+107 e^6\right ) x}{e^8}-\frac {\left (600 d^5+225 d^4 e+444 d^3 e^2+111 d^2 e^3+296 d e^4-65 e^5\right ) x^2}{2 e^7}+\frac {\left (500 d^4+180 d^3 e+333 d^2 e^2+74 d e^3+148 e^4\right ) x^3}{3 e^6}-\frac {\left (400 d^3+135 d^2 e+222 d e^2+37 e^3\right ) x^4}{4 e^5}+\frac {3 \left (100 d^2+30 d e+37 e^2\right ) x^5}{5 e^4}-\frac {5 (40 d+9 e) x^6}{6 e^3}+\frac {100 x^7}{7 e^2}-\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e^9 (d+e x)}-\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right ) \log (d+e x)}{e^9} \]
(700*d^6+270*d^5*e+555*d^4*e^2+148*d^3*e^3+444*d^2*e^4-130*d*e^5+107*e^6)* x/e^8-1/2*(600*d^5+225*d^4*e+444*d^3*e^2+111*d^2*e^3+296*d*e^4-65*e^5)*x^2 /e^7+1/3*(500*d^4+180*d^3*e+333*d^2*e^2+74*d*e^3+148*e^4)*x^3/e^6-1/4*(400 *d^3+135*d^2*e+222*d*e^2+37*e^3)*x^4/e^5+3/5*(100*d^2+30*d*e+37*e^2)*x^5/e ^4-5/6*(40*d+9*e)*x^6/e^3+100/7*x^7/e^2-(5*d^2-2*d*e+3*e^2)^2*(4*d^4+5*d^3 *e+3*d^2*e^2-d*e^3+2*e^4)/e^9/(e*x+d)-(5*d^2-2*d*e+3*e^2)*(160*d^5+127*d^4 *e+88*d^3*e^2-4*d^2*e^3+64*d*e^4-11*e^5)*ln(e*x+d)/e^9
Time = 0.09 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.97 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^2} \, dx=\frac {420 e \left (700 d^6+270 d^5 e+555 d^4 e^2+148 d^3 e^3+444 d^2 e^4-130 d e^5+107 e^6\right ) x-210 e^2 \left (600 d^5+225 d^4 e+444 d^3 e^2+111 d^2 e^3+296 d e^4-65 e^5\right ) x^2+140 e^3 \left (500 d^4+180 d^3 e+333 d^2 e^2+74 d e^3+148 e^4\right ) x^3-105 e^4 \left (400 d^3+135 d^2 e+222 d e^2+37 e^3\right ) x^4+252 e^5 \left (100 d^2+30 d e+37 e^2\right ) x^5-350 e^6 (40 d+9 e) x^6+6000 e^7 x^7-\frac {420 \left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{d+e x}-420 \left (800 d^7+315 d^6 e+666 d^5 e^2+185 d^4 e^3+592 d^3 e^4-195 d^2 e^5+214 d e^6-33 e^7\right ) \log (d+e x)}{420 e^9} \]
(420*e*(700*d^6 + 270*d^5*e + 555*d^4*e^2 + 148*d^3*e^3 + 444*d^2*e^4 - 13 0*d*e^5 + 107*e^6)*x - 210*e^2*(600*d^5 + 225*d^4*e + 444*d^3*e^2 + 111*d^ 2*e^3 + 296*d*e^4 - 65*e^5)*x^2 + 140*e^3*(500*d^4 + 180*d^3*e + 333*d^2*e ^2 + 74*d*e^3 + 148*e^4)*x^3 - 105*e^4*(400*d^3 + 135*d^2*e + 222*d*e^2 + 37*e^3)*x^4 + 252*e^5*(100*d^2 + 30*d*e + 37*e^2)*x^5 - 350*e^6*(40*d + 9* e)*x^6 + 6000*e^7*x^7 - (420*(5*d^2 - 2*d*e + 3*e^2)^2*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(d + e*x) - 420*(800*d^7 + 315*d^6*e + 666*d^5 *e^2 + 185*d^4*e^3 + 592*d^3*e^4 - 195*d^2*e^5 + 214*d*e^6 - 33*e^7)*Log[d + e*x])/(420*e^9)
Time = 0.62 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2159, 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.
| \(\displaystyle \int \frac {\left (5 x^2+2 x+3\right )^2 \left (4 x^4-5 x^3+3 x^2+x+2\right )}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (\frac {3 x^4 \left (100 d^2+30 d e+37 e^2\right )}{e^4}-\frac {x^3 \left (400 d^3+135 d^2 e+222 d e^2+37 e^3\right )}{e^5}+\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e^8 (d+e x)^2}+\frac {x^2 \left (500 d^4+180 d^3 e+333 d^2 e^2+74 d e^3+148 e^4\right )}{e^6}+\frac {x \left (-600 d^5-225 d^4 e-444 d^3 e^2-111 d^2 e^3-296 d e^4+65 e^5\right )}{e^7}+\frac {700 d^6+270 d^5 e+555 d^4 e^2+148 d^3 e^3+444 d^2 e^4-130 d e^5+107 e^6}{e^8}+\frac {-800 d^7-315 d^6 e-666 d^5 e^2-185 d^4 e^3-592 d^3 e^4+195 d^2 e^5-214 d e^6+33 e^7}{e^8 (d+e x)}-\frac {5 x^5 (40 d+9 e)}{e^3}+\frac {100 x^6}{e^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 x^5 \left (100 d^2+30 d e+37 e^2\right )}{5 e^4}-\frac {x^4 \left (400 d^3+135 d^2 e+222 d e^2+37 e^3\right )}{4 e^5}-\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e^9 (d+e x)}+\frac {x^3 \left (500 d^4+180 d^3 e+333 d^2 e^2+74 d e^3+148 e^4\right )}{3 e^6}-\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right ) \log (d+e x)}{e^9}-\frac {x^2 \left (600 d^5+225 d^4 e+444 d^3 e^2+111 d^2 e^3+296 d e^4-65 e^5\right )}{2 e^7}+\frac {x \left (700 d^6+270 d^5 e+555 d^4 e^2+148 d^3 e^3+444 d^2 e^4-130 d e^5+107 e^6\right )}{e^8}-\frac {5 x^6 (40 d+9 e)}{6 e^3}+\frac {100 x^7}{7 e^2}\) |
((700*d^6 + 270*d^5*e + 555*d^4*e^2 + 148*d^3*e^3 + 444*d^2*e^4 - 130*d*e^ 5 + 107*e^6)*x)/e^8 - ((600*d^5 + 225*d^4*e + 444*d^3*e^2 + 111*d^2*e^3 + 296*d*e^4 - 65*e^5)*x^2)/(2*e^7) + ((500*d^4 + 180*d^3*e + 333*d^2*e^2 + 7 4*d*e^3 + 148*e^4)*x^3)/(3*e^6) - ((400*d^3 + 135*d^2*e + 222*d*e^2 + 37*e ^3)*x^4)/(4*e^5) + (3*(100*d^2 + 30*d*e + 37*e^2)*x^5)/(5*e^4) - (5*(40*d + 9*e)*x^6)/(6*e^3) + (100*x^7)/(7*e^2) - ((5*d^2 - 2*d*e + 3*e^2)^2*(4*d^ 4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(e^9*(d + e*x)) - ((5*d^2 - 2*d* e + 3*e^2)*(160*d^5 + 127*d^4*e + 88*d^3*e^2 - 4*d^2*e^3 + 64*d*e^4 - 11*e ^5)*Log[d + e*x])/e^9
3.4.1.3.1 Defintions of rubi rules used
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.56 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.04
| method | result | size |
| norman | \(\frac {\frac {\left (800 d^{8}+315 d^{7} e +666 d^{6} e^{2}+185 d^{5} e^{3}+592 d^{4} e^{4}-195 d^{3} e^{5}+214 d^{2} e^{6}-33 d \,e^{7}+18 e^{8}\right ) x}{e^{8} d}+\frac {100 x^{8}}{7 e}-\frac {5 \left (160 d +63 e \right ) x^{7}}{42 e^{2}}+\frac {\left (800 d^{2}+315 d e +666 e^{2}\right ) x^{6}}{30 e^{3}}-\frac {\left (800 d^{3}+315 d^{2} e +666 d \,e^{2}+185 e^{3}\right ) x^{5}}{20 e^{4}}+\frac {\left (800 d^{4}+315 d^{3} e +666 d^{2} e^{2}+185 d \,e^{3}+592 e^{4}\right ) x^{4}}{12 e^{5}}-\frac {\left (800 d^{5}+315 d^{4} e +666 d^{3} e^{2}+185 d^{2} e^{3}+592 d \,e^{4}-195 e^{5}\right ) x^{3}}{6 e^{6}}+\frac {\left (800 d^{6}+315 d^{5} e +666 d^{4} e^{2}+185 d^{3} e^{3}+592 d^{2} e^{4}-195 d \,e^{5}+214 e^{6}\right ) x^{2}}{2 e^{7}}}{e x +d}-\frac {\left (800 d^{7}+315 d^{6} e +666 d^{5} e^{2}+185 d^{4} e^{3}+592 d^{3} e^{4}-195 d^{2} e^{5}+214 d \,e^{6}-33 e^{7}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(366\) |
| default | \(\frac {\frac {111}{5} e^{6} x^{5}-\frac {15}{2} e^{6} x^{6}+60 d^{3} x^{3} e^{3}+270 d^{5} e x -\frac {225}{2} d^{4} e^{2} x^{2}-\frac {135}{4} d^{2} e^{4} x^{4}+18 d \,e^{5} x^{5}+\frac {100}{7} e^{6} x^{7}-\frac {37}{4} x^{4} e^{6}+\frac {148}{3} x^{3} e^{6}+\frac {65}{2} x^{2} e^{6}+107 x \,e^{6}+700 d^{6} x -\frac {111}{2} d \,e^{5} x^{4}+111 d^{2} e^{4} x^{3}-222 d^{3} e^{3} x^{2}+555 d^{4} e^{2} x -300 d^{5} e \,x^{2}+\frac {500}{3} d^{4} e^{2} x^{3}+\frac {74}{3} x^{3} d \,e^{5}-\frac {111}{2} x^{2} d^{2} e^{4}-148 x^{2} d \,e^{5}+148 x \,d^{3} e^{3}+444 x \,d^{2} e^{4}-130 x d \,e^{5}-100 d^{3} x^{4} e^{3}+60 d^{2} e^{4} x^{5}-\frac {100}{3} d \,e^{5} x^{6}}{e^{8}}-\frac {100 d^{8}+45 d^{7} e +111 d^{6} e^{2}+37 d^{5} e^{3}+148 d^{4} e^{4}-65 d^{3} e^{5}+107 d^{2} e^{6}-33 d \,e^{7}+18 e^{8}}{e^{9} \left (e x +d \right )}+\frac {\left (-800 d^{7}-315 d^{6} e -666 d^{5} e^{2}-185 d^{4} e^{3}-592 d^{3} e^{4}+195 d^{2} e^{5}-214 d \,e^{6}+33 e^{7}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(401\) |
| risch | \(-\frac {111 d^{6}}{e^{7} \left (e x +d \right )}-\frac {37 d^{5}}{e^{6} \left (e x +d \right )}-\frac {148 d^{4}}{e^{5} \left (e x +d \right )}+\frac {65 d^{3}}{e^{4} \left (e x +d \right )}-\frac {107 d^{2}}{e^{3} \left (e x +d \right )}+\frac {33 d}{e^{2} \left (e x +d \right )}-\frac {15 x^{6}}{2 e^{2}}+\frac {60 d^{3} x^{3}}{e^{5}}+\frac {270 d^{5} x}{e^{7}}-\frac {225 d^{4} x^{2}}{2 e^{6}}-\frac {135 d^{2} x^{4}}{4 e^{4}}+\frac {18 d \,x^{5}}{e^{3}}+\frac {700 d^{6} x}{e^{8}}-\frac {111 d \,x^{4}}{2 e^{3}}+\frac {111 d^{2} x^{3}}{e^{4}}-\frac {222 d^{3} x^{2}}{e^{5}}+\frac {555 d^{4} x}{e^{6}}-\frac {300 d^{5} x^{2}}{e^{7}}+\frac {500 d^{4} x^{3}}{3 e^{6}}+\frac {74 x^{3} d}{3 e^{3}}-\frac {37 x^{4}}{4 e^{2}}+\frac {148 x^{3}}{3 e^{2}}+\frac {65 x^{2}}{2 e^{2}}-\frac {18}{e \left (e x +d \right )}+\frac {33 \ln \left (e x +d \right )}{e^{2}}-\frac {111 x^{2} d^{2}}{2 e^{4}}-\frac {148 x^{2} d}{e^{3}}+\frac {148 x \,d^{3}}{e^{5}}+\frac {444 x \,d^{2}}{e^{4}}-\frac {130 x d}{e^{3}}-\frac {100 d^{3} x^{4}}{e^{5}}+\frac {60 d^{2} x^{5}}{e^{4}}-\frac {100 d \,x^{6}}{3 e^{3}}-\frac {100 d^{8}}{e^{9} \left (e x +d \right )}-\frac {45 d^{7}}{e^{8} \left (e x +d \right )}-\frac {800 \ln \left (e x +d \right ) d^{7}}{e^{9}}-\frac {315 \ln \left (e x +d \right ) d^{6}}{e^{8}}-\frac {666 \ln \left (e x +d \right ) d^{5}}{e^{7}}-\frac {185 \ln \left (e x +d \right ) d^{4}}{e^{6}}-\frac {592 \ln \left (e x +d \right ) d^{3}}{e^{5}}+\frac {195 \ln \left (e x +d \right ) d^{2}}{e^{4}}-\frac {214 \ln \left (e x +d \right ) d}{e^{3}}+\frac {111 x^{5}}{5 e^{2}}+\frac {100 x^{7}}{7 e^{2}}+\frac {107 x}{e^{2}}\) | \(500\) |
| parallelrisch | \(-\frac {336000 \ln \left (e x +d \right ) d^{8}-9324 x^{6} e^{8}+3885 x^{5} e^{8}-20720 x^{4} e^{8}-13650 x^{3} e^{8}-44940 x^{2} e^{8}-13860 d \,e^{7}-81900 d^{3} e^{5}+89880 d^{2} e^{6}+77700 d^{5} e^{3}+248640 d^{4} e^{4}+279720 d^{6} e^{2}-23310 x^{4} d^{2} e^{6}+7560 e^{8}+336000 d^{8}+77700 \ln \left (e x +d \right ) x \,d^{4} e^{4}+248640 \ln \left (e x +d \right ) x \,d^{3} e^{5}-81900 \ln \left (e x +d \right ) x \,d^{2} e^{6}+89880 \ln \left (e x +d \right ) x d \,e^{7}+336000 \ln \left (e x +d \right ) x \,d^{7} e +132300 \ln \left (e x +d \right ) x \,d^{6} e^{2}+279720 \ln \left (e x +d \right ) x \,d^{5} e^{3}-168000 d^{6} e^{2} x^{2}+56000 d^{5} e^{3} x^{3}+16800 d^{3} e^{5} x^{5}-11200 d^{2} e^{6} x^{6}+8000 d \,e^{7} x^{7}+46620 x^{3} d^{3} e^{5}+12950 x^{3} d^{2} e^{6}+41440 x^{3} d \,e^{7}-139860 x^{2} d^{4} e^{4}-38850 x^{2} d^{3} e^{5}+3150 e^{8} x^{7}+132300 d^{7} e -4410 d \,e^{7} x^{6}+6615 d^{2} e^{6} x^{5}-11025 d^{3} e^{5} x^{4}+22050 d^{4} e^{4} x^{3}-66150 d^{5} e^{3} x^{2}-6000 e^{8} x^{8}-124320 x^{2} d^{2} e^{6}+40950 x^{2} d \,e^{7}+13986 x^{5} d \,e^{7}-13860 \ln \left (e x +d \right ) x \,e^{8}+132300 \ln \left (e x +d \right ) d^{7} e +279720 \ln \left (e x +d \right ) d^{6} e^{2}+77700 \ln \left (e x +d \right ) d^{5} e^{3}+248640 \ln \left (e x +d \right ) d^{4} e^{4}-81900 \ln \left (e x +d \right ) d^{3} e^{5}+89880 \ln \left (e x +d \right ) d^{2} e^{6}-13860 \ln \left (e x +d \right ) d \,e^{7}-6475 x^{4} d \,e^{7}-28000 d^{4} x^{4} e^{4}}{420 e^{9} \left (e x +d \right )}\) | \(569\) |
((800*d^8+315*d^7*e+666*d^6*e^2+185*d^5*e^3+592*d^4*e^4-195*d^3*e^5+214*d^ 2*e^6-33*d*e^7+18*e^8)/e^8/d*x+100/7*x^8/e-5/42*(160*d+63*e)/e^2*x^7+1/30* (800*d^2+315*d*e+666*e^2)/e^3*x^6-1/20*(800*d^3+315*d^2*e+666*d*e^2+185*e^ 3)/e^4*x^5+1/12*(800*d^4+315*d^3*e+666*d^2*e^2+185*d*e^3+592*e^4)/e^5*x^4- 1/6*(800*d^5+315*d^4*e+666*d^3*e^2+185*d^2*e^3+592*d*e^4-195*e^5)/e^6*x^3+ 1/2*(800*d^6+315*d^5*e+666*d^4*e^2+185*d^3*e^3+592*d^2*e^4-195*d*e^5+214*e ^6)/e^7*x^2)/(e*x+d)-(800*d^7+315*d^6*e+666*d^5*e^2+185*d^4*e^3+592*d^3*e^ 4-195*d^2*e^5+214*d*e^6-33*e^7)/e^9*ln(e*x+d)
Time = 0.24 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.39 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^2} \, dx=\frac {6000 \, e^{8} x^{8} - 42000 \, d^{8} - 18900 \, d^{7} e - 46620 \, d^{6} e^{2} - 15540 \, d^{5} e^{3} - 62160 \, d^{4} e^{4} + 27300 \, d^{3} e^{5} - 44940 \, d^{2} e^{6} + 13860 \, d e^{7} - 7560 \, e^{8} - 50 \, {\left (160 \, d e^{7} + 63 \, e^{8}\right )} x^{7} + 14 \, {\left (800 \, d^{2} e^{6} + 315 \, d e^{7} + 666 \, e^{8}\right )} x^{6} - 21 \, {\left (800 \, d^{3} e^{5} + 315 \, d^{2} e^{6} + 666 \, d e^{7} + 185 \, e^{8}\right )} x^{5} + 35 \, {\left (800 \, d^{4} e^{4} + 315 \, d^{3} e^{5} + 666 \, d^{2} e^{6} + 185 \, d e^{7} + 592 \, e^{8}\right )} x^{4} - 70 \, {\left (800 \, d^{5} e^{3} + 315 \, d^{4} e^{4} + 666 \, d^{3} e^{5} + 185 \, d^{2} e^{6} + 592 \, d e^{7} - 195 \, e^{8}\right )} x^{3} + 210 \, {\left (800 \, d^{6} e^{2} + 315 \, d^{5} e^{3} + 666 \, d^{4} e^{4} + 185 \, d^{3} e^{5} + 592 \, d^{2} e^{6} - 195 \, d e^{7} + 214 \, e^{8}\right )} x^{2} + 420 \, {\left (700 \, d^{7} e + 270 \, d^{6} e^{2} + 555 \, d^{5} e^{3} + 148 \, d^{4} e^{4} + 444 \, d^{3} e^{5} - 130 \, d^{2} e^{6} + 107 \, d e^{7}\right )} x - 420 \, {\left (800 \, d^{8} + 315 \, d^{7} e + 666 \, d^{6} e^{2} + 185 \, d^{5} e^{3} + 592 \, d^{4} e^{4} - 195 \, d^{3} e^{5} + 214 \, d^{2} e^{6} - 33 \, d e^{7} + {\left (800 \, d^{7} e + 315 \, d^{6} e^{2} + 666 \, d^{5} e^{3} + 185 \, d^{4} e^{4} + 592 \, d^{3} e^{5} - 195 \, d^{2} e^{6} + 214 \, d e^{7} - 33 \, e^{8}\right )} x\right )} \log \left (e x + d\right )}{420 \, {\left (e^{10} x + d e^{9}\right )}} \]
1/420*(6000*e^8*x^8 - 42000*d^8 - 18900*d^7*e - 46620*d^6*e^2 - 15540*d^5* e^3 - 62160*d^4*e^4 + 27300*d^3*e^5 - 44940*d^2*e^6 + 13860*d*e^7 - 7560*e ^8 - 50*(160*d*e^7 + 63*e^8)*x^7 + 14*(800*d^2*e^6 + 315*d*e^7 + 666*e^8)* x^6 - 21*(800*d^3*e^5 + 315*d^2*e^6 + 666*d*e^7 + 185*e^8)*x^5 + 35*(800*d ^4*e^4 + 315*d^3*e^5 + 666*d^2*e^6 + 185*d*e^7 + 592*e^8)*x^4 - 70*(800*d^ 5*e^3 + 315*d^4*e^4 + 666*d^3*e^5 + 185*d^2*e^6 + 592*d*e^7 - 195*e^8)*x^3 + 210*(800*d^6*e^2 + 315*d^5*e^3 + 666*d^4*e^4 + 185*d^3*e^5 + 592*d^2*e^ 6 - 195*d*e^7 + 214*e^8)*x^2 + 420*(700*d^7*e + 270*d^6*e^2 + 555*d^5*e^3 + 148*d^4*e^4 + 444*d^3*e^5 - 130*d^2*e^6 + 107*d*e^7)*x - 420*(800*d^8 + 315*d^7*e + 666*d^6*e^2 + 185*d^5*e^3 + 592*d^4*e^4 - 195*d^3*e^5 + 214*d^ 2*e^6 - 33*d*e^7 + (800*d^7*e + 315*d^6*e^2 + 666*d^5*e^3 + 185*d^4*e^4 + 592*d^3*e^5 - 195*d^2*e^6 + 214*d*e^7 - 33*e^8)*x)*log(e*x + d))/(e^10*x + d*e^9)
Time = 0.70 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.11 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^2} \, dx=x^{6} \left (- \frac {100 d}{3 e^{3}} - \frac {15}{2 e^{2}}\right ) + x^{5} \cdot \left (\frac {60 d^{2}}{e^{4}} + \frac {18 d}{e^{3}} + \frac {111}{5 e^{2}}\right ) + x^{4} \left (- \frac {100 d^{3}}{e^{5}} - \frac {135 d^{2}}{4 e^{4}} - \frac {111 d}{2 e^{3}} - \frac {37}{4 e^{2}}\right ) + x^{3} \cdot \left (\frac {500 d^{4}}{3 e^{6}} + \frac {60 d^{3}}{e^{5}} + \frac {111 d^{2}}{e^{4}} + \frac {74 d}{3 e^{3}} + \frac {148}{3 e^{2}}\right ) + x^{2} \left (- \frac {300 d^{5}}{e^{7}} - \frac {225 d^{4}}{2 e^{6}} - \frac {222 d^{3}}{e^{5}} - \frac {111 d^{2}}{2 e^{4}} - \frac {148 d}{e^{3}} + \frac {65}{2 e^{2}}\right ) + x \left (\frac {700 d^{6}}{e^{8}} + \frac {270 d^{5}}{e^{7}} + \frac {555 d^{4}}{e^{6}} + \frac {148 d^{3}}{e^{5}} + \frac {444 d^{2}}{e^{4}} - \frac {130 d}{e^{3}} + \frac {107}{e^{2}}\right ) + \frac {- 100 d^{8} - 45 d^{7} e - 111 d^{6} e^{2} - 37 d^{5} e^{3} - 148 d^{4} e^{4} + 65 d^{3} e^{5} - 107 d^{2} e^{6} + 33 d e^{7} - 18 e^{8}}{d e^{9} + e^{10} x} + \frac {100 x^{7}}{7 e^{2}} - \frac {\left (5 d^{2} - 2 d e + 3 e^{2}\right ) \left (160 d^{5} + 127 d^{4} e + 88 d^{3} e^{2} - 4 d^{2} e^{3} + 64 d e^{4} - 11 e^{5}\right ) \log {\left (d + e x \right )}}{e^{9}} \]
x**6*(-100*d/(3*e**3) - 15/(2*e**2)) + x**5*(60*d**2/e**4 + 18*d/e**3 + 11 1/(5*e**2)) + x**4*(-100*d**3/e**5 - 135*d**2/(4*e**4) - 111*d/(2*e**3) - 37/(4*e**2)) + x**3*(500*d**4/(3*e**6) + 60*d**3/e**5 + 111*d**2/e**4 + 74 *d/(3*e**3) + 148/(3*e**2)) + x**2*(-300*d**5/e**7 - 225*d**4/(2*e**6) - 2 22*d**3/e**5 - 111*d**2/(2*e**4) - 148*d/e**3 + 65/(2*e**2)) + x*(700*d**6 /e**8 + 270*d**5/e**7 + 555*d**4/e**6 + 148*d**3/e**5 + 444*d**2/e**4 - 13 0*d/e**3 + 107/e**2) + (-100*d**8 - 45*d**7*e - 111*d**6*e**2 - 37*d**5*e* *3 - 148*d**4*e**4 + 65*d**3*e**5 - 107*d**2*e**6 + 33*d*e**7 - 18*e**8)/( d*e**9 + e**10*x) + 100*x**7/(7*e**2) - (5*d**2 - 2*d*e + 3*e**2)*(160*d** 5 + 127*d**4*e + 88*d**3*e**2 - 4*d**2*e**3 + 64*d*e**4 - 11*e**5)*log(d + e*x)/e**9
Time = 0.19 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.05 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^2} \, dx=-\frac {100 \, d^{8} + 45 \, d^{7} e + 111 \, d^{6} e^{2} + 37 \, d^{5} e^{3} + 148 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + 107 \, d^{2} e^{6} - 33 \, d e^{7} + 18 \, e^{8}}{e^{10} x + d e^{9}} + \frac {6000 \, e^{6} x^{7} - 350 \, {\left (40 \, d e^{5} + 9 \, e^{6}\right )} x^{6} + 252 \, {\left (100 \, d^{2} e^{4} + 30 \, d e^{5} + 37 \, e^{6}\right )} x^{5} - 105 \, {\left (400 \, d^{3} e^{3} + 135 \, d^{2} e^{4} + 222 \, d e^{5} + 37 \, e^{6}\right )} x^{4} + 140 \, {\left (500 \, d^{4} e^{2} + 180 \, d^{3} e^{3} + 333 \, d^{2} e^{4} + 74 \, d e^{5} + 148 \, e^{6}\right )} x^{3} - 210 \, {\left (600 \, d^{5} e + 225 \, d^{4} e^{2} + 444 \, d^{3} e^{3} + 111 \, d^{2} e^{4} + 296 \, d e^{5} - 65 \, e^{6}\right )} x^{2} + 420 \, {\left (700 \, d^{6} + 270 \, d^{5} e + 555 \, d^{4} e^{2} + 148 \, d^{3} e^{3} + 444 \, d^{2} e^{4} - 130 \, d e^{5} + 107 \, e^{6}\right )} x}{420 \, e^{8}} - \frac {{\left (800 \, d^{7} + 315 \, d^{6} e + 666 \, d^{5} e^{2} + 185 \, d^{4} e^{3} + 592 \, d^{3} e^{4} - 195 \, d^{2} e^{5} + 214 \, d e^{6} - 33 \, e^{7}\right )} \log \left (e x + d\right )}{e^{9}} \]
-(100*d^8 + 45*d^7*e + 111*d^6*e^2 + 37*d^5*e^3 + 148*d^4*e^4 - 65*d^3*e^5 + 107*d^2*e^6 - 33*d*e^7 + 18*e^8)/(e^10*x + d*e^9) + 1/420*(6000*e^6*x^7 - 350*(40*d*e^5 + 9*e^6)*x^6 + 252*(100*d^2*e^4 + 30*d*e^5 + 37*e^6)*x^5 - 105*(400*d^3*e^3 + 135*d^2*e^4 + 222*d*e^5 + 37*e^6)*x^4 + 140*(500*d^4* e^2 + 180*d^3*e^3 + 333*d^2*e^4 + 74*d*e^5 + 148*e^6)*x^3 - 210*(600*d^5*e + 225*d^4*e^2 + 444*d^3*e^3 + 111*d^2*e^4 + 296*d*e^5 - 65*e^6)*x^2 + 420 *(700*d^6 + 270*d^5*e + 555*d^4*e^2 + 148*d^3*e^3 + 444*d^2*e^4 - 130*d*e^ 5 + 107*e^6)*x)/e^8 - (800*d^7 + 315*d^6*e + 666*d^5*e^2 + 185*d^4*e^3 + 5 92*d^3*e^4 - 195*d^2*e^5 + 214*d*e^6 - 33*e^7)*log(e*x + d)/e^9
Time = 0.27 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.39 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^2} \, dx=-\frac {{\left (e x + d\right )}^{7} {\left (\frac {350 \, {\left (160 \, d e + 9 \, e^{2}\right )}}{{\left (e x + d\right )} e} - \frac {84 \, {\left (2800 \, d^{2} e^{2} + 315 \, d e^{3} + 111 \, e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} + \frac {105 \, {\left (5600 \, d^{3} e^{3} + 945 \, d^{2} e^{4} + 666 \, d e^{5} + 37 \, e^{6}\right )}}{{\left (e x + d\right )}^{3} e^{3}} - \frac {140 \, {\left (7000 \, d^{4} e^{4} + 1575 \, d^{3} e^{5} + 1665 \, d^{2} e^{6} + 185 \, d e^{7} + 148 \, e^{8}\right )}}{{\left (e x + d\right )}^{4} e^{4}} + \frac {210 \, {\left (5600 \, d^{5} e^{5} + 1575 \, d^{4} e^{6} + 2220 \, d^{3} e^{7} + 370 \, d^{2} e^{8} + 592 \, d e^{9} - 65 \, e^{10}\right )}}{{\left (e x + d\right )}^{5} e^{5}} - \frac {420 \, {\left (2800 \, d^{6} e^{6} + 945 \, d^{5} e^{7} + 1665 \, d^{4} e^{8} + 370 \, d^{3} e^{9} + 888 \, d^{2} e^{10} - 195 \, d e^{11} + 107 \, e^{12}\right )}}{{\left (e x + d\right )}^{6} e^{6}} - 6000\right )}}{420 \, e^{9}} + \frac {{\left (800 \, d^{7} + 315 \, d^{6} e + 666 \, d^{5} e^{2} + 185 \, d^{4} e^{3} + 592 \, d^{3} e^{4} - 195 \, d^{2} e^{5} + 214 \, d e^{6} - 33 \, e^{7}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{9}} - \frac {\frac {100 \, d^{8} e^{7}}{e x + d} + \frac {45 \, d^{7} e^{8}}{e x + d} + \frac {111 \, d^{6} e^{9}}{e x + d} + \frac {37 \, d^{5} e^{10}}{e x + d} + \frac {148 \, d^{4} e^{11}}{e x + d} - \frac {65 \, d^{3} e^{12}}{e x + d} + \frac {107 \, d^{2} e^{13}}{e x + d} - \frac {33 \, d e^{14}}{e x + d} + \frac {18 \, e^{15}}{e x + d}}{e^{16}} \]
-1/420*(e*x + d)^7*(350*(160*d*e + 9*e^2)/((e*x + d)*e) - 84*(2800*d^2*e^2 + 315*d*e^3 + 111*e^4)/((e*x + d)^2*e^2) + 105*(5600*d^3*e^3 + 945*d^2*e^ 4 + 666*d*e^5 + 37*e^6)/((e*x + d)^3*e^3) - 140*(7000*d^4*e^4 + 1575*d^3*e ^5 + 1665*d^2*e^6 + 185*d*e^7 + 148*e^8)/((e*x + d)^4*e^4) + 210*(5600*d^5 *e^5 + 1575*d^4*e^6 + 2220*d^3*e^7 + 370*d^2*e^8 + 592*d*e^9 - 65*e^10)/(( e*x + d)^5*e^5) - 420*(2800*d^6*e^6 + 945*d^5*e^7 + 1665*d^4*e^8 + 370*d^3 *e^9 + 888*d^2*e^10 - 195*d*e^11 + 107*e^12)/((e*x + d)^6*e^6) - 6000)/e^9 + (800*d^7 + 315*d^6*e + 666*d^5*e^2 + 185*d^4*e^3 + 592*d^3*e^4 - 195*d^ 2*e^5 + 214*d*e^6 - 33*e^7)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^9 - ( 100*d^8*e^7/(e*x + d) + 45*d^7*e^8/(e*x + d) + 111*d^6*e^9/(e*x + d) + 37* d^5*e^10/(e*x + d) + 148*d^4*e^11/(e*x + d) - 65*d^3*e^12/(e*x + d) + 107* d^2*e^13/(e*x + d) - 33*d*e^14/(e*x + d) + 18*e^15/(e*x + d))/e^16
Time = 13.41 (sec) , antiderivative size = 939, normalized size of antiderivative = 2.66 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^2} \, dx=x^2\,\left (\frac {65}{2\,e^2}-\frac {d\,\left (\frac {148}{e^2}+\frac {2\,d\,\left (\frac {37}{e^2}+\frac {2\,d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e^2}\right )}{e}+\frac {d^2\,\left (\frac {37}{e^2}+\frac {2\,d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e^2}\right )}{2\,e^2}\right )+x^3\,\left (\frac {148}{3\,e^2}+\frac {2\,d\,\left (\frac {37}{e^2}+\frac {2\,d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e^2}\right )}{3\,e}-\frac {d^2\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{3\,e^2}\right )-x^4\,\left (\frac {37}{4\,e^2}+\frac {d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{2\,e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{4\,e^2}\right )+x^5\,\left (\frac {111}{5\,e^2}-\frac {20\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{5\,e}\right )-x^6\,\left (\frac {100\,d}{3\,e^3}+\frac {15}{2\,e^2}\right )-x\,\left (\frac {2\,d\,\left (\frac {65}{e^2}-\frac {2\,d\,\left (\frac {148}{e^2}+\frac {2\,d\,\left (\frac {37}{e^2}+\frac {2\,d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e^2}\right )}{e}+\frac {d^2\,\left (\frac {37}{e^2}+\frac {2\,d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e^2}\right )}{e^2}\right )}{e}-\frac {107}{e^2}+\frac {d^2\,\left (\frac {148}{e^2}+\frac {2\,d\,\left (\frac {37}{e^2}+\frac {2\,d\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {111}{e^2}-\frac {100\,d^2}{e^4}+\frac {2\,d\,\left (\frac {200\,d}{e^3}+\frac {45}{e^2}\right )}{e}\right )}{e^2}\right )}{e^2}\right )+\frac {100\,x^7}{7\,e^2}-\frac {100\,d^8+45\,d^7\,e+111\,d^6\,e^2+37\,d^5\,e^3+148\,d^4\,e^4-65\,d^3\,e^5+107\,d^2\,e^6-33\,d\,e^7+18\,e^8}{e\,\left (x\,e^9+d\,e^8\right )}-\frac {\ln \left (d+e\,x\right )\,\left (800\,d^7+315\,d^6\,e+666\,d^5\,e^2+185\,d^4\,e^3+592\,d^3\,e^4-195\,d^2\,e^5+214\,d\,e^6-33\,e^7\right )}{e^9} \]
x^2*(65/(2*e^2) - (d*(148/e^2 + (2*d*(37/e^2 + (2*d*(111/e^2 - (100*d^2)/e ^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/e - (d^2*((200*d)/e^3 + 45/e^2))/e^2 ))/e - (d^2*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/e^ 2))/e + (d^2*(37/e^2 + (2*d*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/e - (d^2*((200*d)/e^3 + 45/e^2))/e^2))/(2*e^2)) + x^3*(148/( 3*e^2) + (2*d*(37/e^2 + (2*d*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/e - (d^2*((200*d)/e^3 + 45/e^2))/e^2))/(3*e) - (d^2*(111/e^ 2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/(3*e^2)) - x^4*(37/(4 *e^2) + (d*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/(2* e) - (d^2*((200*d)/e^3 + 45/e^2))/(4*e^2)) + x^5*(111/(5*e^2) - (20*d^2)/e ^4 + (2*d*((200*d)/e^3 + 45/e^2))/(5*e)) - x^6*((100*d)/(3*e^3) + 15/(2*e^ 2)) - x*((2*d*(65/e^2 - (2*d*(148/e^2 + (2*d*(37/e^2 + (2*d*(111/e^2 - (10 0*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/e - (d^2*((200*d)/e^3 + 45/e ^2))/e^2))/e - (d^2*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2) )/e))/e^2))/e + (d^2*(37/e^2 + (2*d*(111/e^2 - (100*d^2)/e^4 + (2*d*((200* d)/e^3 + 45/e^2))/e))/e - (d^2*((200*d)/e^3 + 45/e^2))/e^2))/e^2))/e - 107 /e^2 + (d^2*(148/e^2 + (2*d*(37/e^2 + (2*d*(111/e^2 - (100*d^2)/e^4 + (2*d *((200*d)/e^3 + 45/e^2))/e))/e - (d^2*((200*d)/e^3 + 45/e^2))/e^2))/e - (d ^2*(111/e^2 - (100*d^2)/e^4 + (2*d*((200*d)/e^3 + 45/e^2))/e))/e^2))/e^2) + (100*x^7)/(7*e^2) - (45*d^7*e - 33*d*e^7 + 100*d^8 + 18*e^8 + 107*d^2...