Integrand size = 38, antiderivative size = 352 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{d+e x} \, dx=-\frac {\left (100 d^7+45 d^6 e+111 d^5 e^2+37 d^4 e^3+148 d^3 e^4-65 d^2 e^5+107 d e^6-33 e^7\right ) x}{e^8}+\frac {\left (100 d^6+45 d^5 e+111 d^4 e^2+37 d^3 e^3+148 d^2 e^4-65 d e^5+107 e^6\right ) x^2}{2 e^7}-\frac {\left (100 d^5+45 d^4 e+111 d^3 e^2+37 d^2 e^3+148 d e^4-65 e^5\right ) x^3}{3 e^6}+\frac {\left (100 d^4+45 d^3 e+111 d^2 e^2+37 d e^3+148 e^4\right ) x^4}{4 e^5}-\frac {\left (100 d^3+45 d^2 e+111 d e^2+37 e^3\right ) x^5}{5 e^4}+\frac {\left (100 d^2+45 d e+111 e^2\right ) x^6}{6 e^3}-\frac {5 (20 d+9 e) x^7}{7 e^2}+\frac {25 x^8}{2 e}+\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^9} \] Output:
-(100*d^7+45*d^6*e+111*d^5*e^2+37*d^4*e^3+148*d^3*e^4-65*d^2*e^5+107*d*e^6 -33*e^7)*x/e^8+1/2*(100*d^6+45*d^5*e+111*d^4*e^2+37*d^3*e^3+148*d^2*e^4-65 *d*e^5+107*e^6)*x^2/e^7-1/3*(100*d^5+45*d^4*e+111*d^3*e^2+37*d^2*e^3+148*d *e^4-65*e^5)*x^3/e^6+1/4*(100*d^4+45*d^3*e+111*d^2*e^2+37*d*e^3+148*e^4)*x ^4/e^5-1/5*(100*d^3+45*d^2*e+111*d*e^2+37*e^3)*x^5/e^4+1/6*(100*d^2+45*d*e +111*e^2)*x^6/e^3-5/7*(20*d+9*e)*x^7/e^2+25/2*x^8/e+(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)*ln(e*x+d)/e^9
Time = 0.13 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.74 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{d+e x} \, dx=\frac {x \left (-42000 d^7+2100 d^6 e (-9+10 x)-70 d^5 e^2 \left (666-135 x+200 x^2\right )+210 d^4 e^3 \left (-74+111 x-30 x^2+50 x^3\right )-105 d^3 e^4 \left (592-74 x+148 x^2-45 x^3+80 x^4\right )+35 d^2 e^5 \left (780+888 x-148 x^2+333 x^3-108 x^4+200 x^5\right )-d e^6 \left (44940+13650 x+20720 x^2-3885 x^3+9324 x^4-3150 x^5+6000 x^6\right )+2 e^7 \left (6930+11235 x+4550 x^2+7770 x^3-1554 x^4+3885 x^5-1350 x^6+2625 x^7\right )\right )}{420 e^8}+\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^9} \] Input:
Integrate[((3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x), x]
Output:
(x*(-42000*d^7 + 2100*d^6*e*(-9 + 10*x) - 70*d^5*e^2*(666 - 135*x + 200*x^ 2) + 210*d^4*e^3*(-74 + 111*x - 30*x^2 + 50*x^3) - 105*d^3*e^4*(592 - 74*x + 148*x^2 - 45*x^3 + 80*x^4) + 35*d^2*e^5*(780 + 888*x - 148*x^2 + 333*x^ 3 - 108*x^4 + 200*x^5) - d*e^6*(44940 + 13650*x + 20720*x^2 - 3885*x^3 + 9 324*x^4 - 3150*x^5 + 6000*x^6) + 2*e^7*(6930 + 11235*x + 4550*x^2 + 7770*x ^3 - 1554*x^4 + 3885*x^5 - 1350*x^6 + 2625*x^7)))/(420*e^8) + ((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)*Log[d + e*x]) /e^9
Time = 1.10 (sec) , antiderivative size = 352, 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.
| \(\displaystyle \int \frac {\left (5 x^2+2 x+3\right )^2 \left (4 x^4-5 x^3+3 x^2+x+2\right )}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (\frac {x^5 \left (100 d^2+45 d e+111 e^2\right )}{e^3}-\frac {x^4 \left (100 d^3+45 d^2 e+111 d e^2+37 e^3\right )}{e^4}+\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e^8 (d+e x)}+\frac {x^3 \left (100 d^4+45 d^3 e+111 d^2 e^2+37 d e^3+148 e^4\right )}{e^5}+\frac {x^2 \left (-100 d^5-45 d^4 e-111 d^3 e^2-37 d^2 e^3-148 d e^4+65 e^5\right )}{e^6}+\frac {x \left (100 d^6+45 d^5 e+111 d^4 e^2+37 d^3 e^3+148 d^2 e^4-65 d e^5+107 e^6\right )}{e^7}+\frac {-100 d^7-45 d^6 e-111 d^5 e^2-37 d^4 e^3-148 d^3 e^4+65 d^2 e^5-107 d e^6+33 e^7}{e^8}-\frac {5 x^6 (20 d+9 e)}{e^2}+\frac {100 x^7}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^6 \left (100 d^2+45 d e+111 e^2\right )}{6 e^3}-\frac {x^5 \left (100 d^3+45 d^2 e+111 d e^2+37 e^3\right )}{5 e^4}+\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^9}+\frac {x^4 \left (100 d^4+45 d^3 e+111 d^2 e^2+37 d e^3+148 e^4\right )}{4 e^5}-\frac {x^3 \left (100 d^5+45 d^4 e+111 d^3 e^2+37 d^2 e^3+148 d e^4-65 e^5\right )}{3 e^6}+\frac {x^2 \left (100 d^6+45 d^5 e+111 d^4 e^2+37 d^3 e^3+148 d^2 e^4-65 d e^5+107 e^6\right )}{2 e^7}-\frac {x \left (100 d^7+45 d^6 e+111 d^5 e^2+37 d^4 e^3+148 d^3 e^4-65 d^2 e^5+107 d e^6-33 e^7\right )}{e^8}-\frac {5 x^7 (20 d+9 e)}{7 e^2}+\frac {25 x^8}{2 e}\) |
Input:
Int[((3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x),x]
Output:
-(((100*d^7 + 45*d^6*e + 111*d^5*e^2 + 37*d^4*e^3 + 148*d^3*e^4 - 65*d^2*e ^5 + 107*d*e^6 - 33*e^7)*x)/e^8) + ((100*d^6 + 45*d^5*e + 111*d^4*e^2 + 37 *d^3*e^3 + 148*d^2*e^4 - 65*d*e^5 + 107*e^6)*x^2)/(2*e^7) - ((100*d^5 + 45 *d^4*e + 111*d^3*e^2 + 37*d^2*e^3 + 148*d*e^4 - 65*e^5)*x^3)/(3*e^6) + ((1 00*d^4 + 45*d^3*e + 111*d^2*e^2 + 37*d*e^3 + 148*e^4)*x^4)/(4*e^5) - ((100 *d^3 + 45*d^2*e + 111*d*e^2 + 37*e^3)*x^5)/(5*e^4) + ((100*d^2 + 45*d*e + 111*e^2)*x^6)/(6*e^3) - (5*(20*d + 9*e)*x^7)/(7*e^2) + (25*x^8)/(2*e) + (( 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)*Log [d + e*x])/e^9
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 = 0.10 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.01
| method | result | size |
| norman | \(\frac {25 x^{8}}{2 e}-\frac {5 \left (20 d +9 e \right ) x^{7}}{7 e^{2}}+\frac {\left (100 d^{2}+45 d e +111 e^{2}\right ) x^{6}}{6 e^{3}}-\frac {\left (100 d^{3}+45 d^{2} e +111 d \,e^{2}+37 e^{3}\right ) x^{5}}{5 e^{4}}+\frac {\left (100 d^{4}+45 d^{3} e +111 d^{2} e^{2}+37 d \,e^{3}+148 e^{4}\right ) x^{4}}{4 e^{5}}-\frac {\left (100 d^{5}+45 d^{4} e +111 d^{3} e^{2}+37 d^{2} e^{3}+148 d \,e^{4}-65 e^{5}\right ) x^{3}}{3 e^{6}}+\frac {\left (100 d^{6}+45 d^{5} e +111 d^{4} e^{2}+37 d^{3} e^{3}+148 d^{2} e^{4}-65 d \,e^{5}+107 e^{6}\right ) x^{2}}{2 e^{7}}-\frac {\left (100 d^{7}+45 d^{6} e +111 d^{5} e^{2}+37 d^{4} e^{3}+148 d^{3} e^{4}-65 d^{2} e^{5}+107 d \,e^{6}-33 e^{7}\right ) x}{e^{8}}+\frac {\left (100 d^{8}+45 d^{7} e +111 d^{6} e^{2}+37 d^{5} e^{3}+148 d^{4} e^{4}-65 d^{3} e^{5}+107 d^{2} e^{6}-33 d \,e^{7}+18 e^{8}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(354\) |
| default | \(-\frac {-33 e^{7} x +100 x \,d^{7}-\frac {107}{2} e^{7} x^{2}-\frac {65}{3} e^{7} x^{3}+\frac {37}{5} e^{7} x^{5}-37 e^{7} x^{4}-\frac {25}{2} x^{8} e^{7}-\frac {37}{2} x^{6} e^{7}+107 d \,e^{6} x +37 d^{4} e^{3} x +148 d^{3} e^{4} x -65 d^{2} e^{5} x -50 d^{6} e \,x^{2}+111 d^{5} e^{2} x +\frac {65}{2} d \,e^{6} x^{2}+45 d^{6} e x -74 d^{2} e^{5} x^{2}-\frac {45}{2} d^{5} e^{2} x^{2}+\frac {100}{3} d^{5} e^{2} x^{3}-\frac {111}{2} d^{4} e^{3} x^{2}-\frac {37}{2} d^{3} e^{4} x^{2}+\frac {37}{3} d^{2} e^{5} x^{3}+\frac {148}{3} d \,e^{6} x^{3}+20 d^{3} e^{4} x^{5}-\frac {111}{4} d^{2} e^{5} x^{4}-\frac {37}{4} d \,e^{6} x^{4}+15 d^{4} e^{3} x^{3}-25 d^{4} e^{3} x^{4}+37 d^{3} e^{4} x^{3}-\frac {45}{4} d^{3} e^{4} x^{4}+\frac {111}{5} d \,e^{6} x^{5}+\frac {100}{7} x^{7} e^{6} d +9 d^{2} e^{5} x^{5}-\frac {50}{3} d^{2} e^{5} x^{6}-\frac {15}{2} x^{6} e^{6} d +\frac {45}{7} x^{7} e^{7}}{e^{8}}+\frac {\left (100 d^{8}+45 d^{7} e +111 d^{6} e^{2}+37 d^{5} e^{3}+148 d^{4} e^{4}-65 d^{3} e^{5}+107 d^{2} e^{6}-33 d \,e^{7}+18 e^{8}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(416\) |
| parallelrisch | \(\frac {7560 \ln \left (e x +d \right ) e^{8}+5250 x^{8} e^{8}-2700 x^{7} e^{8}+7770 x^{6} e^{8}-3108 x^{5} e^{8}+15540 x^{4} e^{8}+9100 x^{3} e^{8}+22470 x^{2} e^{8}+13860 x \,e^{8}+42000 \ln \left (e x +d \right ) d^{8}+3885 x^{4} d \,e^{7}-14000 x^{3} d^{5} e^{3}-6300 x^{3} d^{4} e^{4}-15540 x^{3} d^{3} e^{5}-5180 x^{3} d^{2} e^{6}-20720 x^{3} d \,e^{7}+21000 x^{2} d^{6} e^{2}+9450 x^{2} d^{5} e^{3}+23310 x^{2} d^{4} e^{4}+7770 x^{2} d^{3} e^{5}+31080 x^{2} d^{2} e^{6}-13650 x^{2} d \,e^{7}-42000 x \,d^{7} e -18900 x \,d^{6} e^{2}-46620 x \,d^{5} e^{3}-15540 x \,d^{4} e^{4}-62160 x \,d^{3} e^{5}+27300 x \,d^{2} e^{6}-44940 x d \,e^{7}-6000 x^{7} e^{7} d +7000 x^{6} d^{2} e^{6}+3150 x^{6} d \,e^{7}+46620 \ln \left (e x +d \right ) d^{6} e^{2}+15540 \ln \left (e x +d \right ) d^{5} e^{3}+62160 \ln \left (e x +d \right ) d^{4} e^{4}-27300 \ln \left (e x +d \right ) d^{3} e^{5}+44940 \ln \left (e x +d \right ) d^{2} e^{6}-13860 \ln \left (e x +d \right ) d \,e^{7}-8400 x^{5} d^{3} e^{5}-3780 x^{5} d^{2} e^{6}-9324 x^{5} d \,e^{7}+10500 x^{4} d^{4} e^{4}+4725 x^{4} d^{3} e^{5}+11655 x^{4} d^{2} e^{6}+18900 \ln \left (e x +d \right ) d^{7} e}{420 e^{9}}\) | \(463\) |
| risch | \(-\frac {37 x^{5}}{5 e}+\frac {107 x^{2}}{2 e}+\frac {25 x^{8}}{2 e}+\frac {33 x}{e}+\frac {18 \ln \left (e x +d \right )}{e}+\frac {37 x^{6}}{2 e}+\frac {37 x^{4}}{e}+\frac {65 x^{3}}{3 e}-\frac {45 x^{7}}{7 e}-\frac {111 x^{5} d}{5 e^{2}}+\frac {111 d^{2} x^{4}}{4 e^{3}}+\frac {37 x^{4} d}{4 e^{2}}-\frac {37 d^{3} x^{3}}{e^{4}}-\frac {37 d^{2} x^{3}}{3 e^{3}}-\frac {148 d \,x^{3}}{3 e^{2}}+\frac {111 d^{4} x^{2}}{2 e^{5}}+\frac {37 d^{3} x^{2}}{2 e^{4}}+\frac {74 d^{2} x^{2}}{e^{3}}-\frac {65 d \,x^{2}}{2 e^{2}}-\frac {111 x \,d^{5}}{e^{6}}-\frac {37 d^{4} x}{e^{5}}-\frac {148 d^{3} x}{e^{4}}+\frac {65 d^{2} x}{e^{3}}-\frac {107 d x}{e^{2}}+\frac {111 \ln \left (e x +d \right ) d^{6}}{e^{7}}+\frac {37 \ln \left (e x +d \right ) d^{5}}{e^{6}}+\frac {148 \ln \left (e x +d \right ) d^{4}}{e^{5}}-\frac {65 \ln \left (e x +d \right ) d^{3}}{e^{4}}+\frac {107 \ln \left (e x +d \right ) d^{2}}{e^{3}}-\frac {33 \ln \left (e x +d \right ) d}{e^{2}}+\frac {100 \ln \left (e x +d \right ) d^{8}}{e^{9}}+\frac {45 \ln \left (e x +d \right ) d^{7}}{e^{8}}-\frac {100 d^{5} x^{3}}{3 e^{6}}-\frac {20 d^{3} x^{5}}{e^{4}}-\frac {15 d^{4} x^{3}}{e^{5}}+\frac {25 d^{4} x^{4}}{e^{5}}+\frac {45 d^{3} x^{4}}{4 e^{4}}-\frac {100 x^{7} d}{7 e^{2}}-\frac {9 d^{2} x^{5}}{e^{3}}+\frac {50 d^{2} x^{6}}{3 e^{3}}+\frac {15 x^{6} d}{2 e^{2}}-\frac {100 x \,d^{7}}{e^{8}}+\frac {50 d^{6} x^{2}}{e^{7}}-\frac {45 d^{6} x}{e^{7}}+\frac {45 d^{5} x^{2}}{2 e^{6}}\) | \(465\) |
Input:
int((5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d),x,method=_RETURNVERBOS E)
Output:
25/2*x^8/e-5/7*(20*d+9*e)*x^7/e^2+1/6*(100*d^2+45*d*e+111*e^2)*x^6/e^3-1/5 *(100*d^3+45*d^2*e+111*d*e^2+37*e^3)*x^5/e^4+1/4*(100*d^4+45*d^3*e+111*d^2 *e^2+37*d*e^3+148*e^4)*x^4/e^5-1/3*(100*d^5+45*d^4*e+111*d^3*e^2+37*d^2*e^ 3+148*d*e^4-65*e^5)*x^3/e^6+1/2*(100*d^6+45*d^5*e+111*d^4*e^2+37*d^3*e^3+1 48*d^2*e^4-65*d*e^5+107*e^6)*x^2/e^7-(100*d^7+45*d^6*e+111*d^5*e^2+37*d^4* e^3+148*d^3*e^4-65*d^2*e^5+107*d*e^6-33*e^7)*x/e^8+(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* ln(e*x+d)
Time = 0.07 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.05 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{d+e x} \, dx=\frac {5250 \, e^{8} x^{8} - 300 \, {\left (20 \, d e^{7} + 9 \, e^{8}\right )} x^{7} + 70 \, {\left (100 \, d^{2} e^{6} + 45 \, d e^{7} + 111 \, e^{8}\right )} x^{6} - 84 \, {\left (100 \, d^{3} e^{5} + 45 \, d^{2} e^{6} + 111 \, d e^{7} + 37 \, e^{8}\right )} x^{5} + 105 \, {\left (100 \, d^{4} e^{4} + 45 \, d^{3} e^{5} + 111 \, d^{2} e^{6} + 37 \, d e^{7} + 148 \, e^{8}\right )} x^{4} - 140 \, {\left (100 \, d^{5} e^{3} + 45 \, d^{4} e^{4} + 111 \, d^{3} e^{5} + 37 \, d^{2} e^{6} + 148 \, d e^{7} - 65 \, e^{8}\right )} x^{3} + 210 \, {\left (100 \, d^{6} e^{2} + 45 \, d^{5} e^{3} + 111 \, d^{4} e^{4} + 37 \, d^{3} e^{5} + 148 \, d^{2} e^{6} - 65 \, d e^{7} + 107 \, e^{8}\right )} x^{2} - 420 \, {\left (100 \, d^{7} e + 45 \, d^{6} e^{2} + 111 \, d^{5} e^{3} + 37 \, d^{4} e^{4} + 148 \, d^{3} e^{5} - 65 \, d^{2} e^{6} + 107 \, d e^{7} - 33 \, e^{8}\right )} x + 420 \, {\left (100 \, d^{8} + 45 \, d^{7} e + 111 \, d^{6} e^{2} + 37 \, d^{5} e^{3} + 148 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + 107 \, d^{2} e^{6} - 33 \, d e^{7} + 18 \, e^{8}\right )} \log \left (e x + d\right )}{420 \, e^{9}} \] Input:
integrate((5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d),x, algorithm="fr icas")
Output:
1/420*(5250*e^8*x^8 - 300*(20*d*e^7 + 9*e^8)*x^7 + 70*(100*d^2*e^6 + 45*d* e^7 + 111*e^8)*x^6 - 84*(100*d^3*e^5 + 45*d^2*e^6 + 111*d*e^7 + 37*e^8)*x^ 5 + 105*(100*d^4*e^4 + 45*d^3*e^5 + 111*d^2*e^6 + 37*d*e^7 + 148*e^8)*x^4 - 140*(100*d^5*e^3 + 45*d^4*e^4 + 111*d^3*e^5 + 37*d^2*e^6 + 148*d*e^7 - 6 5*e^8)*x^3 + 210*(100*d^6*e^2 + 45*d^5*e^3 + 111*d^4*e^4 + 37*d^3*e^5 + 14 8*d^2*e^6 - 65*d*e^7 + 107*e^8)*x^2 - 420*(100*d^7*e + 45*d^6*e^2 + 111*d^ 5*e^3 + 37*d^4*e^4 + 148*d^3*e^5 - 65*d^2*e^6 + 107*d*e^7 - 33*e^8)*x + 42 0*(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)*log(e*x + d))/e^9
Time = 0.51 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.06 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{d+e x} \, dx=x^{7} \left (- \frac {100 d}{7 e^{2}} - \frac {45}{7 e}\right ) + x^{6} \cdot \left (\frac {50 d^{2}}{3 e^{3}} + \frac {15 d}{2 e^{2}} + \frac {37}{2 e}\right ) + x^{5} \left (- \frac {20 d^{3}}{e^{4}} - \frac {9 d^{2}}{e^{3}} - \frac {111 d}{5 e^{2}} - \frac {37}{5 e}\right ) + x^{4} \cdot \left (\frac {25 d^{4}}{e^{5}} + \frac {45 d^{3}}{4 e^{4}} + \frac {111 d^{2}}{4 e^{3}} + \frac {37 d}{4 e^{2}} + \frac {37}{e}\right ) + x^{3} \left (- \frac {100 d^{5}}{3 e^{6}} - \frac {15 d^{4}}{e^{5}} - \frac {37 d^{3}}{e^{4}} - \frac {37 d^{2}}{3 e^{3}} - \frac {148 d}{3 e^{2}} + \frac {65}{3 e}\right ) + x^{2} \cdot \left (\frac {50 d^{6}}{e^{7}} + \frac {45 d^{5}}{2 e^{6}} + \frac {111 d^{4}}{2 e^{5}} + \frac {37 d^{3}}{2 e^{4}} + \frac {74 d^{2}}{e^{3}} - \frac {65 d}{2 e^{2}} + \frac {107}{2 e}\right ) + x \left (- \frac {100 d^{7}}{e^{8}} - \frac {45 d^{6}}{e^{7}} - \frac {111 d^{5}}{e^{6}} - \frac {37 d^{4}}{e^{5}} - \frac {148 d^{3}}{e^{4}} + \frac {65 d^{2}}{e^{3}} - \frac {107 d}{e^{2}} + \frac {33}{e}\right ) + \frac {25 x^{8}}{2 e} + \frac {\left (5 d^{2} - 2 d e + 3 e^{2}\right )^{2} \cdot \left (4 d^{4} + 5 d^{3} e + 3 d^{2} e^{2} - d e^{3} + 2 e^{4}\right ) \log {\left (d + e x \right )}}{e^{9}} \] Input:
integrate((5*x**2+2*x+3)**2*(4*x**4-5*x**3+3*x**2+x+2)/(e*x+d),x)
Output:
x**7*(-100*d/(7*e**2) - 45/(7*e)) + x**6*(50*d**2/(3*e**3) + 15*d/(2*e**2) + 37/(2*e)) + x**5*(-20*d**3/e**4 - 9*d**2/e**3 - 111*d/(5*e**2) - 37/(5* e)) + x**4*(25*d**4/e**5 + 45*d**3/(4*e**4) + 111*d**2/(4*e**3) + 37*d/(4* e**2) + 37/e) + x**3*(-100*d**5/(3*e**6) - 15*d**4/e**5 - 37*d**3/e**4 - 3 7*d**2/(3*e**3) - 148*d/(3*e**2) + 65/(3*e)) + x**2*(50*d**6/e**7 + 45*d** 5/(2*e**6) + 111*d**4/(2*e**5) + 37*d**3/(2*e**4) + 74*d**2/e**3 - 65*d/(2 *e**2) + 107/(2*e)) + x*(-100*d**7/e**8 - 45*d**6/e**7 - 111*d**5/e**6 - 3 7*d**4/e**5 - 148*d**3/e**4 + 65*d**2/e**3 - 107*d/e**2 + 33/e) + 25*x**8/ (2*e) + (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)*log(d + e*x)/e**9
Time = 0.03 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.04 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{d+e x} \, dx=\frac {5250 \, e^{7} x^{8} - 300 \, {\left (20 \, d e^{6} + 9 \, e^{7}\right )} x^{7} + 70 \, {\left (100 \, d^{2} e^{5} + 45 \, d e^{6} + 111 \, e^{7}\right )} x^{6} - 84 \, {\left (100 \, d^{3} e^{4} + 45 \, d^{2} e^{5} + 111 \, d e^{6} + 37 \, e^{7}\right )} x^{5} + 105 \, {\left (100 \, d^{4} e^{3} + 45 \, d^{3} e^{4} + 111 \, d^{2} e^{5} + 37 \, d e^{6} + 148 \, e^{7}\right )} x^{4} - 140 \, {\left (100 \, d^{5} e^{2} + 45 \, d^{4} e^{3} + 111 \, d^{3} e^{4} + 37 \, d^{2} e^{5} + 148 \, d e^{6} - 65 \, e^{7}\right )} x^{3} + 210 \, {\left (100 \, d^{6} e + 45 \, d^{5} e^{2} + 111 \, d^{4} e^{3} + 37 \, d^{3} e^{4} + 148 \, d^{2} e^{5} - 65 \, d e^{6} + 107 \, e^{7}\right )} x^{2} - 420 \, {\left (100 \, d^{7} + 45 \, d^{6} e + 111 \, d^{5} e^{2} + 37 \, d^{4} e^{3} + 148 \, d^{3} e^{4} - 65 \, d^{2} e^{5} + 107 \, d e^{6} - 33 \, e^{7}\right )} x}{420 \, e^{8}} + \frac {{\left (100 \, d^{8} + 45 \, d^{7} e + 111 \, d^{6} e^{2} + 37 \, d^{5} e^{3} + 148 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + 107 \, d^{2} e^{6} - 33 \, d e^{7} + 18 \, e^{8}\right )} \log \left (e x + d\right )}{e^{9}} \] Input:
integrate((5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d),x, algorithm="ma xima")
Output:
1/420*(5250*e^7*x^8 - 300*(20*d*e^6 + 9*e^7)*x^7 + 70*(100*d^2*e^5 + 45*d* e^6 + 111*e^7)*x^6 - 84*(100*d^3*e^4 + 45*d^2*e^5 + 111*d*e^6 + 37*e^7)*x^ 5 + 105*(100*d^4*e^3 + 45*d^3*e^4 + 111*d^2*e^5 + 37*d*e^6 + 148*e^7)*x^4 - 140*(100*d^5*e^2 + 45*d^4*e^3 + 111*d^3*e^4 + 37*d^2*e^5 + 148*d*e^6 - 6 5*e^7)*x^3 + 210*(100*d^6*e + 45*d^5*e^2 + 111*d^4*e^3 + 37*d^3*e^4 + 148* d^2*e^5 - 65*d*e^6 + 107*e^7)*x^2 - 420*(100*d^7 + 45*d^6*e + 111*d^5*e^2 + 37*d^4*e^3 + 148*d^3*e^4 - 65*d^2*e^5 + 107*d*e^6 - 33*e^7)*x)/e^8 + (10 0*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 + 1 07*d^2*e^6 - 33*d*e^7 + 18*e^8)*log(e*x + d)/e^9
Time = 0.16 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.18 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{d+e x} \, dx=\frac {5250 \, e^{7} x^{8} - 6000 \, d e^{6} x^{7} - 2700 \, e^{7} x^{7} + 7000 \, d^{2} e^{5} x^{6} + 3150 \, d e^{6} x^{6} + 7770 \, e^{7} x^{6} - 8400 \, d^{3} e^{4} x^{5} - 3780 \, d^{2} e^{5} x^{5} - 9324 \, d e^{6} x^{5} - 3108 \, e^{7} x^{5} + 10500 \, d^{4} e^{3} x^{4} + 4725 \, d^{3} e^{4} x^{4} + 11655 \, d^{2} e^{5} x^{4} + 3885 \, d e^{6} x^{4} + 15540 \, e^{7} x^{4} - 14000 \, d^{5} e^{2} x^{3} - 6300 \, d^{4} e^{3} x^{3} - 15540 \, d^{3} e^{4} x^{3} - 5180 \, d^{2} e^{5} x^{3} - 20720 \, d e^{6} x^{3} + 9100 \, e^{7} x^{3} + 21000 \, d^{6} e x^{2} + 9450 \, d^{5} e^{2} x^{2} + 23310 \, d^{4} e^{3} x^{2} + 7770 \, d^{3} e^{4} x^{2} + 31080 \, d^{2} e^{5} x^{2} - 13650 \, d e^{6} x^{2} + 22470 \, e^{7} x^{2} - 42000 \, d^{7} x - 18900 \, d^{6} e x - 46620 \, d^{5} e^{2} x - 15540 \, d^{4} e^{3} x - 62160 \, d^{3} e^{4} x + 27300 \, d^{2} e^{5} x - 44940 \, d e^{6} x + 13860 \, e^{7} x}{420 \, e^{8}} + \frac {{\left (100 \, d^{8} + 45 \, d^{7} e + 111 \, d^{6} e^{2} + 37 \, d^{5} e^{3} + 148 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + 107 \, d^{2} e^{6} - 33 \, d e^{7} + 18 \, e^{8}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{9}} \] Input:
integrate((5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d),x, algorithm="gi ac")
Output:
1/420*(5250*e^7*x^8 - 6000*d*e^6*x^7 - 2700*e^7*x^7 + 7000*d^2*e^5*x^6 + 3 150*d*e^6*x^6 + 7770*e^7*x^6 - 8400*d^3*e^4*x^5 - 3780*d^2*e^5*x^5 - 9324* d*e^6*x^5 - 3108*e^7*x^5 + 10500*d^4*e^3*x^4 + 4725*d^3*e^4*x^4 + 11655*d^ 2*e^5*x^4 + 3885*d*e^6*x^4 + 15540*e^7*x^4 - 14000*d^5*e^2*x^3 - 6300*d^4* e^3*x^3 - 15540*d^3*e^4*x^3 - 5180*d^2*e^5*x^3 - 20720*d*e^6*x^3 + 9100*e^ 7*x^3 + 21000*d^6*e*x^2 + 9450*d^5*e^2*x^2 + 23310*d^4*e^3*x^2 + 7770*d^3* e^4*x^2 + 31080*d^2*e^5*x^2 - 13650*d*e^6*x^2 + 22470*e^7*x^2 - 42000*d^7* x - 18900*d^6*e*x - 46620*d^5*e^2*x - 15540*d^4*e^3*x - 62160*d^3*e^4*x + 27300*d^2*e^5*x - 44940*d*e^6*x + 13860*e^7*x)/e^8 + (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)*log(abs(e*x + d))/e^9
Time = 0.08 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.23 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{d+e x} \, dx=x\,\left (\frac {33}{e}-\frac {d\,\left (\frac {107}{e}-\frac {d\,\left (\frac {65}{e}-\frac {d\,\left (\frac {148}{e}+\frac {d\,\left (\frac {37}{e}+\frac {d\,\left (\frac {111}{e}+\frac {d\,\left (\frac {100\,d}{e^2}+\frac {45}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )-x^7\,\left (\frac {100\,d}{7\,e^2}+\frac {45}{7\,e}\right )+x^6\,\left (\frac {37}{2\,e}+\frac {d\,\left (\frac {100\,d}{e^2}+\frac {45}{e}\right )}{6\,e}\right )-x^5\,\left (\frac {37}{5\,e}+\frac {d\,\left (\frac {111}{e}+\frac {d\,\left (\frac {100\,d}{e^2}+\frac {45}{e}\right )}{e}\right )}{5\,e}\right )+x^4\,\left (\frac {37}{e}+\frac {d\,\left (\frac {37}{e}+\frac {d\,\left (\frac {111}{e}+\frac {d\,\left (\frac {100\,d}{e^2}+\frac {45}{e}\right )}{e}\right )}{e}\right )}{4\,e}\right )+x^3\,\left (\frac {65}{3\,e}-\frac {d\,\left (\frac {148}{e}+\frac {d\,\left (\frac {37}{e}+\frac {d\,\left (\frac {111}{e}+\frac {d\,\left (\frac {100\,d}{e^2}+\frac {45}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{3\,e}\right )+x^2\,\left (\frac {107}{2\,e}-\frac {d\,\left (\frac {65}{e}-\frac {d\,\left (\frac {148}{e}+\frac {d\,\left (\frac {37}{e}+\frac {d\,\left (\frac {111}{e}+\frac {d\,\left (\frac {100\,d}{e^2}+\frac {45}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{2\,e}\right )+\frac {25\,x^8}{2\,e}+\frac {\ln \left (d+e\,x\right )\,\left (100\,d^8+45\,d^7\,e+111\,d^6\,e^2+37\,d^5\,e^3+148\,d^4\,e^4-65\,d^3\,e^5+107\,d^2\,e^6-33\,d\,e^7+18\,e^8\right )}{e^9} \] Input:
int(((2*x + 5*x^2 + 3)^2*(x + 3*x^2 - 5*x^3 + 4*x^4 + 2))/(d + e*x),x)
Output:
x*(33/e - (d*(107/e - (d*(65/e - (d*(148/e + (d*(37/e + (d*(111/e + (d*((1 00*d)/e^2 + 45/e))/e))/e))/e))/e))/e))/e) - x^7*((100*d)/(7*e^2) + 45/(7*e )) + x^6*(37/(2*e) + (d*((100*d)/e^2 + 45/e))/(6*e)) - x^5*(37/(5*e) + (d* (111/e + (d*((100*d)/e^2 + 45/e))/e))/(5*e)) + x^4*(37/e + (d*(37/e + (d*( 111/e + (d*((100*d)/e^2 + 45/e))/e))/e))/(4*e)) + x^3*(65/(3*e) - (d*(148/ e + (d*(37/e + (d*(111/e + (d*((100*d)/e^2 + 45/e))/e))/e))/e))/(3*e)) + x ^2*(107/(2*e) - (d*(65/e - (d*(148/e + (d*(37/e + (d*(111/e + (d*((100*d)/ e^2 + 45/e))/e))/e))/e))/e))/(2*e)) + (25*x^8)/(2*e) + (log(d + e*x)*(45*d ^7*e - 33*d*e^7 + 100*d^8 + 18*e^8 + 107*d^2*e^6 - 65*d^3*e^5 + 148*d^4*e^ 4 + 37*d^5*e^3 + 111*d^6*e^2))/e^9
Time = 0.16 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.31 \[ \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{d+e x} \, dx=\frac {-15540 d^{4} e^{4} x -8400 d^{3} e^{5} x^{5}+4725 d^{3} e^{5} x^{4}-15540 d^{3} e^{5} x^{3}+7770 d^{3} e^{5} x^{2}-62160 d^{3} e^{5} x +7000 d^{2} e^{6} x^{6}-3780 d^{2} e^{6} x^{5}+11655 d^{2} e^{6} x^{4}-5180 d^{2} e^{6} x^{3}+31080 d^{2} e^{6} x^{2}+27300 d^{2} e^{6} x -6000 d \,e^{7} x^{7}+3150 d \,e^{7} x^{6}-9324 d \,e^{7} x^{5}+3885 d \,e^{7} x^{4}-20720 d \,e^{7} x^{3}-13650 d \,e^{7} x^{2}-44940 d \,e^{7} x +18900 \,\mathrm {log}\left (e x +d \right ) d^{7} e +46620 \,\mathrm {log}\left (e x +d \right ) d^{6} e^{2}+15540 \,\mathrm {log}\left (e x +d \right ) d^{5} e^{3}+62160 \,\mathrm {log}\left (e x +d \right ) d^{4} e^{4}-27300 \,\mathrm {log}\left (e x +d \right ) d^{3} e^{5}+44940 \,\mathrm {log}\left (e x +d \right ) d^{2} e^{6}-13860 \,\mathrm {log}\left (e x +d \right ) d \,e^{7}-42000 d^{7} e x +21000 d^{6} e^{2} x^{2}-18900 d^{6} e^{2} x -14000 d^{5} e^{3} x^{3}+9450 d^{5} e^{3} x^{2}-46620 d^{5} e^{3} x +10500 d^{4} e^{4} x^{4}-6300 d^{4} e^{4} x^{3}+23310 d^{4} e^{4} x^{2}+42000 \,\mathrm {log}\left (e x +d \right ) d^{8}+7560 \,\mathrm {log}\left (e x +d \right ) e^{8}+5250 e^{8} x^{8}-2700 e^{8} x^{7}+7770 e^{8} x^{6}-3108 e^{8} x^{5}+15540 e^{8} x^{4}+9100 e^{8} x^{3}+22470 e^{8} x^{2}+13860 e^{8} x}{420 e^{9}} \] Input:
int((5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d),x)
Output:
(42000*log(d + e*x)*d**8 + 18900*log(d + e*x)*d**7*e + 46620*log(d + e*x)* d**6*e**2 + 15540*log(d + e*x)*d**5*e**3 + 62160*log(d + e*x)*d**4*e**4 - 27300*log(d + e*x)*d**3*e**5 + 44940*log(d + e*x)*d**2*e**6 - 13860*log(d + e*x)*d*e**7 + 7560*log(d + e*x)*e**8 - 42000*d**7*e*x + 21000*d**6*e**2* x**2 - 18900*d**6*e**2*x - 14000*d**5*e**3*x**3 + 9450*d**5*e**3*x**2 - 46 620*d**5*e**3*x + 10500*d**4*e**4*x**4 - 6300*d**4*e**4*x**3 + 23310*d**4* e**4*x**2 - 15540*d**4*e**4*x - 8400*d**3*e**5*x**5 + 4725*d**3*e**5*x**4 - 15540*d**3*e**5*x**3 + 7770*d**3*e**5*x**2 - 62160*d**3*e**5*x + 7000*d* *2*e**6*x**6 - 3780*d**2*e**6*x**5 + 11655*d**2*e**6*x**4 - 5180*d**2*e**6 *x**3 + 31080*d**2*e**6*x**2 + 27300*d**2*e**6*x - 6000*d*e**7*x**7 + 3150 *d*e**7*x**6 - 9324*d*e**7*x**5 + 3885*d*e**7*x**4 - 20720*d*e**7*x**3 - 1 3650*d*e**7*x**2 - 44940*d*e**7*x + 5250*e**8*x**8 - 2700*e**8*x**7 + 7770 *e**8*x**6 - 3108*e**8*x**5 + 15540*e**8*x**4 + 9100*e**8*x**3 + 22470*e** 8*x**2 + 13860*e**8*x)/(420*e**9)