Integrand size = 38, antiderivative size = 313 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 \left (3+2 x+5 x^2\right )^2} \, dx=-\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}-\frac {1367 d^2-586 d e-703 e^2+\left (423 d^2-2734 d e+293 e^2\right ) x}{140 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3+2 x+5 x^2\right )}+\frac {\left (1313 d^4-10044 d^3 e+4290 d^2 e^2+156 d e^3-271 e^4\right ) \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )}{28 \sqrt {14} \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}-\frac {\left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) \log \left (3+2 x+5 x^2\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^3} \]
(-4*d^4-5*d^3*e-3*d^2*e^2+d*e^3-2*e^4)/e/(5*d^2-2*d*e+3*e^2)^2/(e*x+d)+1/1 40*(-1367*d^2+586*d*e+703*e^2-(423*d^2-2734*d*e+293*e^2)*x)/(5*d^2-2*d*e+3 *e^2)^2/(5*x^2+2*x+3)+(41*d^4-8*d^3*e-60*d^2*e^2+24*d*e^3-5*e^4)*ln(e*x+d) /(5*d^2-2*d*e+3*e^2)^3-1/2*(41*d^4-8*d^3*e-60*d^2*e^2+24*d*e^3-5*e^4)*ln(5 *x^2+2*x+3)/(5*d^2-2*d*e+3*e^2)^3+1/392*(1313*d^4-10044*d^3*e+4290*d^2*e^2 +156*d*e^3-271*e^4)*arctan(1/14*(1+5*x)*14^(1/2))/(5*d^2-2*d*e+3*e^2)^3*14 ^(1/2)
Time = 0.15 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.86 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 \left (3+2 x+5 x^2\right )^2} \, dx=\frac {-\frac {1960 \left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e (d+e x)}-\frac {14 \left (5 d^2-2 d e+3 e^2\right ) \left (e^2 (-703+293 x)+d^2 (1367+423 x)-2 d e (293+1367 x)\right )}{3+2 x+5 x^2}+5 \sqrt {14} \left (1313 d^4-10044 d^3 e+4290 d^2 e^2+156 d e^3-271 e^4\right ) \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )+1960 \left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) \log (d+e x)+980 \left (-41 d^4+8 d^3 e+60 d^2 e^2-24 d e^3+5 e^4\right ) \log \left (3+2 x+5 x^2\right )}{1960 \left (5 d^2-2 d e+3 e^2\right )^3} \]
((-1960*(5*d^2 - 2*d*e + 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e ^4))/(e*(d + e*x)) - (14*(5*d^2 - 2*d*e + 3*e^2)*(e^2*(-703 + 293*x) + d^2 *(1367 + 423*x) - 2*d*e*(293 + 1367*x)))/(3 + 2*x + 5*x^2) + 5*Sqrt[14]*(1 313*d^4 - 10044*d^3*e + 4290*d^2*e^2 + 156*d*e^3 - 271*e^4)*ArcTan[(1 + 5* x)/Sqrt[14]] + 1960*(41*d^4 - 8*d^3*e - 60*d^2*e^2 + 24*d*e^3 - 5*e^4)*Log [d + e*x] + 980*(-41*d^4 + 8*d^3*e + 60*d^2*e^2 - 24*d*e^3 + 5*e^4)*Log[3 + 2*x + 5*x^2])/(1960*(5*d^2 - 2*d*e + 3*e^2)^3)
Time = 0.88 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2177, 27, 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.
| \(\displaystyle \int \frac {4 x^4-5 x^3+3 x^2+x+2}{\left (5 x^2+2 x+3\right )^2 (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2177 |
| \(\displaystyle \frac {1}{56} \int \frac {2 \left (\frac {\left (560 d^4-448 e d^3+677 e^2 d^2+278 e^3 d+143 e^4\right ) x^2}{\left (5 d^2-2 e d+3 e^2\right )^2}-\frac {2 \left (462 d^4-285 e d^3+338 e^2 d^2-171 e^3 d+14 e^4\right ) x}{\left (5 d^2-2 e d+3 e^2\right )^2}+\frac {369 d^4-842 e d^3+787 e^2 d^2-224 e^3 d+168 e^4}{\left (5 d^2-2 e d+3 e^2\right )^2}\right )}{(d+e x)^2 \left (5 x^2+2 x+3\right )}dx-\frac {x \left (423 d^2-2734 d e+293 e^2\right )+1367 d^2-586 d e-703 e^2}{140 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \int \frac {\frac {\left (560 d^4-448 e d^3+677 e^2 d^2+278 e^3 d+143 e^4\right ) x^2}{\left (5 d^2-2 e d+3 e^2\right )^2}-\frac {2 \left (462 d^4-285 e d^3+338 e^2 d^2-171 e^3 d+14 e^4\right ) x}{\left (5 d^2-2 e d+3 e^2\right )^2}+\frac {369 d^4-842 e d^3+787 e^2 d^2-224 e^3 d+168 e^4}{\left (5 d^2-2 e d+3 e^2\right )^2}}{(d+e x)^2 \left (5 x^2+2 x+3\right )}dx-\frac {x \left (423 d^2-2734 d e+293 e^2\right )+1367 d^2-586 d e-703 e^2}{140 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^2}\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \frac {1}{28} \int \left (\frac {28 \left (4 d^4+5 e d^3+3 e^2 d^2-e^3 d+2 e^4\right )}{\left (5 d^2-2 e d+3 e^2\right )^2 (d+e x)^2}-\frac {28 e \left (-41 d^4+8 e d^3+60 e^2 d^2-24 e^3 d+5 e^4\right )}{\left (5 d^2-2 e d+3 e^2\right )^3 (d+e x)}+\frac {165 d^4-9820 e d^3+5970 e^2 d^2-516 e^3 d-131 e^4-140 \left (41 d^4-8 e d^3-60 e^2 d^2+24 e^3 d-5 e^4\right ) x}{\left (5 d^2-2 e d+3 e^2\right )^3 \left (5 x^2+2 x+3\right )}\right )dx-\frac {x \left (423 d^2-2734 d e+293 e^2\right )+1367 d^2-586 d e-703 e^2}{140 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{28} \left (\frac {\arctan \left (\frac {5 x+1}{\sqrt {14}}\right ) \left (1313 d^4-10044 d^3 e+4290 d^2 e^2+156 d e^3-271 e^4\right )}{\sqrt {14} \left (5 d^2-2 d e+3 e^2\right )^3}-\frac {14 \left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) \log \left (5 x^2+2 x+3\right )}{\left (5 d^2-2 d e+3 e^2\right )^3}-\frac {28 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac {28 \left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^3}\right )-\frac {x \left (423 d^2-2734 d e+293 e^2\right )+1367 d^2-586 d e-703 e^2}{140 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^2}\) |
-1/140*(1367*d^2 - 586*d*e - 703*e^2 + (423*d^2 - 2734*d*e + 293*e^2)*x)/( (5*d^2 - 2*d*e + 3*e^2)^2*(3 + 2*x + 5*x^2)) + ((-28*(4*d^4 + 5*d^3*e + 3* d^2*e^2 - d*e^3 + 2*e^4))/(e*(5*d^2 - 2*d*e + 3*e^2)^2*(d + e*x)) + ((1313 *d^4 - 10044*d^3*e + 4290*d^2*e^2 + 156*d*e^3 - 271*e^4)*ArcTan[(1 + 5*x)/ Sqrt[14]])/(Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^3) + (28*(41*d^4 - 8*d^3*e - 60*d^2*e^2 + 24*d*e^3 - 5*e^4)*Log[d + e*x])/(5*d^2 - 2*d*e + 3*e^2)^3 - ( 14*(41*d^4 - 8*d^3*e - 60*d^2*e^2 + 24*d*e^3 - 5*e^4)*Log[3 + 2*x + 5*x^2] )/(5*d^2 - 2*d*e + 3*e^2)^3)/28
3.4.16.3.1 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]]
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]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x + c* x^2, x], R = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + (2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Qx)/(d + e*x )^m - ((2*p + 3)*(2*c*R - b*S))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
Time = 0.87 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {\frac {\left (\frac {423}{140} d^{4}-\frac {3629}{175} d^{3} e +\frac {4101}{350} d^{2} e^{2}-\frac {2197}{175} d \,e^{3}+\frac {879}{700} e^{4}\right ) x +\frac {1367 d^{4}}{140}-\frac {1416 d^{3} e}{175}+\frac {879 d^{2} e^{2}}{350}-\frac {88 d \,e^{3}}{175}-\frac {2109 e^{4}}{700}}{x^{2}+\frac {2}{5} x +\frac {3}{5}}+\frac {\left (5740 d^{4}-1120 d^{3} e -8400 d^{2} e^{2}+3360 d \,e^{3}-700 e^{4}\right ) \ln \left (5 x^{2}+2 x +3\right )}{280}+\frac {\left (-1313 d^{4}+10044 d^{3} e -4290 d^{2} e^{2}-156 d \,e^{3}+271 e^{4}\right ) \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{392}}{\left (5 d^{2}-2 d e +3 e^{2}\right )^{3}}-\frac {4 d^{4}+5 d^{3} e +3 d^{2} e^{2}-d \,e^{3}+2 e^{4}}{\left (5 d^{2}-2 d e +3 e^{2}\right )^{2} e \left (e x +d \right )}+\frac {\left (41 d^{4}-8 d^{3} e -60 d^{2} e^{2}+24 d \,e^{3}-5 e^{4}\right ) \ln \left (e x +d \right )}{\left (5 d^{2}-2 d e +3 e^{2}\right )^{3}}\) | \(303\) |
| risch | \(\text {Expression too large to display}\) | \(19150\) |
-1/(5*d^2-2*d*e+3*e^2)^3*(((423/140*d^4-3629/175*d^3*e+4101/350*d^2*e^2-21 97/175*d*e^3+879/700*e^4)*x+1367/140*d^4-1416/175*d^3*e+879/350*d^2*e^2-88 /175*d*e^3-2109/700*e^4)/(x^2+2/5*x+3/5)+1/280*(5740*d^4-1120*d^3*e-8400*d ^2*e^2+3360*d*e^3-700*e^4)*ln(5*x^2+2*x+3)+1/392*(-1313*d^4+10044*d^3*e-42 90*d^2*e^2-156*d*e^3+271*e^4)*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2)))-(4* d^4+5*d^3*e+3*d^2*e^2-d*e^3+2*e^4)/(5*d^2-2*d*e+3*e^2)^2/e/(e*x+d)+(41*d^4 -8*d^3*e-60*d^2*e^2+24*d*e^3-5*e^4)*ln(e*x+d)/(5*d^2-2*d*e+3*e^2)^3
Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (304) = 608\).
Time = 0.35 (sec) , antiderivative size = 910, normalized size of antiderivative = 2.91 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 \left (3+2 x+5 x^2\right )^2} \, dx=-\frac {117600 \, d^{6} + 195650 \, d^{5} e + 20664 \, d^{4} e^{2} + 48132 \, d^{3} e^{3} + 118552 \, d^{2} e^{4} - 70686 \, d e^{5} + 35280 \, e^{6} + 14 \, {\left (14000 \, d^{6} + 11900 \, d^{5} e + 14015 \, d^{4} e^{2} - 11716 \, d^{3} e^{3} + 22902 \, d^{2} e^{4} - 13688 \, d e^{5} + 5079 \, e^{6}\right )} x^{2} - 5 \, \sqrt {14} {\left (3939 \, d^{5} e - 30132 \, d^{4} e^{2} + 12870 \, d^{3} e^{3} + 468 \, d^{2} e^{4} - 813 \, d e^{5} + 5 \, {\left (1313 \, d^{4} e^{2} - 10044 \, d^{3} e^{3} + 4290 \, d^{2} e^{4} + 156 \, d e^{5} - 271 \, e^{6}\right )} x^{3} + {\left (6565 \, d^{5} e - 47594 \, d^{4} e^{2} + 1362 \, d^{3} e^{3} + 9360 \, d^{2} e^{4} - 1043 \, d e^{5} - 542 \, e^{6}\right )} x^{2} + {\left (2626 \, d^{5} e - 16149 \, d^{4} e^{2} - 21552 \, d^{3} e^{3} + 13182 \, d^{2} e^{4} - 74 \, d e^{5} - 813 \, e^{6}\right )} x\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + 14 \, {\left (5600 \, d^{6} + 6875 \, d^{5} e - 2921 \, d^{4} e^{2} + 3658 \, d^{3} e^{3} - 1150 \, d^{2} e^{4} - 1433 \, d e^{5} - 429 \, e^{6}\right )} x - 1960 \, {\left (123 \, d^{5} e - 24 \, d^{4} e^{2} - 180 \, d^{3} e^{3} + 72 \, d^{2} e^{4} - 15 \, d e^{5} + 5 \, {\left (41 \, d^{4} e^{2} - 8 \, d^{3} e^{3} - 60 \, d^{2} e^{4} + 24 \, d e^{5} - 5 \, e^{6}\right )} x^{3} + {\left (205 \, d^{5} e + 42 \, d^{4} e^{2} - 316 \, d^{3} e^{3} + 23 \, d e^{5} - 10 \, e^{6}\right )} x^{2} + {\left (82 \, d^{5} e + 107 \, d^{4} e^{2} - 144 \, d^{3} e^{3} - 132 \, d^{2} e^{4} + 62 \, d e^{5} - 15 \, e^{6}\right )} x\right )} \log \left (e x + d\right ) + 980 \, {\left (123 \, d^{5} e - 24 \, d^{4} e^{2} - 180 \, d^{3} e^{3} + 72 \, d^{2} e^{4} - 15 \, d e^{5} + 5 \, {\left (41 \, d^{4} e^{2} - 8 \, d^{3} e^{3} - 60 \, d^{2} e^{4} + 24 \, d e^{5} - 5 \, e^{6}\right )} x^{3} + {\left (205 \, d^{5} e + 42 \, d^{4} e^{2} - 316 \, d^{3} e^{3} + 23 \, d e^{5} - 10 \, e^{6}\right )} x^{2} + {\left (82 \, d^{5} e + 107 \, d^{4} e^{2} - 144 \, d^{3} e^{3} - 132 \, d^{2} e^{4} + 62 \, d e^{5} - 15 \, e^{6}\right )} x\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{1960 \, {\left (375 \, d^{7} e - 450 \, d^{6} e^{2} + 855 \, d^{5} e^{3} - 564 \, d^{4} e^{4} + 513 \, d^{3} e^{5} - 162 \, d^{2} e^{6} + 81 \, d e^{7} + 5 \, {\left (125 \, d^{6} e^{2} - 150 \, d^{5} e^{3} + 285 \, d^{4} e^{4} - 188 \, d^{3} e^{5} + 171 \, d^{2} e^{6} - 54 \, d e^{7} + 27 \, e^{8}\right )} x^{3} + {\left (625 \, d^{7} e - 500 \, d^{6} e^{2} + 1125 \, d^{5} e^{3} - 370 \, d^{4} e^{4} + 479 \, d^{3} e^{5} + 72 \, d^{2} e^{6} + 27 \, d e^{7} + 54 \, e^{8}\right )} x^{2} + {\left (250 \, d^{7} e + 75 \, d^{6} e^{2} + 120 \, d^{5} e^{3} + 479 \, d^{4} e^{4} - 222 \, d^{3} e^{5} + 405 \, d^{2} e^{6} - 108 \, d e^{7} + 81 \, e^{8}\right )} x\right )}} \]
-1/1960*(117600*d^6 + 195650*d^5*e + 20664*d^4*e^2 + 48132*d^3*e^3 + 11855 2*d^2*e^4 - 70686*d*e^5 + 35280*e^6 + 14*(14000*d^6 + 11900*d^5*e + 14015* d^4*e^2 - 11716*d^3*e^3 + 22902*d^2*e^4 - 13688*d*e^5 + 5079*e^6)*x^2 - 5* sqrt(14)*(3939*d^5*e - 30132*d^4*e^2 + 12870*d^3*e^3 + 468*d^2*e^4 - 813*d *e^5 + 5*(1313*d^4*e^2 - 10044*d^3*e^3 + 4290*d^2*e^4 + 156*d*e^5 - 271*e^ 6)*x^3 + (6565*d^5*e - 47594*d^4*e^2 + 1362*d^3*e^3 + 9360*d^2*e^4 - 1043* d*e^5 - 542*e^6)*x^2 + (2626*d^5*e - 16149*d^4*e^2 - 21552*d^3*e^3 + 13182 *d^2*e^4 - 74*d*e^5 - 813*e^6)*x)*arctan(1/14*sqrt(14)*(5*x + 1)) + 14*(56 00*d^6 + 6875*d^5*e - 2921*d^4*e^2 + 3658*d^3*e^3 - 1150*d^2*e^4 - 1433*d* e^5 - 429*e^6)*x - 1960*(123*d^5*e - 24*d^4*e^2 - 180*d^3*e^3 + 72*d^2*e^4 - 15*d*e^5 + 5*(41*d^4*e^2 - 8*d^3*e^3 - 60*d^2*e^4 + 24*d*e^5 - 5*e^6)*x ^3 + (205*d^5*e + 42*d^4*e^2 - 316*d^3*e^3 + 23*d*e^5 - 10*e^6)*x^2 + (82* d^5*e + 107*d^4*e^2 - 144*d^3*e^3 - 132*d^2*e^4 + 62*d*e^5 - 15*e^6)*x)*lo g(e*x + d) + 980*(123*d^5*e - 24*d^4*e^2 - 180*d^3*e^3 + 72*d^2*e^4 - 15*d *e^5 + 5*(41*d^4*e^2 - 8*d^3*e^3 - 60*d^2*e^4 + 24*d*e^5 - 5*e^6)*x^3 + (2 05*d^5*e + 42*d^4*e^2 - 316*d^3*e^3 + 23*d*e^5 - 10*e^6)*x^2 + (82*d^5*e + 107*d^4*e^2 - 144*d^3*e^3 - 132*d^2*e^4 + 62*d*e^5 - 15*e^6)*x)*log(5*x^2 + 2*x + 3))/(375*d^7*e - 450*d^6*e^2 + 855*d^5*e^3 - 564*d^4*e^4 + 513*d^ 3*e^5 - 162*d^2*e^6 + 81*d*e^7 + 5*(125*d^6*e^2 - 150*d^5*e^3 + 285*d^4*e^ 4 - 188*d^3*e^5 + 171*d^2*e^6 - 54*d*e^7 + 27*e^8)*x^3 + (625*d^7*e - 5...
Timed out. \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 \left (3+2 x+5 x^2\right )^2} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.75 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 \left (3+2 x+5 x^2\right )^2} \, dx=\frac {\sqrt {14} {\left (1313 \, d^{4} - 10044 \, d^{3} e + 4290 \, d^{2} e^{2} + 156 \, d e^{3} - 271 \, e^{4}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right )}{392 \, {\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac {{\left (41 \, d^{4} - 8 \, d^{3} e - 60 \, d^{2} e^{2} + 24 \, d e^{3} - 5 \, e^{4}\right )} \log \left (e x + d\right )}{125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}} - \frac {{\left (41 \, d^{4} - 8 \, d^{3} e - 60 \, d^{2} e^{2} + 24 \, d e^{3} - 5 \, e^{4}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{2 \, {\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} - \frac {1680 \, d^{4} + 3467 \, d^{3} e + 674 \, d^{2} e^{2} - 1123 \, d e^{3} + 840 \, e^{4} + {\left (2800 \, d^{4} + 3500 \, d^{3} e + 2523 \, d^{2} e^{2} - 3434 \, d e^{3} + 1693 \, e^{4}\right )} x^{2} + {\left (1120 \, d^{4} + 1823 \, d^{3} e - 527 \, d^{2} e^{2} - 573 \, d e^{3} - 143 \, e^{4}\right )} x}{140 \, {\left (75 \, d^{5} e - 60 \, d^{4} e^{2} + 102 \, d^{3} e^{3} - 36 \, d^{2} e^{4} + 27 \, d e^{5} + 5 \, {\left (25 \, d^{4} e^{2} - 20 \, d^{3} e^{3} + 34 \, d^{2} e^{4} - 12 \, d e^{5} + 9 \, e^{6}\right )} x^{3} + {\left (125 \, d^{5} e - 50 \, d^{4} e^{2} + 130 \, d^{3} e^{3} + 8 \, d^{2} e^{4} + 21 \, d e^{5} + 18 \, e^{6}\right )} x^{2} + {\left (50 \, d^{5} e + 35 \, d^{4} e^{2} + 8 \, d^{3} e^{3} + 78 \, d^{2} e^{4} - 18 \, d e^{5} + 27 \, e^{6}\right )} x\right )}} \]
1/392*sqrt(14)*(1313*d^4 - 10044*d^3*e + 4290*d^2*e^2 + 156*d*e^3 - 271*e^ 4)*arctan(1/14*sqrt(14)*(5*x + 1))/(125*d^6 - 150*d^5*e + 285*d^4*e^2 - 18 8*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6) + (41*d^4 - 8*d^3*e - 60*d^2* e^2 + 24*d*e^3 - 5*e^4)*log(e*x + d)/(125*d^6 - 150*d^5*e + 285*d^4*e^2 - 188*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6) - 1/2*(41*d^4 - 8*d^3*e - 6 0*d^2*e^2 + 24*d*e^3 - 5*e^4)*log(5*x^2 + 2*x + 3)/(125*d^6 - 150*d^5*e + 285*d^4*e^2 - 188*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6) - 1/140*(1680 *d^4 + 3467*d^3*e + 674*d^2*e^2 - 1123*d*e^3 + 840*e^4 + (2800*d^4 + 3500* d^3*e + 2523*d^2*e^2 - 3434*d*e^3 + 1693*e^4)*x^2 + (1120*d^4 + 1823*d^3*e - 527*d^2*e^2 - 573*d*e^3 - 143*e^4)*x)/(75*d^5*e - 60*d^4*e^2 + 102*d^3* e^3 - 36*d^2*e^4 + 27*d*e^5 + 5*(25*d^4*e^2 - 20*d^3*e^3 + 34*d^2*e^4 - 12 *d*e^5 + 9*e^6)*x^3 + (125*d^5*e - 50*d^4*e^2 + 130*d^3*e^3 + 8*d^2*e^4 + 21*d*e^5 + 18*e^6)*x^2 + (50*d^5*e + 35*d^4*e^2 + 8*d^3*e^3 + 78*d^2*e^4 - 18*d*e^5 + 27*e^6)*x)
Time = 0.27 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.87 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 \left (3+2 x+5 x^2\right )^2} \, dx=-\frac {{\left (41 \, d^{4} - 8 \, d^{3} e - 60 \, d^{2} e^{2} + 24 \, d e^{3} - 5 \, e^{4}\right )} \log \left (-\frac {10 \, d}{e x + d} + \frac {5 \, d^{2}}{{\left (e x + d\right )}^{2}} + \frac {2 \, e}{e x + d} - \frac {2 \, d e}{{\left (e x + d\right )}^{2}} + \frac {3 \, e^{2}}{{\left (e x + d\right )}^{2}} + 5\right )}{2 \, {\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} - \frac {\frac {4 \, d^{4} e^{3}}{e x + d} + \frac {5 \, d^{3} e^{4}}{e x + d} + \frac {3 \, d^{2} e^{5}}{e x + d} - \frac {d e^{6}}{e x + d} + \frac {2 \, e^{7}}{e x + d}}{25 \, d^{4} e^{4} - 20 \, d^{3} e^{5} + 34 \, d^{2} e^{6} - 12 \, d e^{7} + 9 \, e^{8}} + \frac {\sqrt {14} {\left (1313 \, d^{4} e^{2} - 10044 \, d^{3} e^{3} + 4290 \, d^{2} e^{4} + 156 \, d e^{5} - 271 \, e^{6}\right )} \arctan \left (\frac {\sqrt {14} {\left (5 \, d - \frac {5 \, d^{2}}{e x + d} + \frac {2 \, d e}{e x + d} - e - \frac {3 \, e^{2}}{e x + d}\right )}}{14 \, e}\right )}{392 \, {\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )} e^{2}} + \frac {\frac {423 \, d^{3} e - 4101 \, d^{2} e^{2} + 879 \, d e^{3} + 703 \, e^{4}}{5 \, d^{2} - 2 \, d e + 3 \, e^{2}} - \frac {423 \, d^{4} e^{2} - 5468 \, d^{3} e^{3} + 1758 \, d^{2} e^{4} + 2812 \, d e^{5} - 457 \, e^{6}}{{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )} {\left (e x + d\right )} e}}{28 \, {\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}^{2} {\left (\frac {10 \, d}{e x + d} - \frac {5 \, d^{2}}{{\left (e x + d\right )}^{2}} - \frac {2 \, e}{e x + d} + \frac {2 \, d e}{{\left (e x + d\right )}^{2}} - \frac {3 \, e^{2}}{{\left (e x + d\right )}^{2}} - 5\right )}} \]
-1/2*(41*d^4 - 8*d^3*e - 60*d^2*e^2 + 24*d*e^3 - 5*e^4)*log(-10*d/(e*x + d ) + 5*d^2/(e*x + d)^2 + 2*e/(e*x + d) - 2*d*e/(e*x + d)^2 + 3*e^2/(e*x + d )^2 + 5)/(125*d^6 - 150*d^5*e + 285*d^4*e^2 - 188*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6) - (4*d^4*e^3/(e*x + d) + 5*d^3*e^4/(e*x + d) + 3*d^2*e^ 5/(e*x + d) - d*e^6/(e*x + d) + 2*e^7/(e*x + d))/(25*d^4*e^4 - 20*d^3*e^5 + 34*d^2*e^6 - 12*d*e^7 + 9*e^8) + 1/392*sqrt(14)*(1313*d^4*e^2 - 10044*d^ 3*e^3 + 4290*d^2*e^4 + 156*d*e^5 - 271*e^6)*arctan(1/14*sqrt(14)*(5*d - 5* d^2/(e*x + d) + 2*d*e/(e*x + d) - e - 3*e^2/(e*x + d))/e)/((125*d^6 - 150* d^5*e + 285*d^4*e^2 - 188*d^3*e^3 + 171*d^2*e^4 - 54*d*e^5 + 27*e^6)*e^2) + 1/28*((423*d^3*e - 4101*d^2*e^2 + 879*d*e^3 + 703*e^4)/(5*d^2 - 2*d*e + 3*e^2) - (423*d^4*e^2 - 5468*d^3*e^3 + 1758*d^2*e^4 + 2812*d*e^5 - 457*e^6 )/((5*d^2 - 2*d*e + 3*e^2)*(e*x + d)*e))/((5*d^2 - 2*d*e + 3*e^2)^2*(10*d/ (e*x + d) - 5*d^2/(e*x + d)^2 - 2*e/(e*x + d) + 2*d*e/(e*x + d)^2 - 3*e^2/ (e*x + d)^2 - 5))
Time = 14.28 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.92 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 \left (3+2 x+5 x^2\right )^2} \, dx=\ln \left (d+e\,x\right )\,\left (\frac {41}{25\,\left (5\,d^2-2\,d\,e+3\,e^2\right )}-\frac {4\,e^3\,\left (423\,d-1367\,e\right )}{125\,{\left (5\,d^2-2\,d\,e+3\,e^2\right )}^3}+\frac {2\,e\,\left (310\,d-1323\,e\right )}{125\,{\left (5\,d^2-2\,d\,e+3\,e^2\right )}^2}\right )-\frac {\frac {1680\,d^4+3467\,d^3\,e+674\,d^2\,e^2-1123\,d\,e^3+840\,e^4}{140\,e\,\left (25\,d^4-20\,d^3\,e+34\,d^2\,e^2-12\,d\,e^3+9\,e^4\right )}-\frac {x\,\left (-1120\,d^4-1823\,d^3\,e+527\,d^2\,e^2+573\,d\,e^3+143\,e^4\right )}{140\,e\,\left (25\,d^4-20\,d^3\,e+34\,d^2\,e^2-12\,d\,e^3+9\,e^4\right )}+\frac {x^2\,\left (2800\,d^4+3500\,d^3\,e+2523\,d^2\,e^2-3434\,d\,e^3+1693\,e^4\right )}{140\,e\,\left (25\,d^4-20\,d^3\,e+34\,d^2\,e^2-12\,d\,e^3+9\,e^4\right )}}{5\,e\,x^3+\left (5\,d+2\,e\right )\,x^2+\left (2\,d+3\,e\right )\,x+3\,d}+\frac {\ln \left (x+\frac {1}{5}-\frac {\sqrt {14}\,1{}\mathrm {i}}{5}\right )\,\left (\left (\frac {1313\,\sqrt {14}}{784}-\frac {41}{2}{}\mathrm {i}\right )\,d^4+\left (-\frac {2511\,\sqrt {14}}{196}+4{}\mathrm {i}\right )\,d^3\,e+\left (\frac {2145\,\sqrt {14}}{392}+30{}\mathrm {i}\right )\,d^2\,e^2+\left (\frac {39\,\sqrt {14}}{196}-12{}\mathrm {i}\right )\,d\,e^3+\left (-\frac {271\,\sqrt {14}}{784}+\frac {5}{2}{}\mathrm {i}\right )\,e^4\right )}{d^6\,125{}\mathrm {i}-d^5\,e\,150{}\mathrm {i}+d^4\,e^2\,285{}\mathrm {i}-d^3\,e^3\,188{}\mathrm {i}+d^2\,e^4\,171{}\mathrm {i}-d\,e^5\,54{}\mathrm {i}+e^6\,27{}\mathrm {i}}-\frac {\ln \left (x+\frac {1}{5}+\frac {\sqrt {14}\,1{}\mathrm {i}}{5}\right )\,\left (\left (\frac {1313\,\sqrt {14}}{784}+\frac {41}{2}{}\mathrm {i}\right )\,d^4+\left (-\frac {2511\,\sqrt {14}}{196}-4{}\mathrm {i}\right )\,d^3\,e+\left (\frac {2145\,\sqrt {14}}{392}-30{}\mathrm {i}\right )\,d^2\,e^2+\left (\frac {39\,\sqrt {14}}{196}+12{}\mathrm {i}\right )\,d\,e^3+\left (-\frac {271\,\sqrt {14}}{784}-\frac {5}{2}{}\mathrm {i}\right )\,e^4\right )}{d^6\,125{}\mathrm {i}-d^5\,e\,150{}\mathrm {i}+d^4\,e^2\,285{}\mathrm {i}-d^3\,e^3\,188{}\mathrm {i}+d^2\,e^4\,171{}\mathrm {i}-d\,e^5\,54{}\mathrm {i}+e^6\,27{}\mathrm {i}} \]
log(d + e*x)*(41/(25*(5*d^2 - 2*d*e + 3*e^2)) - (4*e^3*(423*d - 1367*e))/( 125*(5*d^2 - 2*d*e + 3*e^2)^3) + (2*e*(310*d - 1323*e))/(125*(5*d^2 - 2*d* e + 3*e^2)^2)) - ((3467*d^3*e - 1123*d*e^3 + 1680*d^4 + 840*e^4 + 674*d^2* e^2)/(140*e*(25*d^4 - 20*d^3*e - 12*d*e^3 + 9*e^4 + 34*d^2*e^2)) - (x*(573 *d*e^3 - 1823*d^3*e - 1120*d^4 + 143*e^4 + 527*d^2*e^2))/(140*e*(25*d^4 - 20*d^3*e - 12*d*e^3 + 9*e^4 + 34*d^2*e^2)) + (x^2*(3500*d^3*e - 3434*d*e^3 + 2800*d^4 + 1693*e^4 + 2523*d^2*e^2))/(140*e*(25*d^4 - 20*d^3*e - 12*d*e ^3 + 9*e^4 + 34*d^2*e^2)))/(3*d + x^2*(5*d + 2*e) + 5*e*x^3 + x*(2*d + 3*e )) + (log(x - (14^(1/2)*1i)/5 + 1/5)*(d^4*((1313*14^(1/2))/784 - 41i/2) - e^4*((271*14^(1/2))/784 - 5i/2) + d^2*e^2*((2145*14^(1/2))/392 + 30i) + d* e^3*((39*14^(1/2))/196 - 12i) - d^3*e*((2511*14^(1/2))/196 - 4i)))/(d^6*12 5i - d^5*e*150i - d*e^5*54i + e^6*27i + d^2*e^4*171i - d^3*e^3*188i + d^4* e^2*285i) - (log(x + (14^(1/2)*1i)/5 + 1/5)*(d^4*((1313*14^(1/2))/784 + 41 i/2) - e^4*((271*14^(1/2))/784 + 5i/2) + d^2*e^2*((2145*14^(1/2))/392 - 30 i) + d*e^3*((39*14^(1/2))/196 + 12i) - d^3*e*((2511*14^(1/2))/196 + 4i)))/ (d^6*125i - d^5*e*150i - d*e^5*54i + e^6*27i + d^2*e^4*171i - d^3*e^3*188i + d^4*e^2*285i)