Integrand size = 38, antiderivative size = 412 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 \left (3+2 x+5 x^2\right )^2} \, dx=-\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)^2}-\frac {41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4}{\left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}-\frac {1367 d^3-879 d^2 e-2109 d e^2+457 e^3+\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) x}{28 \left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}+\frac {\left (6565 d^5-74017 d^4 e+35022 d^3 e^2+42858 d^2 e^3-17247 d e^4+579 e^5\right ) \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )}{28 \sqrt {14} \left (5 d^2-2 d e+3 e^2\right )^4}+\frac {\left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}-\frac {\left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) \log \left (3+2 x+5 x^2\right )}{2 \left (5 d^2-2 d e+3 e^2\right )^4} \]
1/2*(-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) ^2+(-41*d^4+8*d^3*e+60*d^2*e^2-24*d*e^3+5*e^4)/(5*d^2-2*d*e+3*e^2)^3/(e*x+ d)+1/28*(-1367*d^3+879*d^2*e+2109*d*e^2-457*e^3-(423*d^3-4101*d^2*e+879*d* e^2+703*e^3)*x)/(5*d^2-2*d*e+3*e^2)^3/(5*x^2+2*x+3)+(205*d^5-19*d^4*e-846* d^3*e^2+396*d^2*e^3+57*d*e^4-21*e^5)*ln(e*x+d)/(5*d^2-2*d*e+3*e^2)^4-1/2*( 205*d^5-19*d^4*e-846*d^3*e^2+396*d^2*e^3+57*d*e^4-21*e^5)*ln(5*x^2+2*x+3)/ (5*d^2-2*d*e+3*e^2)^4+1/392*(6565*d^5-74017*d^4*e+35022*d^3*e^2+42858*d^2* e^3-17247*d*e^4+579*e^5)*arctan(1/14*(1+5*x)*14^(1/2))/(5*d^2-2*d*e+3*e^2) ^4*14^(1/2)
Time = 0.22 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.88 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 \left (3+2 x+5 x^2\right )^2} \, dx=\frac {-\frac {196 \left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e (d+e x)^2}+\frac {392 \left (5 d^2-2 d e+3 e^2\right ) \left (-41 d^4+8 d^3 e+60 d^2 e^2-24 d e^3+5 e^4\right )}{d+e x}-\frac {14 \left (5 d^2-2 d e+3 e^2\right ) \left (3 d e^2 (-703+293 x)+d^3 (1367+423 x)+e^3 (457+703 x)-3 d^2 e (293+1367 x)\right )}{3+2 x+5 x^2}+\sqrt {14} \left (6565 d^5-74017 d^4 e+35022 d^3 e^2+42858 d^2 e^3-17247 d e^4+579 e^5\right ) \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )+392 \left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) \log (d+e x)+196 \left (-205 d^5+19 d^4 e+846 d^3 e^2-396 d^2 e^3-57 d e^4+21 e^5\right ) \log \left (3+2 x+5 x^2\right )}{392 \left (5 d^2-2 d e+3 e^2\right )^4} \]
((-196*(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*(d + e*x)^2) + (392*(5*d^2 - 2*d*e + 3*e^2)*(-41*d^4 + 8*d^3*e + 60*d^2*e^2 - 24*d*e^3 + 5*e^4))/(d + e*x) - (14*(5*d^2 - 2*d*e + 3*e^2)*(3 *d*e^2*(-703 + 293*x) + d^3*(1367 + 423*x) + e^3*(457 + 703*x) - 3*d^2*e*( 293 + 1367*x)))/(3 + 2*x + 5*x^2) + Sqrt[14]*(6565*d^5 - 74017*d^4*e + 350 22*d^3*e^2 + 42858*d^2*e^3 - 17247*d*e^4 + 579*e^5)*ArcTan[(1 + 5*x)/Sqrt[ 14]] + 392*(205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21 *e^5)*Log[d + e*x] + 196*(-205*d^5 + 19*d^4*e + 846*d^3*e^2 - 396*d^2*e^3 - 57*d*e^4 + 21*e^5)*Log[3 + 2*x + 5*x^2])/(392*(5*d^2 - 2*d*e + 3*e^2)^4)
Time = 1.18 (sec) , antiderivative size = 411, 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.
| \(\displaystyle \int \frac {4 x^4-5 x^3+3 x^2+x+2}{\left (5 x^2+2 x+3\right )^2 (d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 2177 |
| \(\displaystyle \frac {1}{56} \int \frac {2 \left (-\frac {e^3 \left (423 d^3-4101 e d^2+879 e^2 d+703 e^3\right ) x^3}{\left (5 d^2-2 e d+3 e^2\right )^3}+\frac {\left (2800 d^6-3360 e d^5+5115 e^2 d^4+5527 e^3 d^3+1311 e^4 d^2+1251 e^5 d-28 e^6\right ) x^2}{\left (5 d^2-2 e d+3 e^2\right )^3}-\frac {\left (4620 d^6-4275 e d^5+5925 e^2 d^4-5651 e^3 d^3-663 e^4 d^2-168 e^5 d+84 e^6\right ) x}{\left (5 d^2-2 e d+3 e^2\right )^3}+\frac {3 \left (615 d^6-2105 e d^5+2535 e^2 d^4-1037 e^3 d^3+1064 e^4 d^2-336 e^5 d+168 e^6\right )}{\left (5 d^2-2 e d+3 e^2\right )^3}\right )}{(d+e x)^3 \left (5 x^2+2 x+3\right )}dx-\frac {1367 d^3-879 d^2 e+x \left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right )-2109 d e^2+457 e^3}{28 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \int \frac {-\frac {e^3 \left (423 d^3-4101 e d^2+879 e^2 d+703 e^3\right ) x^3}{\left (5 d^2-2 e d+3 e^2\right )^3}+\frac {\left (2800 d^6-3360 e d^5+5115 e^2 d^4+5527 e^3 d^3+1311 e^4 d^2+1251 e^5 d-28 e^6\right ) x^2}{\left (5 d^2-2 e d+3 e^2\right )^3}-\frac {\left (4620 d^6-4275 e d^5+5925 e^2 d^4-5651 e^3 d^3-663 e^4 d^2-168 e^5 d+84 e^6\right ) x}{\left (5 d^2-2 e d+3 e^2\right )^3}+\frac {3 \left (615 d^6-2105 e d^5+2535 e^2 d^4-1037 e^3 d^3+1064 e^4 d^2-336 e^5 d+168 e^6\right )}{\left (5 d^2-2 e d+3 e^2\right )^3}}{(d+e x)^3 \left (5 x^2+2 x+3\right )}dx-\frac {1367 d^3-879 d^2 e+x \left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right )-2109 d e^2+457 e^3}{28 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^3}\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \frac {1}{28} \int \left (\frac {28 \left (4 d^4+5 e d^3+3 e^2 d^2-e^3 d+2 e^4\right )}{\left (5 d^2-2 e d+3 e^2\right )^2 (d+e x)^3}-\frac {28 e \left (-205 d^5+19 e d^4+846 e^2 d^3-396 e^3 d^2-57 e^4 d+21 e^5\right )}{\left (5 d^2-2 e d+3 e^2\right )^4 (d+e x)}+\frac {3 \left (275 d^5-24495 e d^4+19570 e^2 d^3+10590 e^3 d^2-6281 e^4 d+389 e^5\right )-140 \left (205 d^5-19 e d^4-846 e^2 d^3+396 e^3 d^2+57 e^4 d-21 e^5\right ) x}{\left (5 d^2-2 e d+3 e^2\right )^4 \left (5 x^2+2 x+3\right )}-\frac {28 e \left (-41 d^4+8 e d^3+60 e^2 d^2-24 e^3 d+5 e^4\right )}{\left (5 d^2-2 e d+3 e^2\right )^3 (d+e x)^2}\right )dx-\frac {1367 d^3-879 d^2 e+x \left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right )-2109 d e^2+457 e^3}{28 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{28} \left (\frac {\arctan \left (\frac {5 x+1}{\sqrt {14}}\right ) \left (6565 d^5-74017 d^4 e+35022 d^3 e^2+42858 d^2 e^3-17247 d e^4+579 e^5\right )}{\sqrt {14} \left (5 d^2-2 d e+3 e^2\right )^4}-\frac {28 \left (41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right )}{\left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}-\frac {14 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)^2}-\frac {14 \left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) \log \left (5 x^2+2 x+3\right )}{\left (5 d^2-2 d e+3 e^2\right )^4}+\frac {28 \left (205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right ) \log (d+e x)}{\left (5 d^2-2 d e+3 e^2\right )^4}\right )-\frac {1367 d^3-879 d^2 e+x \left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right )-2109 d e^2+457 e^3}{28 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )^3}\) |
-1/28*(1367*d^3 - 879*d^2*e - 2109*d*e^2 + 457*e^3 + (423*d^3 - 4101*d^2*e + 879*d*e^2 + 703*e^3)*x)/((5*d^2 - 2*d*e + 3*e^2)^3*(3 + 2*x + 5*x^2)) + ((-14*(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)^2) - (28*(41*d^4 - 8*d^3*e - 60*d^2*e^2 + 24*d*e^3 - 5* e^4))/((5*d^2 - 2*d*e + 3*e^2)^3*(d + e*x)) + ((6565*d^5 - 74017*d^4*e + 3 5022*d^3*e^2 + 42858*d^2*e^3 - 17247*d*e^4 + 579*e^5)*ArcTan[(1 + 5*x)/Sqr t[14]])/(Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^4) + (28*(205*d^5 - 19*d^4*e - 8 46*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21*e^5)*Log[d + e*x])/(5*d^2 - 2*d*e + 3*e^2)^4 - (14*(205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e ^4 - 21*e^5)*Log[3 + 2*x + 5*x^2])/(5*d^2 - 2*d*e + 3*e^2)^4)/28
3.4.17.3.1 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 = 0.86 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {\frac {\left (\frac {423}{28} d^{5}-\frac {21351}{140} d^{4} e +\frac {6933}{70} d^{3} e^{2}-\frac {5273}{70} d^{2} e^{3}+\frac {1231}{140} d \,e^{4}+\frac {2109}{140} e^{5}\right ) x +\frac {1367 d^{5}}{28}-\frac {7129 d^{4} e}{140}-\frac {2343 d^{3} e^{2}}{70}+\frac {1933 d^{2} e^{3}}{70}-\frac {7241 d \,e^{4}}{140}+\frac {1371 e^{5}}{140}}{x^{2}+\frac {2}{5} x +\frac {3}{5}}+\frac {\left (28700 d^{5}-2660 d^{4} e -118440 d^{3} e^{2}+55440 d^{2} e^{3}+7980 d \,e^{4}-2940 e^{5}\right ) \ln \left (5 x^{2}+2 x +3\right )}{280}+\frac {\left (-6565 d^{5}+74017 d^{4} e -35022 d^{3} e^{2}-42858 d^{2} e^{3}+17247 d \,e^{4}-579 e^{5}\right ) \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{392}}{\left (5 d^{2}-2 d e +3 e^{2}\right )^{4}}-\frac {4 d^{4}+5 d^{3} e +3 d^{2} e^{2}-d \,e^{3}+2 e^{4}}{2 \left (5 d^{2}-2 d e +3 e^{2}\right )^{2} e \left (e x +d \right )^{2}}-\frac {41 d^{4}-8 d^{3} e -60 d^{2} e^{2}+24 d \,e^{3}-5 e^{4}}{\left (5 d^{2}-2 d e +3 e^{2}\right )^{3} \left (e x +d \right )}+\frac {\left (205 d^{5}-19 d^{4} e -846 d^{3} e^{2}+396 d^{2} e^{3}+57 d \,e^{4}-21 e^{5}\right ) \ln \left (e x +d \right )}{\left (5 d^{2}-2 d e +3 e^{2}\right )^{4}}\) | \(400\) |
| risch | \(\text {Expression too large to display}\) | \(1549\) |
-1/(5*d^2-2*d*e+3*e^2)^4*(((423/28*d^5-21351/140*d^4*e+6933/70*d^3*e^2-527 3/70*d^2*e^3+1231/140*d*e^4+2109/140*e^5)*x+1367/28*d^5-7129/140*d^4*e-234 3/70*d^3*e^2+1933/70*d^2*e^3-7241/140*d*e^4+1371/140*e^5)/(x^2+2/5*x+3/5)+ 1/280*(28700*d^5-2660*d^4*e-118440*d^3*e^2+55440*d^2*e^3+7980*d*e^4-2940*e ^5)*ln(5*x^2+2*x+3)+1/392*(-6565*d^5+74017*d^4*e-35022*d^3*e^2-42858*d^2*e ^3+17247*d*e^4-579*e^5)*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2)))-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)^2-(41*d^4 -8*d^3*e-60*d^2*e^2+24*d*e^3-5*e^4)/(5*d^2-2*d*e+3*e^2)^3/(e*x+d)+(205*d^5 -19*d^4*e-846*d^3*e^2+396*d^2*e^3+57*d*e^4-21*e^5)*ln(e*x+d)/(5*d^2-2*d*e+ 3*e^2)^4
Leaf count of result is larger than twice the leaf count of optimal. 1499 vs. \(2 (401) = 802\).
Time = 0.45 (sec) , antiderivative size = 1499, normalized size of antiderivative = 3.64 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 \left (3+2 x+5 x^2\right )^2} \, dx=\text {Too large to display} \]
-1/392*(58800*d^8 + 363230*d^7*e - 178010*d^6*e^2 - 233184*d^5*e^3 + 39516 4*d^4*e^4 - 437122*d^3*e^5 + 178542*d^2*e^6 - 37044*d*e^7 + 10584*e^8 + 14 *(28700*d^6*e^2 - 14965*d^5*e^3 - 43891*d^4*e^4 + 44106*d^3*e^5 - 45966*d^ 2*e^6 + 12711*d*e^7 + 9*e^8)*x^3 + 14*(7000*d^8 + 31850*d^7*e + 6400*d^6*e ^2 - 62649*d^5*e^3 + 52187*d^4*e^4 - 53652*d^3*e^5 + 11130*d^2*e^6 - 2841* d*e^7 + 1791*e^8)*x^2 - sqrt(14)*(19695*d^7*e - 222051*d^6*e^2 + 105066*d^ 5*e^3 + 128574*d^4*e^4 - 51741*d^3*e^5 + 1737*d^2*e^6 + 5*(6565*d^5*e^3 - 74017*d^4*e^4 + 35022*d^3*e^5 + 42858*d^2*e^6 - 17247*d*e^7 + 579*e^8)*x^4 + 2*(32825*d^6*e^2 - 363520*d^5*e^3 + 101093*d^4*e^4 + 249312*d^3*e^5 - 4 3377*d^2*e^6 - 14352*d*e^7 + 579*e^8)*x^3 + (32825*d^7*e - 343825*d^6*e^2 - 101263*d^5*e^3 + 132327*d^4*e^4 + 190263*d^3*e^5 + 62481*d^2*e^6 - 49425 *d*e^7 + 1737*e^8)*x^2 + 2*(6565*d^7*e - 54322*d^6*e^2 - 187029*d^5*e^3 + 147924*d^4*e^4 + 111327*d^3*e^5 - 51162*d^2*e^6 + 1737*d*e^7)*x)*arctan(1/ 14*sqrt(14)*(5*x + 1)) + 14*(2800*d^8 + 14855*d^7*e + 5815*d^6*e^2 - 18620 *d^5*e^3 - 17202*d^4*e^4 + 11119*d^3*e^5 - 26037*d^2*e^6 + 7866*d*e^7 - 75 6*e^8)*x - 392*(615*d^7*e - 57*d^6*e^2 - 2538*d^5*e^3 + 1188*d^4*e^4 + 171 *d^3*e^5 - 63*d^2*e^6 + 5*(205*d^5*e^3 - 19*d^4*e^4 - 846*d^3*e^5 + 396*d^ 2*e^6 + 57*d*e^7 - 21*e^8)*x^4 + 2*(1025*d^6*e^2 + 110*d^5*e^3 - 4249*d^4* e^4 + 1134*d^3*e^5 + 681*d^2*e^6 - 48*d*e^7 - 21*e^8)*x^3 + (1025*d^7*e + 725*d^6*e^2 - 3691*d^5*e^3 - 1461*d^4*e^4 - 669*d^3*e^5 + 1311*d^2*e^6 ...
Timed out. \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 \left (3+2 x+5 x^2\right )^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 851 vs. \(2 (401) = 802\).
Time = 0.30 (sec) , antiderivative size = 851, normalized size of antiderivative = 2.07 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 \left (3+2 x+5 x^2\right )^2} \, dx=\frac {\sqrt {14} {\left (6565 \, d^{5} - 74017 \, d^{4} e + 35022 \, d^{3} e^{2} + 42858 \, d^{2} e^{3} - 17247 \, d e^{4} + 579 \, e^{5}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right )}{392 \, {\left (625 \, d^{8} - 1000 \, d^{7} e + 2100 \, d^{6} e^{2} - 1960 \, d^{5} e^{3} + 2086 \, d^{4} e^{4} - 1176 \, d^{3} e^{5} + 756 \, d^{2} e^{6} - 216 \, d e^{7} + 81 \, e^{8}\right )}} + \frac {{\left (205 \, d^{5} - 19 \, d^{4} e - 846 \, d^{3} e^{2} + 396 \, d^{2} e^{3} + 57 \, d e^{4} - 21 \, e^{5}\right )} \log \left (e x + d\right )}{625 \, d^{8} - 1000 \, d^{7} e + 2100 \, d^{6} e^{2} - 1960 \, d^{5} e^{3} + 2086 \, d^{4} e^{4} - 1176 \, d^{3} e^{5} + 756 \, d^{2} e^{6} - 216 \, d e^{7} + 81 \, e^{8}} - \frac {{\left (205 \, d^{5} - 19 \, d^{4} e - 846 \, d^{3} e^{2} + 396 \, d^{2} e^{3} + 57 \, d e^{4} - 21 \, e^{5}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{2 \, {\left (625 \, d^{8} - 1000 \, d^{7} e + 2100 \, d^{6} e^{2} - 1960 \, d^{5} e^{3} + 2086 \, d^{4} e^{4} - 1176 \, d^{3} e^{5} + 756 \, d^{2} e^{6} - 216 \, d e^{7} + 81 \, e^{8}\right )}} - \frac {840 \, d^{6} + 5525 \, d^{5} e - 837 \, d^{4} e^{2} - 6981 \, d^{3} e^{3} + 3355 \, d^{2} e^{4} - 714 \, d e^{5} + 252 \, e^{6} + {\left (5740 \, d^{4} e^{2} - 697 \, d^{3} e^{3} - 12501 \, d^{2} e^{4} + 4239 \, d e^{5} + 3 \, e^{6}\right )} x^{3} + {\left (1400 \, d^{6} + 6930 \, d^{5} e + 3212 \, d^{4} e^{2} - 15403 \, d^{3} e^{3} + 2349 \, d^{2} e^{4} - 549 \, d e^{5} + 597 \, e^{6}\right )} x^{2} + {\left (560 \, d^{6} + 3195 \, d^{5} e + 2105 \, d^{4} e^{2} - 4799 \, d^{3} e^{3} - 6623 \, d^{2} e^{4} + 2454 \, d e^{5} - 252 \, e^{6}\right )} x}{28 \, {\left (375 \, d^{8} e - 450 \, d^{7} e^{2} + 855 \, d^{6} e^{3} - 564 \, d^{5} e^{4} + 513 \, d^{4} e^{5} - 162 \, d^{3} e^{6} + 81 \, d^{2} e^{7} + 5 \, {\left (125 \, d^{6} e^{3} - 150 \, d^{5} e^{4} + 285 \, d^{4} e^{5} - 188 \, d^{3} e^{6} + 171 \, d^{2} e^{7} - 54 \, d e^{8} + 27 \, e^{9}\right )} x^{4} + 2 \, {\left (625 \, d^{7} e^{2} - 625 \, d^{6} e^{3} + 1275 \, d^{5} e^{4} - 655 \, d^{4} e^{5} + 667 \, d^{3} e^{6} - 99 \, d^{2} e^{7} + 81 \, d e^{8} + 27 \, e^{9}\right )} x^{3} + {\left (625 \, d^{8} e - 250 \, d^{7} e^{2} + 1200 \, d^{6} e^{3} - 250 \, d^{5} e^{4} + 958 \, d^{4} e^{5} - 150 \, d^{3} e^{6} + 432 \, d^{2} e^{7} - 54 \, d e^{8} + 81 \, e^{9}\right )} x^{2} + 2 \, {\left (125 \, d^{8} e + 225 \, d^{7} e^{2} - 165 \, d^{6} e^{3} + 667 \, d^{5} e^{4} - 393 \, d^{4} e^{5} + 459 \, d^{3} e^{6} - 135 \, d^{2} e^{7} + 81 \, d e^{8}\right )} x\right )}} \]
1/392*sqrt(14)*(6565*d^5 - 74017*d^4*e + 35022*d^3*e^2 + 42858*d^2*e^3 - 1 7247*d*e^4 + 579*e^5)*arctan(1/14*sqrt(14)*(5*x + 1))/(625*d^8 - 1000*d^7* e + 2100*d^6*e^2 - 1960*d^5*e^3 + 2086*d^4*e^4 - 1176*d^3*e^5 + 756*d^2*e^ 6 - 216*d*e^7 + 81*e^8) + (205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21*e^5)*log(e*x + d)/(625*d^8 - 1000*d^7*e + 2100*d^6*e^2 - 1 960*d^5*e^3 + 2086*d^4*e^4 - 1176*d^3*e^5 + 756*d^2*e^6 - 216*d*e^7 + 81*e ^8) - 1/2*(205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21* e^5)*log(5*x^2 + 2*x + 3)/(625*d^8 - 1000*d^7*e + 2100*d^6*e^2 - 1960*d^5* e^3 + 2086*d^4*e^4 - 1176*d^3*e^5 + 756*d^2*e^6 - 216*d*e^7 + 81*e^8) - 1/ 28*(840*d^6 + 5525*d^5*e - 837*d^4*e^2 - 6981*d^3*e^3 + 3355*d^2*e^4 - 714 *d*e^5 + 252*e^6 + (5740*d^4*e^2 - 697*d^3*e^3 - 12501*d^2*e^4 + 4239*d*e^ 5 + 3*e^6)*x^3 + (1400*d^6 + 6930*d^5*e + 3212*d^4*e^2 - 15403*d^3*e^3 + 2 349*d^2*e^4 - 549*d*e^5 + 597*e^6)*x^2 + (560*d^6 + 3195*d^5*e + 2105*d^4* e^2 - 4799*d^3*e^3 - 6623*d^2*e^4 + 2454*d*e^5 - 252*e^6)*x)/(375*d^8*e - 450*d^7*e^2 + 855*d^6*e^3 - 564*d^5*e^4 + 513*d^4*e^5 - 162*d^3*e^6 + 81*d ^2*e^7 + 5*(125*d^6*e^3 - 150*d^5*e^4 + 285*d^4*e^5 - 188*d^3*e^6 + 171*d^ 2*e^7 - 54*d*e^8 + 27*e^9)*x^4 + 2*(625*d^7*e^2 - 625*d^6*e^3 + 1275*d^5*e ^4 - 655*d^4*e^5 + 667*d^3*e^6 - 99*d^2*e^7 + 81*d*e^8 + 27*e^9)*x^3 + (62 5*d^8*e - 250*d^7*e^2 + 1200*d^6*e^3 - 250*d^5*e^4 + 958*d^4*e^5 - 150*d^3 *e^6 + 432*d^2*e^7 - 54*d*e^8 + 81*e^9)*x^2 + 2*(125*d^8*e + 225*d^7*e^...
Time = 0.30 (sec) , antiderivative size = 648, normalized size of antiderivative = 1.57 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 \left (3+2 x+5 x^2\right )^2} \, dx=\frac {\sqrt {14} {\left (6565 \, d^{5} - 74017 \, d^{4} e + 35022 \, d^{3} e^{2} + 42858 \, d^{2} e^{3} - 17247 \, d e^{4} + 579 \, e^{5}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right )}{392 \, {\left (625 \, d^{8} - 1000 \, d^{7} e + 2100 \, d^{6} e^{2} - 1960 \, d^{5} e^{3} + 2086 \, d^{4} e^{4} - 1176 \, d^{3} e^{5} + 756 \, d^{2} e^{6} - 216 \, d e^{7} + 81 \, e^{8}\right )}} - \frac {{\left (205 \, d^{5} - 19 \, d^{4} e - 846 \, d^{3} e^{2} + 396 \, d^{2} e^{3} + 57 \, d e^{4} - 21 \, e^{5}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{2 \, {\left (625 \, d^{8} - 1000 \, d^{7} e + 2100 \, d^{6} e^{2} - 1960 \, d^{5} e^{3} + 2086 \, d^{4} e^{4} - 1176 \, d^{3} e^{5} + 756 \, d^{2} e^{6} - 216 \, d e^{7} + 81 \, e^{8}\right )}} + \frac {{\left (205 \, d^{5} e - 19 \, d^{4} e^{2} - 846 \, d^{3} e^{3} + 396 \, d^{2} e^{4} + 57 \, d e^{5} - 21 \, e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{625 \, d^{8} e - 1000 \, d^{7} e^{2} + 2100 \, d^{6} e^{3} - 1960 \, d^{5} e^{4} + 2086 \, d^{4} e^{5} - 1176 \, d^{3} e^{6} + 756 \, d^{2} e^{7} - 216 \, d e^{8} + 81 \, e^{9}} - \frac {4200 \, d^{8} + 25945 \, d^{7} e - 12715 \, d^{6} e^{2} - 16656 \, d^{5} e^{3} + 28226 \, d^{4} e^{4} - 31223 \, d^{3} e^{5} + 12753 \, d^{2} e^{6} - 2646 \, d e^{7} + 756 \, e^{8} + {\left (28700 \, d^{6} e^{2} - 14965 \, d^{5} e^{3} - 43891 \, d^{4} e^{4} + 44106 \, d^{3} e^{5} - 45966 \, d^{2} e^{6} + 12711 \, d e^{7} + 9 \, e^{8}\right )} x^{3} + {\left (7000 \, d^{8} + 31850 \, d^{7} e + 6400 \, d^{6} e^{2} - 62649 \, d^{5} e^{3} + 52187 \, d^{4} e^{4} - 53652 \, d^{3} e^{5} + 11130 \, d^{2} e^{6} - 2841 \, d e^{7} + 1791 \, e^{8}\right )} x^{2} + {\left (2800 \, d^{8} + 14855 \, d^{7} e + 5815 \, d^{6} e^{2} - 18620 \, d^{5} e^{3} - 17202 \, d^{4} e^{4} + 11119 \, d^{3} e^{5} - 26037 \, d^{2} e^{6} + 7866 \, d e^{7} - 756 \, e^{8}\right )} x}{28 \, {\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}^{4} {\left (e x + d\right )}^{2} {\left (5 \, x^{2} + 2 \, x + 3\right )} e} \]
1/392*sqrt(14)*(6565*d^5 - 74017*d^4*e + 35022*d^3*e^2 + 42858*d^2*e^3 - 1 7247*d*e^4 + 579*e^5)*arctan(1/14*sqrt(14)*(5*x + 1))/(625*d^8 - 1000*d^7* e + 2100*d^6*e^2 - 1960*d^5*e^3 + 2086*d^4*e^4 - 1176*d^3*e^5 + 756*d^2*e^ 6 - 216*d*e^7 + 81*e^8) - 1/2*(205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2* e^3 + 57*d*e^4 - 21*e^5)*log(5*x^2 + 2*x + 3)/(625*d^8 - 1000*d^7*e + 2100 *d^6*e^2 - 1960*d^5*e^3 + 2086*d^4*e^4 - 1176*d^3*e^5 + 756*d^2*e^6 - 216* d*e^7 + 81*e^8) + (205*d^5*e - 19*d^4*e^2 - 846*d^3*e^3 + 396*d^2*e^4 + 57 *d*e^5 - 21*e^6)*log(abs(e*x + d))/(625*d^8*e - 1000*d^7*e^2 + 2100*d^6*e^ 3 - 1960*d^5*e^4 + 2086*d^4*e^5 - 1176*d^3*e^6 + 756*d^2*e^7 - 216*d*e^8 + 81*e^9) - 1/28*(4200*d^8 + 25945*d^7*e - 12715*d^6*e^2 - 16656*d^5*e^3 + 28226*d^4*e^4 - 31223*d^3*e^5 + 12753*d^2*e^6 - 2646*d*e^7 + 756*e^8 + (28 700*d^6*e^2 - 14965*d^5*e^3 - 43891*d^4*e^4 + 44106*d^3*e^5 - 45966*d^2*e^ 6 + 12711*d*e^7 + 9*e^8)*x^3 + (7000*d^8 + 31850*d^7*e + 6400*d^6*e^2 - 62 649*d^5*e^3 + 52187*d^4*e^4 - 53652*d^3*e^5 + 11130*d^2*e^6 - 2841*d*e^7 + 1791*e^8)*x^2 + (2800*d^8 + 14855*d^7*e + 5815*d^6*e^2 - 18620*d^5*e^3 - 17202*d^4*e^4 + 11119*d^3*e^5 - 26037*d^2*e^6 + 7866*d*e^7 - 756*e^8)*x)/( (5*d^2 - 2*d*e + 3*e^2)^4*(e*x + d)^2*(5*x^2 + 2*x + 3)*e)
Time = 14.36 (sec) , antiderivative size = 887, normalized size of antiderivative = 2.15 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 \left (3+2 x+5 x^2\right )^2} \, dx=\ln \left (d+e\,x\right )\,\left (\frac {\frac {41\,d}{5}+\frac {29\,e}{5}}{{\left (5\,d^2-2\,d\,e+3\,e^2\right )}^2}+\frac {168\,e^4\,\left (458\,d-7\,e\right )}{125\,{\left (5\,d^2-2\,d\,e+3\,e^2\right )}^4}-\frac {2\,e^2\,\left (12610\,d+1329\,e\right )}{125\,{\left (5\,d^2-2\,d\,e+3\,e^2\right )}^3}\right )-\frac {\frac {840\,d^6+5525\,d^5\,e-837\,d^4\,e^2-6981\,d^3\,e^3+3355\,d^2\,e^4-714\,d\,e^5+252\,e^6}{28\,e\,\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 {x^3\,\left (5740\,d^4\,e-697\,d^3\,e^2-12501\,d^2\,e^3+4239\,d\,e^4+3\,e^5\right )}{28\,\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 {x^2\,\left (1400\,d^6+6930\,d^5\,e+3212\,d^4\,e^2-15403\,d^3\,e^3+2349\,d^2\,e^4-549\,d\,e^5+597\,e^6\right )}{28\,e\,\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 {x\,\left (560\,d^6+3195\,d^5\,e+2105\,d^4\,e^2-4799\,d^3\,e^3-6623\,d^2\,e^4+2454\,d\,e^5-252\,e^6\right )}{28\,e\,\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 )}}{x^2\,\left (5\,d^2+4\,d\,e+3\,e^2\right )+x\,\left (2\,d^2+6\,e\,d\right )+3\,d^2+x^3\,\left (2\,e^2+10\,d\,e\right )+5\,e^2\,x^4}+\frac {\ln \left (x+\frac {1}{5}-\frac {\sqrt {14}\,1{}\mathrm {i}}{5}\right )\,\left (\left (\frac {6565\,\sqrt {14}}{784}-\frac {205}{2}{}\mathrm {i}\right )\,d^5+\left (-\frac {74017\,\sqrt {14}}{784}+\frac {19}{2}{}\mathrm {i}\right )\,d^4\,e+\left (\frac {17511\,\sqrt {14}}{392}+423{}\mathrm {i}\right )\,d^3\,e^2+\left (\frac {21429\,\sqrt {14}}{392}-198{}\mathrm {i}\right )\,d^2\,e^3+\left (-\frac {17247\,\sqrt {14}}{784}-\frac {57}{2}{}\mathrm {i}\right )\,d\,e^4+\left (\frac {579\,\sqrt {14}}{784}+\frac {21}{2}{}\mathrm {i}\right )\,e^5\right )}{d^8\,625{}\mathrm {i}-d^7\,e\,1000{}\mathrm {i}+d^6\,e^2\,2100{}\mathrm {i}-d^5\,e^3\,1960{}\mathrm {i}+d^4\,e^4\,2086{}\mathrm {i}-d^3\,e^5\,1176{}\mathrm {i}+d^2\,e^6\,756{}\mathrm {i}-d\,e^7\,216{}\mathrm {i}+e^8\,81{}\mathrm {i}}-\frac {\ln \left (x+\frac {1}{5}+\frac {\sqrt {14}\,1{}\mathrm {i}}{5}\right )\,\left (\left (\frac {6565\,\sqrt {14}}{784}+\frac {205}{2}{}\mathrm {i}\right )\,d^5+\left (-\frac {74017\,\sqrt {14}}{784}-\frac {19}{2}{}\mathrm {i}\right )\,d^4\,e+\left (\frac {17511\,\sqrt {14}}{392}-423{}\mathrm {i}\right )\,d^3\,e^2+\left (\frac {21429\,\sqrt {14}}{392}+198{}\mathrm {i}\right )\,d^2\,e^3+\left (-\frac {17247\,\sqrt {14}}{784}+\frac {57}{2}{}\mathrm {i}\right )\,d\,e^4+\left (\frac {579\,\sqrt {14}}{784}-\frac {21}{2}{}\mathrm {i}\right )\,e^5\right )}{d^8\,625{}\mathrm {i}-d^7\,e\,1000{}\mathrm {i}+d^6\,e^2\,2100{}\mathrm {i}-d^5\,e^3\,1960{}\mathrm {i}+d^4\,e^4\,2086{}\mathrm {i}-d^3\,e^5\,1176{}\mathrm {i}+d^2\,e^6\,756{}\mathrm {i}-d\,e^7\,216{}\mathrm {i}+e^8\,81{}\mathrm {i}} \]
log(d + e*x)*(((41*d)/5 + (29*e)/5)/(5*d^2 - 2*d*e + 3*e^2)^2 + (168*e^4*( 458*d - 7*e))/(125*(5*d^2 - 2*d*e + 3*e^2)^4) - (2*e^2*(12610*d + 1329*e)) /(125*(5*d^2 - 2*d*e + 3*e^2)^3)) - ((5525*d^5*e - 714*d*e^5 + 840*d^6 + 2 52*e^6 + 3355*d^2*e^4 - 6981*d^3*e^3 - 837*d^4*e^2)/(28*e*(125*d^6 - 150*d ^5*e - 54*d*e^5 + 27*e^6 + 171*d^2*e^4 - 188*d^3*e^3 + 285*d^4*e^2)) + (x^ 3*(4239*d*e^4 + 5740*d^4*e + 3*e^5 - 12501*d^2*e^3 - 697*d^3*e^2))/(28*(12 5*d^6 - 150*d^5*e - 54*d*e^5 + 27*e^6 + 171*d^2*e^4 - 188*d^3*e^3 + 285*d^ 4*e^2)) + (x^2*(6930*d^5*e - 549*d*e^5 + 1400*d^6 + 597*e^6 + 2349*d^2*e^4 - 15403*d^3*e^3 + 3212*d^4*e^2))/(28*e*(125*d^6 - 150*d^5*e - 54*d*e^5 + 27*e^6 + 171*d^2*e^4 - 188*d^3*e^3 + 285*d^4*e^2)) + (x*(2454*d*e^5 + 3195 *d^5*e + 560*d^6 - 252*e^6 - 6623*d^2*e^4 - 4799*d^3*e^3 + 2105*d^4*e^2))/ (28*e*(125*d^6 - 150*d^5*e - 54*d*e^5 + 27*e^6 + 171*d^2*e^4 - 188*d^3*e^3 + 285*d^4*e^2)))/(x^2*(4*d*e + 5*d^2 + 3*e^2) + x*(6*d*e + 2*d^2) + 3*d^2 + x^3*(10*d*e + 2*e^2) + 5*e^2*x^4) + (log(x - (14^(1/2)*1i)/5 + 1/5)*(d^ 5*((6565*14^(1/2))/784 - 205i/2) + e^5*((579*14^(1/2))/784 + 21i/2) + d^3* e^2*((17511*14^(1/2))/392 + 423i) + d^2*e^3*((21429*14^(1/2))/392 - 198i) - d*e^4*((17247*14^(1/2))/784 + 57i/2) - d^4*e*((74017*14^(1/2))/784 - 19i /2)))/(d^8*625i - d^7*e*1000i - d*e^7*216i + e^8*81i + d^2*e^6*756i - d^3* e^5*1176i + d^4*e^4*2086i - d^5*e^3*1960i + d^6*e^2*2100i) - (log(x + (14^ (1/2)*1i)/5 + 1/5)*(d^5*((6565*14^(1/2))/784 + 205i/2) + e^5*((579*14^(...