Integrand size = 38, antiderivative size = 171 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {(83065 d-126009 e) e^2 x}{980000}+\frac {2 e^3 x^2}{125}-\frac {(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac {(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac {3 \left (353125 d^3-855175 d^2 e+74085 d e^2+556349 e^3\right ) \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )}{4900000 \sqrt {14}}+\frac {3 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (3+2 x+5 x^2\right )}{6250} \]
1/980000*(83065*d-126009*e)*e^2*x+2/125*e^3*x^2-1/7000*(1367+423*x)*(e*x+d )^3/(5*x^2+2*x+3)^2+1/196000*(e*x+d)^2*(34347*d-6315*e+(11015*d+49177*e)*x )/(5*x^2+2*x+3)+3/6250*e*(100*d^2-245*d*e+47*e^2)*ln(5*x^2+2*x+3)+3/686000 00*(353125*d^3-855175*d^2*e+74085*d*e^2+556349*e^3)*arctan(1/14*(1+5*x)*14 ^(1/2))*14^(1/2)
Time = 0.12 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.22 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {548800 (60 d-49 e) e^2 x+5488000 e^3 x^2-\frac {392 \left (e^3 (54969-53189 x)+125 d^3 (1367+423 x)+75 d^2 e (-1269+5989 x)-15 d e^2 (17967+18323 x)\right )}{\left (3+2 x+5 x^2\right )^2}+\frac {14 \left (e^3 (2639639-3109005 x)+125 d^3 (34347+11015 x)+75 d^2 e (-44399+181765 x)-15 d e^2 (809167+647195 x)\right )}{3+2 x+5 x^2}+15 \sqrt {14} \left (353125 d^3-855175 d^2 e+74085 d e^2+556349 e^3\right ) \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )+164640 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (3+2 x+5 x^2\right )}{343000000} \]
(548800*(60*d - 49*e)*e^2*x + 5488000*e^3*x^2 - (392*(e^3*(54969 - 53189*x ) + 125*d^3*(1367 + 423*x) + 75*d^2*e*(-1269 + 5989*x) - 15*d*e^2*(17967 + 18323*x)))/(3 + 2*x + 5*x^2)^2 + (14*(e^3*(2639639 - 3109005*x) + 125*d^3 *(34347 + 11015*x) + 75*d^2*e*(-44399 + 181765*x) - 15*d*e^2*(809167 + 647 195*x)))/(3 + 2*x + 5*x^2) + 15*Sqrt[14]*(353125*d^3 - 855175*d^2*e + 7408 5*d*e^2 + 556349*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + 164640*e*(100*d^2 - 245 *d*e + 47*e^2)*Log[3 + 2*x + 5*x^2])/343000000
Time = 0.64 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2175, 27, 2175, 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.
| \(\displaystyle \int \frac {\left (4 x^4-5 x^3+3 x^2+x+2\right ) (d+e x)^3}{\left (5 x^2+2 x+3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2175 |
| \(\displaystyle \frac {1}{112} \int \frac {2 (d+e x)^2 \left (5600 e x^3+280 (20 d-33 e) x^2-168 (55 d-27 e) x+3 (1089 d+1367 e)\right )}{125 \left (5 x^2+2 x+3\right )^2}dx-\frac {(423 x+1367) (d+e x)^3}{7000 \left (5 x^2+2 x+3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(d+e x)^2 \left (5600 e x^3+280 (20 d-33 e) x^2-168 (55 d-27 e) x+3 (1089 d+1367 e)\right )}{\left (5 x^2+2 x+3\right )^2}dx}{7000}-\frac {(423 x+1367) (d+e x)^3}{7000 \left (5 x^2+2 x+3\right )^2}\) |
| \(\Big \downarrow \) 2175 |
| \(\displaystyle \frac {\frac {1}{56} \int \frac {10 (d+e x) \left (6272 e^2 x^2+(10341 d-22693 e) e x+3 \left (2825 d^2-5587 e d+842 e^2\right )\right )}{5 x^2+2 x+3}dx+\frac {(x (11015 d+49177 e)+3 (11449 d-2105 e)) (d+e x)^2}{28 \left (5 x^2+2 x+3\right )}}{7000}-\frac {(423 x+1367) (d+e x)^3}{7000 \left (5 x^2+2 x+3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5}{28} \int \frac {(d+e x) \left (6272 e^2 x^2+(10341 d-22693 e) e x+3 \left (2825 d^2-5587 e d+842 e^2\right )\right )}{5 x^2+2 x+3}dx+\frac {(x (11015 d+49177 e)+3 (11449 d-2105 e)) (d+e x)^2}{28 \left (5 x^2+2 x+3\right )}}{7000}-\frac {(423 x+1367) (d+e x)^3}{7000 \left (5 x^2+2 x+3\right )^2}\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \frac {\frac {5}{28} \int \left (\frac {6272 x e^3}{5}+\frac {1}{25} (83065 d-126009 e) e^2+\frac {3 \left (70625 d^3-139675 e d^2-62015 e^2 d+126009 e^3+1568 e \left (100 d^2-245 e d+47 e^2\right ) x\right )}{25 \left (5 x^2+2 x+3\right )}\right )dx+\frac {(x (11015 d+49177 e)+3 (11449 d-2105 e)) (d+e x)^2}{28 \left (5 x^2+2 x+3\right )}}{7000}-\frac {(423 x+1367) (d+e x)^3}{7000 \left (5 x^2+2 x+3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {5}{28} \left (\frac {3 \arctan \left (\frac {5 x+1}{\sqrt {14}}\right ) \left (353125 d^3-855175 d^2 e+74085 d e^2+556349 e^3\right )}{125 \sqrt {14}}+\frac {2352}{125} e \left (100 d^2-245 d e+47 e^2\right ) \log \left (5 x^2+2 x+3\right )+\frac {1}{25} e^2 x (83065 d-126009 e)+\frac {3136 e^3 x^2}{5}\right )+\frac {(x (11015 d+49177 e)+3 (11449 d-2105 e)) (d+e x)^2}{28 \left (5 x^2+2 x+3\right )}}{7000}-\frac {(423 x+1367) (d+e x)^3}{7000 \left (5 x^2+2 x+3\right )^2}\) |
-1/7000*((1367 + 423*x)*(d + e*x)^3)/(3 + 2*x + 5*x^2)^2 + (((d + e*x)^2*( 3*(11449*d - 2105*e) + (11015*d + 49177*e)*x))/(28*(3 + 2*x + 5*x^2)) + (5 *(((83065*d - 126009*e)*e^2*x)/25 + (3136*e^3*x^2)/5 + (3*(353125*d^3 - 85 5175*d^2*e + 74085*d*e^2 + 556349*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]])/(125*Sq rt[14]) + (2352*e*(100*d^2 - 245*d*e + 47*e^2)*Log[3 + 2*x + 5*x^2])/125)) /28)/7000
3.4.18.3.1 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[Pq, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], S = Coeff[Polyno mialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*((R*b - 2*a*S + (2*c*R - b*S)*x)/((p + 1)*(b^2 - 4*a*c))), x ] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2 )^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(d + e*x)*Qx + S*(2*a*e*m + b*d *(2*p + 3)) - R*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*R - b*S)*(m + 2*p + 3)*x , 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] && GtQ[m, 0] && (Inte gerQ[p] || !IntegerQ[m] || !RationalQ[a, b, c, d, e]) && !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.84 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.22
| method | result | size |
| default | \(\frac {2 e^{3} x^{2}}{125}+\frac {12 d \,e^{2} x}{125}-\frac {49 e^{3} x}{625}+\frac {\left (\frac {11015}{1568} d^{3}+\frac {109059}{1568} d^{2} e -\frac {388317}{7840} d \,e^{2}-\frac {621801}{39200} e^{3}\right ) x^{3}+\left (\frac {38753}{1568} d^{3}+\frac {84921}{7840} d^{2} e -\frac {640827}{7840} d \,e^{2}+\frac {1396037}{196000} e^{3}\right ) x^{2}+\left (\frac {17979}{1568} d^{3}+\frac {173283}{7840} d^{2} e -\frac {73125}{1568} d \,e^{2}-\frac {511689}{196000} e^{3}\right ) x +\frac {12953 d^{3}}{1568}-\frac {58599 d^{2} e}{7840}-\frac {230931 d \,e^{2}}{7840}+\frac {1275957 e^{3}}{196000}}{25 \left (5 x^{2}+2 x +3\right )^{2}}+\frac {3 \left (156800 d^{2} e -384160 d \,e^{2}+73696 e^{3}\right ) \ln \left (5 x^{2}+2 x +3\right )}{9800000}+\frac {3 \left (70625 d^{3}-171035 d^{2} e +14817 d \,e^{2}+\frac {556349}{5} e^{3}\right ) \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{13720000}\) | \(209\) |
| risch | \(\frac {44451 \sqrt {14}\, d \,e^{2} \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{13720000}+\frac {1669047 \sqrt {14}\, e^{3} \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{68600000}+\frac {6 d^{2} e \ln \left (350 x^{2}+140 x +210\right )}{125}-\frac {147 d \,e^{2} \ln \left (350 x^{2}+140 x +210\right )}{1250}+\frac {339 \sqrt {14}\, d^{3} \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{21952}-\frac {102621 \sqrt {14}\, d^{2} e \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{2744000}+\frac {12 d \,e^{2} x}{125}+\frac {141 e^{3} \ln \left (350 x^{2}+140 x +210\right )}{6250}+\frac {2 e^{3} x^{2}}{125}-\frac {49 e^{3} x}{625}+\frac {\frac {\left (\frac {11015}{1568} d^{3}+\frac {109059}{1568} d^{2} e -\frac {388317}{7840} d \,e^{2}-\frac {621801}{39200} e^{3}\right ) x^{3}}{25}+\frac {\left (\frac {38753}{1568} d^{3}+\frac {84921}{7840} d^{2} e -\frac {640827}{7840} d \,e^{2}+\frac {1396037}{196000} e^{3}\right ) x^{2}}{25}+\frac {\left (\frac {17979}{1568} d^{3}+\frac {173283}{7840} d^{2} e -\frac {73125}{1568} d \,e^{2}-\frac {511689}{196000} e^{3}\right ) x}{25}+\frac {12953 d^{3}}{39200}-\frac {58599 d^{2} e}{196000}-\frac {230931 d \,e^{2}}{196000}+\frac {1275957 e^{3}}{4900000}}{\left (5 x^{2}+2 x +3\right )^{2}}\) | \(278\) |
2/125*e^3*x^2+12/125*d*e^2*x-49/625*e^3*x+1/25*((11015/1568*d^3+109059/156 8*d^2*e-388317/7840*d*e^2-621801/39200*e^3)*x^3+(38753/1568*d^3+84921/7840 *d^2*e-640827/7840*d*e^2+1396037/196000*e^3)*x^2+(17979/1568*d^3+173283/78 40*d^2*e-73125/1568*d*e^2-511689/196000*e^3)*x+12953/1568*d^3-58599/7840*d ^2*e-230931/7840*d*e^2+1275957/196000*e^3)/(5*x^2+2*x+3)^2+3/9800000*(1568 00*d^2*e-384160*d*e^2+73696*e^3)*ln(5*x^2+2*x+3)+3/13720000*(70625*d^3-171 035*d^2*e+14817*d*e^2+556349/5*e^3)*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2) )
Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (153) = 306\).
Time = 0.25 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.58 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {27440000 \, e^{3} x^{6} + 2744000 \, {\left (60 \, d e^{2} - 41 \, e^{3}\right )} x^{5} + 8780800 \, {\left (15 \, d e^{2} - 8 \, e^{3}\right )} x^{4} + 70 \, {\left (275375 \, d^{3} + 2726475 \, d^{2} e + 1257135 \, d e^{2} - 3045929 \, e^{3}\right )} x^{3} + 22667750 \, d^{3} - 20509650 \, d^{2} e - 80825850 \, d e^{2} + 17863398 \, e^{3} + 14 \, {\left (4844125 \, d^{3} + 2123025 \, d^{2} e - 10375875 \, d e^{2} - 2508283 \, e^{3}\right )} x^{2} + 3 \, \sqrt {14} {\left (25 \, {\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x^{4} + 20 \, {\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x^{3} + 3178125 \, d^{3} - 7696575 \, d^{2} e + 666765 \, d e^{2} + 5007141 \, e^{3} + 34 \, {\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x^{2} + 12 \, {\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + 42 \, {\left (749125 \, d^{3} + 1444025 \, d^{2} e - 1635675 \, d e^{2} - 1323043 \, e^{3}\right )} x + 32928 \, {\left (25 \, {\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x^{4} + 20 \, {\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x^{3} + 900 \, d^{2} e - 2205 \, d e^{2} + 423 \, e^{3} + 34 \, {\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x^{2} + 12 \, {\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{68600000 \, {\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \]
1/68600000*(27440000*e^3*x^6 + 2744000*(60*d*e^2 - 41*e^3)*x^5 + 8780800*( 15*d*e^2 - 8*e^3)*x^4 + 70*(275375*d^3 + 2726475*d^2*e + 1257135*d*e^2 - 3 045929*e^3)*x^3 + 22667750*d^3 - 20509650*d^2*e - 80825850*d*e^2 + 1786339 8*e^3 + 14*(4844125*d^3 + 2123025*d^2*e - 10375875*d*e^2 - 2508283*e^3)*x^ 2 + 3*sqrt(14)*(25*(353125*d^3 - 855175*d^2*e + 74085*d*e^2 + 556349*e^3)* x^4 + 20*(353125*d^3 - 855175*d^2*e + 74085*d*e^2 + 556349*e^3)*x^3 + 3178 125*d^3 - 7696575*d^2*e + 666765*d*e^2 + 5007141*e^3 + 34*(353125*d^3 - 85 5175*d^2*e + 74085*d*e^2 + 556349*e^3)*x^2 + 12*(353125*d^3 - 855175*d^2*e + 74085*d*e^2 + 556349*e^3)*x)*arctan(1/14*sqrt(14)*(5*x + 1)) + 42*(7491 25*d^3 + 1444025*d^2*e - 1635675*d*e^2 - 1323043*e^3)*x + 32928*(25*(100*d ^2*e - 245*d*e^2 + 47*e^3)*x^4 + 20*(100*d^2*e - 245*d*e^2 + 47*e^3)*x^3 + 900*d^2*e - 2205*d*e^2 + 423*e^3 + 34*(100*d^2*e - 245*d*e^2 + 47*e^3)*x^ 2 + 12*(100*d^2*e - 245*d*e^2 + 47*e^3)*x)*log(5*x^2 + 2*x + 3))/(25*x^4 + 20*x^3 + 34*x^2 + 12*x + 9)
Result contains complex when optimal does not.
Time = 2.95 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.74 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {2 e^{3} x^{2}}{125} + x \left (\frac {12 d e^{2}}{125} - \frac {49 e^{3}}{625}\right ) + \left (\frac {3 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{6250} - \frac {3 \sqrt {14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{137200000}\right ) \log {\left (x + \frac {211875 d^{3} - 1830225 d^{2} e + 3271395 d e^{2} - 285237 e^{3} + \frac {65856 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{5} - \frac {3 \sqrt {14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{5}}{1059375 d^{3} - 2565525 d^{2} e + 222255 d e^{2} + 1669047 e^{3}} \right )} + \left (\frac {3 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{6250} + \frac {3 \sqrt {14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{137200000}\right ) \log {\left (x + \frac {211875 d^{3} - 1830225 d^{2} e + 3271395 d e^{2} - 285237 e^{3} + \frac {65856 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{5} + \frac {3 \sqrt {14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{5}}{1059375 d^{3} - 2565525 d^{2} e + 222255 d e^{2} + 1669047 e^{3}} \right )} + \frac {1619125 d^{3} - 1464975 d^{2} e - 5773275 d e^{2} + 1275957 e^{3} + x^{3} \cdot \left (1376875 d^{3} + 13632375 d^{2} e - 9707925 d e^{2} - 3109005 e^{3}\right ) + x^{2} \cdot \left (4844125 d^{3} + 2123025 d^{2} e - 16020675 d e^{2} + 1396037 e^{3}\right ) + x \left (2247375 d^{3} + 4332075 d^{2} e - 9140625 d e^{2} - 511689 e^{3}\right )}{122500000 x^{4} + 98000000 x^{3} + 166600000 x^{2} + 58800000 x + 44100000} \]
2*e**3*x**2/125 + x*(12*d*e**2/125 - 49*e**3/625) + (3*e*(100*d**2 - 245*d *e + 47*e**2)/6250 - 3*sqrt(14)*I*(353125*d**3 - 855175*d**2*e + 74085*d*e **2 + 556349*e**3)/137200000)*log(x + (211875*d**3 - 1830225*d**2*e + 3271 395*d*e**2 - 285237*e**3 + 65856*e*(100*d**2 - 245*d*e + 47*e**2)/5 - 3*sq rt(14)*I*(353125*d**3 - 855175*d**2*e + 74085*d*e**2 + 556349*e**3)/5)/(10 59375*d**3 - 2565525*d**2*e + 222255*d*e**2 + 1669047*e**3)) + (3*e*(100*d **2 - 245*d*e + 47*e**2)/6250 + 3*sqrt(14)*I*(353125*d**3 - 855175*d**2*e + 74085*d*e**2 + 556349*e**3)/137200000)*log(x + (211875*d**3 - 1830225*d* *2*e + 3271395*d*e**2 - 285237*e**3 + 65856*e*(100*d**2 - 245*d*e + 47*e** 2)/5 + 3*sqrt(14)*I*(353125*d**3 - 855175*d**2*e + 74085*d*e**2 + 556349*e **3)/5)/(1059375*d**3 - 2565525*d**2*e + 222255*d*e**2 + 1669047*e**3)) + (1619125*d**3 - 1464975*d**2*e - 5773275*d*e**2 + 1275957*e**3 + x**3*(137 6875*d**3 + 13632375*d**2*e - 9707925*d*e**2 - 3109005*e**3) + x**2*(48441 25*d**3 + 2123025*d**2*e - 16020675*d*e**2 + 1396037*e**3) + x*(2247375*d* *3 + 4332075*d**2*e - 9140625*d*e**2 - 511689*e**3))/(122500000*x**4 + 980 00000*x**3 + 166600000*x**2 + 58800000*x + 44100000)
Time = 0.28 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.30 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {2}{125} \, e^{3} x^{2} + \frac {3}{68600000} \, \sqrt {14} {\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{625} \, {\left (60 \, d e^{2} - 49 \, e^{3}\right )} x + \frac {3}{6250} \, {\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac {5 \, {\left (275375 \, d^{3} + 2726475 \, d^{2} e - 1941585 \, d e^{2} - 621801 \, e^{3}\right )} x^{3} + 1619125 \, d^{3} - 1464975 \, d^{2} e - 5773275 \, d e^{2} + 1275957 \, e^{3} + {\left (4844125 \, d^{3} + 2123025 \, d^{2} e - 16020675 \, d e^{2} + 1396037 \, e^{3}\right )} x^{2} + 3 \, {\left (749125 \, d^{3} + 1444025 \, d^{2} e - 3046875 \, d e^{2} - 170563 \, e^{3}\right )} x}{4900000 \, {\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \]
2/125*e^3*x^2 + 3/68600000*sqrt(14)*(353125*d^3 - 855175*d^2*e + 74085*d*e ^2 + 556349*e^3)*arctan(1/14*sqrt(14)*(5*x + 1)) + 1/625*(60*d*e^2 - 49*e^ 3)*x + 3/6250*(100*d^2*e - 245*d*e^2 + 47*e^3)*log(5*x^2 + 2*x + 3) + 1/49 00000*(5*(275375*d^3 + 2726475*d^2*e - 1941585*d*e^2 - 621801*e^3)*x^3 + 1 619125*d^3 - 1464975*d^2*e - 5773275*d*e^2 + 1275957*e^3 + (4844125*d^3 + 2123025*d^2*e - 16020675*d*e^2 + 1396037*e^3)*x^2 + 3*(749125*d^3 + 144402 5*d^2*e - 3046875*d*e^2 - 170563*e^3)*x)/(25*x^4 + 20*x^3 + 34*x^2 + 12*x + 9)
Time = 0.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {2}{125} \, e^{3} x^{2} + \frac {12}{125} \, d e^{2} x - \frac {49}{625} \, e^{3} x + \frac {3}{68600000} \, \sqrt {14} {\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {3}{6250} \, {\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac {5 \, {\left (275375 \, d^{3} + 2726475 \, d^{2} e - 1941585 \, d e^{2} - 621801 \, e^{3}\right )} x^{3} + 1619125 \, d^{3} - 1464975 \, d^{2} e - 5773275 \, d e^{2} + 1275957 \, e^{3} + {\left (4844125 \, d^{3} + 2123025 \, d^{2} e - 16020675 \, d e^{2} + 1396037 \, e^{3}\right )} x^{2} + 3 \, {\left (749125 \, d^{3} + 1444025 \, d^{2} e - 3046875 \, d e^{2} - 170563 \, e^{3}\right )} x}{4900000 \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{2}} \]
2/125*e^3*x^2 + 12/125*d*e^2*x - 49/625*e^3*x + 3/68600000*sqrt(14)*(35312 5*d^3 - 855175*d^2*e + 74085*d*e^2 + 556349*e^3)*arctan(1/14*sqrt(14)*(5*x + 1)) + 3/6250*(100*d^2*e - 245*d*e^2 + 47*e^3)*log(5*x^2 + 2*x + 3) + 1/ 4900000*(5*(275375*d^3 + 2726475*d^2*e - 1941585*d*e^2 - 621801*e^3)*x^3 + 1619125*d^3 - 1464975*d^2*e - 5773275*d*e^2 + 1275957*e^3 + (4844125*d^3 + 2123025*d^2*e - 16020675*d*e^2 + 1396037*e^3)*x^2 + 3*(749125*d^3 + 1444 025*d^2*e - 3046875*d*e^2 - 170563*e^3)*x)/(5*x^2 + 2*x + 3)^2
Time = 0.16 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.75 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^3} \, dx=x\,\left (\frac {e^2\,\left (12\,d-5\,e\right )}{125}-\frac {24\,e^3}{625}\right )-\frac {\frac {1154655\,d\,e^2}{1568}+\frac {292995\,d^2\,e}{1568}+x\,\left (-\frac {449475\,d^3}{1568}-\frac {866415\,d^2\,e}{1568}+\frac {1828125\,d\,e^2}{1568}+\frac {511689\,e^3}{7840}\right )-\frac {323825\,d^3}{1568}-\frac {1275957\,e^3}{7840}+x^3\,\left (-\frac {275375\,d^3}{1568}-\frac {2726475\,d^2\,e}{1568}+\frac {1941585\,d\,e^2}{1568}+\frac {621801\,e^3}{1568}\right )-x^2\,\left (\frac {968825\,d^3}{1568}+\frac {424605\,d^2\,e}{1568}-\frac {3204135\,d\,e^2}{1568}+\frac {1396037\,e^3}{7840}\right )}{15625\,x^4+12500\,x^3+21250\,x^2+7500\,x+5625}+\ln \left (5\,x^2+2\,x+3\right )\,\left (\frac {6\,d^2\,e}{125}-\frac {147\,d\,e^2}{1250}+\frac {141\,e^3}{6250}\right )+\frac {2\,e^3\,x^2}{125}+\frac {3\,\sqrt {14}\,\mathrm {atan}\left (\frac {\frac {3\,\sqrt {14}\,\left (353125\,d^3-855175\,d^2\,e+74085\,d\,e^2+556349\,e^3\right )}{68600000}+\frac {3\,\sqrt {14}\,x\,\left (353125\,d^3-855175\,d^2\,e+74085\,d\,e^2+556349\,e^3\right )}{13720000}}{\frac {339\,d^3}{1568}-\frac {102621\,d^2\,e}{196000}+\frac {44451\,d\,e^2}{980000}+\frac {1669047\,e^3}{4900000}}\right )\,\left (353125\,d^3-855175\,d^2\,e+74085\,d\,e^2+556349\,e^3\right )}{68600000} \]
x*((e^2*(12*d - 5*e))/125 - (24*e^3)/625) - ((1154655*d*e^2)/1568 + (29299 5*d^2*e)/1568 + x*((1828125*d*e^2)/1568 - (866415*d^2*e)/1568 - (449475*d^ 3)/1568 + (511689*e^3)/7840) - (323825*d^3)/1568 - (1275957*e^3)/7840 + x^ 3*((1941585*d*e^2)/1568 - (2726475*d^2*e)/1568 - (275375*d^3)/1568 + (6218 01*e^3)/1568) - x^2*((424605*d^2*e)/1568 - (3204135*d*e^2)/1568 + (968825* d^3)/1568 + (1396037*e^3)/7840))/(7500*x + 21250*x^2 + 12500*x^3 + 15625*x ^4 + 5625) + log(2*x + 5*x^2 + 3)*((6*d^2*e)/125 - (147*d*e^2)/1250 + (141 *e^3)/6250) + (2*e^3*x^2)/125 + (3*14^(1/2)*atan(((3*14^(1/2)*(74085*d*e^2 - 855175*d^2*e + 353125*d^3 + 556349*e^3))/68600000 + (3*14^(1/2)*x*(7408 5*d*e^2 - 855175*d^2*e + 353125*d^3 + 556349*e^3))/13720000)/((44451*d*e^2 )/980000 - (102621*d^2*e)/196000 + (339*d^3)/1568 + (1669047*e^3)/4900000) )*(74085*d*e^2 - 855175*d^2*e + 353125*d^3 + 556349*e^3))/68600000