Integrand size = 38, antiderivative size = 221 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {\left (10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3\right ) x}{15625}-\frac {\left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right ) x^2}{6250}+\frac {\left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right ) x^3}{1875}+\frac {3}{500} e \left (100 d^2-165 d e+27 e^2\right ) x^4+\frac {3}{125} (20 d-11 e) e^2 x^5+\frac {2 e^3 x^6}{15}-\frac {\left (52875 d^3+449175 d^2 e-274845 d e^2-53189 e^3\right ) \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )}{78125 \sqrt {14}}+\frac {\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) \log \left (3+2 x+5 x^2\right )}{156250} \] Output:
1/15625*(10125*d^3+34350*d^2*e-13215*d*e^2-5108*e^3)*x-1/6250*(4125*d^3-60 75*d^2*e-6870*d*e^2+881*e^3)*x^2+1/1875*(500*d^3-2475*d^2*e+1215*d*e^2+458 *e^3)*x^3+3/500*e*(100*d^2-165*d*e+27*e^2)*x^4+3/125*(20*d-11*e)*e^2*x^5+2 /15*e^3*x^6-1/1093750*(52875*d^3+449175*d^2*e-274845*d*e^2-53189*e^3)*arct an(1/14*(1+5*x)*14^(1/2))*14^(1/2)+1/156250*(57250*d^3-66075*d^2*e-76620*d *e^2+23431*e^3)*ln(5*x^2+2*x+3)
Time = 0.13 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {35 x \left (250 d^3 \left (486-495 x+200 x^2\right )+450 d^2 e \left (916+405 x-550 x^2+250 x^3\right )+45 d e^2 \left (-3524+4580 x+2700 x^2-4125 x^3+2000 x^4\right )+e^3 \left (-61296-26430 x+45800 x^2+30375 x^3-49500 x^4+25000 x^5\right )\right )-6 \sqrt {14} \left (52875 d^3+449175 d^2 e-274845 d e^2-53189 e^3\right ) \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )+42 \left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) \log \left (3+2 x+5 x^2\right )}{6562500} \] Input:
Integrate[((d + e*x)^3*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2), x]
Output:
(35*x*(250*d^3*(486 - 495*x + 200*x^2) + 450*d^2*e*(916 + 405*x - 550*x^2 + 250*x^3) + 45*d*e^2*(-3524 + 4580*x + 2700*x^2 - 4125*x^3 + 2000*x^4) + e^3*(-61296 - 26430*x + 45800*x^2 + 30375*x^3 - 49500*x^4 + 25000*x^5)) - 6*Sqrt[14]*(52875*d^3 + 449175*d^2*e - 274845*d*e^2 - 53189*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + 42*(57250*d^3 - 66075*d^2*e - 76620*d*e^2 + 23431*e^3) *Log[3 + 2*x + 5*x^2])/6562500
Time = 0.74 (sec) , antiderivative size = 221, 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.
| \(\displaystyle \int \frac {\left (4 x^4-5 x^3+3 x^2+x+2\right ) (d+e x)^3}{5 x^2+2 x+3} \, dx\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (\frac {3}{125} e x^3 \left (100 d^2-165 d e+27 e^2\right )+\frac {1}{625} x^2 \left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right )+\frac {875 d^3-103050 d^2 e+x \left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right )+39645 d e^2+15324 e^3}{15625 \left (5 x^2+2 x+3\right )}-\frac {x \left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right )}{3125}+\frac {10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3}{15625}+\frac {3}{25} e^2 x^4 (20 d-11 e)+\frac {4 e^3 x^5}{5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {5 x+1}{\sqrt {14}}\right ) \left (52875 d^3+449175 d^2 e-274845 d e^2-53189 e^3\right )}{78125 \sqrt {14}}+\frac {3}{500} e x^4 \left (100 d^2-165 d e+27 e^2\right )+\frac {x^3 \left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right )}{1875}-\frac {x^2 \left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right )}{6250}+\frac {\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) \log \left (5 x^2+2 x+3\right )}{156250}+\frac {x \left (10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3\right )}{15625}+\frac {3}{125} e^2 x^5 (20 d-11 e)+\frac {2 e^3 x^6}{15}\) |
Input:
Int[((d + e*x)^3*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2),x]
Output:
((10125*d^3 + 34350*d^2*e - 13215*d*e^2 - 5108*e^3)*x)/15625 - ((4125*d^3 - 6075*d^2*e - 6870*d*e^2 + 881*e^3)*x^2)/6250 + ((500*d^3 - 2475*d^2*e + 1215*d*e^2 + 458*e^3)*x^3)/1875 + (3*e*(100*d^2 - 165*d*e + 27*e^2)*x^4)/5 00 + (3*(20*d - 11*e)*e^2*x^5)/125 + (2*e^3*x^6)/15 - ((52875*d^3 + 449175 *d^2*e - 274845*d*e^2 - 53189*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]])/(78125*Sqrt [14]) + ((57250*d^3 - 66075*d^2*e - 76620*d*e^2 + 23431*e^3)*Log[3 + 2*x + 5*x^2])/156250
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 = 0.28 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(\frac {2 e^{3} x^{6}}{15}+\frac {12 x^{5} d \,e^{2}}{25}-\frac {33 x^{5} e^{3}}{125}+\frac {3 x^{4} d^{2} e}{5}-\frac {99 x^{4} d \,e^{2}}{100}+\frac {81 e^{3} x^{4}}{500}+\frac {4 d^{3} x^{3}}{15}-\frac {33 d^{2} e \,x^{3}}{25}+\frac {81 d \,e^{2} x^{3}}{125}+\frac {458 e^{3} x^{3}}{1875}-\frac {33 d^{3} x^{2}}{50}+\frac {243 x^{2} d^{2} e}{250}+\frac {687 d \,e^{2} x^{2}}{625}-\frac {881 e^{3} x^{2}}{6250}+\frac {81 d^{3} x}{125}+\frac {1374 d^{2} e x}{625}-\frac {2643 d \,e^{2} x}{3125}-\frac {5108 e^{3} x}{15625}+\frac {\left (57250 d^{3}-66075 d^{2} e -76620 d \,e^{2}+23431 e^{3}\right ) \ln \left (5 x^{2}+2 x +3\right )}{156250}+\frac {\left (-10575 d^{3}-89835 d^{2} e +54969 d \,e^{2}+\frac {53189}{5} e^{3}\right ) \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{218750}\) | \(222\) |
| risch | \(-\frac {33 x^{5} e^{3}}{125}+\frac {4 d^{3} x^{3}}{15}+\frac {81 d^{3} x}{125}-\frac {2643 d^{2} e \ln \left (350 x^{2}+140 x +210\right )}{6250}-\frac {7662 d \,e^{2} \ln \left (350 x^{2}+140 x +210\right )}{15625}-\frac {423 \sqrt {14}\, d^{3} \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{8750}+\frac {53189 \sqrt {14}\, e^{3} \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{1093750}-\frac {881 e^{3} x^{2}}{6250}+\frac {81 e^{3} x^{4}}{500}+\frac {2 e^{3} x^{6}}{15}-\frac {33 d^{3} x^{2}}{50}+\frac {12 x^{5} d \,e^{2}}{25}+\frac {3 x^{4} d^{2} e}{5}-\frac {99 x^{4} d \,e^{2}}{100}+\frac {243 x^{2} d^{2} e}{250}-\frac {33 d^{2} e \,x^{3}}{25}+\frac {81 d \,e^{2} x^{3}}{125}-\frac {17967 \sqrt {14}\, d^{2} e \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{43750}+\frac {54969 \sqrt {14}\, d \,e^{2} \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{218750}-\frac {5108 e^{3} x}{15625}+\frac {1374 d^{2} e x}{625}+\frac {687 d \,e^{2} x^{2}}{625}+\frac {229 d^{3} \ln \left (350 x^{2}+140 x +210\right )}{625}+\frac {23431 e^{3} \ln \left (350 x^{2}+140 x +210\right )}{156250}-\frac {2643 d \,e^{2} x}{3125}+\frac {458 e^{3} x^{3}}{1875}\) | \(299\) |
Input:
int((e*x+d)^3*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x,method=_RETURNVERBOS E)
Output:
2/15*e^3*x^6+12/25*x^5*d*e^2-33/125*x^5*e^3+3/5*x^4*d^2*e-99/100*x^4*d*e^2 +81/500*e^3*x^4+4/15*d^3*x^3-33/25*d^2*e*x^3+81/125*d*e^2*x^3+458/1875*e^3 *x^3-33/50*d^3*x^2+243/250*x^2*d^2*e+687/625*d*e^2*x^2-881/6250*e^3*x^2+81 /125*d^3*x+1374/625*d^2*e*x-2643/3125*d*e^2*x-5108/15625*e^3*x+1/156250*(5 7250*d^3-66075*d^2*e-76620*d*e^2+23431*e^3)*ln(5*x^2+2*x+3)+1/218750*(-105 75*d^3-89835*d^2*e+54969*d*e^2+53189/5*e^3)*14^(1/2)*arctan(1/28*(10*x+2)* 14^(1/2))
Time = 0.07 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {2}{15} \, e^{3} x^{6} + \frac {3}{125} \, {\left (20 \, d e^{2} - 11 \, e^{3}\right )} x^{5} + \frac {3}{500} \, {\left (100 \, d^{2} e - 165 \, d e^{2} + 27 \, e^{3}\right )} x^{4} + \frac {1}{1875} \, {\left (500 \, d^{3} - 2475 \, d^{2} e + 1215 \, d e^{2} + 458 \, e^{3}\right )} x^{3} - \frac {1}{6250} \, {\left (4125 \, d^{3} - 6075 \, d^{2} e - 6870 \, d e^{2} + 881 \, e^{3}\right )} x^{2} - \frac {1}{1093750} \, \sqrt {14} {\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{15625} \, {\left (10125 \, d^{3} + 34350 \, d^{2} e - 13215 \, d e^{2} - 5108 \, e^{3}\right )} x + \frac {1}{156250} \, {\left (57250 \, d^{3} - 66075 \, d^{2} e - 76620 \, d e^{2} + 23431 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \] Input:
integrate((e*x+d)^3*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x, algorithm="fr icas")
Output:
2/15*e^3*x^6 + 3/125*(20*d*e^2 - 11*e^3)*x^5 + 3/500*(100*d^2*e - 165*d*e^ 2 + 27*e^3)*x^4 + 1/1875*(500*d^3 - 2475*d^2*e + 1215*d*e^2 + 458*e^3)*x^3 - 1/6250*(4125*d^3 - 6075*d^2*e - 6870*d*e^2 + 881*e^3)*x^2 - 1/1093750*s qrt(14)*(52875*d^3 + 449175*d^2*e - 274845*d*e^2 - 53189*e^3)*arctan(1/14* sqrt(14)*(5*x + 1)) + 1/15625*(10125*d^3 + 34350*d^2*e - 13215*d*e^2 - 510 8*e^3)*x + 1/156250*(57250*d^3 - 66075*d^2*e - 76620*d*e^2 + 23431*e^3)*lo g(5*x^2 + 2*x + 3)
Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 450, normalized size of antiderivative = 2.04 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {2 e^{3} x^{6}}{15} + x^{5} \cdot \left (\frac {12 d e^{2}}{25} - \frac {33 e^{3}}{125}\right ) + x^{4} \cdot \left (\frac {3 d^{2} e}{5} - \frac {99 d e^{2}}{100} + \frac {81 e^{3}}{500}\right ) + x^{3} \cdot \left (\frac {4 d^{3}}{15} - \frac {33 d^{2} e}{25} + \frac {81 d e^{2}}{125} + \frac {458 e^{3}}{1875}\right ) + x^{2} \left (- \frac {33 d^{3}}{50} + \frac {243 d^{2} e}{250} + \frac {687 d e^{2}}{625} - \frac {881 e^{3}}{6250}\right ) + x \left (\frac {81 d^{3}}{125} + \frac {1374 d^{2} e}{625} - \frac {2643 d e^{2}}{3125} - \frac {5108 e^{3}}{15625}\right ) + \left (\frac {229 d^{3}}{625} - \frac {2643 d^{2} e}{6250} - \frac {7662 d e^{2}}{15625} + \frac {23431 e^{3}}{156250} - \frac {\sqrt {14} i \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{2187500}\right ) \log {\left (x + \frac {10575 d^{3} + 89835 d^{2} e - 54969 d e^{2} - \frac {53189 e^{3}}{5} + \frac {\sqrt {14} i \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{5}}{52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}} \right )} + \left (\frac {229 d^{3}}{625} - \frac {2643 d^{2} e}{6250} - \frac {7662 d e^{2}}{15625} + \frac {23431 e^{3}}{156250} + \frac {\sqrt {14} i \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{2187500}\right ) \log {\left (x + \frac {10575 d^{3} + 89835 d^{2} e - 54969 d e^{2} - \frac {53189 e^{3}}{5} - \frac {\sqrt {14} i \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{5}}{52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}} \right )} \] Input:
integrate((e*x+d)**3*(4*x**4-5*x**3+3*x**2+x+2)/(5*x**2+2*x+3),x)
Output:
2*e**3*x**6/15 + x**5*(12*d*e**2/25 - 33*e**3/125) + x**4*(3*d**2*e/5 - 99 *d*e**2/100 + 81*e**3/500) + x**3*(4*d**3/15 - 33*d**2*e/25 + 81*d*e**2/12 5 + 458*e**3/1875) + x**2*(-33*d**3/50 + 243*d**2*e/250 + 687*d*e**2/625 - 881*e**3/6250) + x*(81*d**3/125 + 1374*d**2*e/625 - 2643*d*e**2/3125 - 51 08*e**3/15625) + (229*d**3/625 - 2643*d**2*e/6250 - 7662*d*e**2/15625 + 23 431*e**3/156250 - sqrt(14)*I*(52875*d**3 + 449175*d**2*e - 274845*d*e**2 - 53189*e**3)/2187500)*log(x + (10575*d**3 + 89835*d**2*e - 54969*d*e**2 - 53189*e**3/5 + sqrt(14)*I*(52875*d**3 + 449175*d**2*e - 274845*d*e**2 - 53 189*e**3)/5)/(52875*d**3 + 449175*d**2*e - 274845*d*e**2 - 53189*e**3)) + (229*d**3/625 - 2643*d**2*e/6250 - 7662*d*e**2/15625 + 23431*e**3/156250 + sqrt(14)*I*(52875*d**3 + 449175*d**2*e - 274845*d*e**2 - 53189*e**3)/2187 500)*log(x + (10575*d**3 + 89835*d**2*e - 54969*d*e**2 - 53189*e**3/5 - sq rt(14)*I*(52875*d**3 + 449175*d**2*e - 274845*d*e**2 - 53189*e**3)/5)/(528 75*d**3 + 449175*d**2*e - 274845*d*e**2 - 53189*e**3))
Time = 0.11 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {2}{15} \, e^{3} x^{6} + \frac {3}{125} \, {\left (20 \, d e^{2} - 11 \, e^{3}\right )} x^{5} + \frac {3}{500} \, {\left (100 \, d^{2} e - 165 \, d e^{2} + 27 \, e^{3}\right )} x^{4} + \frac {1}{1875} \, {\left (500 \, d^{3} - 2475 \, d^{2} e + 1215 \, d e^{2} + 458 \, e^{3}\right )} x^{3} - \frac {1}{6250} \, {\left (4125 \, d^{3} - 6075 \, d^{2} e - 6870 \, d e^{2} + 881 \, e^{3}\right )} x^{2} - \frac {1}{1093750} \, \sqrt {14} {\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{15625} \, {\left (10125 \, d^{3} + 34350 \, d^{2} e - 13215 \, d e^{2} - 5108 \, e^{3}\right )} x + \frac {1}{156250} \, {\left (57250 \, d^{3} - 66075 \, d^{2} e - 76620 \, d e^{2} + 23431 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \] Input:
integrate((e*x+d)^3*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x, algorithm="ma xima")
Output:
2/15*e^3*x^6 + 3/125*(20*d*e^2 - 11*e^3)*x^5 + 3/500*(100*d^2*e - 165*d*e^ 2 + 27*e^3)*x^4 + 1/1875*(500*d^3 - 2475*d^2*e + 1215*d*e^2 + 458*e^3)*x^3 - 1/6250*(4125*d^3 - 6075*d^2*e - 6870*d*e^2 + 881*e^3)*x^2 - 1/1093750*s qrt(14)*(52875*d^3 + 449175*d^2*e - 274845*d*e^2 - 53189*e^3)*arctan(1/14* sqrt(14)*(5*x + 1)) + 1/15625*(10125*d^3 + 34350*d^2*e - 13215*d*e^2 - 510 8*e^3)*x + 1/156250*(57250*d^3 - 66075*d^2*e - 76620*d*e^2 + 23431*e^3)*lo g(5*x^2 + 2*x + 3)
Time = 0.15 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {2}{15} \, e^{3} x^{6} + \frac {12}{25} \, d e^{2} x^{5} - \frac {33}{125} \, e^{3} x^{5} + \frac {3}{5} \, d^{2} e x^{4} - \frac {99}{100} \, d e^{2} x^{4} + \frac {81}{500} \, e^{3} x^{4} + \frac {4}{15} \, d^{3} x^{3} - \frac {33}{25} \, d^{2} e x^{3} + \frac {81}{125} \, d e^{2} x^{3} + \frac {458}{1875} \, e^{3} x^{3} - \frac {33}{50} \, d^{3} x^{2} + \frac {243}{250} \, d^{2} e x^{2} + \frac {687}{625} \, d e^{2} x^{2} - \frac {881}{6250} \, e^{3} x^{2} + \frac {81}{125} \, d^{3} x + \frac {1374}{625} \, d^{2} e x - \frac {2643}{3125} \, d e^{2} x - \frac {5108}{15625} \, e^{3} x - \frac {1}{1093750} \, \sqrt {14} {\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{156250} \, {\left (57250 \, d^{3} - 66075 \, d^{2} e - 76620 \, d e^{2} + 23431 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \] Input:
integrate((e*x+d)^3*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x, algorithm="gi ac")
Output:
2/15*e^3*x^6 + 12/25*d*e^2*x^5 - 33/125*e^3*x^5 + 3/5*d^2*e*x^4 - 99/100*d *e^2*x^4 + 81/500*e^3*x^4 + 4/15*d^3*x^3 - 33/25*d^2*e*x^3 + 81/125*d*e^2* x^3 + 458/1875*e^3*x^3 - 33/50*d^3*x^2 + 243/250*d^2*e*x^2 + 687/625*d*e^2 *x^2 - 881/6250*e^3*x^2 + 81/125*d^3*x + 1374/625*d^2*e*x - 2643/3125*d*e^ 2*x - 5108/15625*e^3*x - 1/1093750*sqrt(14)*(52875*d^3 + 449175*d^2*e - 27 4845*d*e^2 - 53189*e^3)*arctan(1/14*sqrt(14)*(5*x + 1)) + 1/156250*(57250* d^3 - 66075*d^2*e - 76620*d*e^2 + 23431*e^3)*log(5*x^2 + 2*x + 3)
Time = 17.21 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.80 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=x^2\,\left (\frac {26\,e^2\,\left (12\,d-5\,e\right )}{625}-\frac {33\,e\,\left (4\,d^2-5\,d\,e+e^2\right )}{250}-\frac {3\,d\,e^2}{50}+\frac {3\,d^2\,e}{2}-\frac {33\,d^3}{50}+\frac {622\,e^3}{3125}\right )-x^3\,\left (\frac {11\,e^2\,\left (12\,d-5\,e\right )}{375}+\frac {2\,e\,\left (4\,d^2-5\,d\,e+e^2\right )}{25}-\frac {3\,d\,e^2}{5}+d^2\,e-\frac {4\,d^3}{15}-\frac {111\,e^3}{625}\right )+x^5\,\left (\frac {e^2\,\left (12\,d-5\,e\right )}{25}-\frac {8\,e^3}{125}\right )-\ln \left (5\,x^2+2\,x+3\right )\,\left (-\frac {229\,d^3}{625}+\frac {2643\,d^2\,e}{6250}+\frac {7662\,d\,e^2}{15625}-\frac {23431\,e^3}{156250}\right )-x^4\,\left (\frac {e^2\,\left (12\,d-5\,e\right )}{50}-\frac {3\,e\,\left (4\,d^2-5\,d\,e+e^2\right )}{20}+\frac {11\,e^3}{125}\right )+\frac {2\,e^3\,x^6}{15}+x\,\left (\frac {61\,e^2\,\left (12\,d-5\,e\right )}{3125}+\frac {3\,d\,\left (d^2+d\,e+2\,e^2\right )}{5}+\frac {156\,e\,\left (4\,d^2-5\,d\,e+e^2\right )}{625}-\frac {129\,d\,e^2}{125}+\frac {3\,d^2\,e}{5}+\frac {6\,d^3}{125}-\frac {7483\,e^3}{15625}\right )+\frac {\sqrt {14}\,\mathrm {atan}\left (\frac {\frac {\sqrt {14}\,\left (-52875\,d^3-449175\,d^2\,e+274845\,d\,e^2+53189\,e^3\right )}{1093750}+\frac {\sqrt {14}\,x\,\left (-52875\,d^3-449175\,d^2\,e+274845\,d\,e^2+53189\,e^3\right )}{218750}}{-\frac {423\,d^3}{625}-\frac {17967\,d^2\,e}{3125}+\frac {54969\,d\,e^2}{15625}+\frac {53189\,e^3}{78125}}\right )\,\left (-52875\,d^3-449175\,d^2\,e+274845\,d\,e^2+53189\,e^3\right )}{1093750} \] Input:
int(((d + e*x)^3*(x + 3*x^2 - 5*x^3 + 4*x^4 + 2))/(2*x + 5*x^2 + 3),x)
Output:
x^2*((26*e^2*(12*d - 5*e))/625 - (33*e*(4*d^2 - 5*d*e + e^2))/250 - (3*d*e ^2)/50 + (3*d^2*e)/2 - (33*d^3)/50 + (622*e^3)/3125) - x^3*((11*e^2*(12*d - 5*e))/375 + (2*e*(4*d^2 - 5*d*e + e^2))/25 - (3*d*e^2)/5 + d^2*e - (4*d^ 3)/15 - (111*e^3)/625) + x^5*((e^2*(12*d - 5*e))/25 - (8*e^3)/125) - log(2 *x + 5*x^2 + 3)*((7662*d*e^2)/15625 + (2643*d^2*e)/6250 - (229*d^3)/625 - (23431*e^3)/156250) - x^4*((e^2*(12*d - 5*e))/50 - (3*e*(4*d^2 - 5*d*e + e ^2))/20 + (11*e^3)/125) + (2*e^3*x^6)/15 + x*((61*e^2*(12*d - 5*e))/3125 + (3*d*(d*e + d^2 + 2*e^2))/5 + (156*e*(4*d^2 - 5*d*e + e^2))/625 - (129*d* e^2)/125 + (3*d^2*e)/5 + (6*d^3)/125 - (7483*e^3)/15625) + (14^(1/2)*atan( ((14^(1/2)*(274845*d*e^2 - 449175*d^2*e - 52875*d^3 + 53189*e^3))/1093750 + (14^(1/2)*x*(274845*d*e^2 - 449175*d^2*e - 52875*d^3 + 53189*e^3))/21875 0)/((54969*d*e^2)/15625 - (17967*d^2*e)/3125 - (423*d^3)/625 + (53189*e^3) /78125))*(274845*d*e^2 - 449175*d^2*e - 52875*d^3 + 53189*e^3))/1093750
Time = 0.20 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {229 \,\mathrm {log}\left (5 x^{2}+2 x +3\right ) d^{3}}{625}+\frac {23431 \,\mathrm {log}\left (5 x^{2}+2 x +3\right ) e^{3}}{156250}-\frac {17967 \sqrt {14}\, \mathit {atan} \left (\frac {5 x +1}{\sqrt {14}}\right ) d^{2} e}{43750}+\frac {54969 \sqrt {14}\, \mathit {atan} \left (\frac {5 x +1}{\sqrt {14}}\right ) d \,e^{2}}{218750}-\frac {881 e^{3} x^{2}}{6250}-\frac {5108 e^{3} x}{15625}+\frac {3 d^{2} e \,x^{4}}{5}-\frac {33 d^{2} e \,x^{3}}{25}+\frac {243 d^{2} e \,x^{2}}{250}+\frac {1374 d^{2} e x}{625}+\frac {12 d \,e^{2} x^{5}}{25}-\frac {99 d \,e^{2} x^{4}}{100}+\frac {81 d \,e^{2} x^{3}}{125}+\frac {687 d \,e^{2} x^{2}}{625}-\frac {423 \sqrt {14}\, \mathit {atan} \left (\frac {5 x +1}{\sqrt {14}}\right ) d^{3}}{8750}+\frac {53189 \sqrt {14}\, \mathit {atan} \left (\frac {5 x +1}{\sqrt {14}}\right ) e^{3}}{1093750}-\frac {2643 \,\mathrm {log}\left (5 x^{2}+2 x +3\right ) d^{2} e}{6250}-\frac {7662 \,\mathrm {log}\left (5 x^{2}+2 x +3\right ) d \,e^{2}}{15625}-\frac {2643 d \,e^{2} x}{3125}+\frac {4 d^{3} x^{3}}{15}-\frac {33 d^{3} x^{2}}{50}+\frac {81 d^{3} x}{125}+\frac {2 e^{3} x^{6}}{15}-\frac {33 e^{3} x^{5}}{125}+\frac {81 e^{3} x^{4}}{500}+\frac {458 e^{3} x^{3}}{1875} \] Input:
int((e*x+d)^3*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x)
Output:
( - 317250*sqrt(14)*atan((5*x + 1)/sqrt(14))*d**3 - 2695050*sqrt(14)*atan( (5*x + 1)/sqrt(14))*d**2*e + 1649070*sqrt(14)*atan((5*x + 1)/sqrt(14))*d*e **2 + 319134*sqrt(14)*atan((5*x + 1)/sqrt(14))*e**3 + 2404500*log(5*x**2 + 2*x + 3)*d**3 - 2775150*log(5*x**2 + 2*x + 3)*d**2*e - 3218040*log(5*x**2 + 2*x + 3)*d*e**2 + 984102*log(5*x**2 + 2*x + 3)*e**3 + 1750000*d**3*x**3 - 4331250*d**3*x**2 + 4252500*d**3*x + 3937500*d**2*e*x**4 - 8662500*d**2 *e*x**3 + 6378750*d**2*e*x**2 + 14427000*d**2*e*x + 3150000*d*e**2*x**5 - 6496875*d*e**2*x**4 + 4252500*d*e**2*x**3 + 7213500*d*e**2*x**2 - 5550300* d*e**2*x + 875000*e**3*x**6 - 1732500*e**3*x**5 + 1063125*e**3*x**4 + 1603 000*e**3*x**3 - 925050*e**3*x**2 - 2145360*e**3*x)/6562500