Integrand size = 36, antiderivative size = 99 \[ \int \frac {(d+e x) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {1}{625} (405 d+458 e) x-\frac {3}{250} (55 d-27 e) x^2+\frac {1}{75} (20 d-33 e) x^3+\frac {e x^4}{5}-\frac {(2115 d+5989 e) \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )}{3125 \sqrt {14}}+\frac {(2290 d-881 e) \log \left (3+2 x+5 x^2\right )}{6250} \] Output:
1/625*(405*d+458*e)*x-3/250*(55*d-27*e)*x^2+1/75*(20*d-33*e)*x^3+1/5*e*x^4 -1/43750*(2115*d+5989*e)*arctan(1/14*(1+5*x)*14^(1/2))*14^(1/2)+1/6250*(22 90*d-881*e)*ln(5*x^2+2*x+3)
Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {35 x \left (5 d \left (486-495 x+200 x^2\right )+3 e \left (916+405 x-550 x^2+250 x^3\right )\right )-3 \sqrt {14} (2115 d+5989 e) \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )+21 (2290 d-881 e) \log \left (3+2 x+5 x^2\right )}{131250} \] Input:
Integrate[((d + e*x)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2),x]
Output:
(35*x*(5*d*(486 - 495*x + 200*x^2) + 3*e*(916 + 405*x - 550*x^2 + 250*x^3) ) - 3*Sqrt[14]*(2115*d + 5989*e)*ArcTan[(1 + 5*x)/Sqrt[14]] + 21*(2290*d - 881*e)*Log[3 + 2*x + 5*x^2])/131250
Time = 0.49 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, 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.
\(\displaystyle \int \frac {\left (4 x^4-5 x^3+3 x^2+x+2\right ) (d+e x)}{5 x^2+2 x+3} \, dx\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle \int \left (\frac {1}{25} x^2 (20 d-33 e)+\frac {x (2290 d-881 e)+35 d-1374 e}{625 \left (5 x^2+2 x+3\right )}-\frac {3}{125} x (55 d-27 e)+\frac {1}{625} (405 d+458 e)+\frac {4 e x^3}{5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\arctan \left (\frac {5 x+1}{\sqrt {14}}\right ) (2115 d+5989 e)}{3125 \sqrt {14}}+\frac {1}{75} x^3 (20 d-33 e)-\frac {3}{250} x^2 (55 d-27 e)+\frac {(2290 d-881 e) \log \left (5 x^2+2 x+3\right )}{6250}+\frac {1}{625} x (405 d+458 e)+\frac {e x^4}{5}\) |
Input:
Int[((d + e*x)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2),x]
Output:
((405*d + 458*e)*x)/625 - (3*(55*d - 27*e)*x^2)/250 + ((20*d - 33*e)*x^3)/ 75 + (e*x^4)/5 - ((2115*d + 5989*e)*ArcTan[(1 + 5*x)/Sqrt[14]])/(3125*Sqrt [14]) + ((2290*d - 881*e)*Log[3 + 2*x + 5*x^2])/6250
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 = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {e \,x^{4}}{5}+\frac {4 d \,x^{3}}{15}-\frac {11 e \,x^{3}}{25}-\frac {33 d \,x^{2}}{50}+\frac {81 e \,x^{2}}{250}+\frac {81 d x}{125}+\frac {458 e x}{625}+\frac {\left (2290 d -881 e \right ) \ln \left (5 x^{2}+2 x +3\right )}{6250}+\frac {\left (-423 d -\frac {5989 e}{5}\right ) \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{8750}\) | \(83\) |
risch | \(\frac {e \,x^{4}}{5}+\frac {4 d \,x^{3}}{15}-\frac {11 e \,x^{3}}{25}-\frac {33 d \,x^{2}}{50}+\frac {81 e \,x^{2}}{250}+\frac {81 d x}{125}+\frac {458 e x}{625}+\frac {229 d \ln \left (350 x^{2}+140 x +210\right )}{625}-\frac {881 e \ln \left (350 x^{2}+140 x +210\right )}{6250}-\frac {423 \sqrt {14}\, d \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{8750}-\frac {5989 \sqrt {14}\, e \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{43750}\) | \(106\) |
Input:
int((e*x+d)*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x,method=_RETURNVERBOSE)
Output:
1/5*e*x^4+4/15*d*x^3-11/25*e*x^3-33/50*d*x^2+81/250*e*x^2+81/125*d*x+458/6 25*e*x+1/6250*(2290*d-881*e)*ln(5*x^2+2*x+3)+1/8750*(-423*d-5989/5*e)*14^( 1/2)*arctan(1/28*(10*x+2)*14^(1/2))
Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {1}{5} \, e x^{4} + \frac {1}{75} \, {\left (20 \, d - 33 \, e\right )} x^{3} - \frac {3}{250} \, {\left (55 \, d - 27 \, e\right )} x^{2} - \frac {1}{43750} \, \sqrt {14} {\left (2115 \, d + 5989 \, e\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{625} \, {\left (405 \, d + 458 \, e\right )} x + \frac {1}{6250} \, {\left (2290 \, d - 881 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \] Input:
integrate((e*x+d)*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x, algorithm="fric as")
Output:
1/5*e*x^4 + 1/75*(20*d - 33*e)*x^3 - 3/250*(55*d - 27*e)*x^2 - 1/43750*sqr t(14)*(2115*d + 5989*e)*arctan(1/14*sqrt(14)*(5*x + 1)) + 1/625*(405*d + 4 58*e)*x + 1/6250*(2290*d - 881*e)*log(5*x^2 + 2*x + 3)
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.65 \[ \int \frac {(d+e x) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {e x^{4}}{5} + x^{3} \cdot \left (\frac {4 d}{15} - \frac {11 e}{25}\right ) + x^{2} \left (- \frac {33 d}{50} + \frac {81 e}{250}\right ) + x \left (\frac {81 d}{125} + \frac {458 e}{625}\right ) + \left (\frac {229 d}{625} - \frac {881 e}{6250} - \frac {\sqrt {14} i \left (2115 d + 5989 e\right )}{87500}\right ) \log {\left (x + \frac {423 d + \frac {5989 e}{5} + \frac {\sqrt {14} i \left (2115 d + 5989 e\right )}{5}}{2115 d + 5989 e} \right )} + \left (\frac {229 d}{625} - \frac {881 e}{6250} + \frac {\sqrt {14} i \left (2115 d + 5989 e\right )}{87500}\right ) \log {\left (x + \frac {423 d + \frac {5989 e}{5} - \frac {\sqrt {14} i \left (2115 d + 5989 e\right )}{5}}{2115 d + 5989 e} \right )} \] Input:
integrate((e*x+d)*(4*x**4-5*x**3+3*x**2+x+2)/(5*x**2+2*x+3),x)
Output:
e*x**4/5 + x**3*(4*d/15 - 11*e/25) + x**2*(-33*d/50 + 81*e/250) + x*(81*d/ 125 + 458*e/625) + (229*d/625 - 881*e/6250 - sqrt(14)*I*(2115*d + 5989*e)/ 87500)*log(x + (423*d + 5989*e/5 + sqrt(14)*I*(2115*d + 5989*e)/5)/(2115*d + 5989*e)) + (229*d/625 - 881*e/6250 + sqrt(14)*I*(2115*d + 5989*e)/87500 )*log(x + (423*d + 5989*e/5 - sqrt(14)*I*(2115*d + 5989*e)/5)/(2115*d + 59 89*e))
Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {1}{5} \, e x^{4} + \frac {1}{75} \, {\left (20 \, d - 33 \, e\right )} x^{3} - \frac {3}{250} \, {\left (55 \, d - 27 \, e\right )} x^{2} - \frac {1}{43750} \, \sqrt {14} {\left (2115 \, d + 5989 \, e\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{625} \, {\left (405 \, d + 458 \, e\right )} x + \frac {1}{6250} \, {\left (2290 \, d - 881 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \] Input:
integrate((e*x+d)*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x, algorithm="maxi ma")
Output:
1/5*e*x^4 + 1/75*(20*d - 33*e)*x^3 - 3/250*(55*d - 27*e)*x^2 - 1/43750*sqr t(14)*(2115*d + 5989*e)*arctan(1/14*sqrt(14)*(5*x + 1)) + 1/625*(405*d + 4 58*e)*x + 1/6250*(2290*d - 881*e)*log(5*x^2 + 2*x + 3)
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=\frac {1}{5} \, e x^{4} + \frac {4}{15} \, d x^{3} - \frac {11}{25} \, e x^{3} - \frac {33}{50} \, d x^{2} + \frac {81}{250} \, e x^{2} - \frac {1}{43750} \, \sqrt {14} {\left (2115 \, d + 5989 \, e\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {81}{125} \, d x + \frac {458}{625} \, e x + \frac {1}{6250} \, {\left (2290 \, d - 881 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \] Input:
integrate((e*x+d)*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x, algorithm="giac ")
Output:
1/5*e*x^4 + 4/15*d*x^3 - 11/25*e*x^3 - 33/50*d*x^2 + 81/250*e*x^2 - 1/4375 0*sqrt(14)*(2115*d + 5989*e)*arctan(1/14*sqrt(14)*(5*x + 1)) + 81/125*d*x + 458/625*e*x + 1/6250*(2290*d - 881*e)*log(5*x^2 + 2*x + 3)
Time = 0.07 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=x^3\,\left (\frac {4\,d}{15}-\frac {11\,e}{25}\right )-x^2\,\left (\frac {33\,d}{50}-\frac {81\,e}{250}\right )+\ln \left (5\,x^2+2\,x+3\right )\,\left (\frac {229\,d}{625}-\frac {881\,e}{6250}\right )+\frac {e\,x^4}{5}+x\,\left (\frac {81\,d}{125}+\frac {458\,e}{625}\right )-\frac {\sqrt {14}\,\mathrm {atan}\left (\frac {\frac {\sqrt {14}\,\left (2115\,d+5989\,e\right )}{43750}+\frac {\sqrt {14}\,x\,\left (2115\,d+5989\,e\right )}{8750}}{\frac {423\,d}{625}+\frac {5989\,e}{3125}}\right )\,\left (2115\,d+5989\,e\right )}{43750} \] Input:
int(((d + e*x)*(x + 3*x^2 - 5*x^3 + 4*x^4 + 2))/(2*x + 5*x^2 + 3),x)
Output:
x^3*((4*d)/15 - (11*e)/25) - x^2*((33*d)/50 - (81*e)/250) + log(2*x + 5*x^ 2 + 3)*((229*d)/625 - (881*e)/6250) + (e*x^4)/5 + x*((81*d)/125 + (458*e)/ 625) - (14^(1/2)*atan(((14^(1/2)*(2115*d + 5989*e))/43750 + (14^(1/2)*x*(2 115*d + 5989*e))/8750)/((423*d)/625 + (5989*e)/3125))*(2115*d + 5989*e))/4 3750
Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx=-\frac {423 \sqrt {14}\, \mathit {atan} \left (\frac {5 x +1}{\sqrt {14}}\right ) d}{8750}-\frac {5989 \sqrt {14}\, \mathit {atan} \left (\frac {5 x +1}{\sqrt {14}}\right ) e}{43750}+\frac {229 \,\mathrm {log}\left (5 x^{2}+2 x +3\right ) d}{625}-\frac {881 \,\mathrm {log}\left (5 x^{2}+2 x +3\right ) e}{6250}+\frac {4 d \,x^{3}}{15}-\frac {33 d \,x^{2}}{50}+\frac {81 d x}{125}+\frac {e \,x^{4}}{5}-\frac {11 e \,x^{3}}{25}+\frac {81 e \,x^{2}}{250}+\frac {458 e x}{625} \] Input:
int((e*x+d)*(4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x)
Output:
( - 6345*sqrt(14)*atan((5*x + 1)/sqrt(14))*d - 17967*sqrt(14)*atan((5*x + 1)/sqrt(14))*e + 48090*log(5*x**2 + 2*x + 3)*d - 18501*log(5*x**2 + 2*x + 3)*e + 35000*d*x**3 - 86625*d*x**2 + 85050*d*x + 26250*e*x**4 - 57750*e*x* *3 + 42525*e*x**2 + 96180*e*x)/131250