Integrand size = 31, antiderivative size = 56 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{3+2 x+5 x^2} \, dx=\frac {81 x}{125}-\frac {33 x^2}{50}+\frac {4 x^3}{15}-\frac {423 \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )}{625 \sqrt {14}}+\frac {229}{625} \log \left (3+2 x+5 x^2\right ) \] Output:
81/125*x-33/50*x^2+4/15*x^3-423/8750*arctan(1/14*(1+5*x)*14^(1/2))*14^(1/2 )+229/625*ln(5*x^2+2*x+3)
Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{3+2 x+5 x^2} \, dx=\frac {35 x \left (486-495 x+200 x^2\right )-1269 \sqrt {14} \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )+9618 \log \left (3+2 x+5 x^2\right )}{26250} \] Input:
Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/(3 + 2*x + 5*x^2),x]
Output:
(35*x*(486 - 495*x + 200*x^2) - 1269*Sqrt[14]*ArcTan[(1 + 5*x)/Sqrt[14]] + 9618*Log[3 + 2*x + 5*x^2])/26250
Time = 0.36 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2188, 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.
\(\displaystyle \int \frac {4 x^4-5 x^3+3 x^2+x+2}{5 x^2+2 x+3} \, dx\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \int \left (\frac {4 x^2}{5}+\frac {458 x+7}{125 \left (5 x^2+2 x+3\right )}-\frac {33 x}{25}+\frac {81}{125}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {423 \arctan \left (\frac {5 x+1}{\sqrt {14}}\right )}{625 \sqrt {14}}+\frac {4 x^3}{15}-\frac {33 x^2}{50}+\frac {229}{625} \log \left (5 x^2+2 x+3\right )+\frac {81 x}{125}\) |
Input:
Int[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/(3 + 2*x + 5*x^2),x]
Output:
(81*x)/125 - (33*x^2)/50 + (4*x^3)/15 - (423*ArcTan[(1 + 5*x)/Sqrt[14]])/( 625*Sqrt[14]) + (229*Log[3 + 2*x + 5*x^2])/625
Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {4 x^{3}}{15}-\frac {33 x^{2}}{50}+\frac {81 x}{125}+\frac {229 \ln \left (5 x^{2}+2 x +3\right )}{625}-\frac {423 \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{8750}\) | \(44\) |
risch | \(\frac {4 x^{3}}{15}-\frac {33 x^{2}}{50}+\frac {81 x}{125}+\frac {229 \ln \left (25 x^{2}+10 x +15\right )}{625}-\frac {423 \arctan \left (\frac {\left (1+5 x \right ) \sqrt {14}}{14}\right ) \sqrt {14}}{8750}\) | \(44\) |
Input:
int((4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x,method=_RETURNVERBOSE)
Output:
4/15*x^3-33/50*x^2+81/125*x+229/625*ln(5*x^2+2*x+3)-423/8750*14^(1/2)*arct an(1/28*(10*x+2)*14^(1/2))
Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{3+2 x+5 x^2} \, dx=\frac {4}{15} \, x^{3} - \frac {33}{50} \, x^{2} - \frac {423}{8750} \, \sqrt {14} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {81}{125} \, x + \frac {229}{625} \, \log \left (5 \, x^{2} + 2 \, x + 3\right ) \] Input:
integrate((4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x, algorithm="fricas")
Output:
4/15*x^3 - 33/50*x^2 - 423/8750*sqrt(14)*arctan(1/14*sqrt(14)*(5*x + 1)) + 81/125*x + 229/625*log(5*x^2 + 2*x + 3)
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{3+2 x+5 x^2} \, dx=\frac {4 x^{3}}{15} - \frac {33 x^{2}}{50} + \frac {81 x}{125} + \frac {229 \log {\left (x^{2} + \frac {2 x}{5} + \frac {3}{5} \right )}}{625} - \frac {423 \sqrt {14} \operatorname {atan}{\left (\frac {5 \sqrt {14} x}{14} + \frac {\sqrt {14}}{14} \right )}}{8750} \] Input:
integrate((4*x**4-5*x**3+3*x**2+x+2)/(5*x**2+2*x+3),x)
Output:
4*x**3/15 - 33*x**2/50 + 81*x/125 + 229*log(x**2 + 2*x/5 + 3/5)/625 - 423* sqrt(14)*atan(5*sqrt(14)*x/14 + sqrt(14)/14)/8750
Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{3+2 x+5 x^2} \, dx=\frac {4}{15} \, x^{3} - \frac {33}{50} \, x^{2} - \frac {423}{8750} \, \sqrt {14} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {81}{125} \, x + \frac {229}{625} \, \log \left (5 \, x^{2} + 2 \, x + 3\right ) \] Input:
integrate((4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x, algorithm="maxima")
Output:
4/15*x^3 - 33/50*x^2 - 423/8750*sqrt(14)*arctan(1/14*sqrt(14)*(5*x + 1)) + 81/125*x + 229/625*log(5*x^2 + 2*x + 3)
Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{3+2 x+5 x^2} \, dx=\frac {4}{15} \, x^{3} - \frac {33}{50} \, x^{2} - \frac {423}{8750} \, \sqrt {14} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {81}{125} \, x + \frac {229}{625} \, \log \left (5 \, x^{2} + 2 \, x + 3\right ) \] Input:
integrate((4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x, algorithm="giac")
Output:
4/15*x^3 - 33/50*x^2 - 423/8750*sqrt(14)*arctan(1/14*sqrt(14)*(5*x + 1)) + 81/125*x + 229/625*log(5*x^2 + 2*x + 3)
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{3+2 x+5 x^2} \, dx=\frac {81\,x}{125}+\frac {229\,\ln \left (5\,x^2+2\,x+3\right )}{625}-\frac {423\,\sqrt {14}\,\mathrm {atan}\left (\frac {5\,\sqrt {14}\,x}{14}+\frac {\sqrt {14}}{14}\right )}{8750}-\frac {33\,x^2}{50}+\frac {4\,x^3}{15} \] Input:
int((x + 3*x^2 - 5*x^3 + 4*x^4 + 2)/(2*x + 5*x^2 + 3),x)
Output:
(81*x)/125 + (229*log(2*x + 5*x^2 + 3))/625 - (423*14^(1/2)*atan((5*14^(1/ 2)*x)/14 + 14^(1/2)/14))/8750 - (33*x^2)/50 + (4*x^3)/15
Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.75 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{3+2 x+5 x^2} \, dx=-\frac {423 \sqrt {14}\, \mathit {atan} \left (\frac {5 x +1}{\sqrt {14}}\right )}{8750}+\frac {229 \,\mathrm {log}\left (5 x^{2}+2 x +3\right )}{625}+\frac {4 x^{3}}{15}-\frac {33 x^{2}}{50}+\frac {81 x}{125} \] Input:
int((4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3),x)
Output:
( - 1269*sqrt(14)*atan((5*x + 1)/sqrt(14)) + 9618*log(5*x**2 + 2*x + 3) + 7000*x**3 - 17325*x**2 + 17010*x)/26250