Integrand size = 25, antiderivative size = 84 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{3-x+2 x^2} \, dx=\frac {122691 x}{128}-\frac {28747 x^2}{128}-\frac {21229 x^3}{96}+\frac {6245 x^4}{64}+\frac {1855 x^5}{8}+\frac {3625 x^6}{24}+\frac {625 x^7}{14}+\frac {1156639 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{256 \sqrt {23}}+\frac {307461}{512} \log \left (3-x+2 x^2\right ) \] Output:
122691/128*x-28747/128*x^2-21229/96*x^3+6245/64*x^4+1855/8*x^5+3625/24*x^6 +625/14*x^7+1156639/5888*arctan(1/23*(1-4*x)*23^(1/2))*23^(1/2)+307461/512 *ln(2*x^2-x+3)
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.86 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{3-x+2 x^2} \, dx=\frac {x \left (2576511-603687 x-594412 x^2+262290 x^3+623280 x^4+406000 x^5+120000 x^6\right )}{2688}-\frac {1156639 \arctan \left (\frac {-1+4 x}{\sqrt {23}}\right )}{256 \sqrt {23}}+\frac {307461}{512} \log \left (3-x+2 x^2\right ) \] Input:
Integrate[(2 + 3*x + 5*x^2)^4/(3 - x + 2*x^2),x]
Output:
(x*(2576511 - 603687*x - 594412*x^2 + 262290*x^3 + 623280*x^4 + 406000*x^5 + 120000*x^6))/2688 - (1156639*ArcTan[(-1 + 4*x)/Sqrt[23]])/(256*Sqrt[23] ) + (307461*Log[3 - x + 2*x^2])/512
Time = 0.24 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
\(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^4}{2 x^2-x+3} \, dx\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \int \left (\frac {625 x^6}{2}+\frac {3625 x^5}{4}+\frac {9275 x^4}{8}+\frac {6245 x^3}{16}-\frac {21229 x^2}{32}-\frac {14641 (25-21 x)}{128 \left (2 x^2-x+3\right )}-\frac {28747 x}{64}+\frac {122691}{128}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1156639 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{256 \sqrt {23}}+\frac {625 x^7}{14}+\frac {3625 x^6}{24}+\frac {1855 x^5}{8}+\frac {6245 x^4}{64}-\frac {21229 x^3}{96}-\frac {28747 x^2}{128}+\frac {307461}{512} \log \left (2 x^2-x+3\right )+\frac {122691 x}{128}\) |
Input:
Int[(2 + 3*x + 5*x^2)^4/(3 - x + 2*x^2),x]
Output:
(122691*x)/128 - (28747*x^2)/128 - (21229*x^3)/96 + (6245*x^4)/64 + (1855* x^5)/8 + (3625*x^6)/24 + (625*x^7)/14 + (1156639*ArcTan[(1 - 4*x)/Sqrt[23] ])/(256*Sqrt[23]) + (307461*Log[3 - x + 2*x^2])/512
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 = 4.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {625 x^{7}}{14}+\frac {3625 x^{6}}{24}+\frac {1855 x^{5}}{8}+\frac {6245 x^{4}}{64}-\frac {21229 x^{3}}{96}-\frac {28747 x^{2}}{128}+\frac {122691 x}{128}+\frac {307461 \ln \left (2 x^{2}-x +3\right )}{512}-\frac {1156639 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{5888}\) | \(64\) |
risch | \(\frac {625 x^{7}}{14}+\frac {3625 x^{6}}{24}+\frac {1855 x^{5}}{8}+\frac {6245 x^{4}}{64}-\frac {21229 x^{3}}{96}-\frac {28747 x^{2}}{128}+\frac {122691 x}{128}+\frac {307461 \ln \left (16 x^{2}-8 x +24\right )}{512}-\frac {1156639 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{5888}\) | \(64\) |
Input:
int((5*x^2+3*x+2)^4/(2*x^2-x+3),x,method=_RETURNVERBOSE)
Output:
625/14*x^7+3625/24*x^6+1855/8*x^5+6245/64*x^4-21229/96*x^3-28747/128*x^2+1 22691/128*x+307461/512*ln(2*x^2-x+3)-1156639/5888*23^(1/2)*arctan(1/23*(4* x-1)*23^(1/2))
Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{3-x+2 x^2} \, dx=\frac {625}{14} \, x^{7} + \frac {3625}{24} \, x^{6} + \frac {1855}{8} \, x^{5} + \frac {6245}{64} \, x^{4} - \frac {21229}{96} \, x^{3} - \frac {28747}{128} \, x^{2} - \frac {1156639}{5888} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {122691}{128} \, x + \frac {307461}{512} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate((5*x^2+3*x+2)^4/(2*x^2-x+3),x, algorithm="fricas")
Output:
625/14*x^7 + 3625/24*x^6 + 1855/8*x^5 + 6245/64*x^4 - 21229/96*x^3 - 28747 /128*x^2 - 1156639/5888*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 122691/ 128*x + 307461/512*log(2*x^2 - x + 3)
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.04 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{3-x+2 x^2} \, dx=\frac {625 x^{7}}{14} + \frac {3625 x^{6}}{24} + \frac {1855 x^{5}}{8} + \frac {6245 x^{4}}{64} - \frac {21229 x^{3}}{96} - \frac {28747 x^{2}}{128} + \frac {122691 x}{128} + \frac {307461 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{512} - \frac {1156639 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{5888} \] Input:
integrate((5*x**2+3*x+2)**4/(2*x**2-x+3),x)
Output:
625*x**7/14 + 3625*x**6/24 + 1855*x**5/8 + 6245*x**4/64 - 21229*x**3/96 - 28747*x**2/128 + 122691*x/128 + 307461*log(x**2 - x/2 + 3/2)/512 - 1156639 *sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/5888
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{3-x+2 x^2} \, dx=\frac {625}{14} \, x^{7} + \frac {3625}{24} \, x^{6} + \frac {1855}{8} \, x^{5} + \frac {6245}{64} \, x^{4} - \frac {21229}{96} \, x^{3} - \frac {28747}{128} \, x^{2} - \frac {1156639}{5888} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {122691}{128} \, x + \frac {307461}{512} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate((5*x^2+3*x+2)^4/(2*x^2-x+3),x, algorithm="maxima")
Output:
625/14*x^7 + 3625/24*x^6 + 1855/8*x^5 + 6245/64*x^4 - 21229/96*x^3 - 28747 /128*x^2 - 1156639/5888*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 122691/ 128*x + 307461/512*log(2*x^2 - x + 3)
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{3-x+2 x^2} \, dx=\frac {625}{14} \, x^{7} + \frac {3625}{24} \, x^{6} + \frac {1855}{8} \, x^{5} + \frac {6245}{64} \, x^{4} - \frac {21229}{96} \, x^{3} - \frac {28747}{128} \, x^{2} - \frac {1156639}{5888} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {122691}{128} \, x + \frac {307461}{512} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate((5*x^2+3*x+2)^4/(2*x^2-x+3),x, algorithm="giac")
Output:
625/14*x^7 + 3625/24*x^6 + 1855/8*x^5 + 6245/64*x^4 - 21229/96*x^3 - 28747 /128*x^2 - 1156639/5888*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 122691/ 128*x + 307461/512*log(2*x^2 - x + 3)
Time = 16.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.77 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{3-x+2 x^2} \, dx=\frac {122691\,x}{128}+\frac {307461\,\ln \left (2\,x^2-x+3\right )}{512}-\frac {1156639\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{5888}-\frac {28747\,x^2}{128}-\frac {21229\,x^3}{96}+\frac {6245\,x^4}{64}+\frac {1855\,x^5}{8}+\frac {3625\,x^6}{24}+\frac {625\,x^7}{14} \] Input:
int((3*x + 5*x^2 + 2)^4/(2*x^2 - x + 3),x)
Output:
(122691*x)/128 + (307461*log(2*x^2 - x + 3))/512 - (1156639*23^(1/2)*atan( (4*23^(1/2)*x)/23 - 23^(1/2)/23))/5888 - (28747*x^2)/128 - (21229*x^3)/96 + (6245*x^4)/64 + (1855*x^5)/8 + (3625*x^6)/24 + (625*x^7)/14
Time = 0.17 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.74 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{3-x+2 x^2} \, dx=-\frac {1156639 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right )}{5888}+\frac {307461 \,\mathrm {log}\left (2 x^{2}-x +3\right )}{512}+\frac {625 x^{7}}{14}+\frac {3625 x^{6}}{24}+\frac {1855 x^{5}}{8}+\frac {6245 x^{4}}{64}-\frac {21229 x^{3}}{96}-\frac {28747 x^{2}}{128}+\frac {122691 x}{128} \] Input:
int((5*x^2+3*x+2)^4/(2*x^2-x+3),x)
Output:
( - 48578838*sqrt(23)*atan((4*x - 1)/sqrt(23)) + 148503663*log(2*x**2 - x + 3) + 11040000*x**7 + 37352000*x**6 + 57341760*x**5 + 24130680*x**4 - 546 85904*x**3 - 55539204*x**2 + 237039012*x)/247296