Integrand size = 25, antiderivative size = 63 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^2} \, dx=\frac {25 x}{4}+\frac {121 (19-7 x)}{184 \left (3-x+2 x^2\right )}+\frac {1859 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{92 \sqrt {23}}+\frac {55}{8} \log \left (3-x+2 x^2\right ) \] Output:
25/4*x+121*(19-7*x)/(368*x^2-184*x+552)+1859/2116*arctan(1/23*(1-4*x)*23^( 1/2))*23^(1/2)+55/8*ln(2*x^2-x+3)
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^2} \, dx=\frac {25 x}{4}-\frac {121 (-19+7 x)}{184 \left (3-x+2 x^2\right )}-\frac {1859 \arctan \left (\frac {-1+4 x}{\sqrt {23}}\right )}{92 \sqrt {23}}+\frac {55}{8} \log \left (3-x+2 x^2\right ) \] Input:
Integrate[(2 + 3*x + 5*x^2)^2/(3 - x + 2*x^2)^2,x]
Output:
(25*x)/4 - (121*(-19 + 7*x))/(184*(3 - x + 2*x^2)) - (1859*ArcTan[(-1 + 4* x)/Sqrt[23]])/(92*Sqrt[23]) + (55*Log[3 - x + 2*x^2])/8
Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2191, 27, 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.
\(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^2}{\left (2 x^2-x+3\right )^2} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{23} \int \frac {1150 x^2+1955 x+163}{4 \left (2 x^2-x+3\right )}dx+\frac {121 (19-7 x)}{184 \left (2 x^2-x+3\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{92} \int \frac {1150 x^2+1955 x+163}{2 x^2-x+3}dx+\frac {121 (19-7 x)}{184 \left (2 x^2-x+3\right )}\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \frac {1}{92} \int \left (575-\frac {22 (71-115 x)}{2 x^2-x+3}\right )dx+\frac {121 (19-7 x)}{184 \left (2 x^2-x+3\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{92} \left (\frac {1859 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{\sqrt {23}}+\frac {1265}{2} \log \left (2 x^2-x+3\right )+575 x\right )+\frac {121 (19-7 x)}{184 \left (2 x^2-x+3\right )}\) |
Input:
Int[(2 + 3*x + 5*x^2)^2/(3 - x + 2*x^2)^2,x]
Output:
(121*(19 - 7*x))/(184*(3 - x + 2*x^2)) + (575*x + (1859*ArcTan[(1 - 4*x)/S qrt[23]])/Sqrt[23] + (1265*Log[3 - x + 2*x^2])/2)/92
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Time = 2.36 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {25 x}{4}+\frac {-\frac {847 x}{368}+\frac {2299}{368}}{x^{2}-\frac {1}{2} x +\frac {3}{2}}+\frac {55 \ln \left (16 x^{2}-8 x +24\right )}{8}-\frac {1859 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{2116}\) | \(50\) |
default | \(\frac {25 x}{4}+\frac {-\frac {847 x}{368}+\frac {2299}{368}}{x^{2}-\frac {1}{2} x +\frac {3}{2}}+\frac {55 \ln \left (2 x^{2}-x +3\right )}{8}-\frac {1859 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{2116}\) | \(51\) |
Input:
int((5*x^2+3*x+2)^2/(2*x^2-x+3)^2,x,method=_RETURNVERBOSE)
Output:
25/4*x+(-847/368*x+2299/368)/(x^2-1/2*x+3/2)+55/8*ln(16*x^2-8*x+24)-1859/2 116*23^(1/2)*arctan(1/23*(4*x-1)*23^(1/2))
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.24 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^2} \, dx=\frac {52900 \, x^{3} - 3718 \, \sqrt {23} {\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - 26450 \, x^{2} + 29095 \, {\left (2 \, x^{2} - x + 3\right )} \log \left (2 \, x^{2} - x + 3\right ) + 59869 \, x + 52877}{4232 \, {\left (2 \, x^{2} - x + 3\right )}} \] Input:
integrate((5*x^2+3*x+2)^2/(2*x^2-x+3)^2,x, algorithm="fricas")
Output:
1/4232*(52900*x^3 - 3718*sqrt(23)*(2*x^2 - x + 3)*arctan(1/23*sqrt(23)*(4* x - 1)) - 26450*x^2 + 29095*(2*x^2 - x + 3)*log(2*x^2 - x + 3) + 59869*x + 52877)/(2*x^2 - x + 3)
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^2} \, dx=\frac {25 x}{4} + \frac {2299 - 847 x}{368 x^{2} - 184 x + 552} + \frac {55 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{8} - \frac {1859 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{2116} \] Input:
integrate((5*x**2+3*x+2)**2/(2*x**2-x+3)**2,x)
Output:
25*x/4 + (2299 - 847*x)/(368*x**2 - 184*x + 552) + 55*log(x**2 - x/2 + 3/2 )/8 - 1859*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/2116
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^2} \, dx=-\frac {1859}{2116} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {25}{4} \, x - \frac {121 \, {\left (7 \, x - 19\right )}}{184 \, {\left (2 \, x^{2} - x + 3\right )}} + \frac {55}{8} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate((5*x^2+3*x+2)^2/(2*x^2-x+3)^2,x, algorithm="maxima")
Output:
-1859/2116*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 25/4*x - 121/184*(7* x - 19)/(2*x^2 - x + 3) + 55/8*log(2*x^2 - x + 3)
Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^2} \, dx=-\frac {1859}{2116} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {25}{4} \, x - \frac {121 \, {\left (7 \, x - 19\right )}}{184 \, {\left (2 \, x^{2} - x + 3\right )}} + \frac {55}{8} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate((5*x^2+3*x+2)^2/(2*x^2-x+3)^2,x, algorithm="giac")
Output:
-1859/2116*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 25/4*x - 121/184*(7* x - 19)/(2*x^2 - x + 3) + 55/8*log(2*x^2 - x + 3)
Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^2} \, dx=\frac {25\,x}{4}+\frac {55\,\ln \left (2\,x^2-x+3\right )}{8}-\frac {\frac {847\,x}{368}-\frac {2299}{368}}{x^2-\frac {x}{2}+\frac {3}{2}}-\frac {1859\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{2116} \] Input:
int((3*x + 5*x^2 + 2)^2/(2*x^2 - x + 3)^2,x)
Output:
(25*x)/4 + (55*log(2*x^2 - x + 3))/8 - ((847*x)/368 - 2299/368)/(x^2 - x/2 + 3/2) - (1859*23^(1/2)*atan((4*23^(1/2)*x)/23 - 23^(1/2)/23))/2116
Time = 0.20 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.86 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^2} \, dx=\frac {-7436 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{2}+3718 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x -11154 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right )+58190 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x^{2}-29095 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x +87285 \,\mathrm {log}\left (2 x^{2}-x +3\right )+52900 x^{3}+93288 x^{2}+232484}{8464 x^{2}-4232 x +12696} \] Input:
int((5*x^2+3*x+2)^2/(2*x^2-x+3)^2,x)
Output:
( - 7436*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**2 + 3718*sqrt(23)*atan((4*x - 1)/sqrt(23))*x - 11154*sqrt(23)*atan((4*x - 1)/sqrt(23)) + 58190*log(2*x **2 - x + 3)*x**2 - 29095*log(2*x**2 - x + 3)*x + 87285*log(2*x**2 - x + 3 ) + 52900*x**3 + 93288*x**2 + 232484)/(4232*(2*x**2 - x + 3))