Integrand size = 25, antiderivative size = 64 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {121 (19-7 x)}{368 \left (3-x+2 x^2\right )^2}-\frac {55 (975+332 x)}{8464 \left (3-x+2 x^2\right )}-\frac {4330 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{529 \sqrt {23}} \] Output:
121/368*(19-7*x)/(2*x^2-x+3)^2-55*(975+332*x)/(16928*x^2-8464*x+25392)-433 0/12167*arctan(1/23*(1-4*x)*23^(1/2))*23^(1/2)
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^3} \, dx=-\frac {11 \left (4909+938 x+4045 x^2+1660 x^3\right )}{4232 \left (-3+x-2 x^2\right )^2}+\frac {4330 \arctan \left (\frac {-1+4 x}{\sqrt {23}}\right )}{529 \sqrt {23}} \] Input:
Integrate[(2 + 3*x + 5*x^2)^2/(3 - x + 2*x^2)^3,x]
Output:
(-11*(4909 + 938*x + 4045*x^2 + 1660*x^3))/(4232*(-3 + x - 2*x^2)^2) + (43 30*ArcTan[(-1 + 4*x)/Sqrt[23]])/(529*Sqrt[23])
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2191, 27, 2191, 27, 1083, 217}
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.
\(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^2}{\left (2 x^2-x+3\right )^3} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{46} \int -\frac {5 \left (-920 x^2-1564 x+39\right )}{8 \left (2 x^2-x+3\right )^2}dx+\frac {121 (19-7 x)}{368 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {121 (19-7 x)}{368 \left (2 x^2-x+3\right )^2}-\frac {5}{368} \int \frac {-920 x^2-1564 x+39}{\left (2 x^2-x+3\right )^2}dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {121 (19-7 x)}{368 \left (2 x^2-x+3\right )^2}-\frac {5}{368} \left (\frac {1}{23} \int -\frac {6928}{2 x^2-x+3}dx+\frac {11 (332 x+975)}{23 \left (2 x^2-x+3\right )}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {121 (19-7 x)}{368 \left (2 x^2-x+3\right )^2}-\frac {5}{368} \left (\frac {11 (332 x+975)}{23 \left (2 x^2-x+3\right )}-\frac {6928}{23} \int \frac {1}{2 x^2-x+3}dx\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {121 (19-7 x)}{368 \left (2 x^2-x+3\right )^2}-\frac {5}{368} \left (\frac {13856}{23} \int \frac {1}{-(4 x-1)^2-23}d(4 x-1)+\frac {11 (332 x+975)}{23 \left (2 x^2-x+3\right )}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {121 (19-7 x)}{368 \left (2 x^2-x+3\right )^2}-\frac {5}{368} \left (\frac {11 (332 x+975)}{23 \left (2 x^2-x+3\right )}-\frac {13856 \arctan \left (\frac {4 x-1}{\sqrt {23}}\right )}{23 \sqrt {23}}\right )\) |
Input:
Int[(2 + 3*x + 5*x^2)^2/(3 - x + 2*x^2)^3,x]
Output:
(121*(19 - 7*x))/(368*(3 - x + 2*x^2)^2) - (5*((11*(975 + 332*x))/(23*(3 - x + 2*x^2)) - (13856*ArcTan[(-1 + 4*x)/Sqrt[23]])/(23*Sqrt[23])))/368
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
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 = 2.40 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {-\frac {4565}{1058} x^{3}-\frac {44495}{4232} x^{2}-\frac {5159}{2116} x -\frac {53999}{4232}}{\left (2 x^{2}-x +3\right )^{2}}+\frac {4330 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{12167}\) | \(47\) |
risch | \(\frac {-\frac {4565}{1058} x^{3}-\frac {44495}{4232} x^{2}-\frac {5159}{2116} x -\frac {53999}{4232}}{\left (2 x^{2}-x +3\right )^{2}}+\frac {4330 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{12167}\) | \(47\) |
Input:
int((5*x^2+3*x+2)^2/(2*x^2-x+3)^3,x,method=_RETURNVERBOSE)
Output:
4*(-4565/4232*x^3-44495/16928*x^2-5159/8464*x-53999/16928)/(2*x^2-x+3)^2+4 330/12167*23^(1/2)*arctan(1/23*(4*x-1)*23^(1/2))
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.17 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^3} \, dx=-\frac {419980 \, x^{3} - 34640 \, \sqrt {23} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 1023385 \, x^{2} + 237314 \, x + 1241977}{97336 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \] Input:
integrate((5*x^2+3*x+2)^2/(2*x^2-x+3)^3,x, algorithm="fricas")
Output:
-1/97336*(419980*x^3 - 34640*sqrt(23)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*a rctan(1/23*sqrt(23)*(4*x - 1)) + 1023385*x^2 + 237314*x + 1241977)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.98 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {- 18260 x^{3} - 44495 x^{2} - 10318 x - 53999}{16928 x^{4} - 16928 x^{3} + 55016 x^{2} - 25392 x + 38088} + \frac {4330 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{12167} \] Input:
integrate((5*x**2+3*x+2)**2/(2*x**2-x+3)**3,x)
Output:
(-18260*x**3 - 44495*x**2 - 10318*x - 53999)/(16928*x**4 - 16928*x**3 + 55 016*x**2 - 25392*x + 38088) + 4330*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23 )/23)/12167
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {4330}{12167} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {11 \, {\left (1660 \, x^{3} + 4045 \, x^{2} + 938 \, x + 4909\right )}}{4232 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \] Input:
integrate((5*x^2+3*x+2)^2/(2*x^2-x+3)^3,x, algorithm="maxima")
Output:
4330/12167*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) - 11/4232*(1660*x^3 + 4045*x^2 + 938*x + 4909)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.72 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {4330}{12167} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {11 \, {\left (1660 \, x^{3} + 4045 \, x^{2} + 938 \, x + 4909\right )}}{4232 \, {\left (2 \, x^{2} - x + 3\right )}^{2}} \] Input:
integrate((5*x^2+3*x+2)^2/(2*x^2-x+3)^3,x, algorithm="giac")
Output:
4330/12167*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) - 11/4232*(1660*x^3 + 4045*x^2 + 938*x + 4909)/(2*x^2 - x + 3)^2
Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {4330\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{12167}-\frac {\frac {4565\,x^3}{4232}+\frac {44495\,x^2}{16928}+\frac {5159\,x}{8464}+\frac {53999}{16928}}{x^4-x^3+\frac {13\,x^2}{4}-\frac {3\,x}{2}+\frac {9}{4}} \] Input:
int((3*x + 5*x^2 + 2)^2/(2*x^2 - x + 3)^3,x)
Output:
(4330*23^(1/2)*atan((4*23^(1/2)*x)/23 - 23^(1/2)/23))/12167 - ((5159*x)/84 64 + (44495*x^2)/16928 + (4565*x^3)/4232 + 53999/16928)/((13*x^2)/4 - (3*x )/2 - x^3 + x^4 + 9/4)
Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.92 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {34640 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{4}-34640 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{3}+112580 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{2}-51960 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x +77940 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right )-104995 x^{4}-597080 x^{2}+98164 x -546733}{97336 x^{4}-97336 x^{3}+316342 x^{2}-146004 x +219006} \] Input:
int((5*x^2+3*x+2)^2/(2*x^2-x+3)^3,x)
Output:
(34640*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**4 - 34640*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**3 + 112580*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**2 - 51960 *sqrt(23)*atan((4*x - 1)/sqrt(23))*x + 77940*sqrt(23)*atan((4*x - 1)/sqrt( 23)) - 104995*x**4 - 597080*x**2 + 98164*x - 546733)/(24334*(4*x**4 - 4*x* *3 + 13*x**2 - 6*x + 9))