Integrand size = 23, antiderivative size = 64 \[ \int \frac {2+3 x+5 x^2}{\left (3-x+2 x^2\right )^3} \, dx=-\frac {11 (5+3 x)}{92 \left (3-x+2 x^2\right )^2}-\frac {131 (1-4 x)}{2116 \left (3-x+2 x^2\right )}-\frac {262 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{529 \sqrt {23}} \] Output:
1/92*(-55-33*x)/(2*x^2-x+3)^2-131*(1-4*x)/(4232*x^2-2116*x+6348)-262/12167 *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 \[ \int \frac {2+3 x+5 x^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {\frac {46 \left (-829+472 x-393 x^2+524 x^3\right )}{\left (-3+x-2 x^2\right )^2}+1048 \sqrt {23} \arctan \left (\frac {-1+4 x}{\sqrt {23}}\right )}{48668} \] Input:
Integrate[(2 + 3*x + 5*x^2)/(3 - x + 2*x^2)^3,x]
Output:
((46*(-829 + 472*x - 393*x^2 + 524*x^3))/(-3 + x - 2*x^2)^2 + 1048*Sqrt[23 ]*ArcTan[(-1 + 4*x)/Sqrt[23]])/48668
Time = 0.22 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2191, 27, 1086, 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.
\(\displaystyle \int \frac {5 x^2+3 x+2}{\left (2 x^2-x+3\right )^3} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{46} \int \frac {131}{2 \left (2 x^2-x+3\right )^2}dx-\frac {11 (3 x+5)}{92 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {131}{92} \int \frac {1}{\left (2 x^2-x+3\right )^2}dx-\frac {11 (3 x+5)}{92 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 1086 |
\(\displaystyle \frac {131}{92} \left (\frac {4}{23} \int \frac {1}{2 x^2-x+3}dx-\frac {1-4 x}{23 \left (2 x^2-x+3\right )}\right )-\frac {11 (3 x+5)}{92 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {131}{92} \left (-\frac {8}{23} \int \frac {1}{-(4 x-1)^2-23}d(4 x-1)-\frac {1-4 x}{23 \left (2 x^2-x+3\right )}\right )-\frac {11 (3 x+5)}{92 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {131}{92} \left (\frac {8 \arctan \left (\frac {4 x-1}{\sqrt {23}}\right )}{23 \sqrt {23}}-\frac {1-4 x}{23 \left (2 x^2-x+3\right )}\right )-\frac {11 (3 x+5)}{92 \left (2 x^2-x+3\right )^2}\) |
Input:
Int[(2 + 3*x + 5*x^2)/(3 - x + 2*x^2)^3,x]
Output:
(-11*(5 + 3*x))/(92*(3 - x + 2*x^2)^2) + (131*(-1/23*(1 - 4*x)/(3 - x + 2* x^2) + (8*ArcTan[(-1 + 4*x)/Sqrt[23]])/(23*Sqrt[23])))/92
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
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 = 1.66 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {\frac {262}{529} x^{3}-\frac {393}{1058} x^{2}+\frac {236}{529} x -\frac {829}{1058}}{\left (2 x^{2}-x +3\right )^{2}}+\frac {262 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{12167}\) | \(47\) |
risch | \(\frac {\frac {262}{529} x^{3}-\frac {393}{1058} x^{2}+\frac {236}{529} x -\frac {829}{1058}}{\left (2 x^{2}-x +3\right )^{2}}+\frac {262 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{12167}\) | \(47\) |
Input:
int((5*x^2+3*x+2)/(2*x^2-x+3)^3,x,method=_RETURNVERBOSE)
Output:
4*(131/1058*x^3-393/4232*x^2+59/529*x-829/4232)/(2*x^2-x+3)^2+262/12167*23 ^(1/2)*arctan(1/23*(4*x-1)*23^(1/2))
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.17 \[ \int \frac {2+3 x+5 x^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {12052 \, x^{3} + 524 \, \sqrt {23} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - 9039 \, x^{2} + 10856 \, x - 19067}{24334 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \] Input:
integrate((5*x^2+3*x+2)/(2*x^2-x+3)^3,x, algorithm="fricas")
Output:
1/24334*(12052*x^3 + 524*sqrt(23)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*arcta n(1/23*sqrt(23)*(4*x - 1)) - 9039*x^2 + 10856*x - 19067)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \frac {2+3 x+5 x^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {524 x^{3} - 393 x^{2} + 472 x - 829}{4232 x^{4} - 4232 x^{3} + 13754 x^{2} - 6348 x + 9522} + \frac {262 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{12167} \] Input:
integrate((5*x**2+3*x+2)/(2*x**2-x+3)**3,x)
Output:
(524*x**3 - 393*x**2 + 472*x - 829)/(4232*x**4 - 4232*x**3 + 13754*x**2 - 6348*x + 9522) + 262*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 \[ \int \frac {2+3 x+5 x^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {262}{12167} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {524 \, x^{3} - 393 \, x^{2} + 472 \, x - 829}{1058 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \] Input:
integrate((5*x^2+3*x+2)/(2*x^2-x+3)^3,x, algorithm="maxima")
Output:
262/12167*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/1058*(524*x^3 - 393 *x^2 + 472*x - 829)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.72 \[ \int \frac {2+3 x+5 x^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {262}{12167} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {524 \, x^{3} - 393 \, x^{2} + 472 \, x - 829}{1058 \, {\left (2 \, x^{2} - x + 3\right )}^{2}} \] Input:
integrate((5*x^2+3*x+2)/(2*x^2-x+3)^3,x, algorithm="giac")
Output:
262/12167*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/1058*(524*x^3 - 393 *x^2 + 472*x - 829)/(2*x^2 - x + 3)^2
Time = 15.33 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \frac {2+3 x+5 x^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {262\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{12167}+\frac {\frac {131\,x^3}{1058}-\frac {393\,x^2}{4232}+\frac {59\,x}{529}-\frac {829}{4232}}{x^4-x^3+\frac {13\,x^2}{4}-\frac {3\,x}{2}+\frac {9}{4}} \] Input:
int((3*x + 5*x^2 + 2)/(2*x^2 - x + 3)^3,x)
Output:
(262*23^(1/2)*atan((4*23^(1/2)*x)/23 - 23^(1/2)/23))/12167 + ((59*x)/529 - (393*x^2)/4232 + (131*x^3)/1058 - 829/4232)/((13*x^2)/4 - (3*x)/2 - x^3 + x^4 + 9/4)
Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.92 \[ \int \frac {2+3 x+5 x^2}{\left (3-x+2 x^2\right )^3} \, dx=\frac {1048 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{4}-1048 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{3}+3406 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{2}-1572 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x +2358 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right )+6026 x^{4}+15065 x^{2}-3611 x +4025}{48668 x^{4}-48668 x^{3}+158171 x^{2}-73002 x +109503} \] Input:
int((5*x^2+3*x+2)/(2*x^2-x+3)^3,x)
Output:
(1048*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**4 - 1048*sqrt(23)*atan((4*x - 1 )/sqrt(23))*x**3 + 3406*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**2 - 1572*sqrt (23)*atan((4*x - 1)/sqrt(23))*x + 2358*sqrt(23)*atan((4*x - 1)/sqrt(23)) + 6026*x**4 + 15065*x**2 - 3611*x + 4025)/(12167*(4*x**4 - 4*x**3 + 13*x**2 - 6*x + 9))