Integrand size = 25, antiderivative size = 63 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {4 x}{25}+\frac {121 (61+69 x)}{3875 \left (2+3 x+5 x^2\right )}+\frac {41932 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{3875 \sqrt {31}}-\frac {22}{125} \log \left (2+3 x+5 x^2\right ) \] Output:
4/25*x+121*(61+69*x)/(19375*x^2+11625*x+7750)+41932/120125*arctan(1/31*(3+ 10*x)*31^(1/2))*31^(1/2)-22/125*ln(5*x^2+3*x+2)
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {19220 x+\frac {3751 (61+69 x)}{2+3 x+5 x^2}+41932 \sqrt {31} \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )-21142 \log \left (2+3 x+5 x^2\right )}{120125} \] Input:
Integrate[(3 - x + 2*x^2)^2/(2 + 3*x + 5*x^2)^2,x]
Output:
(19220*x + (3751*(61 + 69*x))/(2 + 3*x + 5*x^2) + 41932*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] - 21142*Log[2 + 3*x + 5*x^2])/120125
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.05, 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.
\(\displaystyle \int \frac {\left (2 x^2-x+3\right )^2}{\left (5 x^2+3 x+2\right )^2} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{31} \int \frac {4 \left (155 x^2-248 x+1008\right )}{25 \left (5 x^2+3 x+2\right )}dx+\frac {121 (69 x+61)}{3875 \left (5 x^2+3 x+2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4}{775} \int \frac {155 x^2-248 x+1008}{5 x^2+3 x+2}dx+\frac {121 (69 x+61)}{3875 \left (5 x^2+3 x+2\right )}\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \frac {4}{775} \int \left (\frac {11 (86-31 x)}{5 x^2+3 x+2}+31\right )dx+\frac {121 (69 x+61)}{3875 \left (5 x^2+3 x+2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4}{775} \left (\frac {10483 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{5 \sqrt {31}}-\frac {341}{10} \log \left (5 x^2+3 x+2\right )+31 x\right )+\frac {121 (69 x+61)}{3875 \left (5 x^2+3 x+2\right )}\) |
Input:
Int[(3 - x + 2*x^2)^2/(2 + 3*x + 5*x^2)^2,x]
Output:
(121*(61 + 69*x))/(3875*(2 + 3*x + 5*x^2)) + (4*(31*x + (10483*ArcTan[(3 + 10*x)/Sqrt[31]])/(5*Sqrt[31]) - (341*Log[2 + 3*x + 5*x^2])/10))/775
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 = 2.38 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {4 x}{25}+\frac {\frac {8349 x}{19375}+\frac {7381}{19375}}{x^{2}+\frac {3}{5} x +\frac {2}{5}}-\frac {22 \ln \left (100 x^{2}+60 x +40\right )}{125}+\frac {41932 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{120125}\) | \(50\) |
default | \(\frac {4 x}{25}-\frac {11 \left (-\frac {759 x}{775}-\frac {671}{775}\right )}{25 \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )}-\frac {22 \ln \left (5 x^{2}+3 x +2\right )}{125}+\frac {41932 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{120125}\) | \(51\) |
Input:
int((2*x^2-x+3)^2/(5*x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
4/25*x+(8349/19375*x+7381/19375)/(x^2+3/5*x+2/5)-22/125*ln(100*x^2+60*x+40 )+41932/120125*arctan(1/31*(10*x+3)*31^(1/2))*31^(1/2)
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.24 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {96100 \, x^{3} + 41932 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 57660 \, x^{2} - 21142 \, {\left (5 \, x^{2} + 3 \, x + 2\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 297259 \, x + 228811}{120125 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \] Input:
integrate((2*x^2-x+3)^2/(5*x^2+3*x+2)^2,x, algorithm="fricas")
Output:
1/120125*(96100*x^3 + 41932*sqrt(31)*(5*x^2 + 3*x + 2)*arctan(1/31*sqrt(31 )*(10*x + 3)) + 57660*x^2 - 21142*(5*x^2 + 3*x + 2)*log(5*x^2 + 3*x + 2) + 297259*x + 228811)/(5*x^2 + 3*x + 2)
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {4 x}{25} + \frac {8349 x + 7381}{19375 x^{2} + 11625 x + 7750} - \frac {22 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{125} + \frac {41932 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{120125} \] Input:
integrate((2*x**2-x+3)**2/(5*x**2+3*x+2)**2,x)
Output:
4*x/25 + (8349*x + 7381)/(19375*x**2 + 11625*x + 7750) - 22*log(x**2 + 3*x /5 + 2/5)/125 + 41932*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/1201 25
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {41932}{120125} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {4}{25} \, x + \frac {121 \, {\left (69 \, x + 61\right )}}{3875 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac {22}{125} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \] Input:
integrate((2*x^2-x+3)^2/(5*x^2+3*x+2)^2,x, algorithm="maxima")
Output:
41932/120125*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 4/25*x + 121/3875 *(69*x + 61)/(5*x^2 + 3*x + 2) - 22/125*log(5*x^2 + 3*x + 2)
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {41932}{120125} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {4}{25} \, x + \frac {121 \, {\left (69 \, x + 61\right )}}{3875 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac {22}{125} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \] Input:
integrate((2*x^2-x+3)^2/(5*x^2+3*x+2)^2,x, algorithm="giac")
Output:
41932/120125*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 4/25*x + 121/3875 *(69*x + 61)/(5*x^2 + 3*x + 2) - 22/125*log(5*x^2 + 3*x + 2)
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {4\,x}{25}-\frac {22\,\ln \left (5\,x^2+3\,x+2\right )}{125}+\frac {\frac {8349\,x}{19375}+\frac {7381}{19375}}{x^2+\frac {3\,x}{5}+\frac {2}{5}}+\frac {41932\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{120125} \] Input:
int((2*x^2 - x + 3)^2/(3*x + 5*x^2 + 2)^2,x)
Output:
(4*x)/25 - (22*log(3*x + 5*x^2 + 2))/125 + ((8349*x)/19375 + 7381/19375)/( (3*x)/5 + x^2 + 2/5) + (41932*31^(1/2)*atan((10*31^(1/2)*x)/31 + (3*31^(1/ 2))/31))/120125
Time = 0.19 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.86 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {628980 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{2}+377388 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x +251592 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right )-317130 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{2}-190278 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x -126852 \,\mathrm {log}\left (5 x^{2}+3 x +2\right )+288300 x^{3}-1313315 x^{2}+91915}{1801875 x^{2}+1081125 x +720750} \] Input:
int((2*x^2-x+3)^2/(5*x^2+3*x+2)^2,x)
Output:
(628980*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**2 + 377388*sqrt(31)*atan((10 *x + 3)/sqrt(31))*x + 251592*sqrt(31)*atan((10*x + 3)/sqrt(31)) - 317130*l og(5*x**2 + 3*x + 2)*x**2 - 190278*log(5*x**2 + 3*x + 2)*x - 126852*log(5* x**2 + 3*x + 2) + 288300*x**3 - 1313315*x**2 + 91915)/(360375*(5*x**2 + 3* x + 2))