Integrand size = 25, antiderivative size = 84 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {8 x}{125}+\frac {1331 (443+247 x)}{193750 \left (2+3 x+5 x^2\right )^2}+\frac {121 (188381+342840 x)}{6006250 \left (2+3 x+5 x^2\right )}+\frac {11341176 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{600625 \sqrt {31}}-\frac {66}{625} \log \left (2+3 x+5 x^2\right ) \] Output:
8/125*x+1331/193750*(443+247*x)/(5*x^2+3*x+2)^2+121*(188381+342840*x)/(300 31250*x^2+18018750*x+12012500)+11341176/18619375*arctan(1/31*(3+10*x)*31^( 1/2))*31^(1/2)-66/625*ln(5*x^2+3*x+2)
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.93 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {11916400 x+\frac {1279091 (443+247 x)}{\left (2+3 x+5 x^2\right )^2}+\frac {3751 (188381+342840 x)}{2+3 x+5 x^2}+113411760 \sqrt {31} \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )-19662060 \log \left (2+3 x+5 x^2\right )}{186193750} \] Input:
Integrate[(3 - x + 2*x^2)^3/(2 + 3*x + 5*x^2)^3,x]
Output:
(11916400*x + (1279091*(443 + 247*x))/(2 + 3*x + 5*x^2)^2 + (3751*(188381 + 342840*x))/(2 + 3*x + 5*x^2) + 113411760*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt [31]] - 19662060*Log[2 + 3*x + 5*x^2])/186193750
Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2191, 27, 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.
| \(\displaystyle \int \frac {\left (2 x^2-x+3\right )^3}{\left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {1}{62} \int \frac {310000 x^4-651000 x^3+1894100 x^2-2309810 x+4055767}{3125 \left (5 x^2+3 x+2\right )^2}dx+\frac {1331 (247 x+443)}{193750 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {310000 x^4-651000 x^3+1894100 x^2-2309810 x+4055767}{\left (5 x^2+3 x+2\right )^2}dx}{193750}+\frac {1331 (247 x+443)}{193750 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {\frac {1}{31} \int \frac {100 \left (19220 x^2-51894 x+555719\right )}{5 x^2+3 x+2}dx+\frac {121 (342840 x+188381)}{31 \left (5 x^2+3 x+2\right )}}{193750}+\frac {1331 (247 x+443)}{193750 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {100}{31} \int \frac {19220 x^2-51894 x+555719}{5 x^2+3 x+2}dx+\frac {121 (342840 x+188381)}{31 \left (5 x^2+3 x+2\right )}}{193750}+\frac {1331 (247 x+443)}{193750 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2188 |
| \(\displaystyle \frac {\frac {100}{31} \int \left (\frac {33 (16607-1922 x)}{5 x^2+3 x+2}+3844\right )dx+\frac {121 (342840 x+188381)}{31 \left (5 x^2+3 x+2\right )}}{193750}+\frac {1331 (247 x+443)}{193750 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {100}{31} \left (\frac {5670588 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{5 \sqrt {31}}-\frac {31713}{5} \log \left (5 x^2+3 x+2\right )+3844 x\right )+\frac {121 (342840 x+188381)}{31 \left (5 x^2+3 x+2\right )}}{193750}+\frac {1331 (247 x+443)}{193750 \left (5 x^2+3 x+2\right )^2}\) |
Input:
Int[(3 - x + 2*x^2)^3/(2 + 3*x + 5*x^2)^3,x]
Output:
(1331*(443 + 247*x))/(193750*(2 + 3*x + 5*x^2)^2) + ((121*(188381 + 342840 *x))/(31*(2 + 3*x + 5*x^2)) + (100*(3844*x + (5670588*ArcTan[(3 + 10*x)/Sq rt[31]])/(5*Sqrt[31]) - (31713*Log[2 + 3*x + 5*x^2])/5))/31)/193750
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 = 2.44 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75
| method | result | size |
| default | \(\frac {8 x}{125}-\frac {11 \left (-\frac {377124}{24025} x^{3}-\frac {866987}{48050} x^{2}-\frac {293711}{24025} x -\frac {232243}{48050}\right )}{5 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {66 \ln \left (5 x^{2}+3 x +2\right )}{625}+\frac {11341176 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{18619375}\) | \(63\) |
| risch | \(\frac {8 x}{125}+\frac {\frac {4148364}{120125} x^{3}+\frac {9536857}{240250} x^{2}+\frac {3230821}{120125} x +\frac {2554673}{240250}}{\left (5 x^{2}+3 x +2\right )^{2}}-\frac {66 \ln \left (100 x^{2}+60 x +40\right )}{625}+\frac {11341176 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{18619375}\) | \(63\) |
Input:
int((2*x^2-x+3)^3/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
Output:
8/125*x-11/5*(-377124/24025*x^3-866987/48050*x^2-293711/24025*x-232243/480 50)/(5*x^2+3*x+2)^2-66/625*ln(5*x^2+3*x+2)+11341176/18619375*arctan(1/31*( 10*x+3)*31^(1/2))*31^(1/2)
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.40 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {59582000 \, x^{5} + 71498400 \, x^{4} + 1355107960 \, x^{3} + 22682352 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 1506812195 \, x^{2} - 3932412 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 1011087630 \, x + 395974315}{37238750 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \] Input:
integrate((2*x^2-x+3)^3/(5*x^2+3*x+2)^3,x, algorithm="fricas")
Output:
1/37238750*(59582000*x^5 + 71498400*x^4 + 1355107960*x^3 + 22682352*sqrt(3 1)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1506812195*x^2 - 3932412*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*log(5*x^2 + 3*x + 2) + 1011087630*x + 395974315)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {8 x}{125} + \frac {8296728 x^{3} + 9536857 x^{2} + 6461642 x + 2554673}{6006250 x^{4} + 7207500 x^{3} + 6967250 x^{2} + 2883000 x + 961000} - \frac {66 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{625} + \frac {11341176 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{18619375} \] Input:
integrate((2*x**2-x+3)**3/(5*x**2+3*x+2)**3,x)
Output:
8*x/125 + (8296728*x**3 + 9536857*x**2 + 6461642*x + 2554673)/(6006250*x** 4 + 7207500*x**3 + 6967250*x**2 + 2883000*x + 961000) - 66*log(x**2 + 3*x/ 5 + 2/5)/625 + 11341176*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/18 619375
Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.86 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {11341176}{18619375} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {8}{125} \, x + \frac {121 \, {\left (68568 \, x^{3} + 78817 \, x^{2} + 53402 \, x + 21113\right )}}{240250 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} - \frac {66}{625} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \] Input:
integrate((2*x^2-x+3)^3/(5*x^2+3*x+2)^3,x, algorithm="maxima")
Output:
11341176/18619375*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 8/125*x + 12 1/240250*(68568*x^3 + 78817*x^2 + 53402*x + 21113)/(25*x^4 + 30*x^3 + 29*x ^2 + 12*x + 4) - 66/625*log(5*x^2 + 3*x + 2)
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.74 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {11341176}{18619375} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {8}{125} \, x + \frac {121 \, {\left (68568 \, x^{3} + 78817 \, x^{2} + 53402 \, x + 21113\right )}}{240250 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} - \frac {66}{625} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \] Input:
integrate((2*x^2-x+3)^3/(5*x^2+3*x+2)^3,x, algorithm="giac")
Output:
11341176/18619375*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 8/125*x + 12 1/240250*(68568*x^3 + 78817*x^2 + 53402*x + 21113)/(5*x^2 + 3*x + 2)^2 - 6 6/625*log(5*x^2 + 3*x + 2)
Time = 16.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.85 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {8\,x}{125}-\frac {66\,\ln \left (5\,x^2+3\,x+2\right )}{625}+\frac {11341176\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{18619375}+\frac {\frac {4148364\,x^3}{3003125}+\frac {9536857\,x^2}{6006250}+\frac {3230821\,x}{3003125}+\frac {2554673}{6006250}}{x^4+\frac {6\,x^3}{5}+\frac {29\,x^2}{25}+\frac {12\,x}{25}+\frac {4}{25}} \] Input:
int((2*x^2 - x + 3)^3/(3*x + 5*x^2 + 2)^3,x)
Output:
(8*x)/125 - (66*log(3*x + 5*x^2 + 2))/625 + (11341176*31^(1/2)*atan((10*31 ^(1/2)*x)/31 + (3*31^(1/2))/31))/18619375 + ((3230821*x)/3003125 + (953685 7*x^2)/6006250 + (4148364*x^3)/3003125 + 2554673/6006250)/((12*x)/25 + (29 *x^2)/25 + (6*x^3)/5 + x^4 + 4/25)
Time = 0.19 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.42 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {1701176400 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{4}+2041411680 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{3}+1973364624 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{2}+816564672 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x +272188224 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right )-294930900 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{4}-353917080 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{3}-342119844 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{2}-141566832 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x -47188944 \,\mathrm {log}\left (5 x^{2}+3 x +2\right )+178746000 x^{5}-3173274700 x^{4}+590623501 x^{2}+1407133338 x +645879761}{2792906250 x^{4}+3351487500 x^{3}+3239771250 x^{2}+1340595000 x +446865000} \] Input:
int((2*x^2-x+3)^3/(5*x^2+3*x+2)^3,x)
Output:
(1701176400*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**4 + 2041411680*sqrt(31)* atan((10*x + 3)/sqrt(31))*x**3 + 1973364624*sqrt(31)*atan((10*x + 3)/sqrt( 31))*x**2 + 816564672*sqrt(31)*atan((10*x + 3)/sqrt(31))*x + 272188224*sqr t(31)*atan((10*x + 3)/sqrt(31)) - 294930900*log(5*x**2 + 3*x + 2)*x**4 - 3 53917080*log(5*x**2 + 3*x + 2)*x**3 - 342119844*log(5*x**2 + 3*x + 2)*x**2 - 141566832*log(5*x**2 + 3*x + 2)*x - 47188944*log(5*x**2 + 3*x + 2) + 17 8746000*x**5 - 3173274700*x**4 + 590623501*x**2 + 1407133338*x + 645879761 )/(111716250*(25*x**4 + 30*x**3 + 29*x**2 + 12*x + 4))