Integrand size = 25, antiderivative size = 77 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {1466 x}{625}-\frac {54 x^2}{125}+\frac {8 x^3}{75}+\frac {1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}+\frac {3819607 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{96875 \sqrt {31}}-\frac {10769 \log \left (2+3 x+5 x^2\right )}{6250} \] Output:
1466/625*x-54/125*x^2+8/75*x^3+1331*(443+247*x)/(484375*x^2+290625*x+19375 0)+3819607/3003125*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)-10769/6250*ln(5 *x^2+3*x+2)
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {1466 x}{625}-\frac {54 x^2}{125}+\frac {8 x^3}{75}+\frac {1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}+\frac {3819607 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{96875 \sqrt {31}}-\frac {10769 \log \left (2+3 x+5 x^2\right )}{6250} \] Input:
Integrate[(3 - x + 2*x^2)^3/(2 + 3*x + 5*x^2)^2,x]
Output:
(1466*x)/625 - (54*x^2)/125 + (8*x^3)/75 + (1331*(443 + 247*x))/(96875*(2 + 3*x + 5*x^2)) + (3819607*ArcTan[(3 + 10*x)/Sqrt[31]])/(96875*Sqrt[31]) - (10769*Log[2 + 3*x + 5*x^2])/6250
Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01, 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.
\(\displaystyle \int \frac {\left (2 x^2-x+3\right )^3}{\left (5 x^2+3 x+2\right )^2} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{31} \int \frac {31000 x^4-65100 x^3+189410 x^2-230981 x+372701}{625 \left (5 x^2+3 x+2\right )}dx+\frac {1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {31000 x^4-65100 x^3+189410 x^2-230981 x+372701}{5 x^2+3 x+2}dx}{19375}+\frac {1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \frac {\int \left (6200 x^2-16740 x+\frac {121 (2329-2759 x)}{5 x^2+3 x+2}+45446\right )dx}{19375}+\frac {1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {3819607 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{5 \sqrt {31}}+\frac {6200 x^3}{3}-8370 x^2-\frac {333839}{10} \log \left (5 x^2+3 x+2\right )+45446 x}{19375}+\frac {1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}\) |
Input:
Int[(3 - x + 2*x^2)^3/(2 + 3*x + 5*x^2)^2,x]
Output:
(1331*(443 + 247*x))/(96875*(2 + 3*x + 5*x^2)) + (45446*x - 8370*x^2 + (62 00*x^3)/3 + (3819607*ArcTan[(3 + 10*x)/Sqrt[31]])/(5*Sqrt[31]) - (333839*L og[2 + 3*x + 5*x^2])/10)/19375
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 = 2.42 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {8 x^{3}}{75}-\frac {54 x^{2}}{125}+\frac {1466 x}{625}+\frac {\frac {328757 x}{484375}+\frac {589633}{484375}}{x^{2}+\frac {3}{5} x +\frac {2}{5}}-\frac {10769 \ln \left (100 x^{2}+60 x +40\right )}{6250}+\frac {3819607 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{3003125}\) | \(60\) |
default | \(\frac {8 x^{3}}{75}-\frac {54 x^{2}}{125}+\frac {1466 x}{625}-\frac {121 \left (-\frac {2717 x}{775}-\frac {4873}{775}\right )}{625 \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )}-\frac {10769 \ln \left (5 x^{2}+3 x +2\right )}{6250}+\frac {3819607 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{3003125}\) | \(61\) |
Input:
int((2*x^2-x+3)^3/(5*x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
8/75*x^3-54/125*x^2+1466/625*x+(328757/484375*x+589633/484375)/(x^2+3/5*x+ 2/5)-10769/6250*ln(100*x^2+60*x+40)+3819607/3003125*arctan(1/31*(10*x+3)*3 1^(1/2))*31^(1/2)
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.14 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {9610000 \, x^{5} - 33154500 \, x^{4} + 191815600 \, x^{3} + 22917642 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 111226140 \, x^{2} - 31047027 \, {\left (5 \, x^{2} + 3 \, x + 2\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 145678362 \, x + 109671738}{18018750 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \] Input:
integrate((2*x^2-x+3)^3/(5*x^2+3*x+2)^2,x, algorithm="fricas")
Output:
1/18018750*(9610000*x^5 - 33154500*x^4 + 191815600*x^3 + 22917642*sqrt(31) *(5*x^2 + 3*x + 2)*arctan(1/31*sqrt(31)*(10*x + 3)) + 111226140*x^2 - 3104 7027*(5*x^2 + 3*x + 2)*log(5*x^2 + 3*x + 2) + 145678362*x + 109671738)/(5* x^2 + 3*x + 2)
Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {8 x^{3}}{75} - \frac {54 x^{2}}{125} + \frac {1466 x}{625} + \frac {328757 x + 589633}{484375 x^{2} + 290625 x + 193750} - \frac {10769 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{6250} + \frac {3819607 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{3003125} \] Input:
integrate((2*x**2-x+3)**3/(5*x**2+3*x+2)**2,x)
Output:
8*x**3/75 - 54*x**2/125 + 1466*x/625 + (328757*x + 589633)/(484375*x**2 + 290625*x + 193750) - 10769*log(x**2 + 3*x/5 + 2/5)/6250 + 3819607*sqrt(31) *atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/3003125
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {8}{75} \, x^{3} - \frac {54}{125} \, x^{2} + \frac {3819607}{3003125} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {1466}{625} \, x + \frac {1331 \, {\left (247 \, x + 443\right )}}{96875 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac {10769}{6250} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \] Input:
integrate((2*x^2-x+3)^3/(5*x^2+3*x+2)^2,x, algorithm="maxima")
Output:
8/75*x^3 - 54/125*x^2 + 3819607/3003125*sqrt(31)*arctan(1/31*sqrt(31)*(10* x + 3)) + 1466/625*x + 1331/96875*(247*x + 443)/(5*x^2 + 3*x + 2) - 10769/ 6250*log(5*x^2 + 3*x + 2)
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {8}{75} \, x^{3} - \frac {54}{125} \, x^{2} + \frac {3819607}{3003125} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {1466}{625} \, x + \frac {1331 \, {\left (247 \, x + 443\right )}}{96875 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac {10769}{6250} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \] Input:
integrate((2*x^2-x+3)^3/(5*x^2+3*x+2)^2,x, algorithm="giac")
Output:
8/75*x^3 - 54/125*x^2 + 3819607/3003125*sqrt(31)*arctan(1/31*sqrt(31)*(10* x + 3)) + 1466/625*x + 1331/96875*(247*x + 443)/(5*x^2 + 3*x + 2) - 10769/ 6250*log(5*x^2 + 3*x + 2)
Time = 15.34 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {1466\,x}{625}-\frac {10769\,\ln \left (5\,x^2+3\,x+2\right )}{6250}+\frac {\frac {328757\,x}{484375}+\frac {589633}{484375}}{x^2+\frac {3\,x}{5}+\frac {2}{5}}+\frac {3819607\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{3003125}-\frac {54\,x^2}{125}+\frac {8\,x^3}{75} \] Input:
int((2*x^2 - x + 3)^3/(3*x + 5*x^2 + 2)^2,x)
Output:
(1466*x)/625 - (10769*log(3*x + 5*x^2 + 2))/6250 + ((328757*x)/484375 + 58 9633/484375)/((3*x)/5 + x^2 + 2/5) + (3819607*31^(1/2)*atan((10*31^(1/2)*x )/31 + (3*31^(1/2))/31))/3003125 - (54*x^2)/125 + (8*x^3)/75
Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.65 \[ \int \frac {\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {114588210 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{2}+68752926 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x +45835284 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right )-155235135 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{2}-93141081 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x -62094054 \,\mathrm {log}\left (5 x^{2}+3 x +2\right )+9610000 x^{5}-33154500 x^{4}+191815600 x^{3}-131571130 x^{2}+12552830}{90093750 x^{2}+54056250 x +36037500} \] Input:
int((2*x^2-x+3)^3/(5*x^2+3*x+2)^2,x)
Output:
(114588210*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**2 + 68752926*sqrt(31)*ata n((10*x + 3)/sqrt(31))*x + 45835284*sqrt(31)*atan((10*x + 3)/sqrt(31)) - 1 55235135*log(5*x**2 + 3*x + 2)*x**2 - 93141081*log(5*x**2 + 3*x + 2)*x - 6 2094054*log(5*x**2 + 3*x + 2) + 9610000*x**5 - 33154500*x**4 + 191815600*x **3 - 131571130*x**2 + 12552830)/(18018750*(5*x**2 + 3*x + 2))