Integrand size = 25, antiderivative size = 91 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^2} \, dx=-\frac {89359 x}{64}-\frac {1185 x^2}{8}+\frac {9775 x^3}{48}+\frac {2125 x^4}{16}+\frac {125 x^5}{4}-\frac {14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}-\frac {13292697 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{1472 \sqrt {23}}-\frac {30613}{128} \log \left (3-x+2 x^2\right ) \] Output:
-89359/64*x-1185/8*x^2+9775/48*x^3+2125/16*x^4+125/4*x^5-14641*(101+79*x)/ (5888*x^2-2944*x+8832)-13292697/33856*arctan(1/23*(1-4*x)*23^(1/2))*23^(1/ 2)-30613/128*ln(2*x^2-x+3)
Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^2} \, dx=-\frac {89359 x}{64}-\frac {1185 x^2}{8}+\frac {9775 x^3}{48}+\frac {2125 x^4}{16}+\frac {125 x^5}{4}-\frac {14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}+\frac {13292697 \arctan \left (\frac {-1+4 x}{\sqrt {23}}\right )}{1472 \sqrt {23}}-\frac {30613}{128} \log \left (3-x+2 x^2\right ) \] Input:
Integrate[(2 + 3*x + 5*x^2)^4/(3 - x + 2*x^2)^2,x]
Output:
(-89359*x)/64 - (1185*x^2)/8 + (9775*x^3)/48 + (2125*x^4)/16 + (125*x^5)/4 - (14641*(101 + 79*x))/(2944*(3 - x + 2*x^2)) + (13292697*ArcTan[(-1 + 4* x)/Sqrt[23]])/(1472*Sqrt[23]) - (30613*Log[3 - x + 2*x^2])/128
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.95, 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.
\(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^4}{\left (2 x^2-x+3\right )^2} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{23} \int \frac {460000 x^6+1334000 x^5+1706600 x^4+574540 x^3-976534 x^2-661181 x+832627}{64 \left (2 x^2-x+3\right )}dx-\frac {14641 (79 x+101)}{2944 \left (2 x^2-x+3\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {460000 x^6+1334000 x^5+1706600 x^4+574540 x^3-976534 x^2-661181 x+832627}{2 x^2-x+3}dx}{1472}-\frac {14641 (79 x+101)}{2944 \left (2 x^2-x+3\right )}\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \frac {\int \left (230000 x^4+782000 x^3+899300 x^2-436080 x+\frac {2662 (2629-529 x)}{2 x^2-x+3}-2055257\right )dx}{1472}-\frac {14641 (79 x+101)}{2944 \left (2 x^2-x+3\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {13292697 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{\sqrt {23}}+46000 x^5+195500 x^4+\frac {899300 x^3}{3}-218040 x^2-\frac {704099}{2} \log \left (2 x^2-x+3\right )-2055257 x}{1472}-\frac {14641 (79 x+101)}{2944 \left (2 x^2-x+3\right )}\) |
Input:
Int[(2 + 3*x + 5*x^2)^4/(3 - x + 2*x^2)^2,x]
Output:
(-14641*(101 + 79*x))/(2944*(3 - x + 2*x^2)) + (-2055257*x - 218040*x^2 + (899300*x^3)/3 + 195500*x^4 + 46000*x^5 - (13292697*ArcTan[(1 - 4*x)/Sqrt[ 23]])/Sqrt[23] - (704099*Log[3 - x + 2*x^2])/2)/1472
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 = 2.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {125 x^{5}}{4}+\frac {2125 x^{4}}{16}+\frac {9775 x^{3}}{48}-\frac {1185 x^{2}}{8}-\frac {89359 x}{64}+\frac {-\frac {1156639 x}{5888}-\frac {1478741}{5888}}{x^{2}-\frac {1}{2} x +\frac {3}{2}}-\frac {30613 \ln \left (16 x^{2}-8 x +24\right )}{128}+\frac {13292697 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{33856}\) | \(70\) |
default | \(\frac {125 x^{5}}{4}+\frac {2125 x^{4}}{16}+\frac {9775 x^{3}}{48}-\frac {1185 x^{2}}{8}-\frac {89359 x}{64}-\frac {1331 \left (\frac {869 x}{92}+\frac {1111}{92}\right )}{64 \left (x^{2}-\frac {1}{2} x +\frac {3}{2}\right )}-\frac {30613 \ln \left (2 x^{2}-x +3\right )}{128}+\frac {13292697 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{33856}\) | \(71\) |
Input:
int((5*x^2+3*x+2)^4/(2*x^2-x+3)^2,x,method=_RETURNVERBOSE)
Output:
125/4*x^5+2125/16*x^4+9775/48*x^3-1185/8*x^2-89359/64*x+(-1156639/5888*x-1 478741/5888)/(x^2-1/2*x+3/2)-30613/128*ln(16*x^2-8*x+24)+13292697/33856*23 ^(1/2)*arctan(1/23*(4*x-1)*23^(1/2))
Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.08 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^2} \, dx=\frac {12696000 \, x^{7} + 47610000 \, x^{6} + 74800600 \, x^{5} - 20609840 \, x^{4} - 413058012 \, x^{3} + 79756182 \, \sqrt {23} {\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 193356906 \, x^{2} - 48582831 \, {\left (2 \, x^{2} - x + 3\right )} \log \left (2 \, x^{2} - x + 3\right ) - 930684489 \, x - 102033129}{203136 \, {\left (2 \, x^{2} - x + 3\right )}} \] Input:
integrate((5*x^2+3*x+2)^4/(2*x^2-x+3)^2,x, algorithm="fricas")
Output:
1/203136*(12696000*x^7 + 47610000*x^6 + 74800600*x^5 - 20609840*x^4 - 4130 58012*x^3 + 79756182*sqrt(23)*(2*x^2 - x + 3)*arctan(1/23*sqrt(23)*(4*x - 1)) + 193356906*x^2 - 48582831*(2*x^2 - x + 3)*log(2*x^2 - x + 3) - 930684 489*x - 102033129)/(2*x^2 - x + 3)
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^2} \, dx=\frac {125 x^{5}}{4} + \frac {2125 x^{4}}{16} + \frac {9775 x^{3}}{48} - \frac {1185 x^{2}}{8} - \frac {89359 x}{64} + \frac {- 1156639 x - 1478741}{5888 x^{2} - 2944 x + 8832} - \frac {30613 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{128} + \frac {13292697 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{33856} \] Input:
integrate((5*x**2+3*x+2)**4/(2*x**2-x+3)**2,x)
Output:
125*x**5/4 + 2125*x**4/16 + 9775*x**3/48 - 1185*x**2/8 - 89359*x/64 + (-11 56639*x - 1478741)/(5888*x**2 - 2944*x + 8832) - 30613*log(x**2 - x/2 + 3/ 2)/128 + 13292697*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/33856
Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.79 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^2} \, dx=\frac {125}{4} \, x^{5} + \frac {2125}{16} \, x^{4} + \frac {9775}{48} \, x^{3} - \frac {1185}{8} \, x^{2} + \frac {13292697}{33856} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {89359}{64} \, x - \frac {14641 \, {\left (79 \, x + 101\right )}}{2944 \, {\left (2 \, x^{2} - x + 3\right )}} - \frac {30613}{128} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate((5*x^2+3*x+2)^4/(2*x^2-x+3)^2,x, algorithm="maxima")
Output:
125/4*x^5 + 2125/16*x^4 + 9775/48*x^3 - 1185/8*x^2 + 13292697/33856*sqrt(2 3)*arctan(1/23*sqrt(23)*(4*x - 1)) - 89359/64*x - 14641/2944*(79*x + 101)/ (2*x^2 - x + 3) - 30613/128*log(2*x^2 - x + 3)
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.79 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^2} \, dx=\frac {125}{4} \, x^{5} + \frac {2125}{16} \, x^{4} + \frac {9775}{48} \, x^{3} - \frac {1185}{8} \, x^{2} + \frac {13292697}{33856} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {89359}{64} \, x - \frac {14641 \, {\left (79 \, x + 101\right )}}{2944 \, {\left (2 \, x^{2} - x + 3\right )}} - \frac {30613}{128} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate((5*x^2+3*x+2)^4/(2*x^2-x+3)^2,x, algorithm="giac")
Output:
125/4*x^5 + 2125/16*x^4 + 9775/48*x^3 - 1185/8*x^2 + 13292697/33856*sqrt(2 3)*arctan(1/23*sqrt(23)*(4*x - 1)) - 89359/64*x - 14641/2944*(79*x + 101)/ (2*x^2 - x + 3) - 30613/128*log(2*x^2 - x + 3)
Time = 15.78 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.79 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^2} \, dx=\frac {13292697\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{33856}-\frac {30613\,\ln \left (2\,x^2-x+3\right )}{128}-\frac {\frac {1156639\,x}{5888}+\frac {1478741}{5888}}{x^2-\frac {x}{2}+\frac {3}{2}}-\frac {89359\,x}{64}-\frac {1185\,x^2}{8}+\frac {9775\,x^3}{48}+\frac {2125\,x^4}{16}+\frac {125\,x^5}{4} \] Input:
int((3*x + 5*x^2 + 2)^4/(2*x^2 - x + 3)^2,x)
Output:
(13292697*23^(1/2)*atan((4*23^(1/2)*x)/23 - 23^(1/2)/23))/33856 - (30613*l og(2*x^2 - x + 3))/128 - ((1156639*x)/5888 + 1478741/5888)/(x^2 - x/2 + 3/ 2) - (89359*x)/64 - (1185*x^2)/8 + (9775*x^3)/48 + (2125*x^4)/16 + (125*x^ 5)/4
Time = 0.18 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.51 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^2} \, dx=\frac {159512364 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{2}-79756182 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x +239268546 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right )-97165662 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x^{2}+48582831 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x -145748493 \,\mathrm {log}\left (2 x^{2}-x +3\right )+12696000 x^{7}+47610000 x^{6}+74800600 x^{5}-20609840 x^{4}-413058012 x^{3}-1668012072 x^{2}-2894086596}{406272 x^{2}-203136 x +609408} \] Input:
int((5*x^2+3*x+2)^4/(2*x^2-x+3)^2,x)
Output:
(159512364*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**2 - 79756182*sqrt(23)*atan ((4*x - 1)/sqrt(23))*x + 239268546*sqrt(23)*atan((4*x - 1)/sqrt(23)) - 971 65662*log(2*x**2 - x + 3)*x**2 + 48582831*log(2*x**2 - x + 3)*x - 14574849 3*log(2*x**2 - x + 3) + 12696000*x**7 + 47610000*x**6 + 74800600*x**5 - 20 609840*x**4 - 413058012*x**3 - 1668012072*x**2 - 2894086596)/(203136*(2*x* *2 - x + 3))