Integrand size = 25, antiderivative size = 77 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^2} \, dx=\frac {915 x}{16}+\frac {175 x^2}{4}+\frac {125 x^3}{12}-\frac {1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}+\frac {223971 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{368 \sqrt {23}}-\frac {2057}{32} \log \left (3-x+2 x^2\right ) \] Output:
915/16*x+175/4*x^2+125/12*x^3-1331*(17-45*x)/(1472*x^2-736*x+2208)+223971/ 8464*arctan(1/23*(1-4*x)*23^(1/2))*23^(1/2)-2057/32*ln(2*x^2-x+3)
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^2} \, dx=\frac {915 x}{16}+\frac {175 x^2}{4}+\frac {125 x^3}{12}+\frac {1331 (-17+45 x)}{736 \left (3-x+2 x^2\right )}-\frac {223971 \arctan \left (\frac {-1+4 x}{\sqrt {23}}\right )}{368 \sqrt {23}}-\frac {2057}{32} \log \left (3-x+2 x^2\right ) \] Input:
Integrate[(2 + 3*x + 5*x^2)^3/(3 - x + 2*x^2)^2,x]
Output:
(915*x)/16 + (175*x^2)/4 + (125*x^3)/12 + (1331*(-17 + 45*x))/(736*(3 - x + 2*x^2)) - (223971*ArcTan[(-1 + 4*x)/Sqrt[23]])/(368*Sqrt[23]) - (2057*Lo g[3 - x + 2*x^2])/32
Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99, 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.
\(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^3}{\left (2 x^2-x+3\right )^2} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{23} \int -\frac {-23000 x^4-52900 x^3-44390 x^2+19067 x+25195}{16 \left (2 x^2-x+3\right )}dx-\frac {1331 (17-45 x)}{736 \left (2 x^2-x+3\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{368} \int \frac {-23000 x^4-52900 x^3-44390 x^2+19067 x+25195}{2 x^2-x+3}dx-\frac {1331 (17-45 x)}{736 \left (2 x^2-x+3\right )}\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle -\frac {1}{368} \int \left (-11500 x^2-32200 x+\frac {242 (391 x+365)}{2 x^2-x+3}-21045\right )dx-\frac {1331 (17-45 x)}{736 \left (2 x^2-x+3\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{368} \left (\frac {223971 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{\sqrt {23}}+\frac {11500 x^3}{3}+16100 x^2-\frac {47311}{2} \log \left (2 x^2-x+3\right )+21045 x\right )-\frac {1331 (17-45 x)}{736 \left (2 x^2-x+3\right )}\) |
Input:
Int[(2 + 3*x + 5*x^2)^3/(3 - x + 2*x^2)^2,x]
Output:
(-1331*(17 - 45*x))/(736*(3 - x + 2*x^2)) + (21045*x + 16100*x^2 + (11500* x^3)/3 + (223971*ArcTan[(1 - 4*x)/Sqrt[23]])/Sqrt[23] - (47311*Log[3 - x + 2*x^2])/2)/368
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 = 2.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {125 x^{3}}{12}+\frac {175 x^{2}}{4}+\frac {915 x}{16}+\frac {\frac {59895 x}{1472}-\frac {22627}{1472}}{x^{2}-\frac {1}{2} x +\frac {3}{2}}-\frac {2057 \ln \left (16 x^{2}-8 x +24\right )}{32}-\frac {223971 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{8464}\) | \(60\) |
default | \(\frac {125 x^{3}}{12}+\frac {175 x^{2}}{4}+\frac {915 x}{16}-\frac {121 \left (-\frac {495 x}{92}+\frac {187}{92}\right )}{16 \left (x^{2}-\frac {1}{2} x +\frac {3}{2}\right )}-\frac {2057 \ln \left (2 x^{2}-x +3\right )}{32}-\frac {223971 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{8464}\) | \(61\) |
Input:
int((5*x^2+3*x+2)^3/(2*x^2-x+3)^2,x,method=_RETURNVERBOSE)
Output:
125/12*x^3+175/4*x^2+915/16*x+(59895/1472*x-22627/1472)/(x^2-1/2*x+3/2)-20 57/32*ln(16*x^2-8*x+24)-223971/8464*23^(1/2)*arctan(1/23*(4*x-1)*23^(1/2))
Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.14 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^2} \, dx=\frac {1058000 \, x^{5} + 3914600 \, x^{4} + 5173620 \, x^{3} - 1343826 \, \sqrt {23} {\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 3761190 \, x^{2} - 3264459 \, {\left (2 \, x^{2} - x + 3\right )} \log \left (2 \, x^{2} - x + 3\right ) + 12845385 \, x - 1561263}{50784 \, {\left (2 \, x^{2} - x + 3\right )}} \] Input:
integrate((5*x^2+3*x+2)^3/(2*x^2-x+3)^2,x, algorithm="fricas")
Output:
1/50784*(1058000*x^5 + 3914600*x^4 + 5173620*x^3 - 1343826*sqrt(23)*(2*x^2 - x + 3)*arctan(1/23*sqrt(23)*(4*x - 1)) + 3761190*x^2 - 3264459*(2*x^2 - x + 3)*log(2*x^2 - x + 3) + 12845385*x - 1561263)/(2*x^2 - x + 3)
Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^2} \, dx=\frac {125 x^{3}}{12} + \frac {175 x^{2}}{4} + \frac {915 x}{16} + \frac {59895 x - 22627}{1472 x^{2} - 736 x + 2208} - \frac {2057 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{32} - \frac {223971 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{8464} \] Input:
integrate((5*x**2+3*x+2)**3/(2*x**2-x+3)**2,x)
Output:
125*x**3/12 + 175*x**2/4 + 915*x/16 + (59895*x - 22627)/(1472*x**2 - 736*x + 2208) - 2057*log(x**2 - x/2 + 3/2)/32 - 223971*sqrt(23)*atan(4*sqrt(23) *x/23 - sqrt(23)/23)/8464
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^2} \, dx=\frac {125}{12} \, x^{3} + \frac {175}{4} \, x^{2} - \frac {223971}{8464} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {915}{16} \, x + \frac {1331 \, {\left (45 \, x - 17\right )}}{736 \, {\left (2 \, x^{2} - x + 3\right )}} - \frac {2057}{32} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate((5*x^2+3*x+2)^3/(2*x^2-x+3)^2,x, algorithm="maxima")
Output:
125/12*x^3 + 175/4*x^2 - 223971/8464*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 915/16*x + 1331/736*(45*x - 17)/(2*x^2 - x + 3) - 2057/32*log(2*x^2 - x + 3)
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^2} \, dx=\frac {125}{12} \, x^{3} + \frac {175}{4} \, x^{2} - \frac {223971}{8464} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {915}{16} \, x + \frac {1331 \, {\left (45 \, x - 17\right )}}{736 \, {\left (2 \, x^{2} - x + 3\right )}} - \frac {2057}{32} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate((5*x^2+3*x+2)^3/(2*x^2-x+3)^2,x, algorithm="giac")
Output:
125/12*x^3 + 175/4*x^2 - 223971/8464*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 915/16*x + 1331/736*(45*x - 17)/(2*x^2 - x + 3) - 2057/32*log(2*x^2 - x + 3)
Time = 16.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^2} \, dx=\frac {915\,x}{16}-\frac {2057\,\ln \left (2\,x^2-x+3\right )}{32}+\frac {\frac {59895\,x}{1472}-\frac {22627}{1472}}{x^2-\frac {x}{2}+\frac {3}{2}}-\frac {223971\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{8464}+\frac {175\,x^2}{4}+\frac {125\,x^3}{12} \] Input:
int((3*x + 5*x^2 + 2)^3/(2*x^2 - x + 3)^2,x)
Output:
(915*x)/16 - (2057*log(2*x^2 - x + 3))/32 + ((59895*x)/1472 - 22627/1472)/ (x^2 - x/2 + 3/2) - (223971*23^(1/2)*atan((4*23^(1/2)*x)/23 - 23^(1/2)/23) )/8464 + (175*x^2)/4 + (125*x^3)/12
Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.65 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^2} \, dx=\frac {-2687652 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{2}+1343826 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x -4031478 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right )-6528918 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x^{2}+3264459 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x -9793377 \,\mathrm {log}\left (2 x^{2}-x +3\right )+1058000 x^{5}+3914600 x^{4}+5173620 x^{3}+29451960 x^{2}+36974892}{101568 x^{2}-50784 x +152352} \] Input:
int((5*x^2+3*x+2)^3/(2*x^2-x+3)^2,x)
Output:
( - 2687652*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**2 + 1343826*sqrt(23)*atan ((4*x - 1)/sqrt(23))*x - 4031478*sqrt(23)*atan((4*x - 1)/sqrt(23)) - 65289 18*log(2*x**2 - x + 3)*x**2 + 3264459*log(2*x**2 - x + 3)*x - 9793377*log( 2*x**2 - x + 3) + 1058000*x**5 + 3914600*x**4 + 5173620*x**3 + 29451960*x* *2 + 36974892)/(50784*(2*x**2 - x + 3))