Integrand size = 25, antiderivative size = 84 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^3} \, dx=\frac {125 x}{8}-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}+\frac {165099 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{8464 \sqrt {23}}+\frac {825}{32} \log \left (3-x+2 x^2\right ) \]
125/8*x-1331/1472*(17-45*x)/(2*x^2-x+3)^2+121/33856*(21193-12828*x)/(2*x^2 -x+3)+825/32*ln(2*x^2-x+3)+165099/194672*arctan(1/23*(1-4*x)*23^(1/2))*23^ (1/2)
Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^3} \, dx=\frac {125 x}{8}+\frac {1331 (-17+45 x)}{1472 \left (3-x+2 x^2\right )^2}-\frac {121 (-21193+12828 x)}{33856 \left (3-x+2 x^2\right )}-\frac {165099 \arctan \left (\frac {-1+4 x}{\sqrt {23}}\right )}{8464 \sqrt {23}}+\frac {825}{32} \log \left (3-x+2 x^2\right ) \]
(125*x)/8 + (1331*(-17 + 45*x))/(1472*(3 - x + 2*x^2)^2) - (121*(-21193 + 12828*x))/(33856*(3 - x + 2*x^2)) - (165099*ArcTan[(-1 + 4*x)/Sqrt[23]])/( 8464*Sqrt[23]) + (825*Log[3 - x + 2*x^2])/32
Time = 0.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.12, 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.
\(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^3}{\left (2 x^2-x+3\right )^3} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{46} \int -\frac {-92000 x^4-211600 x^3-177560 x^2+76268 x+40885}{32 \left (2 x^2-x+3\right )^2}dx-\frac {1331 (17-45 x)}{1472 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {-92000 x^4-211600 x^3-177560 x^2+76268 x+40885}{\left (2 x^2-x+3\right )^2}dx}{1472}-\frac {1331 (17-45 x)}{1472 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {\frac {121 (21193-12828 x)}{23 \left (2 x^2-x+3\right )}-\frac {1}{23} \int -\frac {16 \left (66125 x^2+185150 x+23997\right )}{2 x^2-x+3}dx}{1472}-\frac {1331 (17-45 x)}{1472 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {16}{23} \int \frac {66125 x^2+185150 x+23997}{2 x^2-x+3}dx+\frac {121 (21193-12828 x)}{23 \left (2 x^2-x+3\right )}}{1472}-\frac {1331 (17-45 x)}{1472 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \frac {\frac {16}{23} \int \left (\frac {66125}{2}-\frac {33 (4557-13225 x)}{2 \left (2 x^2-x+3\right )}\right )dx+\frac {121 (21193-12828 x)}{23 \left (2 x^2-x+3\right )}}{1472}-\frac {1331 (17-45 x)}{1472 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {16}{23} \left (\frac {165099 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{4 \sqrt {23}}+\frac {436425}{8} \log \left (2 x^2-x+3\right )+\frac {66125 x}{2}\right )+\frac {121 (21193-12828 x)}{23 \left (2 x^2-x+3\right )}}{1472}-\frac {1331 (17-45 x)}{1472 \left (2 x^2-x+3\right )^2}\) |
(-1331*(17 - 45*x))/(1472*(3 - x + 2*x^2)^2) + ((121*(21193 - 12828*x))/(2 3*(3 - x + 2*x^2)) + (16*((66125*x)/2 + (165099*ArcTan[(1 - 4*x)/Sqrt[23]] )/(4*Sqrt[23]) + (436425*Log[3 - x + 2*x^2])/8))/23)/1472
3.1.52.3.1 Defintions of rubi rules used
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 = 0.69 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {125 x}{8}+\frac {-\frac {388047}{4232} x^{3}+\frac {3340447}{16928} x^{2}-\frac {1460833}{8464} x +\frac {3586319}{16928}}{\left (2 x^{2}-x +3\right )^{2}}+\frac {825 \ln \left (2 x^{2}-x +3\right )}{32}-\frac {165099 \sqrt {23}\, \arctan \left (\frac {\left (-1+4 x \right ) \sqrt {23}}{23}\right )}{194672}\) | \(63\) |
risch | \(\frac {125 x}{8}+\frac {-\frac {388047}{4232} x^{3}+\frac {3340447}{16928} x^{2}-\frac {1460833}{8464} x +\frac {3586319}{16928}}{\left (2 x^{2}-x +3\right )^{2}}+\frac {825 \ln \left (16 x^{2}-8 x +24\right )}{32}-\frac {165099 \sqrt {23}\, \arctan \left (\frac {\left (-1+4 x \right ) \sqrt {23}}{23}\right )}{194672}\) | \(63\) |
125/8*x+11/2*(-35277/2116*x^3+303677/8464*x^2-132803/4232*x+326029/8464)/( 2*x^2-x+3)^2+825/32*ln(2*x^2-x+3)-165099/194672*23^(1/2)*arctan(1/23*(-1+4 *x)*23^(1/2))
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.40 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^3} \, dx=\frac {24334000 \, x^{5} - 24334000 \, x^{4} + 43385176 \, x^{3} - 330198 \, \sqrt {23} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 40329281 \, x^{2} + 10037775 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x^{2} - x + 3\right ) - 12446818 \, x + 82485337}{389344 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \]
1/389344*(24334000*x^5 - 24334000*x^4 + 43385176*x^3 - 330198*sqrt(23)*(4* x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*arctan(1/23*sqrt(23)*(4*x - 1)) + 40329281 *x^2 + 10037775*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(2*x^2 - x + 3) - 12 446818*x + 82485337)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.98 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^3} \, dx=\frac {125 x}{8} + \frac {- 1552188 x^{3} + 3340447 x^{2} - 2921666 x + 3586319}{67712 x^{4} - 67712 x^{3} + 220064 x^{2} - 101568 x + 152352} + \frac {825 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{32} - \frac {165099 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{194672} \]
125*x/8 + (-1552188*x**3 + 3340447*x**2 - 2921666*x + 3586319)/(67712*x**4 - 67712*x**3 + 220064*x**2 - 101568*x + 152352) + 825*log(x**2 - x/2 + 3/ 2)/32 - 165099*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/194672
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.86 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^3} \, dx=-\frac {165099}{194672} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {125}{8} \, x - \frac {121 \, {\left (12828 \, x^{3} - 27607 \, x^{2} + 24146 \, x - 29639\right )}}{16928 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} + \frac {825}{32} \, \log \left (2 \, x^{2} - x + 3\right ) \]
-165099/194672*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 125/8*x - 121/16 928*(12828*x^3 - 27607*x^2 + 24146*x - 29639)/(4*x^4 - 4*x^3 + 13*x^2 - 6* x + 9) + 825/32*log(2*x^2 - x + 3)
Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.74 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^3} \, dx=-\frac {165099}{194672} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {125}{8} \, x - \frac {121 \, {\left (12828 \, x^{3} - 27607 \, x^{2} + 24146 \, x - 29639\right )}}{16928 \, {\left (2 \, x^{2} - x + 3\right )}^{2}} + \frac {825}{32} \, \log \left (2 \, x^{2} - x + 3\right ) \]
-165099/194672*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 125/8*x - 121/16 928*(12828*x^3 - 27607*x^2 + 24146*x - 29639)/(2*x^2 - x + 3)^2 + 825/32*l og(2*x^2 - x + 3)
Time = 12.44 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.86 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^3} \, dx=\frac {125\,x}{8}+\frac {825\,\ln \left (2\,x^2-x+3\right )}{32}-\frac {165099\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{194672}-\frac {\frac {388047\,x^3}{16928}-\frac {3340447\,x^2}{67712}+\frac {1460833\,x}{33856}-\frac {3586319}{67712}}{x^4-x^3+\frac {13\,x^2}{4}-\frac {3\,x}{2}+\frac {9}{4}} \]