Integrand size = 27, antiderivative size = 101 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\sqrt {3-x+2 x^2}} \, dx=-\frac {11373 \sqrt {3-x+2 x^2}}{1024}+\frac {3443}{768} x \sqrt {3-x+2 x^2}+\frac {655}{96} x^2 \sqrt {3-x+2 x^2}+\frac {25}{8} x^3 \sqrt {3-x+2 x^2}+\frac {30725 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2048 \sqrt {2}} \]
30725/4096*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-11373/1024*(2*x^2-x+3)^( 1/2)+3443/768*x*(2*x^2-x+3)^(1/2)+655/96*x^2*(2*x^2-x+3)^(1/2)+25/8*x^3*(2 *x^2-x+3)^(1/2)
Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.64 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\sqrt {3-x+2 x^2}} \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (-34119+13772 x+20960 x^2+9600 x^3\right )+92175 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{12288} \]
(4*Sqrt[3 - x + 2*x^2]*(-34119 + 13772*x + 20960*x^2 + 9600*x^3) + 92175*S qrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/12288
Time = 0.33 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2192, 27, 2192, 27, 2192, 27, 1160, 1090, 222}
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.
\(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^2}{\sqrt {2 x^2-x+3}} \, dx\) |
\(\Big \downarrow \) 2192 |
\(\displaystyle \frac {1}{8} \int \frac {655 x^3+14 x^2+192 x+64}{2 \sqrt {2 x^2-x+3}}dx+\frac {25}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{16} \int \frac {655 x^3+14 x^2+192 x+64}{\sqrt {2 x^2-x+3}}dx+\frac {25}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 2192 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{6} \int \frac {3443 x^2-5556 x+768}{2 \sqrt {2 x^2-x+3}}dx+\frac {655}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {25}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{12} \int \frac {3443 x^2-5556 x+768}{\sqrt {2 x^2-x+3}}dx+\frac {655}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {25}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 2192 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{12} \left (\frac {1}{4} \int -\frac {3 (11373 x+4838)}{2 \sqrt {2 x^2-x+3}}dx+\frac {3443}{4} \sqrt {2 x^2-x+3} x\right )+\frac {655}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {25}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{12} \left (\frac {3443}{4} x \sqrt {2 x^2-x+3}-\frac {3}{8} \int \frac {11373 x+4838}{\sqrt {2 x^2-x+3}}dx\right )+\frac {655}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {25}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{12} \left (\frac {3443}{4} x \sqrt {2 x^2-x+3}-\frac {3}{8} \left (\frac {30725}{4} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {11373}{2} \sqrt {2 x^2-x+3}\right )\right )+\frac {655}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {25}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{12} \left (\frac {3443}{4} x \sqrt {2 x^2-x+3}-\frac {3}{8} \left (\frac {30725 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{4 \sqrt {46}}+\frac {11373}{2} \sqrt {2 x^2-x+3}\right )\right )+\frac {655}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {25}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{12} \left (\frac {3443}{4} x \sqrt {2 x^2-x+3}-\frac {3}{8} \left (\frac {30725 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{4 \sqrt {2}}+\frac {11373}{2} \sqrt {2 x^2-x+3}\right )\right )+\frac {655}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {25}{8} \sqrt {2 x^2-x+3} x^3\) |
(25*x^3*Sqrt[3 - x + 2*x^2])/8 + ((655*x^2*Sqrt[3 - x + 2*x^2])/6 + ((3443 *x*Sqrt[3 - x + 2*x^2])/4 - (3*((11373*Sqrt[3 - x + 2*x^2])/2 + (30725*Arc Sinh[(-1 + 4*x)/Sqrt[23]])/(4*Sqrt[2])))/8)/12)/16
3.1.81.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Time = 0.74 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.45
method | result | size |
risch | \(\frac {\left (9600 x^{3}+20960 x^{2}+13772 x -34119\right ) \sqrt {2 x^{2}-x +3}}{3072}-\frac {30725 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4096}\) | \(45\) |
trager | \(\left (\frac {25}{8} x^{3}+\frac {655}{96} x^{2}+\frac {3443}{768} x -\frac {11373}{1024}\right ) \sqrt {2 x^{2}-x +3}-\frac {30725 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{4096}\) | \(71\) |
default | \(-\frac {30725 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4096}-\frac {11373 \sqrt {2 x^{2}-x +3}}{1024}+\frac {25 x^{3} \sqrt {2 x^{2}-x +3}}{8}+\frac {655 x^{2} \sqrt {2 x^{2}-x +3}}{96}+\frac {3443 x \sqrt {2 x^{2}-x +3}}{768}\) | \(79\) |
1/3072*(9600*x^3+20960*x^2+13772*x-34119)*(2*x^2-x+3)^(1/2)-30725/4096*2^( 1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.67 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\sqrt {3-x+2 x^2}} \, dx=\frac {1}{3072} \, {\left (9600 \, x^{3} + 20960 \, x^{2} + 13772 \, x - 34119\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {30725}{8192} \, \sqrt {2} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
1/3072*(9600*x^3 + 20960*x^2 + 13772*x - 34119)*sqrt(2*x^2 - x + 3) + 3072 5/8192*sqrt(2)*log(4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)
Time = 0.38 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.55 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\sqrt {3-x+2 x^2}} \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {25 x^{3}}{8} + \frac {655 x^{2}}{96} + \frac {3443 x}{768} - \frac {11373}{1024}\right ) - \frac {30725 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{4096} \]
sqrt(2*x**2 - x + 3)*(25*x**3/8 + 655*x**2/96 + 3443*x/768 - 11373/1024) - 30725*sqrt(2)*asinh(4*sqrt(23)*(x - 1/4)/23)/4096
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.79 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\sqrt {3-x+2 x^2}} \, dx=\frac {25}{8} \, \sqrt {2 \, x^{2} - x + 3} x^{3} + \frac {655}{96} \, \sqrt {2 \, x^{2} - x + 3} x^{2} + \frac {3443}{768} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {30725}{4096} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {11373}{1024} \, \sqrt {2 \, x^{2} - x + 3} \]
25/8*sqrt(2*x^2 - x + 3)*x^3 + 655/96*sqrt(2*x^2 - x + 3)*x^2 + 3443/768*s qrt(2*x^2 - x + 3)*x - 30725/4096*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 11373/1024*sqrt(2*x^2 - x + 3)
Time = 0.37 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\sqrt {3-x+2 x^2}} \, dx=\frac {1}{3072} \, {\left (4 \, {\left (40 \, {\left (60 \, x + 131\right )} x + 3443\right )} x - 34119\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {30725}{4096} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
1/3072*(4*(40*(60*x + 131)*x + 3443)*x - 34119)*sqrt(2*x^2 - x + 3) + 3072 5/4096*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
Timed out. \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\sqrt {3-x+2 x^2}} \, dx=\int \frac {{\left (5\,x^2+3\,x+2\right )}^2}{\sqrt {2\,x^2-x+3}} \,d x \]