Integrand size = 27, antiderivative size = 143 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\sqrt {3-x+2 x^2}} \, dx=-\frac {203373 \sqrt {3-x+2 x^2}}{32768}-\frac {372783 x \sqrt {3-x+2 x^2}}{8192}-\frac {3387 x^2 \sqrt {3-x+2 x^2}}{1024}+\frac {8185}{256} x^3 \sqrt {3-x+2 x^2}+\frac {1355}{48} x^4 \sqrt {3-x+2 x^2}+\frac {125}{12} x^5 \sqrt {3-x+2 x^2}-\frac {9267707 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{65536 \sqrt {2}} \]
-9267707/131072*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-203373/32768*(2*x^2 -x+3)^(1/2)-372783/8192*x*(2*x^2-x+3)^(1/2)-3387/1024*x^2*(2*x^2-x+3)^(1/2 )+8185/256*x^3*(2*x^2-x+3)^(1/2)+1355/48*x^4*(2*x^2-x+3)^(1/2)+125/12*x^5* (2*x^2-x+3)^(1/2)
Time = 0.45 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.52 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\sqrt {3-x+2 x^2}} \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (-610119-4473396 x-325152 x^2+3143040 x^3+2775040 x^4+1024000 x^5\right )-27803121 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{393216} \]
(4*Sqrt[3 - x + 2*x^2]*(-610119 - 4473396*x - 325152*x^2 + 3143040*x^3 + 2 775040*x^4 + 1024000*x^5) - 27803121*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/393216
Time = 0.50 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {2192, 27, 2192, 27, 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.
| \(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^3}{\sqrt {2 x^2-x+3}} \, dx\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{12} \int \frac {6775 x^5+3090 x^4+4968 x^3+2736 x^2+864 x+192}{2 \sqrt {2 x^2-x+3}}dx+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \int \frac {6775 x^5+3090 x^4+4968 x^3+2736 x^2+864 x+192}{\sqrt {2 x^2-x+3}}dx+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{10} \int \frac {15 \left (8185 x^4-4216 x^3+3648 x^2+1152 x+256\right )}{2 \sqrt {2 x^2-x+3}}dx+\frac {1355}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {3}{4} \int \frac {8185 x^4-4216 x^3+3648 x^2+1152 x+256}{\sqrt {2 x^2-x+3}}dx+\frac {1355}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{8} \int \frac {-10161 x^3-88962 x^2+18432 x+4096}{2 \sqrt {2 x^2-x+3}}dx+\frac {8185}{8} \sqrt {2 x^2-x+3} x^3\right )+\frac {1355}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \int \frac {-10161 x^3-88962 x^2+18432 x+4096}{\sqrt {2 x^2-x+3}}dx+\frac {8185}{8} \sqrt {2 x^2-x+3} x^3\right )+\frac {1355}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{6} \int \frac {3 \left (-372783 x^2+114372 x+16384\right )}{2 \sqrt {2 x^2-x+3}}dx-\frac {3387}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {8185}{8} \sqrt {2 x^2-x+3} x^3\right )+\frac {1355}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{4} \int \frac {-372783 x^2+114372 x+16384}{\sqrt {2 x^2-x+3}}dx-\frac {3387}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {8185}{8} \sqrt {2 x^2-x+3} x^3\right )+\frac {1355}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{4} \left (\frac {1}{4} \int \frac {2367770-203373 x}{2 \sqrt {2 x^2-x+3}}dx-\frac {372783}{4} x \sqrt {2 x^2-x+3}\right )-\frac {3387}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {8185}{8} \sqrt {2 x^2-x+3} x^3\right )+\frac {1355}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{4} \left (\frac {1}{8} \int \frac {2367770-203373 x}{\sqrt {2 x^2-x+3}}dx-\frac {372783}{4} x \sqrt {2 x^2-x+3}\right )-\frac {3387}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {8185}{8} \sqrt {2 x^2-x+3} x^3\right )+\frac {1355}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {9267707}{4} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {203373}{2} \sqrt {2 x^2-x+3}\right )-\frac {372783}{4} x \sqrt {2 x^2-x+3}\right )-\frac {3387}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {8185}{8} \sqrt {2 x^2-x+3} x^3\right )+\frac {1355}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {9267707 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{4 \sqrt {46}}-\frac {203373}{2} \sqrt {2 x^2-x+3}\right )-\frac {372783}{4} x \sqrt {2 x^2-x+3}\right )-\frac {3387}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {8185}{8} \sqrt {2 x^2-x+3} x^3\right )+\frac {1355}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {9267707 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{4 \sqrt {2}}-\frac {203373}{2} \sqrt {2 x^2-x+3}\right )-\frac {372783}{4} x \sqrt {2 x^2-x+3}\right )-\frac {3387}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {8185}{8} \sqrt {2 x^2-x+3} x^3\right )+\frac {1355}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {125}{12} \sqrt {2 x^2-x+3} x^5\) |
(125*x^5*Sqrt[3 - x + 2*x^2])/12 + ((1355*x^4*Sqrt[3 - x + 2*x^2])/2 + (3* ((8185*x^3*Sqrt[3 - x + 2*x^2])/8 + ((-3387*x^2*Sqrt[3 - x + 2*x^2])/2 + ( (-372783*x*Sqrt[3 - x + 2*x^2])/4 + ((-203373*Sqrt[3 - x + 2*x^2])/2 + (92 67707*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(4*Sqrt[2]))/8)/4)/16))/4)/24
3.1.80.3.1 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 = 0.73 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.38
| method | result | size |
| risch | \(\frac {\left (1024000 x^{5}+2775040 x^{4}+3143040 x^{3}-325152 x^{2}-4473396 x -610119\right ) \sqrt {2 x^{2}-x +3}}{98304}+\frac {9267707 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{131072}\) | \(55\) |
| trager | \(\left (\frac {125}{12} x^{5}+\frac {1355}{48} x^{4}+\frac {8185}{256} x^{3}-\frac {3387}{1024} x^{2}-\frac {372783}{8192} x -\frac {203373}{32768}\right ) \sqrt {2 x^{2}-x +3}-\frac {9267707 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{131072}\) | \(79\) |
| default | \(\frac {9267707 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{131072}-\frac {203373 \sqrt {2 x^{2}-x +3}}{32768}+\frac {125 x^{5} \sqrt {2 x^{2}-x +3}}{12}+\frac {1355 x^{4} \sqrt {2 x^{2}-x +3}}{48}+\frac {8185 x^{3} \sqrt {2 x^{2}-x +3}}{256}-\frac {3387 x^{2} \sqrt {2 x^{2}-x +3}}{1024}-\frac {372783 x \sqrt {2 x^{2}-x +3}}{8192}\) | \(113\) |
1/98304*(1024000*x^5+2775040*x^4+3143040*x^3-325152*x^2-4473396*x-610119)* (2*x^2-x+3)^(1/2)+9267707/131072*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.41 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.55 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\sqrt {3-x+2 x^2}} \, dx=\frac {1}{98304} \, {\left (1024000 \, x^{5} + 2775040 \, x^{4} + 3143040 \, x^{3} - 325152 \, x^{2} - 4473396 \, x - 610119\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {9267707}{262144} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
1/98304*(1024000*x^5 + 2775040*x^4 + 3143040*x^3 - 325152*x^2 - 4473396*x - 610119)*sqrt(2*x^2 - x + 3) + 9267707/262144*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 = 70, normalized size of antiderivative = 0.49 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\sqrt {3-x+2 x^2}} \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {125 x^{5}}{12} + \frac {1355 x^{4}}{48} + \frac {8185 x^{3}}{256} - \frac {3387 x^{2}}{1024} - \frac {372783 x}{8192} - \frac {203373}{32768}\right ) + \frac {9267707 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{131072} \]
sqrt(2*x**2 - x + 3)*(125*x**5/12 + 1355*x**4/48 + 8185*x**3/256 - 3387*x* *2/1024 - 372783*x/8192 - 203373/32768) + 9267707*sqrt(2)*asinh(4*sqrt(23) *(x - 1/4)/23)/131072
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.80 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\sqrt {3-x+2 x^2}} \, dx=\frac {125}{12} \, \sqrt {2 \, x^{2} - x + 3} x^{5} + \frac {1355}{48} \, \sqrt {2 \, x^{2} - x + 3} x^{4} + \frac {8185}{256} \, \sqrt {2 \, x^{2} - x + 3} x^{3} - \frac {3387}{1024} \, \sqrt {2 \, x^{2} - x + 3} x^{2} - \frac {372783}{8192} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {9267707}{131072} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {203373}{32768} \, \sqrt {2 \, x^{2} - x + 3} \]
125/12*sqrt(2*x^2 - x + 3)*x^5 + 1355/48*sqrt(2*x^2 - x + 3)*x^4 + 8185/25 6*sqrt(2*x^2 - x + 3)*x^3 - 3387/1024*sqrt(2*x^2 - x + 3)*x^2 - 372783/819 2*sqrt(2*x^2 - x + 3)*x + 9267707/131072*sqrt(2)*arcsinh(1/23*sqrt(23)*(4* x - 1)) - 203373/32768*sqrt(2*x^2 - x + 3)
Time = 0.36 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.51 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\sqrt {3-x+2 x^2}} \, dx=\frac {1}{98304} \, {\left (4 \, {\left (8 \, {\left (20 \, {\left (16 \, {\left (100 \, x + 271\right )} x + 4911\right )} x - 10161\right )} x - 1118349\right )} x - 610119\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {9267707}{131072} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
1/98304*(4*(8*(20*(16*(100*x + 271)*x + 4911)*x - 10161)*x - 1118349)*x - 610119)*sqrt(2*x^2 - x + 3) - 9267707/131072*sqrt(2)*log(-2*sqrt(2)*(sqrt( 2)*x - sqrt(2*x^2 - x + 3)) + 1)
Timed out. \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\sqrt {3-x+2 x^2}} \, dx=\int \frac {{\left (5\,x^2+3\,x+2\right )}^3}{\sqrt {2\,x^2-x+3}} \,d x \]