Integrand size = 33, antiderivative size = 101 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\sqrt {3-x+2 x^2}} \, dx=-\frac {505 \sqrt {3-x+2 x^2}}{1024}-\frac {409}{768} x \sqrt {3-x+2 x^2}+\frac {19}{96} x^2 \sqrt {3-x+2 x^2}+\frac {5}{8} x^3 \sqrt {3-x+2 x^2}-\frac {6863 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2048 \sqrt {2}} \] Output:
-505/1024*(2*x^2-x+3)^(1/2)-409/768*x*(2*x^2-x+3)^(1/2)+19/96*x^2*(2*x^2-x +3)^(1/2)+5/8*x^3*(2*x^2-x+3)^(1/2)-6863/4096*arcsinh(1/23*(1-4*x)*23^(1/2 ))*2^(1/2)
Time = 0.49 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.64 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\sqrt {3-x+2 x^2}} \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (-1515-1636 x+608 x^2+1920 x^3\right )-20589 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{12288} \] Input:
Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/Sqrt[3 - x + 2*x^2],x]
Output:
(4*Sqrt[3 - x + 2*x^2]*(-1515 - 1636*x + 608*x^2 + 1920*x^3) - 20589*Sqrt[ 2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/12288
Time = 0.57 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, 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.
\(\displaystyle \int \frac {5 x^4-x^3+3 x^2+x+2}{\sqrt {2 x^2-x+3}} \, dx\) |
\(\Big \downarrow \) 2192 |
\(\displaystyle \frac {1}{8} \int \frac {19 x^3-42 x^2+16 x+32}{2 \sqrt {2 x^2-x+3}}dx+\frac {5}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{16} \int \frac {19 x^3-42 x^2+16 x+32}{\sqrt {2 x^2-x+3}}dx+\frac {5}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 2192 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{6} \int \frac {-409 x^2-36 x+384}{2 \sqrt {2 x^2-x+3}}dx+\frac {19}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {5}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{12} \int \frac {-409 x^2-36 x+384}{\sqrt {2 x^2-x+3}}dx+\frac {19}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {5}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 2192 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{12} \left (\frac {1}{4} \int \frac {3 (1842-505 x)}{2 \sqrt {2 x^2-x+3}}dx-\frac {409}{4} x \sqrt {2 x^2-x+3}\right )+\frac {19}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {5}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{12} \left (\frac {3}{8} \int \frac {1842-505 x}{\sqrt {2 x^2-x+3}}dx-\frac {409}{4} x \sqrt {2 x^2-x+3}\right )+\frac {19}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {5}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{12} \left (\frac {3}{8} \left (\frac {6863}{4} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {505}{2} \sqrt {2 x^2-x+3}\right )-\frac {409}{4} x \sqrt {2 x^2-x+3}\right )+\frac {19}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {5}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{12} \left (\frac {3}{8} \left (\frac {6863 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{4 \sqrt {46}}-\frac {505}{2} \sqrt {2 x^2-x+3}\right )-\frac {409}{4} x \sqrt {2 x^2-x+3}\right )+\frac {19}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {5}{8} \sqrt {2 x^2-x+3} x^3\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{16} \left (\frac {1}{12} \left (\frac {3}{8} \left (\frac {6863 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{4 \sqrt {2}}-\frac {505}{2} \sqrt {2 x^2-x+3}\right )-\frac {409}{4} x \sqrt {2 x^2-x+3}\right )+\frac {19}{6} \sqrt {2 x^2-x+3} x^2\right )+\frac {5}{8} \sqrt {2 x^2-x+3} x^3\) |
Input:
Int[(2 + x + 3*x^2 - x^3 + 5*x^4)/Sqrt[3 - x + 2*x^2],x]
Output:
(5*x^3*Sqrt[3 - x + 2*x^2])/8 + ((19*x^2*Sqrt[3 - x + 2*x^2])/6 + ((-409*x *Sqrt[3 - x + 2*x^2])/4 + (3*((-505*Sqrt[3 - x + 2*x^2])/2 + (6863*ArcSinh [(-1 + 4*x)/Sqrt[23]])/(4*Sqrt[2])))/8)/12)/16
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 = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.45
method | result | size |
risch | \(\frac {\left (1920 x^{3}+608 x^{2}-1636 x -1515\right ) \sqrt {2 x^{2}-x +3}}{3072}+\frac {6863 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4096}\) | \(45\) |
trager | \(\left (\frac {5}{8} x^{3}+\frac {19}{96} x^{2}-\frac {409}{768} x -\frac {505}{1024}\right ) \sqrt {2 x^{2}-x +3}+\frac {6863 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{4096}\) | \(71\) |
default | \(-\frac {505 \sqrt {2 x^{2}-x +3}}{1024}+\frac {6863 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4096}-\frac {409 x \sqrt {2 x^{2}-x +3}}{768}+\frac {19 x^{2} \sqrt {2 x^{2}-x +3}}{96}+\frac {5 x^{3} \sqrt {2 x^{2}-x +3}}{8}\) | \(79\) |
Input:
int((5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3072*(1920*x^3+608*x^2-1636*x-1515)*(2*x^2-x+3)^(1/2)+6863/4096*2^(1/2)* arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.67 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\sqrt {3-x+2 x^2}} \, dx=\frac {1}{3072} \, {\left (1920 \, x^{3} + 608 \, x^{2} - 1636 \, x - 1515\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {6863}{8192} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \] Input:
integrate((5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(1/2),x, algorithm="fricas")
Output:
1/3072*(1920*x^3 + 608*x^2 - 1636*x - 1515)*sqrt(2*x^2 - x + 3) + 6863/819 2*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 2 5)
Time = 0.41 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.55 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\sqrt {3-x+2 x^2}} \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {5 x^{3}}{8} + \frac {19 x^{2}}{96} - \frac {409 x}{768} - \frac {505}{1024}\right ) + \frac {6863 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{4096} \] Input:
integrate((5*x**4-x**3+3*x**2+x+2)/(2*x**2-x+3)**(1/2),x)
Output:
sqrt(2*x**2 - x + 3)*(5*x**3/8 + 19*x**2/96 - 409*x/768 - 505/1024) + 6863 *sqrt(2)*asinh(4*sqrt(23)*(x - 1/4)/23)/4096
Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.79 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\sqrt {3-x+2 x^2}} \, dx=\frac {5}{8} \, \sqrt {2 \, x^{2} - x + 3} x^{3} + \frac {19}{96} \, \sqrt {2 \, x^{2} - x + 3} x^{2} - \frac {409}{768} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {6863}{4096} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {505}{1024} \, \sqrt {2 \, x^{2} - x + 3} \] Input:
integrate((5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(1/2),x, algorithm="maxima")
Output:
5/8*sqrt(2*x^2 - x + 3)*x^3 + 19/96*sqrt(2*x^2 - x + 3)*x^2 - 409/768*sqrt (2*x^2 - x + 3)*x + 6863/4096*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 5 05/1024*sqrt(2*x^2 - x + 3)
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\sqrt {3-x+2 x^2}} \, dx=\frac {1}{3072} \, {\left (4 \, {\left (8 \, {\left (60 \, x + 19\right )} x - 409\right )} x - 1515\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {6863}{4096} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \] Input:
integrate((5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(1/2),x, algorithm="giac")
Output:
1/3072*(4*(8*(60*x + 19)*x - 409)*x - 1515)*sqrt(2*x^2 - x + 3) - 6863/409 6*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
Timed out. \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\sqrt {3-x+2 x^2}} \, dx=\int \frac {5\,x^4-x^3+3\,x^2+x+2}{\sqrt {2\,x^2-x+3}} \,d x \] Input:
int((x + 3*x^2 - x^3 + 5*x^4 + 2)/(2*x^2 - x + 3)^(1/2),x)
Output:
int((x + 3*x^2 - x^3 + 5*x^4 + 2)/(2*x^2 - x + 3)^(1/2), x)
Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.89 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\sqrt {3-x+2 x^2}} \, dx=\frac {5 \sqrt {2 x^{2}-x +3}\, x^{3}}{8}+\frac {19 \sqrt {2 x^{2}-x +3}\, x^{2}}{96}-\frac {409 \sqrt {2 x^{2}-x +3}\, x}{768}-\frac {505 \sqrt {2 x^{2}-x +3}}{1024}+\frac {6863 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )}{4096} \] Input:
int((5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(1/2),x)
Output:
(7680*sqrt(2*x**2 - x + 3)*x**3 + 2432*sqrt(2*x**2 - x + 3)*x**2 - 6544*sq rt(2*x**2 - x + 3)*x - 6060*sqrt(2*x**2 - x + 3) + 20589*sqrt(2)*log((2*sq rt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23)))/12288