Integrand size = 32, antiderivative size = 83 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2-x+3 x^2}} \, dx=-\frac {\sqrt {2-x+3 x^2}}{13 (1+2 x)}-\frac {\text {arcsinh}\left (\frac {1-6 x}{\sqrt {23}}\right )}{\sqrt {3}}+\frac {9 \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{26 \sqrt {13}} \]
-1/3*arcsinh(1/23*(1-6*x)*23^(1/2))*3^(1/2)+9/338*arctanh(1/26*(9-8*x)*13^ (1/2)/(3*x^2-x+2)^(1/2))*13^(1/2)-1/13*(3*x^2-x+2)^(1/2)/(1+2*x)
Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.19 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2-x+3 x^2}} \, dx=-\frac {\sqrt {2-x+3 x^2}}{13+26 x}-\frac {9 \text {arctanh}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2-x+3 x^2}}{\sqrt {13}}\right )}{13 \sqrt {13}}-\frac {\log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{\sqrt {3}} \]
-(Sqrt[2 - x + 3*x^2]/(13 + 26*x)) - (9*ArcTanh[(Sqrt[3] + 2*Sqrt[3]*x - 2 *Sqrt[2 - x + 3*x^2])/Sqrt[13]])/(13*Sqrt[13]) - Log[1 - 6*x + 2*Sqrt[6 - 3*x + 9*x^2]]/Sqrt[3]
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2181, 27, 1269, 1090, 222, 1154, 219}
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.
\(\displaystyle \int \frac {4 x^2+3 x+1}{(2 x+1)^2 \sqrt {3 x^2-x+2}} \, dx\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle -\frac {1}{13} \int -\frac {52 x+17}{2 (2 x+1) \sqrt {3 x^2-x+2}}dx-\frac {\sqrt {3 x^2-x+2}}{13 (2 x+1)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{26} \int \frac {52 x+17}{(2 x+1) \sqrt {3 x^2-x+2}}dx-\frac {\sqrt {3 x^2-x+2}}{13 (2 x+1)}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {1}{26} \left (26 \int \frac {1}{\sqrt {3 x^2-x+2}}dx-9 \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx\right )-\frac {\sqrt {3 x^2-x+2}}{13 (2 x+1)}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{26} \left (\frac {26 \int \frac {1}{\sqrt {\frac {1}{23} (6 x-1)^2+1}}d(6 x-1)}{\sqrt {69}}-9 \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx\right )-\frac {\sqrt {3 x^2-x+2}}{13 (2 x+1)}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{26} \left (\frac {26 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{\sqrt {3}}-9 \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx\right )-\frac {\sqrt {3 x^2-x+2}}{13 (2 x+1)}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {1}{26} \left (18 \int \frac {1}{52-\frac {(9-8 x)^2}{3 x^2-x+2}}d\frac {9-8 x}{\sqrt {3 x^2-x+2}}+\frac {26 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{\sqrt {3}}\right )-\frac {\sqrt {3 x^2-x+2}}{13 (2 x+1)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{26} \left (\frac {26 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{\sqrt {3}}+\frac {9 \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{\sqrt {13}}\right )-\frac {\sqrt {3 x^2-x+2}}{13 (2 x+1)}\) |
-1/13*Sqrt[2 - x + 3*x^2]/(1 + 2*x) + ((26*ArcSinh[(-1 + 6*x)/Sqrt[23]])/S qrt[3] + (9*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/Sqrt[13]) /26
3.3.44.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) ^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R *(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
Time = 0.67 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {\sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{3}-\frac {\sqrt {3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}}}{26 \left (x +\frac {1}{2}\right )}+\frac {9 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{338}\) | \(67\) |
risch | \(-\frac {\sqrt {3 x^{2}-x +2}}{13 \left (1+2 x \right )}+\frac {\sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{3}+\frac {9 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{338}\) | \(67\) |
trager | \(-\frac {\sqrt {3 x^{2}-x +2}}{13 \left (1+2 x \right )}+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) \ln \left (-\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) x -9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right )-26 \sqrt {3 x^{2}-x +2}}{1+2 x}\right )}{338}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}-x +2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{3}\) | \(114\) |
1/3*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))-1/26/(x+1/2)*(3*(x+1/2)^2-4*x+5 /4)^(1/2)+9/338*13^(1/2)*arctanh(2/13*(9/2-4*x)*13^(1/2)/(12*(x+1/2)^2-16* x+5)^(1/2))
Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.48 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2-x+3 x^2}} \, dx=\frac {338 \, \sqrt {3} {\left (2 \, x + 1\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 27 \, \sqrt {13} {\left (2 \, x + 1\right )} \log \left (\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} - 220 \, x^{2} + 196 \, x - 185}{4 \, x^{2} + 4 \, x + 1}\right ) - 156 \, \sqrt {3 \, x^{2} - x + 2}}{2028 \, {\left (2 \, x + 1\right )}} \]
1/2028*(338*sqrt(3)*(2*x + 1)*log(-4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25) + 27*sqrt(13)*(2*x + 1)*log((4*sqrt(13)*sqrt(3*x^2 - x + 2)*(8*x - 9) - 220*x^2 + 196*x - 185)/(4*x^2 + 4*x + 1)) - 156*sqrt( 3*x^2 - x + 2))/(2*x + 1)
\[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2-x+3 x^2}} \, dx=\int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{2} \sqrt {3 x^{2} - x + 2}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.89 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2-x+3 x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \operatorname {arsinh}\left (\frac {6}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) - \frac {9}{338} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) - \frac {\sqrt {3 \, x^{2} - x + 2}}{13 \, {\left (2 \, x + 1\right )}} \]
1/3*sqrt(3)*arcsinh(6/23*sqrt(23)*x - 1/23*sqrt(23)) - 9/338*sqrt(13)*arcs inh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23)/abs(2*x + 1)) - 1/13*sqrt (3*x^2 - x + 2)/(2*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (66) = 132\).
Time = 0.42 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.30 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2-x+3 x^2}} \, dx=\frac {9 \, \sqrt {13} \log \left (\sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )} - 4\right )}{338 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} - \frac {\sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {2 \, \sqrt {13}}{2 \, x + 1} \right |}}{2 \, {\left (\sqrt {3} + \sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}}\right )}{3 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} - \frac {\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3}}{26 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} \]
9/338*sqrt(13)*log(sqrt(13)*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqr t(13)/(2*x + 1)) - 4)/sgn(1/(2*x + 1)) - 1/3*sqrt(3)*log(1/2*abs(-2*sqrt(3 ) + 2*sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + 2*sqrt(13)/(2*x + 1))/(sqr t(3) + sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1)))/sgn( 1/(2*x + 1)) - 1/26*sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3)/sgn(1/(2*x + 1 ))
Timed out. \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2-x+3 x^2}} \, dx=\int \frac {4\,x^2+3\,x+1}{{\left (2\,x+1\right )}^2\,\sqrt {3\,x^2-x+2}} \,d x \]