Integrand size = 32, antiderivative size = 112 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}} \, dx=\frac {2 (2363+3693 x)}{50531 \sqrt {2-x+3 x^2}}-\frac {2 \sqrt {2-x+3 x^2}}{169 (1+2 x)^2}-\frac {4 \sqrt {2-x+3 x^2}}{2197 (1+2 x)}-\frac {487 \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{2197 \sqrt {13}} \]
-487/28561*arctanh(1/26*(9-8*x)*13^(1/2)/(3*x^2-x+2)^(1/2))*13^(1/2)+2/505 31*(2363+3693*x)/(3*x^2-x+2)^(1/2)-2/169*(3*x^2-x+2)^(1/2)/(1+2*x)^2-4/219 7*(3*x^2-x+2)^(1/2)/(1+2*x)
Time = 0.49 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.78 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}} \, dx=\frac {2 \left (1673+13306 x+23281 x^2+14496 x^3\right )}{50531 (1+2 x)^2 \sqrt {2-x+3 x^2}}+\frac {974 \text {arctanh}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2-x+3 x^2}}{\sqrt {13}}\right )}{2197 \sqrt {13}} \]
(2*(1673 + 13306*x + 23281*x^2 + 14496*x^3))/(50531*(1 + 2*x)^2*Sqrt[2 - x + 3*x^2]) + (974*ArcTanh[(Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 - x + 3*x^2])/ Sqrt[13]])/(2197*Sqrt[13])
Time = 0.38 (sec) , antiderivative size = 116, 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 = {2177, 27, 2181, 27, 1228, 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.
| \(\displaystyle \int \frac {4 x^2+3 x+1}{(2 x+1)^3 \left (3 x^2-x+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2177 |
| \(\displaystyle \frac {2}{23} \int \frac {23 \left (1036 x^2+906 x+363\right )}{2197 (2 x+1)^3 \sqrt {3 x^2-x+2}}dx+\frac {2 (3693 x+2363)}{50531 \sqrt {3 x^2-x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {1036 x^2+906 x+363}{(2 x+1)^3 \sqrt {3 x^2-x+2}}dx}{2197}+\frac {2 (3693 x+2363)}{50531 \sqrt {3 x^2-x+2}}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {2 \left (-\frac {1}{26} \int -\frac {13 (958 x+505)}{(2 x+1)^2 \sqrt {3 x^2-x+2}}dx-\frac {13 \sqrt {3 x^2-x+2}}{(2 x+1)^2}\right )}{2197}+\frac {2 (3693 x+2363)}{50531 \sqrt {3 x^2-x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {958 x+505}{(2 x+1)^2 \sqrt {3 x^2-x+2}}dx-\frac {13 \sqrt {3 x^2-x+2}}{(2 x+1)^2}\right )}{2197}+\frac {2 (3693 x+2363)}{50531 \sqrt {3 x^2-x+2}}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (487 \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx-\frac {4 \sqrt {3 x^2-x+2}}{2 x+1}\right )-\frac {13 \sqrt {3 x^2-x+2}}{(2 x+1)^2}\right )}{2197}+\frac {2 (3693 x+2363)}{50531 \sqrt {3 x^2-x+2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (-974 \int \frac {1}{52-\frac {(9-8 x)^2}{3 x^2-x+2}}d\frac {9-8 x}{\sqrt {3 x^2-x+2}}-\frac {4 \sqrt {3 x^2-x+2}}{2 x+1}\right )-\frac {13 \sqrt {3 x^2-x+2}}{(2 x+1)^2}\right )}{2197}+\frac {2 (3693 x+2363)}{50531 \sqrt {3 x^2-x+2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (-\frac {487 \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{\sqrt {13}}-\frac {4 \sqrt {3 x^2-x+2}}{2 x+1}\right )-\frac {13 \sqrt {3 x^2-x+2}}{(2 x+1)^2}\right )}{2197}+\frac {2 (3693 x+2363)}{50531 \sqrt {3 x^2-x+2}}\) |
(2*(2363 + 3693*x))/(50531*Sqrt[2 - x + 3*x^2]) + (2*((-13*Sqrt[2 - x + 3* x^2])/(1 + 2*x)^2 + ((-4*Sqrt[2 - x + 3*x^2])/(1 + 2*x) - (487*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/Sqrt[13])/2))/2197
3.3.51.3.1 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/(((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[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x + c* x^2, x], R = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + (2*c*R - b*S)*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[(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Qx)/(d + e*x )^m - ((2*p + 3)*(2*c*R - b*S))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* e^2, 0] && LtQ[p, -1] && ILtQ[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 = 0.68 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.61
| method | result | size |
| risch | \(\frac {\frac {28992}{50531} x^{3}+\frac {46562}{50531} x^{2}+\frac {26612}{50531} x +\frac {3346}{50531}}{\left (1+2 x \right )^{2} \sqrt {3 x^{2}-x +2}}-\frac {487 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{28561}\) | \(68\) |
| trager | \(\frac {\frac {28992}{50531} x^{3}+\frac {46562}{50531} x^{2}+\frac {26612}{50531} x +\frac {3346}{50531}}{\left (1+2 x \right )^{2} \sqrt {3 x^{2}-x +2}}+\frac {487 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) x +26 \sqrt {3 x^{2}-x +2}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right )}{1+2 x}\right )}{28561}\) | \(87\) |
| default | \(\frac {487}{4394 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}}}+\frac {-\frac {1208}{50531}+\frac {7248 x}{50531}}{\sqrt {3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}}}-\frac {487 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{28561}+\frac {3}{338 \left (x +\frac {1}{2}\right ) \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}}}-\frac {1}{104 \left (x +\frac {1}{2}\right )^{2} \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}}}\) | \(111\) |
2/50531*(14496*x^3+23281*x^2+13306*x+1673)/(1+2*x)^2/(3*x^2-x+2)^(1/2)-487 /28561*13^(1/2)*arctanh(2/13*(9/2-4*x)*13^(1/2)/(12*(x+1/2)^2-16*x+5)^(1/2 ))
Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.12 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}} \, dx=\frac {11201 \, \sqrt {13} {\left (12 \, x^{4} + 8 \, x^{3} + 7 \, x^{2} + 7 \, x + 2\right )} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 52 \, {\left (14496 \, x^{3} + 23281 \, x^{2} + 13306 \, x + 1673\right )} \sqrt {3 \, x^{2} - x + 2}}{1313806 \, {\left (12 \, x^{4} + 8 \, x^{3} + 7 \, x^{2} + 7 \, x + 2\right )}} \]
1/1313806*(11201*sqrt(13)*(12*x^4 + 8*x^3 + 7*x^2 + 7*x + 2)*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)) + 52*(14496*x^3 + 23281*x^2 + 13306*x + 1673)*sqrt(3*x^2 - x + 2))/(12 *x^4 + 8*x^3 + 7*x^2 + 7*x + 2)
\[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}} \, dx=\int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{3} \left (3 x^{2} - x + 2\right )^{\frac {3}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.29 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}} \, dx=\frac {487}{28561} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {7248 \, x}{50531 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {8785}{101062 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {1}{26 \, {\left (4 \, \sqrt {3 \, x^{2} - x + 2} x^{2} + 4 \, \sqrt {3 \, x^{2} - x + 2} x + \sqrt {3 \, x^{2} - x + 2}\right )}} + \frac {3}{169 \, {\left (2 \, \sqrt {3 \, x^{2} - x + 2} x + \sqrt {3 \, x^{2} - x + 2}\right )}} \]
487/28561*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23)/ab s(2*x + 1)) + 7248/50531*x/sqrt(3*x^2 - x + 2) + 8785/101062/sqrt(3*x^2 - x + 2) - 1/26/(4*sqrt(3*x^2 - x + 2)*x^2 + 4*sqrt(3*x^2 - x + 2)*x + sqrt( 3*x^2 - x + 2)) + 3/169/(2*sqrt(3*x^2 - x + 2)*x + sqrt(3*x^2 - x + 2))
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (90) = 180\).
Time = 0.32 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.99 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}} \, dx=\frac {487}{28561} \, \sqrt {13} \log \left (-\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {13} - 2 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} - x + 2} \right |}}{2 \, {\left (2 \, \sqrt {3} x - \sqrt {13} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} - x + 2}\right )}}\right ) + \frac {2 \, {\left (3693 \, x + 2363\right )}}{50531 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {2 \, {\left (62 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{3} - 37 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{2} + 263 \, \sqrt {3} x - 71 \, \sqrt {3} - 263 \, \sqrt {3 \, x^{2} - x + 2}\right )}}{2197 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{2} + 2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} - 5\right )}^{2}} \]
487/28561*sqrt(13)*log(-1/2*abs(-4*sqrt(3)*x - 2*sqrt(13) - 2*sqrt(3) + 4* sqrt(3*x^2 - x + 2))/(2*sqrt(3)*x - sqrt(13) + sqrt(3) - 2*sqrt(3*x^2 - x + 2))) + 2/50531*(3693*x + 2363)/sqrt(3*x^2 - x + 2) + 2/2197*(62*(sqrt(3) *x - sqrt(3*x^2 - x + 2))^3 - 37*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) ^2 + 263*sqrt(3)*x - 71*sqrt(3) - 263*sqrt(3*x^2 - x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 - x + 2))^2 + 2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) - 5) ^2
Timed out. \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}} \, dx=\int \frac {4\,x^2+3\,x+1}{{\left (2\,x+1\right )}^3\,{\left (3\,x^2-x+2\right )}^{3/2}} \,d x \]