Integrand size = 32, antiderivative size = 110 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (197-837 x)}{11661 \left (2-x+3 x^2\right )^{3/2}}-\frac {24 (841-6633 x)}{1162213 \sqrt {2-x+3 x^2}}-\frac {16 \sqrt {2-x+3 x^2}}{2197 (1+2 x)}-\frac {56 \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{2197 \sqrt {13}} \]
-2/11661*(197-837*x)/(3*x^2-x+2)^(3/2)-56/28561*arctanh(1/26*(9-8*x)*13^(1 /2)/(3*x^2-x+2)^(1/2))*13^(1/2)-24/1162213*(841-6633*x)/(3*x^2-x+2)^(1/2)- 16/2197*(3*x^2-x+2)^(1/2)/(1+2*x)
Time = 0.58 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {2 \left (-170239+569989 x+1021566 x^2+133308 x^3+1318464 x^4\right )}{3486639 (1+2 x) \left (2-x+3 x^2\right )^{3/2}}+\frac {112 \text {arctanh}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2-x+3 x^2}}{\sqrt {13}}\right )}{2197 \sqrt {13}} \]
(2*(-170239 + 569989*x + 1021566*x^2 + 133308*x^3 + 1318464*x^4))/(3486639 *(1 + 2*x)*(2 - x + 3*x^2)^(3/2)) + (112*ArcTanh[(Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 - x + 3*x^2])/Sqrt[13]])/(2197*Sqrt[13])
Time = 0.36 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2177, 27, 2177, 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.
| \(\displaystyle \int \frac {4 x^2+3 x+1}{(2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2177 |
| \(\displaystyle \frac {2}{69} \int \frac {6 \left (1116 x^2+1001 x+371\right )}{169 (2 x+1)^2 \left (3 x^2-x+2\right )^{3/2}}dx-\frac {2 (197-837 x)}{11661 \left (3 x^2-x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \int \frac {1116 x^2+1001 x+371}{(2 x+1)^2 \left (3 x^2-x+2\right )^{3/2}}dx}{3887}-\frac {2 (197-837 x)}{11661 \left (3 x^2-x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2177 |
| \(\displaystyle \frac {4 \left (\frac {2}{23} \int \frac {1058 (3 x+8)}{13 (2 x+1)^2 \sqrt {3 x^2-x+2}}dx-\frac {6 (841-6633 x)}{299 \sqrt {3 x^2-x+2}}\right )}{3887}-\frac {2 (197-837 x)}{11661 \left (3 x^2-x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \left (\frac {92}{13} \int \frac {3 x+8}{(2 x+1)^2 \sqrt {3 x^2-x+2}}dx-\frac {6 (841-6633 x)}{299 \sqrt {3 x^2-x+2}}\right )}{3887}-\frac {2 (197-837 x)}{11661 \left (3 x^2-x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {4 \left (\frac {92}{13} \left (\frac {7}{2} \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx-\frac {\sqrt {3 x^2-x+2}}{2 x+1}\right )-\frac {6 (841-6633 x)}{299 \sqrt {3 x^2-x+2}}\right )}{3887}-\frac {2 (197-837 x)}{11661 \left (3 x^2-x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {4 \left (\frac {92}{13} \left (-7 \int \frac {1}{52-\frac {(9-8 x)^2}{3 x^2-x+2}}d\frac {9-8 x}{\sqrt {3 x^2-x+2}}-\frac {\sqrt {3 x^2-x+2}}{2 x+1}\right )-\frac {6 (841-6633 x)}{299 \sqrt {3 x^2-x+2}}\right )}{3887}-\frac {2 (197-837 x)}{11661 \left (3 x^2-x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {4 \left (\frac {92}{13} \left (-\frac {7 \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{2 \sqrt {13}}-\frac {\sqrt {3 x^2-x+2}}{2 x+1}\right )-\frac {6 (841-6633 x)}{299 \sqrt {3 x^2-x+2}}\right )}{3887}-\frac {2 (197-837 x)}{11661 \left (3 x^2-x+2\right )^{3/2}}\) |
(-2*(197 - 837*x))/(11661*(2 - x + 3*x^2)^(3/2)) + (4*((-6*(841 - 6633*x)) /(299*Sqrt[2 - x + 3*x^2]) + (92*(-(Sqrt[2 - x + 3*x^2]/(1 + 2*x)) - (7*Ar cTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(2*Sqrt[13])))/13))/388 7
3.3.56.3.1 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]
Time = 0.70 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(\frac {\frac {878976}{1162213} x^{4}+\frac {168}{2197} x^{3}+\frac {52388}{89401} x^{2}+\frac {1139978}{3486639} x -\frac {340478}{3486639}}{\left (1+2 x \right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}-\frac {56 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{28561}\) | \(73\) |
| trager | \(\frac {\frac {878976}{1162213} x^{4}+\frac {168}{2197} x^{3}+\frac {52388}{89401} x^{2}+\frac {1139978}{3486639} x -\frac {340478}{3486639}}{\left (1+2 x \right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}+\frac {56 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) x +26 \sqrt {3 x^{2}-x +2}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right )}{1+2 x}\right )}{28561}\) | \(92\) |
| default | \(\frac {-\frac {2}{69}+\frac {4 x}{23}}{\left (3 x^{2}-x +2\right )^{\frac {3}{2}}}+\frac {-\frac {16}{529}+\frac {96 x}{529}}{\sqrt {3 x^{2}-x +2}}-\frac {1}{26 \left (x +\frac {1}{2}\right ) \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {3}{2}}}+\frac {7}{507 \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {3}{2}}}-\frac {128 \left (-1+6 x \right )}{11661 \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {3}{2}}}-\frac {10736 \left (-1+6 x \right )}{1162213 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}}}+\frac {28}{2197 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}}}-\frac {56 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{28561}\) | \(165\) |
2/3486639*(1318464*x^4+133308*x^3+1021566*x^2+569989*x-170239)/(3*x^2-x+2) ^(3/2)/(1+2*x)-56/28561*13^(1/2)*arctanh(2/13*(9/2-4*x)*13^(1/2)/(12*(x+1/ 2)^2-16*x+5)^(1/2))
Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.28 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (22218 \, \sqrt {13} {\left (18 \, x^{5} - 3 \, x^{4} + 20 \, x^{3} + 5 \, x^{2} + 4 \, x + 4\right )} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 13 \, {\left (1318464 \, x^{4} + 133308 \, x^{3} + 1021566 \, x^{2} + 569989 \, x - 170239\right )} \sqrt {3 \, x^{2} - x + 2}\right )}}{45326307 \, {\left (18 \, x^{5} - 3 \, x^{4} + 20 \, x^{3} + 5 \, x^{2} + 4 \, x + 4\right )}} \]
2/45326307*(22218*sqrt(13)*(18*x^5 - 3*x^4 + 20*x^3 + 5*x^2 + 4*x + 4)*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)) + 13*(1318464*x^4 + 133308*x^3 + 1021566*x^2 + 569989*x - 17 0239)*sqrt(3*x^2 - x + 2))/(18*x^5 - 3*x^4 + 20*x^3 + 5*x^2 + 4*x + 4)
\[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}} \, dx=\int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{2} \left (3 x^{2} - x + 2\right )^{\frac {5}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.14 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {56}{28561} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {146496 \, x}{1162213 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {9604}{1162213 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {420 \, x}{3887 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {1}{13 \, {\left (2 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x + {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {49}{11661 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} \]
56/28561*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23)/abs (2*x + 1)) + 146496/1162213*x/sqrt(3*x^2 - x + 2) - 9604/1162213/sqrt(3*x^ 2 - x + 2) + 420/3887*x/(3*x^2 - x + 2)^(3/2) - 1/13/(2*(3*x^2 - x + 2)^(3 /2)*x + (3*x^2 - x + 2)^(3/2)) - 49/11661/(3*x^2 - x + 2)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (88) = 176\).
Time = 0.32 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.12 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}} \, dx=-\frac {56}{15108769} \, \sqrt {13} {\left (872 \, \sqrt {13} \sqrt {3} - 529 \, \log \left (\sqrt {13} \sqrt {3} - 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - \frac {56 \, \sqrt {13} \log \left (\sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )} - 4\right )}{28561 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} + \frac {8 \, {\left (\frac {\frac {\frac {13 \, {\left (\frac {77756}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} + \frac {20631}{{\left (2 \, x + 1\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}\right )}}{2 \, x + 1} - \frac {1399650}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{2 \, x + 1} + \frac {625905}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{2 \, x + 1} - \frac {164808}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}\right )}}{3486639 \, {\left (\frac {8}{2 \, x + 1} - \frac {13}{{\left (2 \, x + 1\right )}^{2}} - 3\right )} \sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3}} \]
-56/15108769*sqrt(13)*(872*sqrt(13)*sqrt(3) - 529*log(sqrt(13)*sqrt(3) - 4 ))*sgn(1/(2*x + 1)) - 56/28561*sqrt(13)*log(sqrt(13)*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1)) - 4)/sgn(1/(2*x + 1)) + 8/348663 9*(((13*(77756/sgn(1/(2*x + 1)) + 20631/((2*x + 1)*sgn(1/(2*x + 1))))/(2*x + 1) - 1399650/sgn(1/(2*x + 1)))/(2*x + 1) + 625905/sgn(1/(2*x + 1)))/(2* x + 1) - 164808/sgn(1/(2*x + 1)))/((8/(2*x + 1) - 13/(2*x + 1)^2 - 3)*sqrt (-8/(2*x + 1) + 13/(2*x + 1)^2 + 3))
Timed out. \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}} \, dx=\int \frac {4\,x^2+3\,x+1}{{\left (2\,x+1\right )}^2\,{\left (3\,x^2-x+2\right )}^{5/2}} \,d x \]