Integrand size = 18, antiderivative size = 136 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=-\frac {4}{9 (1+2 x)^{3/2}}+\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}-\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}}{1+\sqrt {3}+2 x}\right )}{3\ 3^{3/4}} \] Output:
-4/9/(1+2*x)^(3/2)-1/9*2^(1/2)*3^(1/4)*arctan(-1+1/3*2^(1/2)*(1+2*x)^(1/2) *3^(3/4))-1/9*2^(1/2)*3^(1/4)*arctan(1+1/3*2^(1/2)*(1+2*x)^(1/2)*3^(3/4))- 1/9*2^(1/2)*3^(1/4)*arctanh(2^(1/2)*3^(1/4)*(1+2*x)^(1/2)/(1+3^(1/2)+2*x))
Time = 0.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=\frac {1}{9} \left (-\frac {4}{(1+2 x)^{3/2}}-\sqrt {2} \sqrt [4]{3} \arctan \left (\frac {-3+\sqrt {3}+2 \sqrt {3} x}{3^{3/4} \sqrt {2+4 x}}\right )-\sqrt {2} \sqrt [4]{3} \text {arctanh}\left (\frac {3^{3/4} \sqrt {2+4 x}}{3+\sqrt {3}+2 \sqrt {3} x}\right )\right ) \] Input:
Integrate[1/((1 + 2*x)^(5/2)*(1 + x + x^2)),x]
Output:
(-4/(1 + 2*x)^(3/2) - Sqrt[2]*3^(1/4)*ArcTan[(-3 + Sqrt[3] + 2*Sqrt[3]*x)/ (3^(3/4)*Sqrt[2 + 4*x])] - Sqrt[2]*3^(1/4)*ArcTanh[(3^(3/4)*Sqrt[2 + 4*x]) /(3 + Sqrt[3] + 2*Sqrt[3]*x)])/9
Time = 0.38 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.47, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {1117, 1118, 27, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {1}{(2 x+1)^{5/2} \left (x^2+x+1\right )} \, dx\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle -\frac {1}{3} \int \frac {1}{\sqrt {2 x+1} \left (x^2+x+1\right )}dx-\frac {4}{9 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 1118 |
| \(\displaystyle -\frac {1}{6} \int \frac {4}{\sqrt {2 x+1} \left ((2 x+1)^2+3\right )}d(2 x+1)-\frac {4}{9 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{3} \int \frac {1}{\sqrt {2 x+1} \left ((2 x+1)^2+3\right )}d(2 x+1)-\frac {4}{9 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {4}{3} \int \frac {1}{(2 x+1)^2+3}d\sqrt {2 x+1}-\frac {4}{9 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -\frac {4}{3} \left (\frac {\int \frac {-2 x+\sqrt {3}-1}{(2 x+1)^2+3}d\sqrt {2 x+1}}{2 \sqrt {3}}+\frac {\int \frac {2 x+\sqrt {3}+1}{(2 x+1)^2+3}d\sqrt {2 x+1}}{2 \sqrt {3}}\right )-\frac {4}{9 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {4}{3} \left (\frac {\frac {1}{2} \int \frac {1}{2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}+\frac {1}{2} \int \frac {1}{2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt {3}}+\frac {\int \frac {-2 x+\sqrt {3}-1}{(2 x+1)^2+3}d\sqrt {2 x+1}}{2 \sqrt {3}}\right )-\frac {4}{9 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {4}{3} \left (\frac {\frac {\int \frac {1}{-2 x-2}d\left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\int \frac {1}{-2 x-2}d\left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}}{2 \sqrt {3}}+\frac {\int \frac {-2 x+\sqrt {3}-1}{(2 x+1)^2+3}d\sqrt {2 x+1}}{2 \sqrt {3}}\right )-\frac {4}{9 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {4}{3} \left (\frac {\int \frac {-2 x+\sqrt {3}-1}{(2 x+1)^2+3}d\sqrt {2 x+1}}{2 \sqrt {3}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}}{2 \sqrt {3}}\right )-\frac {4}{9 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {4}{3} \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{3}-2 \sqrt {2 x+1}}{2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {2 x+1}+\sqrt [4]{3}\right )}{2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt {2} \sqrt [4]{3}}}{2 \sqrt {3}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}}{2 \sqrt {3}}\right )-\frac {4}{9 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4}{3} \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{3}-2 \sqrt {2 x+1}}{2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt {2} \sqrt [4]{3}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {2 x+1}+\sqrt [4]{3}\right )}{2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt {2} \sqrt [4]{3}}}{2 \sqrt {3}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}}{2 \sqrt {3}}\right )-\frac {4}{9 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4}{3} \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{3}-2 \sqrt {2 x+1}}{2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt {2} \sqrt [4]{3}}+\frac {\int \frac {\sqrt {2} \sqrt {2 x+1}+\sqrt [4]{3}}{2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt [4]{3}}}{2 \sqrt {3}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}}{2 \sqrt {3}}\right )-\frac {4}{9 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {4}{3} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}}{2 \sqrt {3}}+\frac {\frac {\log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{2 \sqrt {2} \sqrt [4]{3}}}{2 \sqrt {3}}\right )-\frac {4}{9 (2 x+1)^{3/2}}\) |
Input:
Int[1/((1 + 2*x)^(5/2)*(1 + x + x^2)),x]
Output:
-4/(9*(1 + 2*x)^(3/2)) - (4*((-(ArcTan[1 - (Sqrt[2]*Sqrt[1 + 2*x])/3^(1/4) ]/(Sqrt[2]*3^(1/4))) + ArcTan[1 + (Sqrt[2]*Sqrt[1 + 2*x])/3^(1/4)]/(Sqrt[2 ]*3^(1/4)))/(2*Sqrt[3]) + (-1/2*Log[1 + Sqrt[3] + 2*x - Sqrt[2]*3^(1/4)*Sq rt[1 + 2*x]]/(Sqrt[2]*3^(1/4)) + Log[1 + Sqrt[3] + 2*x + Sqrt[2]*3^(1/4)*S qrt[1 + 2*x]]/(2*Sqrt[2]*3^(1/4)))/(2*Sqrt[3])))/3
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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[-2*b*d*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m + 1)*(b^2 - 4*a*c))), x] + Simp[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 - 4*a* c))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & & (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) /2])
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[1/e Subst[Int[x^m*(a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 1.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.80
| method | result | size |
| derivativedivides | \(-\frac {3^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}{1+2 x -3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{18}-\frac {4}{9 \left (1+2 x \right )^{\frac {3}{2}}}\) | \(109\) |
| default | \(-\frac {3^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}{1+2 x -3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{18}-\frac {4}{9 \left (1+2 x \right )^{\frac {3}{2}}}\) | \(109\) |
| pseudoelliptic | \(-\frac {8+3^{\frac {1}{4}} \left (1+2 x \right )^{\frac {3}{2}} \left (\ln \left (\frac {1+2 x +3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}{1+2 x -3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right ) \sqrt {2}}{18 \left (1+2 x \right )^{\frac {3}{2}}}\) | \(116\) |
| trager | \(-\frac {4}{9 \left (1+2 x \right )^{\frac {3}{2}}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{4} x +4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right ) x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right )+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right ) x +12 \sqrt {1+2 x}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right )}{x \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}-x -2}\right )}{9}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{5} x +4 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{3}-3 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )+12 \sqrt {1+2 x}-6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )}{x \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}+x +2}\right )}{9}\) | \(242\) |
Input:
int(1/(1+2*x)^(5/2)/(x^2+x+1),x,method=_RETURNVERBOSE)
Output:
-1/18*3^(1/4)*2^(1/2)*(ln((1+2*x+3^(1/4)*(1+2*x)^(1/2)*2^(1/2)+3^(1/2))/(1 +2*x-3^(1/4)*(1+2*x)^(1/2)*2^(1/2)+3^(1/2)))+2*arctan(1+1/3*2^(1/2)*(1+2*x )^(1/2)*3^(3/4))+2*arctan(-1+1/3*2^(1/2)*(1+2*x)^(1/2)*3^(3/4)))-4/9/(1+2* x)^(3/2)
Time = 0.08 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=-\frac {2 \cdot 108^{\frac {3}{4}} {\left (4 \, x^{2} + 4 \, x + 1\right )} \arctan \left (\frac {1}{3} \cdot 108^{\frac {1}{4}} \sqrt {2 \, x + 1} + 1\right ) + 2 \cdot 108^{\frac {3}{4}} {\left (4 \, x^{2} + 4 \, x + 1\right )} \arctan \left (\frac {1}{3} \cdot 108^{\frac {1}{4}} \sqrt {2 \, x + 1} - 1\right ) + 108^{\frac {3}{4}} {\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (108^{\frac {3}{4}} \sqrt {2 \, x + 1} + 36 \, x + 18 \, \sqrt {3} + 18\right ) - 108^{\frac {3}{4}} {\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (-108^{\frac {3}{4}} \sqrt {2 \, x + 1} + 36 \, x + 18 \, \sqrt {3} + 18\right ) + 144 \, \sqrt {2 \, x + 1}}{324 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}} \] Input:
integrate(1/(1+2*x)^(5/2)/(x^2+x+1),x, algorithm="fricas")
Output:
-1/324*(2*108^(3/4)*(4*x^2 + 4*x + 1)*arctan(1/3*108^(1/4)*sqrt(2*x + 1) + 1) + 2*108^(3/4)*(4*x^2 + 4*x + 1)*arctan(1/3*108^(1/4)*sqrt(2*x + 1) - 1 ) + 108^(3/4)*(4*x^2 + 4*x + 1)*log(108^(3/4)*sqrt(2*x + 1) + 36*x + 18*sq rt(3) + 18) - 108^(3/4)*(4*x^2 + 4*x + 1)*log(-108^(3/4)*sqrt(2*x + 1) + 3 6*x + 18*sqrt(3) + 18) + 144*sqrt(2*x + 1))/(4*x^2 + 4*x + 1)
\[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=\int \frac {1}{\left (2 x + 1\right )^{\frac {5}{2}} \left (x^{2} + x + 1\right )}\, dx \] Input:
integrate(1/(1+2*x)**(5/2)/(x**2+x+1),x)
Output:
Integral(1/((2*x + 1)**(5/2)*(x**2 + x + 1)), x)
Time = 0.13 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=-\frac {1}{9} \cdot 3^{\frac {1}{4}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{9} \cdot 3^{\frac {1}{4}} \sqrt {2} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{18} \cdot 3^{\frac {1}{4}} \sqrt {2} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + \frac {1}{18} \cdot 3^{\frac {1}{4}} \sqrt {2} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {4}{9 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}}} \] Input:
integrate(1/(1+2*x)^(5/2)/(x^2+x+1),x, algorithm="maxima")
Output:
-1/9*3^(1/4)*sqrt(2)*arctan(1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) + 2*sqrt( 2*x + 1))) - 1/9*3^(1/4)*sqrt(2)*arctan(-1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt (2) - 2*sqrt(2*x + 1))) - 1/18*3^(1/4)*sqrt(2)*log(3^(1/4)*sqrt(2)*sqrt(2* x + 1) + 2*x + sqrt(3) + 1) + 1/18*3^(1/4)*sqrt(2)*log(-3^(1/4)*sqrt(2)*sq rt(2*x + 1) + 2*x + sqrt(3) + 1) - 4/9/(2*x + 1)^(3/2)
Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=-\frac {1}{9} \cdot 12^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{9} \cdot 12^{\frac {1}{4}} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{18} \cdot 12^{\frac {1}{4}} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + \frac {1}{18} \cdot 12^{\frac {1}{4}} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {4}{9 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}}} \] Input:
integrate(1/(1+2*x)^(5/2)/(x^2+x+1),x, algorithm="giac")
Output:
-1/9*12^(1/4)*arctan(1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) + 2*sqrt(2*x + 1 ))) - 1/9*12^(1/4)*arctan(-1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) - 2*sqrt(2 *x + 1))) - 1/18*12^(1/4)*log(3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt(3 ) + 1) + 1/18*12^(1/4)*log(-3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt(3) + 1) - 4/9/(2*x + 1)^(3/2)
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=-\frac {4}{9\,{\left (2\,x+1\right )}^{3/2}}+\sqrt {2}\,3^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{9}-\frac {1}{9}{}\mathrm {i}\right )+\sqrt {2}\,3^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{9}+\frac {1}{9}{}\mathrm {i}\right ) \] Input:
int(1/((2*x + 1)^(5/2)*(x + x^2 + 1)),x)
Output:
- 4/(9*(2*x + 1)^(3/2)) - 2^(1/2)*3^(1/4)*atan(2^(1/2)*3^(3/4)*(2*x + 1)^( 1/2)*(1/6 - 1i/6))*(1/9 + 1i/9) - 2^(1/2)*3^(1/4)*atan(2^(1/2)*3^(3/4)*(2* x + 1)^(1/2)*(1/6 + 1i/6))*(1/9 - 1i/9)
Time = 0.21 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.28 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=\frac {-4 \sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (2 \sqrt {2 x +1}-\sqrt {2}\, 3^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{3 \sqrt {2}}\right ) x -2 \sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (2 \sqrt {2 x +1}-\sqrt {2}\, 3^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{3 \sqrt {2}}\right )-4 \sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (2 \sqrt {2 x +1}+\sqrt {2}\, 3^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{3 \sqrt {2}}\right ) x -2 \sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (2 \sqrt {2 x +1}+\sqrt {2}\, 3^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{3 \sqrt {2}}\right )+2 \sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}} \mathrm {log}\left (-\sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}}+\sqrt {3}+2 x +1\right ) x +\sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}} \mathrm {log}\left (-\sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}}+\sqrt {3}+2 x +1\right )-2 \sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}} \mathrm {log}\left (\sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}}+\sqrt {3}+2 x +1\right ) x -\sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}} \mathrm {log}\left (\sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}}+\sqrt {3}+2 x +1\right )-8}{18 \sqrt {2 x +1}\, \left (2 x +1\right )} \] Input:
int(1/(1+2*x)^(5/2)/(x^2+x+1),x)
Output:
( - 4*sqrt(2*x + 1)*sqrt(2)*3**(1/4)*atan((2*sqrt(2*x + 1) - sqrt(2)*3**(1 /4))/(sqrt(2)*3**(1/4)))*x - 2*sqrt(2*x + 1)*sqrt(2)*3**(1/4)*atan((2*sqrt (2*x + 1) - sqrt(2)*3**(1/4))/(sqrt(2)*3**(1/4))) - 4*sqrt(2*x + 1)*sqrt(2 )*3**(1/4)*atan((2*sqrt(2*x + 1) + sqrt(2)*3**(1/4))/(sqrt(2)*3**(1/4)))*x - 2*sqrt(2*x + 1)*sqrt(2)*3**(1/4)*atan((2*sqrt(2*x + 1) + sqrt(2)*3**(1/ 4))/(sqrt(2)*3**(1/4))) + 2*sqrt(2*x + 1)*sqrt(2)*3**(1/4)*log( - sqrt(2*x + 1)*sqrt(2)*3**(1/4) + sqrt(3) + 2*x + 1)*x + sqrt(2*x + 1)*sqrt(2)*3**( 1/4)*log( - sqrt(2*x + 1)*sqrt(2)*3**(1/4) + sqrt(3) + 2*x + 1) - 2*sqrt(2 *x + 1)*sqrt(2)*3**(1/4)*log(sqrt(2*x + 1)*sqrt(2)*3**(1/4) + sqrt(3) + 2* x + 1)*x - sqrt(2*x + 1)*sqrt(2)*3**(1/4)*log(sqrt(2*x + 1)*sqrt(2)*3**(1/ 4) + sqrt(3) + 2*x + 1) - 8)/(18*sqrt(2*x + 1)*(2*x + 1))