Integrand size = 18, antiderivative size = 136 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=-\frac {4}{3 \sqrt {1+2 x}}+\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3 \sqrt [4]{3}}-\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3 \sqrt [4]{3}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}}{1+\sqrt {3}+2 x}\right )}{3 \sqrt [4]{3}} \] Output:
-4/3/(1+2*x)^(1/2)-1/9*2^(1/2)*3^(3/4)*arctan(-1+1/3*2^(1/2)*(1+2*x)^(1/2) *3^(3/4))-1/9*2^(1/2)*3^(3/4)*arctan(1+1/3*2^(1/2)*(1+2*x)^(1/2)*3^(3/4))+ 1/9*2^(1/2)*3^(3/4)*arctanh(2^(1/2)*3^(1/4)*(1+2*x)^(1/2)/(1+3^(1/2)+2*x))
Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=\frac {1}{9} \left (-\frac {12}{\sqrt {1+2 x}}-\sqrt {2} 3^{3/4} \arctan \left (\frac {-3+\sqrt {3}+2 \sqrt {3} x}{3^{3/4} \sqrt {2+4 x}}\right )+\sqrt {2} 3^{3/4} \text {arctanh}\left (\frac {3^{3/4} \sqrt {2+4 x}}{3+\sqrt {3}+2 \sqrt {3} x}\right )\right ) \] Input:
Integrate[1/((1 + 2*x)^(3/2)*(1 + x + x^2)),x]
Output:
(-12/Sqrt[1 + 2*x] - Sqrt[2]*3^(3/4)*ArcTan[(-3 + Sqrt[3] + 2*Sqrt[3]*x)/( 3^(3/4)*Sqrt[2 + 4*x])] + Sqrt[2]*3^(3/4)*ArcTanh[(3^(3/4)*Sqrt[2 + 4*x])/ (3 + Sqrt[3] + 2*Sqrt[3]*x)])/9
Time = 0.38 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.40, 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, 826, 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.
| \(\displaystyle \int \frac {1}{(2 x+1)^{3/2} \left (x^2+x+1\right )} \, dx\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle -\frac {1}{3} \int \frac {\sqrt {2 x+1}}{x^2+x+1}dx-\frac {4}{3 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 1118 |
| \(\displaystyle -\frac {1}{6} \int \frac {4 \sqrt {2 x+1}}{(2 x+1)^2+3}d(2 x+1)-\frac {4}{3 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{3} \int \frac {\sqrt {2 x+1}}{(2 x+1)^2+3}d(2 x+1)-\frac {4}{3 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {4}{3} \int \frac {2 x+1}{(2 x+1)^2+3}d\sqrt {2 x+1}-\frac {4}{3 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {4}{3} \left (\frac {1}{2} \int \frac {2 x+\sqrt {3}+1}{(2 x+1)^2+3}d\sqrt {2 x+1}-\frac {1}{2} \int \frac {-2 x+\sqrt {3}-1}{(2 x+1)^2+3}d\sqrt {2 x+1}\right )-\frac {4}{3 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {4}{3} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}+\frac {1}{2} \int \frac {1}{2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}\right )-\frac {1}{2} \int \frac {-2 x+\sqrt {3}-1}{(2 x+1)^2+3}d\sqrt {2 x+1}\right )-\frac {4}{3 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {4}{3} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-2 x-2}d\left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\int \frac {1}{-2 x-2}d\left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}\right )-\frac {1}{2} \int \frac {-2 x+\sqrt {3}-1}{(2 x+1)^2+3}d\sqrt {2 x+1}\right )-\frac {4}{3 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {4}{3} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}\right )-\frac {1}{2} \int \frac {-2 x+\sqrt {3}-1}{(2 x+1)^2+3}d\sqrt {2 x+1}\right )-\frac {4}{3 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {4}{3} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt [4]{3}-2 \sqrt {2 x+1}}{2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt {2} \sqrt [4]{3}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {2 x+1}+\sqrt [4]{3}\right )}{2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt {2} \sqrt [4]{3}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}\right )\right )-\frac {4}{3 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4}{3} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{3}-2 \sqrt {2 x+1}}{2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {2 x+1}+\sqrt [4]{3}\right )}{2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt {2} \sqrt [4]{3}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}\right )\right )-\frac {4}{3 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4}{3} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{3}-2 \sqrt {2 x+1}}{2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\int \frac {\sqrt {2} \sqrt {2 x+1}+\sqrt [4]{3}}{2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1}d\sqrt {2 x+1}}{2 \sqrt [4]{3}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}\right )\right )-\frac {4}{3 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {4}{3} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}\right )+\frac {1}{2} \left (\frac {\log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{2 \sqrt {2} \sqrt [4]{3}}\right )\right )-\frac {4}{3 \sqrt {2 x+1}}\) |
Input:
Int[1/((1 + 2*x)^(3/2)*(1 + x + x^2)),x]
Output:
-4/(3*Sqrt[1 + 2*x]) - (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 + (Log[1 + Sqrt[3] + 2*x - Sqrt[2]*3^(1/4)*Sqrt[1 + 2*x]]/(2*S qrt[2]*3^(1/4)) - Log[1 + Sqrt[3] + 2*x + Sqrt[2]*3^(1/4)*Sqrt[1 + 2*x]]/( 2*Sqrt[2]*3^(1/4)))/2))/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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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 = 1.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.80
| method | result | size |
| derivativedivides | \(-\frac {4}{3 \sqrt {1+2 x}}-\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x -3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}{1+2 x +3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{18}\) | \(109\) |
| default | \(-\frac {4}{3 \sqrt {1+2 x}}-\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x -3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}{1+2 x +3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{18}\) | \(109\) |
| risch | \(-\frac {4}{3 \sqrt {1+2 x}}-\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x -3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}{1+2 x +3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{18}\) | \(109\) |
| pseudoelliptic | \(-\frac {\ln \left (\frac {1+2 x -3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}{1+2 x +3^{\frac {1}{4}} \sqrt {1+2 x}\, \sqrt {2}+\sqrt {3}}\right ) 3^{\frac {3}{4}} \sqrt {2}\, \sqrt {1+2 x}+2 \sqrt {2}\, 3^{\frac {3}{4}} \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right ) \sqrt {1+2 x}+2 \sqrt {2}\, 3^{\frac {3}{4}} \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right ) \sqrt {1+2 x}+24}{18 \sqrt {1+2 x}}\) | \(141\) |
| trager | \(-\frac {4}{3 \sqrt {1+2 x}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right ) \ln \left (-\frac {x \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{5}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{3}-9 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )+108 \sqrt {1+2 x}-18 \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )}{-6+x \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2}-3 x}\right )}{9}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2}\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{4} x -6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2}\right )-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2}\right ) x +108 \sqrt {1+2 x}-18 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2}\right )}{6+x \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2}+3 x}\right )}{9}\) | \(208\) |
Input:
int(1/(1+2*x)^(3/2)/(x^2+x+1),x,method=_RETURNVERBOSE)
Output:
-4/3/(1+2*x)^(1/2)-1/18*3^(3/4)*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)))
Time = 0.09 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=-\frac {2 \cdot 12^{\frac {3}{4}} {\left (2 \, x + 1\right )} \arctan \left (\frac {1}{6} \cdot 12^{\frac {3}{4}} \sqrt {2 \, x + 1} + 1\right ) + 2 \cdot 12^{\frac {3}{4}} {\left (2 \, x + 1\right )} \arctan \left (\frac {1}{6} \cdot 12^{\frac {3}{4}} \sqrt {2 \, x + 1} - 1\right ) - 12^{\frac {3}{4}} {\left (2 \, x + 1\right )} \log \left (4 \, x + 2 \cdot 12^{\frac {1}{4}} \sqrt {2 \, x + 1} + 2 \, \sqrt {3} + 2\right ) + 12^{\frac {3}{4}} {\left (2 \, x + 1\right )} \log \left (4 \, x - 2 \cdot 12^{\frac {1}{4}} \sqrt {2 \, x + 1} + 2 \, \sqrt {3} + 2\right ) + 48 \, \sqrt {2 \, x + 1}}{36 \, {\left (2 \, x + 1\right )}} \] Input:
integrate(1/(1+2*x)^(3/2)/(x^2+x+1),x, algorithm="fricas")
Output:
-1/36*(2*12^(3/4)*(2*x + 1)*arctan(1/6*12^(3/4)*sqrt(2*x + 1) + 1) + 2*12^ (3/4)*(2*x + 1)*arctan(1/6*12^(3/4)*sqrt(2*x + 1) - 1) - 12^(3/4)*(2*x + 1 )*log(4*x + 2*12^(1/4)*sqrt(2*x + 1) + 2*sqrt(3) + 2) + 12^(3/4)*(2*x + 1) *log(4*x - 2*12^(1/4)*sqrt(2*x + 1) + 2*sqrt(3) + 2) + 48*sqrt(2*x + 1))/( 2*x + 1)
\[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=\int \frac {1}{\left (2 x + 1\right )^{\frac {3}{2}} \left (x^{2} + x + 1\right )}\, dx \] Input:
integrate(1/(1+2*x)**(3/2)/(x**2+x+1),x)
Output:
Integral(1/((2*x + 1)**(3/2)*(x**2 + x + 1)), x)
Time = 0.13 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=-\frac {1}{9} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{9} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) + \frac {1}{18} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {1}{18} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {4}{3 \, \sqrt {2 \, x + 1}} \] Input:
integrate(1/(1+2*x)^(3/2)/(x^2+x+1),x, algorithm="maxima")
Output:
-1/9*3^(3/4)*sqrt(2)*arctan(1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) + 2*sqrt( 2*x + 1))) - 1/9*3^(3/4)*sqrt(2)*arctan(-1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt (2) - 2*sqrt(2*x + 1))) + 1/18*3^(3/4)*sqrt(2)*log(3^(1/4)*sqrt(2)*sqrt(2* x + 1) + 2*x + sqrt(3) + 1) - 1/18*3^(3/4)*sqrt(2)*log(-3^(1/4)*sqrt(2)*sq rt(2*x + 1) + 2*x + sqrt(3) + 1) - 4/3/sqrt(2*x + 1)
Time = 0.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=-\frac {1}{9} \cdot 108^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{9} \cdot 108^{\frac {1}{4}} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) + \frac {1}{18} \cdot 108^{\frac {1}{4}} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {1}{18} \cdot 108^{\frac {1}{4}} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {4}{3 \, \sqrt {2 \, x + 1}} \] Input:
integrate(1/(1+2*x)^(3/2)/(x^2+x+1),x, algorithm="giac")
Output:
-1/9*108^(1/4)*arctan(1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) + 2*sqrt(2*x + 1))) - 1/9*108^(1/4)*arctan(-1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) - 2*sqrt (2*x + 1))) + 1/18*108^(1/4)*log(3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqr t(3) + 1) - 1/18*108^(1/4)*log(-3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt (3) + 1) - 4/3/sqrt(2*x + 1)
Time = 5.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=-\frac {4}{3\,\sqrt {2\,x+1}}+\sqrt {2}\,3^{3/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{9}+\frac {1}{9}{}\mathrm {i}\right )+\sqrt {2}\,3^{3/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{9}-\frac {1}{9}{}\mathrm {i}\right ) \] Input:
int(1/((2*x + 1)^(3/2)*(x + x^2 + 1)),x)
Output:
- 4/(3*(2*x + 1)^(1/2)) - 2^(1/2)*3^(3/4)*atan(2^(1/2)*3^(3/4)*(2*x + 1)^( 1/2)*(1/6 - 1i/6))*(1/9 - 1i/9) - 2^(1/2)*3^(3/4)*atan(2^(1/2)*3^(3/4)*(2* x + 1)^(1/2)*(1/6 + 1i/6))*(1/9 + 1i/9)
Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=\frac {-2 \sqrt {2 x +1}\, \sqrt {6}\, 3^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (2 \sqrt {2 x +1}-\sqrt {2}\, 3^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{3 \sqrt {2}}\right )-2 \sqrt {2 x +1}\, \sqrt {6}\, 3^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (2 \sqrt {2 x +1}+\sqrt {2}\, 3^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{3 \sqrt {2}}\right )-\sqrt {2 x +1}\, \sqrt {6}\, 3^{\frac {1}{4}} \mathrm {log}\left (-\sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}}+\sqrt {3}+2 x +1\right )+\sqrt {2 x +1}\, \sqrt {6}\, 3^{\frac {1}{4}} \mathrm {log}\left (\sqrt {2 x +1}\, \sqrt {2}\, 3^{\frac {1}{4}}+\sqrt {3}+2 x +1\right )-24}{18 \sqrt {2 x +1}} \] Input:
int(1/(1+2*x)^(3/2)/(x^2+x+1),x)
Output:
( - 2*sqrt(2*x + 1)*sqrt(6)*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(6)*3**(1/4)*atan((2*sqrt(2 *x + 1) + sqrt(2)*3**(1/4))/(sqrt(2)*3**(1/4))) - sqrt(2*x + 1)*sqrt(6)*3* *(1/4)*log( - sqrt(2*x + 1)*sqrt(2)*3**(1/4) + sqrt(3) + 2*x + 1) + sqrt(2 *x + 1)*sqrt(6)*3**(1/4)*log(sqrt(2*x + 1)*sqrt(2)*3**(1/4) + sqrt(3) + 2* x + 1) - 24)/(18*sqrt(2*x + 1))