Integrand size = 17, antiderivative size = 214 \[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=-\frac {3 \arctan \left (\frac {\sqrt {2 \left (2+\sqrt {13}\right )}-2 \sqrt {2+3 x}}{\sqrt {2 \left (-2+\sqrt {13}\right )}}\right )}{\sqrt {2 \left (-2+\sqrt {13}\right )}}+\frac {3 \arctan \left (\frac {\sqrt {2 \left (2+\sqrt {13}\right )}+2 \sqrt {2+3 x}}{\sqrt {2 \left (-2+\sqrt {13}\right )}}\right )}{\sqrt {2 \left (-2+\sqrt {13}\right )}}+\frac {3 \log \left (2+\sqrt {13}+3 x-\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}-\frac {3 \log \left (2+\sqrt {13}+3 x+\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}} \]
-3*arctan((-2*(2+3*x)^(1/2)+(4+2*13^(1/2))^(1/2))/(-4+2*13^(1/2))^(1/2))/( -4+2*13^(1/2))^(1/2)+3*arctan((2*(2+3*x)^(1/2)+(4+2*13^(1/2))^(1/2))/(-4+2 *13^(1/2))^(1/2))/(-4+2*13^(1/2))^(1/2)+3/2*ln(2+3*x+13^(1/2)-(2+3*x)^(1/2 )*(4+2*13^(1/2))^(1/2))/(4+2*13^(1/2))^(1/2)-3/2*ln(2+3*x+13^(1/2)+(2+3*x) ^(1/2)*(4+2*13^(1/2))^(1/2))/(4+2*13^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=\sqrt {2-3 i} \arctan \left (\sqrt {-\frac {2}{13}-\frac {3 i}{13}} \sqrt {2+3 x}\right )+\sqrt {2+3 i} \arctan \left (\sqrt {-\frac {2}{13}+\frac {3 i}{13}} \sqrt {2+3 x}\right ) \]
Sqrt[2 - 3*I]*ArcTan[Sqrt[-2/13 - (3*I)/13]*Sqrt[2 + 3*x]] + Sqrt[2 + 3*I] *ArcTan[Sqrt[-2/13 + (3*I)/13]*Sqrt[2 + 3*x]]
Time = 0.44 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {483, 1447, 1475, 1083, 217, 1478, 25, 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.
\(\displaystyle \int \frac {\sqrt {3 x+2}}{x^2+1} \, dx\) |
\(\Big \downarrow \) 483 |
\(\displaystyle 6 \int \frac {3 x+2}{(3 x+2)^2-4 (3 x+2)+13}d\sqrt {3 x+2}\) |
\(\Big \downarrow \) 1447 |
\(\displaystyle 6 \left (\frac {1}{2} \int \frac {3 x+\sqrt {13}+2}{(3 x+2)^2-4 (3 x+2)+13}d\sqrt {3 x+2}-\frac {1}{2} \int \frac {-3 x+\sqrt {13}-2}{(3 x+2)^2-4 (3 x+2)+13}d\sqrt {3 x+2}\right )\) |
\(\Big \downarrow \) 1475 |
\(\displaystyle 6 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{3 x-\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {3 x+2}+\sqrt {13}+2}d\sqrt {3 x+2}+\frac {1}{2} \int \frac {1}{3 x+\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {3 x+2}+\sqrt {13}+2}d\sqrt {3 x+2}\right )-\frac {1}{2} \int \frac {-3 x+\sqrt {13}-2}{(3 x+2)^2-4 (3 x+2)+13}d\sqrt {3 x+2}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle 6 \left (\frac {1}{2} \left (-\int \frac {1}{-3 x+2 \left (2-\sqrt {13}\right )-2}d\left (2 \sqrt {3 x+2}-\sqrt {2 \left (2+\sqrt {13}\right )}\right )-\int \frac {1}{-3 x+2 \left (2-\sqrt {13}\right )-2}d\left (2 \sqrt {3 x+2}+\sqrt {2 \left (2+\sqrt {13}\right )}\right )\right )-\frac {1}{2} \int \frac {-3 x+\sqrt {13}-2}{(3 x+2)^2-4 (3 x+2)+13}d\sqrt {3 x+2}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle 6 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {3 x+2}-\sqrt {2 \left (2+\sqrt {13}\right )}}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )}{\sqrt {2 \left (\sqrt {13}-2\right )}}+\frac {\arctan \left (\frac {2 \sqrt {3 x+2}+\sqrt {2 \left (2+\sqrt {13}\right )}}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )-\frac {1}{2} \int \frac {-3 x+\sqrt {13}-2}{(3 x+2)^2-4 (3 x+2)+13}d\sqrt {3 x+2}\right )\) |
\(\Big \downarrow \) 1478 |
\(\displaystyle 6 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2 \left (2+\sqrt {13}\right )}-2 \sqrt {3 x+2}}{3 x-\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {3 x+2}+\sqrt {13}+2}d\sqrt {3 x+2}}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}+\frac {\int -\frac {2 \sqrt {3 x+2}+\sqrt {2 \left (2+\sqrt {13}\right )}}{3 x+\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {3 x+2}+\sqrt {13}+2}d\sqrt {3 x+2}}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {3 x+2}-\sqrt {2 \left (2+\sqrt {13}\right )}}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )}{\sqrt {2 \left (\sqrt {13}-2\right )}}+\frac {\arctan \left (\frac {2 \sqrt {3 x+2}+\sqrt {2 \left (2+\sqrt {13}\right )}}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 6 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2 \left (2+\sqrt {13}\right )}-2 \sqrt {3 x+2}}{3 x-\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {3 x+2}+\sqrt {13}+2}d\sqrt {3 x+2}}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}-\frac {\int \frac {2 \sqrt {3 x+2}+\sqrt {2 \left (2+\sqrt {13}\right )}}{3 x+\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {3 x+2}+\sqrt {13}+2}d\sqrt {3 x+2}}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {3 x+2}-\sqrt {2 \left (2+\sqrt {13}\right )}}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )}{\sqrt {2 \left (\sqrt {13}-2\right )}}+\frac {\arctan \left (\frac {2 \sqrt {3 x+2}+\sqrt {2 \left (2+\sqrt {13}\right )}}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle 6 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {3 x+2}-\sqrt {2 \left (2+\sqrt {13}\right )}}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )}{\sqrt {2 \left (\sqrt {13}-2\right )}}+\frac {\arctan \left (\frac {2 \sqrt {3 x+2}+\sqrt {2 \left (2+\sqrt {13}\right )}}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )+\frac {1}{2} \left (\frac {\log \left (3 x-\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {3 x+2}+\sqrt {13}+2\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}-\frac {\log \left (3 x+\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {3 x+2}+\sqrt {13}+2\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}\right )\right )\) |
6*((ArcTan[(-Sqrt[2*(2 + Sqrt[13])] + 2*Sqrt[2 + 3*x])/Sqrt[2*(-2 + Sqrt[1 3])]]/Sqrt[2*(-2 + Sqrt[13])] + ArcTan[(Sqrt[2*(2 + Sqrt[13])] + 2*Sqrt[2 + 3*x])/Sqrt[2*(-2 + Sqrt[13])]]/Sqrt[2*(-2 + Sqrt[13])])/2 + (Log[2 + Sqr t[13] + 3*x - Sqrt[2*(2 + Sqrt[13])]*Sqrt[2 + 3*x]]/(2*Sqrt[2*(2 + Sqrt[13 ])]) - Log[2 + Sqrt[13] + 3*x + Sqrt[2*(2 + Sqrt[13])]*Sqrt[2 + 3*x]]/(2*S qrt[2*(2 + Sqrt[13])]))/2)
3.7.48.3.1 Defintions of rubi rules used
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[Sqrt[(c_) + (d_.)*(x_)]/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[2*d Subst[Int[x^2/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], x ] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a/c, 2]}, Simp[1/2 Int[(q + x^2)/(a + b*x^2 + c*x^4), x], x] - Simp[1/2 Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && LtQ[b ^2 - 4*a*c, 0] && PosQ[a*c]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[2*(d/e) - b/c, 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]] /; F reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] , 0]))
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[-2*(d/e) - b/c, 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]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ [c*d^2 - a*e^2, 0] && !GtQ[b^2 - 4*a*c, 0]
Time = 4.27 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(-\frac {\sqrt {4+2 \sqrt {13}}\, \left (-2+\sqrt {13}\right ) \left (\frac {\ln \left (2+3 x +\sqrt {13}+\sqrt {2+3 x}\, \sqrt {4+2 \sqrt {13}}\right )}{2}-\frac {\sqrt {4+2 \sqrt {13}}\, \arctan \left (\frac {2 \sqrt {2+3 x}+\sqrt {4+2 \sqrt {13}}}{\sqrt {-4+2 \sqrt {13}}}\right )}{\sqrt {-4+2 \sqrt {13}}}\right )}{6}+\frac {\sqrt {4+2 \sqrt {13}}\, \left (-2+\sqrt {13}\right ) \left (\frac {\ln \left (2+3 x +\sqrt {13}-\sqrt {2+3 x}\, \sqrt {4+2 \sqrt {13}}\right )}{2}+\frac {\sqrt {4+2 \sqrt {13}}\, \arctan \left (\frac {2 \sqrt {2+3 x}-\sqrt {4+2 \sqrt {13}}}{\sqrt {-4+2 \sqrt {13}}}\right )}{\sqrt {-4+2 \sqrt {13}}}\right )}{6}\) | \(194\) |
default | \(-\frac {\sqrt {4+2 \sqrt {13}}\, \left (-2+\sqrt {13}\right ) \left (\frac {\ln \left (2+3 x +\sqrt {13}+\sqrt {2+3 x}\, \sqrt {4+2 \sqrt {13}}\right )}{2}-\frac {\sqrt {4+2 \sqrt {13}}\, \arctan \left (\frac {2 \sqrt {2+3 x}+\sqrt {4+2 \sqrt {13}}}{\sqrt {-4+2 \sqrt {13}}}\right )}{\sqrt {-4+2 \sqrt {13}}}\right )}{6}+\frac {\sqrt {4+2 \sqrt {13}}\, \left (-2+\sqrt {13}\right ) \left (\frac {\ln \left (2+3 x +\sqrt {13}-\sqrt {2+3 x}\, \sqrt {4+2 \sqrt {13}}\right )}{2}+\frac {\sqrt {4+2 \sqrt {13}}\, \arctan \left (\frac {2 \sqrt {2+3 x}-\sqrt {4+2 \sqrt {13}}}{\sqrt {-4+2 \sqrt {13}}}\right )}{\sqrt {-4+2 \sqrt {13}}}\right )}{6}\) | \(194\) |
pseudoelliptic | \(\frac {-\sqrt {13}\, \ln \left (2+3 x +\sqrt {13}+\sqrt {2+3 x}\, \sqrt {4+2 \sqrt {13}}\right )+\sqrt {13}\, \ln \left (2+3 x +\sqrt {13}-\sqrt {2+3 x}\, \sqrt {4+2 \sqrt {13}}\right )+6 \arctan \left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (2+\sqrt {13}\right )+6 \sqrt {2+3 x}}{3 \sqrt {-4+2 \sqrt {13}}}\right )-6 \arctan \left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (2+\sqrt {13}\right )-6 \sqrt {2+3 x}}{3 \sqrt {-4+2 \sqrt {13}}}\right )+2 \ln \left (2+3 x +\sqrt {13}+\sqrt {2+3 x}\, \sqrt {4+2 \sqrt {13}}\right )-2 \ln \left (2+3 x +\sqrt {13}-\sqrt {2+3 x}\, \sqrt {4+2 \sqrt {13}}\right )}{2 \sqrt {-4+2 \sqrt {13}}}\) | \(210\) |
trager | \(-\operatorname {RootOf}\left (\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+1\right ) \ln \left (\frac {816 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{4} x \operatorname {RootOf}\left (\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+1\right )-216 \operatorname {RootOf}\left (\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+1\right ) x \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}-1700 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2} \operatorname {RootOf}\left (\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+1\right )-816 \sqrt {2+3 x}\, \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}-96 \operatorname {RootOf}\left (\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+1\right ) x -400 \operatorname {RootOf}\left (\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+1\right )-1167 \sqrt {2+3 x}}{4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2} x +2 x +3}\right )+\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right ) \ln \left (\frac {-816 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{5} x -1848 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{3} x +816 \sqrt {2+3 x}\, \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}-1700 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{3}-936 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right ) x -351 \sqrt {2+3 x}-1300 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )}{4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2} x +2 x -3}\right )\) | \(408\) |
-1/6*(4+2*13^(1/2))^(1/2)*(-2+13^(1/2))*(1/2*ln(2+3*x+13^(1/2)+(2+3*x)^(1/ 2)*(4+2*13^(1/2))^(1/2))-(4+2*13^(1/2))^(1/2)/(-4+2*13^(1/2))^(1/2)*arctan ((2*(2+3*x)^(1/2)+(4+2*13^(1/2))^(1/2))/(-4+2*13^(1/2))^(1/2)))+1/6*(4+2*1 3^(1/2))^(1/2)*(-2+13^(1/2))*(1/2*ln(2+3*x+13^(1/2)-(2+3*x)^(1/2)*(4+2*13^ (1/2))^(1/2))+(4+2*13^(1/2))^(1/2)/(-4+2*13^(1/2))^(1/2)*arctan((2*(2+3*x) ^(1/2)-(4+2*13^(1/2))^(1/2))/(-4+2*13^(1/2))^(1/2)))
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.36 \[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=\frac {1}{2} \, \sqrt {3 i - 2} \log \left (i \, \sqrt {3 i - 2} + \sqrt {3 \, x + 2}\right ) - \frac {1}{2} \, \sqrt {3 i - 2} \log \left (-i \, \sqrt {3 i - 2} + \sqrt {3 \, x + 2}\right ) - \frac {1}{2} \, \sqrt {-3 i - 2} \log \left (i \, \sqrt {-3 i - 2} + \sqrt {3 \, x + 2}\right ) + \frac {1}{2} \, \sqrt {-3 i - 2} \log \left (-i \, \sqrt {-3 i - 2} + \sqrt {3 \, x + 2}\right ) \]
1/2*sqrt(3*I - 2)*log(I*sqrt(3*I - 2) + sqrt(3*x + 2)) - 1/2*sqrt(3*I - 2) *log(-I*sqrt(3*I - 2) + sqrt(3*x + 2)) - 1/2*sqrt(-3*I - 2)*log(I*sqrt(-3* I - 2) + sqrt(3*x + 2)) + 1/2*sqrt(-3*I - 2)*log(-I*sqrt(-3*I - 2) + sqrt( 3*x + 2))
\[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=\int \frac {\sqrt {3 x + 2}}{x^{2} + 1}\, dx \]
\[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=\int { \frac {\sqrt {3 \, x + 2}}{x^{2} + 1} \,d x } \]
Time = 0.71 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=\frac {1}{2} \, \sqrt {2 \, \sqrt {13} + 4} \arctan \left (\frac {13^{\frac {3}{4}} {\left (13^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + \sqrt {3 \, x + 2}\right )}}{13 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {13} + 4} \arctan \left (-\frac {13^{\frac {3}{4}} {\left (13^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} - \sqrt {3 \, x + 2}\right )}}{13 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {13} - 4} \log \left (2 \cdot 13^{\frac {1}{4}} \sqrt {3 \, x + 2} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + 3 \, x + \sqrt {13} + 2\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {13} - 4} \log \left (-2 \cdot 13^{\frac {1}{4}} \sqrt {3 \, x + 2} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + 3 \, x + \sqrt {13} + 2\right ) \]
1/2*sqrt(2*sqrt(13) + 4)*arctan(1/13*13^(3/4)*(13^(1/4)*sqrt(1/13*sqrt(13) + 1/2) + sqrt(3*x + 2))/sqrt(-1/13*sqrt(13) + 1/2)) + 1/2*sqrt(2*sqrt(13) + 4)*arctan(-1/13*13^(3/4)*(13^(1/4)*sqrt(1/13*sqrt(13) + 1/2) - sqrt(3*x + 2))/sqrt(-1/13*sqrt(13) + 1/2)) - 1/4*sqrt(2*sqrt(13) - 4)*log(2*13^(1/ 4)*sqrt(3*x + 2)*sqrt(1/13*sqrt(13) + 1/2) + 3*x + sqrt(13) + 2) + 1/4*sqr t(2*sqrt(13) - 4)*log(-2*13^(1/4)*sqrt(3*x + 2)*sqrt(1/13*sqrt(13) + 1/2) + 3*x + sqrt(13) + 2)
Time = 0.14 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=-\mathrm {atanh}\left (-\frac {\left (1152\,\sqrt {3\,x+2}\,{\left (\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}-\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )}^2-720\,\sqrt {3\,x+2}\right )\,\left (\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}-\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )}{2808}\right )\,\left (2\,\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}-2\,\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )-\mathrm {atanh}\left (\frac {\left (720\,\sqrt {3\,x+2}-1152\,\sqrt {3\,x+2}\,{\left (\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}+\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )}^2\right )\,\left (\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}+\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )}{2808}\right )\,\left (2\,\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}+2\,\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right ) \]
- atanh(-((1152*(3*x + 2)^(1/2)*((- 13^(1/2)/8 - 1/4)^(1/2) - (13^(1/2)/8 - 1/4)^(1/2))^2 - 720*(3*x + 2)^(1/2))*((- 13^(1/2)/8 - 1/4)^(1/2) - (13^( 1/2)/8 - 1/4)^(1/2)))/2808)*(2*(- 13^(1/2)/8 - 1/4)^(1/2) - 2*(13^(1/2)/8 - 1/4)^(1/2)) - atanh(((720*(3*x + 2)^(1/2) - 1152*(3*x + 2)^(1/2)*((- 13^ (1/2)/8 - 1/4)^(1/2) + (13^(1/2)/8 - 1/4)^(1/2))^2)*((- 13^(1/2)/8 - 1/4)^ (1/2) + (13^(1/2)/8 - 1/4)^(1/2)))/2808)*(2*(- 13^(1/2)/8 - 1/4)^(1/2) + 2 *(13^(1/2)/8 - 1/4)^(1/2))