\(\int \frac {\sqrt {2+3 x}}{1+x^2} \, dx\) [648]

Optimal result

Integrand size = 17, antiderivative size = 214 \[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=-\frac {3 \arctan \left (\frac {\sqrt {2 \left (2+\sqrt {13}\right )}-2 \sqrt {2+3 x}}{\sqrt {2 \left (-2+\sqrt {13}\right )}}\right )}{\sqrt {2 \left (-2+\sqrt {13}\right )}}+\frac {3 \arctan \left (\frac {\sqrt {2 \left (2+\sqrt {13}\right )}+2 \sqrt {2+3 x}}{\sqrt {2 \left (-2+\sqrt {13}\right )}}\right )}{\sqrt {2 \left (-2+\sqrt {13}\right )}}+\frac {3 \log \left (2+\sqrt {13}+3 x-\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}-\frac {3 \log \left (2+\sqrt {13}+3 x+\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {714, 1141, 1175, 632, 210, 1178, 642} \[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=-\frac {3 \arctan \left (\frac {\sqrt {2 \left (2+\sqrt {13}\right )}-2 \sqrt {3 x+2}}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )}{\sqrt {2 \left (\sqrt {13}-2\right )}}+\frac {3 \arctan \left (\frac {2 \sqrt {3 x+2}+\sqrt {2 \left (2+\sqrt {13}\right )}}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )}{\sqrt {2 \left (\sqrt {13}-2\right )}}+\frac {3 \log \left (3 x-\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {3 x+2}+\sqrt {13}+2\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}-\frac {3 \log \left (3 x+\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {3 x+2}+\sqrt {13}+2\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}} \]

[In]

Int[Sqrt[2 + 3*x]/(1 + x^2),x]

[Out]

(-3*ArcTan[(Sqrt[2*(2 + Sqrt[13])] - 2*Sqrt[2 + 3*x])/Sqrt[2*(-2 + Sqrt[13])]])/Sqrt[2*(-2 + Sqrt[13])] + (3*A
rcTan[(Sqrt[2*(2 + Sqrt[13])] + 2*Sqrt[2 + 3*x])/Sqrt[2*(-2 + Sqrt[13])]])/Sqrt[2*(-2 + Sqrt[13])] + (3*Log[2
+ Sqrt[13] + 3*x - Sqrt[2*(2 + Sqrt[13])]*Sqrt[2 + 3*x]])/(2*Sqrt[2*(2 + Sqrt[13])]) - (3*Log[2 + Sqrt[13] + 3
*x + Sqrt[2*(2 + Sqrt[13])]*Sqrt[2 + 3*x]])/(2*Sqrt[2*(2 + Sqrt[13])])

Rule 210

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])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 714

Int[Sqrt[(d_) + (e_.)*(x_)]/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[x^2/(c*d^2 + a*e^2 - 2*c*d
*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1141

Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, Dist[1/2, Int[(q + x^2)/(
a + b*x^2 + c*x^4), x], x] - Dist[1/2, Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && Lt
Q[b^2 - 4*a*c, 0] && PosQ[a*c]

Rule 1175

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e) - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/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[2*(d/e) - b/c, 0] || ( !Lt
Q[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rule 1178

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e) - b/c, 2]},
Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = 6 \text {Subst}\left (\int \frac {x^2}{13-4 x^2+x^4} \, dx,x,\sqrt {2+3 x}\right ) \\ & = -\left (3 \text {Subst}\left (\int \frac {\sqrt {13}-x^2}{13-4 x^2+x^4} \, dx,x,\sqrt {2+3 x}\right )\right )+3 \text {Subst}\left (\int \frac {\sqrt {13}+x^2}{13-4 x^2+x^4} \, dx,x,\sqrt {2+3 x}\right ) \\ & = \frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {13}-\sqrt {2 \left (2+\sqrt {13}\right )} x+x^2} \, dx,x,\sqrt {2+3 x}\right )+\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {13}+\sqrt {2 \left (2+\sqrt {13}\right )} x+x^2} \, dx,x,\sqrt {2+3 x}\right )+\frac {3 \text {Subst}\left (\int \frac {\sqrt {2 \left (2+\sqrt {13}\right )}+2 x}{-\sqrt {13}-\sqrt {2 \left (2+\sqrt {13}\right )} x-x^2} \, dx,x,\sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {2 \left (2+\sqrt {13}\right )}-2 x}{-\sqrt {13}+\sqrt {2 \left (2+\sqrt {13}\right )} x-x^2} \, dx,x,\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 )}}-\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 \text {Subst}\left (\int \frac {1}{2 \left (2-\sqrt {13}\right )-x^2} \, dx,x,-\sqrt {2 \left (2+\sqrt {13}\right )}+2 \sqrt {2+3 x}\right )-3 \text {Subst}\left (\int \frac {1}{2 \left (2-\sqrt {13}\right )-x^2} \, dx,x,\sqrt {2 \left (2+\sqrt {13}\right )}+2 \sqrt {2+3 x}\right ) \\ & = -\frac {3 \tan ^{-1}\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 \tan ^{-1}\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 )}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=\sqrt {2-3 i} \arctan \left (\sqrt {-\frac {2}{13}-\frac {3 i}{13}} \sqrt {2+3 x}\right )+\sqrt {2+3 i} \arctan \left (\sqrt {-\frac {2}{13}+\frac {3 i}{13}} \sqrt {2+3 x}\right ) \]

[In]

Integrate[Sqrt[2 + 3*x]/(1 + x^2),x]

[Out]

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]]

Maple [A] (verified)

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 )\)	\(408\)

[In]

int((2+3*x)^(1/2)/(x^2+1),x,method=_RETURNVERBOSE)

[Out]

-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
*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)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.36 \[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=\frac {1}{2} \, \sqrt {3 i - 2} \log \left (i \, \sqrt {3 i - 2} + \sqrt {3 \, x + 2}\right ) - \frac {1}{2} \, \sqrt {3 i - 2} \log \left (-i \, \sqrt {3 i - 2} + \sqrt {3 \, x + 2}\right ) - \frac {1}{2} \, \sqrt {-3 i - 2} \log \left (i \, \sqrt {-3 i - 2} + \sqrt {3 \, x + 2}\right ) + \frac {1}{2} \, \sqrt {-3 i - 2} \log \left (-i \, \sqrt {-3 i - 2} + \sqrt {3 \, x + 2}\right ) \]

[In]

integrate((2+3*x)^(1/2)/(x^2+1),x, algorithm="fricas")

[Out]

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) + sqr
t(3*x + 2))

Sympy [F]

\[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=\int \frac {\sqrt {3 x + 2}}{x^{2} + 1}\, dx \]

[In]

integrate((2+3*x)**(1/2)/(x**2+1),x)

[Out]

Integral(sqrt(3*x + 2)/(x**2 + 1), x)

Maxima [F]

\[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=\int { \frac {\sqrt {3 \, x + 2}}{x^{2} + 1} \,d x } \]

[In]

integrate((2+3*x)^(1/2)/(x^2+1),x, algorithm="maxima")

[Out]

integrate(sqrt(3*x + 2)/(x^2 + 1), x)

Giac [A] (verification not implemented)

none

Time = 0.71 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=\frac {1}{2} \, \sqrt {2 \, \sqrt {13} + 4} \arctan \left (\frac {13^{\frac {3}{4}} {\left (13^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + \sqrt {3 \, x + 2}\right )}}{13 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {13} + 4} \arctan \left (-\frac {13^{\frac {3}{4}} {\left (13^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} - \sqrt {3 \, x + 2}\right )}}{13 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {13} - 4} \log \left (2 \cdot 13^{\frac {1}{4}} \sqrt {3 \, x + 2} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + 3 \, x + \sqrt {13} + 2\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {13} - 4} \log \left (-2 \cdot 13^{\frac {1}{4}} \sqrt {3 \, x + 2} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + 3 \, x + \sqrt {13} + 2\right ) \]

[In]

integrate((2+3*x)^(1/2)/(x^2+1),x, algorithm="giac")

[Out]

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*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)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx=-\mathrm {atanh}\left (-\frac {\left (1152\,\sqrt {3\,x+2}\,{\left (\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}-\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )}^2-720\,\sqrt {3\,x+2}\right )\,\left (\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}-\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )}{2808}\right )\,\left (2\,\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}-2\,\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )-\mathrm {atanh}\left (\frac {\left (720\,\sqrt {3\,x+2}-1152\,\sqrt {3\,x+2}\,{\left (\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}+\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )}^2\right )\,\left (\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}+\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )}{2808}\right )\,\left (2\,\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}+2\,\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right ) \]

[In]

int((3*x + 2)^(1/2)/(x^2 + 1),x)

[Out]

- 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))