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

Optimal. Leaf size=214 \[ \frac{3 \log \left (3 x-\sqrt{2 \left (2+\sqrt{13}\right )} \sqrt{3 x+2}+\sqrt{13}+2\right )}{2 \sqrt{2 \left (2+\sqrt{13}\right )}}-\frac{3 \log \left (3 x+\sqrt{2 \left (2+\sqrt{13}\right )} \sqrt{3 x+2}+\sqrt{13}+2\right )}{2 \sqrt{2 \left (2+\sqrt{13}\right )}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2 \left (2+\sqrt{13}\right )}-2 \sqrt{3 x+2}}{\sqrt{2 \left (\sqrt{13}-2\right )}}\right )}{\sqrt{2 \left (\sqrt{13}-2\right )}}+\frac{3 \tan ^{-1}\left (\frac{2 \sqrt{3 x+2}+\sqrt{2 \left (2+\sqrt{13}\right )}}{\sqrt{2 \left (\sqrt{13}-2\right )}}\right )}{\sqrt{2 \left (\sqrt{13}-2\right )}} \]

[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])])

________________________________________________________________________________________

Rubi [A]  time = 0.230092, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {700, 1127, 1161, 618, 204, 1164, 628} \[ \frac{3 \log \left (3 x-\sqrt{2 \left (2+\sqrt{13}\right )} \sqrt{3 x+2}+\sqrt{13}+2\right )}{2 \sqrt{2 \left (2+\sqrt{13}\right )}}-\frac{3 \log \left (3 x+\sqrt{2 \left (2+\sqrt{13}\right )} \sqrt{3 x+2}+\sqrt{13}+2\right )}{2 \sqrt{2 \left (2+\sqrt{13}\right )}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2 \left (2+\sqrt{13}\right )}-2 \sqrt{3 x+2}}{\sqrt{2 \left (\sqrt{13}-2\right )}}\right )}{\sqrt{2 \left (\sqrt{13}-2\right )}}+\frac{3 \tan ^{-1}\left (\frac{2 \sqrt{3 x+2}+\sqrt{2 \left (2+\sqrt{13}\right )}}{\sqrt{2 \left (\sqrt{13}-2\right )}}\right )}{\sqrt{2 \left (\sqrt{13}-2\right )}} \]

Antiderivative was successfully verified.

[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 700

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 1127

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 1161

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 618

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1164

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]

Rule 628

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]

Rubi steps

\begin{align*} \int \frac{\sqrt{2+3 x}}{1+x^2} \, dx &=6 \operatorname{Subst}\left (\int \frac{x^2}{13-4 x^2+x^4} \, dx,x,\sqrt{2+3 x}\right )\\ &=-\left (3 \operatorname{Subst}\left (\int \frac{\sqrt{13}-x^2}{13-4 x^2+x^4} \, dx,x,\sqrt{2+3 x}\right )\right )+3 \operatorname{Subst}\left (\int \frac{\sqrt{13}+x^2}{13-4 x^2+x^4} \, dx,x,\sqrt{2+3 x}\right )\\ &=\frac{3}{2} \operatorname{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} \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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]  time = 0.0302046, size = 59, normalized size = 0.28 \[ i \sqrt{2+3 i} \tanh ^{-1}\left (\frac{\sqrt{3 x+2}}{\sqrt{2+3 i}}\right )-i \sqrt{2-3 i} \tanh ^{-1}\left (\frac{\sqrt{3 x+2}}{\sqrt{2-3 i}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + 3*x]/(1 + x^2),x]

[Out]

(-I)*Sqrt[2 - 3*I]*ArcTanh[Sqrt[2 + 3*x]/Sqrt[2 - 3*I]] + I*Sqrt[2 + 3*I]*ArcTanh[Sqrt[2 + 3*x]/Sqrt[2 + 3*I]]

________________________________________________________________________________________

Maple [B]  time = 0.173, size = 360, normalized size = 1.7 \begin{align*} -{\frac{\sqrt{4+2\,\sqrt{13}}}{6}\ln \left ( 2+3\,x+\sqrt{13}-\sqrt{2+3\,x}\sqrt{4+2\,\sqrt{13}} \right ) }-{\frac{4+2\,\sqrt{13}}{3\,\sqrt{-4+2\,\sqrt{13}}}\arctan \left ({\frac{1}{\sqrt{-4+2\,\sqrt{13}}} \left ( 2\,\sqrt{2+3\,x}-\sqrt{4+2\,\sqrt{13}} \right ) } \right ) }+{\frac{\sqrt{4+2\,\sqrt{13}}\sqrt{13}}{12}\ln \left ( 2+3\,x+\sqrt{13}-\sqrt{2+3\,x}\sqrt{4+2\,\sqrt{13}} \right ) }+{\frac{\sqrt{13} \left ( 4+2\,\sqrt{13} \right ) }{6\,\sqrt{-4+2\,\sqrt{13}}}\arctan \left ({\frac{1}{\sqrt{-4+2\,\sqrt{13}}} \left ( 2\,\sqrt{2+3\,x}-\sqrt{4+2\,\sqrt{13}} \right ) } \right ) }+{\frac{\sqrt{4+2\,\sqrt{13}}}{6}\ln \left ( 2+3\,x+\sqrt{13}+\sqrt{2+3\,x}\sqrt{4+2\,\sqrt{13}} \right ) }-{\frac{4+2\,\sqrt{13}}{3\,\sqrt{-4+2\,\sqrt{13}}}\arctan \left ({\frac{1}{\sqrt{-4+2\,\sqrt{13}}} \left ( 2\,\sqrt{2+3\,x}+\sqrt{4+2\,\sqrt{13}} \right ) } \right ) }-{\frac{\sqrt{4+2\,\sqrt{13}}\sqrt{13}}{12}\ln \left ( 2+3\,x+\sqrt{13}+\sqrt{2+3\,x}\sqrt{4+2\,\sqrt{13}} \right ) }+{\frac{\sqrt{13} \left ( 4+2\,\sqrt{13} \right ) }{6\,\sqrt{-4+2\,\sqrt{13}}}\arctan \left ({\frac{1}{\sqrt{-4+2\,\sqrt{13}}} \left ( 2\,\sqrt{2+3\,x}+\sqrt{4+2\,\sqrt{13}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(4+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))-1/3*(4+2*13^(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))+1/12*(4+2*13^(1/2))^(1/2)*13^(
1/2)*ln(2+3*x+13^(1/2)-(2+3*x)^(1/2)*(4+2*13^(1/2))^(1/2))+1/6*13^(1/2)*(4+2*13^(1/2))/(-4+2*13^(1/2))^(1/2)*a
rctan((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)*ln(2+3*x+13^(1/2)
+(2+3*x)^(1/2)*(4+2*13^(1/2))^(1/2))-1/3*(4+2*13^(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))-1/12*(4+2*13^(1/2))^(1/2)*13^(1/2)*ln(2+3*x+13^(1/2)+(2+3*x)^(1/2)*(4+2*1
3^(1/2))^(1/2))+1/6*13^(1/2)*(4+2*13^(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))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{3 \, x + 2}}{x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(3*x + 2)/(x^2 + 1), x)

________________________________________________________________________________________

Fricas [A]  time = 2.62328, size = 990, normalized size = 4.63 \begin{align*} \frac{1}{156} \cdot 13^{\frac{1}{4}} \sqrt{4 \, \sqrt{13} + 26}{\left (2 \, \sqrt{13} - 13\right )} \log \left (\frac{1}{13} \cdot 13^{\frac{3}{4}} \sqrt{3 \, x + 2} \sqrt{4 \, \sqrt{13} + 26} + 3 \, x + \sqrt{13} + 2\right ) - \frac{1}{156} \cdot 13^{\frac{1}{4}} \sqrt{4 \, \sqrt{13} + 26}{\left (2 \, \sqrt{13} - 13\right )} \log \left (-\frac{1}{13} \cdot 13^{\frac{3}{4}} \sqrt{3 \, x + 2} \sqrt{4 \, \sqrt{13} + 26} + 3 \, x + \sqrt{13} + 2\right ) - \frac{1}{13} \cdot 13^{\frac{3}{4}} \sqrt{4 \, \sqrt{13} + 26} \arctan \left (-\frac{1}{39} \cdot 13^{\frac{3}{4}} \sqrt{3 \, x + 2} \sqrt{4 \, \sqrt{13} + 26} + \frac{1}{39} \cdot 13^{\frac{1}{4}} \sqrt{13^{\frac{3}{4}} \sqrt{3 \, x + 2} \sqrt{4 \, \sqrt{13} + 26} + 39 \, x + 13 \, \sqrt{13} + 26} \sqrt{4 \, \sqrt{13} + 26} - \frac{1}{3} \, \sqrt{13} - \frac{2}{3}\right ) - \frac{1}{13} \cdot 13^{\frac{3}{4}} \sqrt{4 \, \sqrt{13} + 26} \arctan \left (-\frac{1}{39} \cdot 13^{\frac{3}{4}} \sqrt{3 \, x + 2} \sqrt{4 \, \sqrt{13} + 26} + \frac{1}{39} \cdot 13^{\frac{1}{4}} \sqrt{-13^{\frac{3}{4}} \sqrt{3 \, x + 2} \sqrt{4 \, \sqrt{13} + 26} + 39 \, x + 13 \, \sqrt{13} + 26} \sqrt{4 \, \sqrt{13} + 26} + \frac{1}{3} \, \sqrt{13} + \frac{2}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/156*13^(1/4)*sqrt(4*sqrt(13) + 26)*(2*sqrt(13) - 13)*log(1/13*13^(3/4)*sqrt(3*x + 2)*sqrt(4*sqrt(13) + 26) +
3*x + sqrt(13) + 2) - 1/156*13^(1/4)*sqrt(4*sqrt(13) + 26)*(2*sqrt(13) - 13)*log(-1/13*13^(3/4)*sqrt(3*x + 2)
*sqrt(4*sqrt(13) + 26) + 3*x + sqrt(13) + 2) - 1/13*13^(3/4)*sqrt(4*sqrt(13) + 26)*arctan(-1/39*13^(3/4)*sqrt(
3*x + 2)*sqrt(4*sqrt(13) + 26) + 1/39*13^(1/4)*sqrt(13^(3/4)*sqrt(3*x + 2)*sqrt(4*sqrt(13) + 26) + 39*x + 13*s
qrt(13) + 26)*sqrt(4*sqrt(13) + 26) - 1/3*sqrt(13) - 2/3) - 1/13*13^(3/4)*sqrt(4*sqrt(13) + 26)*arctan(-1/39*1
3^(3/4)*sqrt(3*x + 2)*sqrt(4*sqrt(13) + 26) + 1/39*13^(1/4)*sqrt(-13^(3/4)*sqrt(3*x + 2)*sqrt(4*sqrt(13) + 26)
+ 39*x + 13*sqrt(13) + 26)*sqrt(4*sqrt(13) + 26) + 1/3*sqrt(13) + 2/3)

________________________________________________________________________________________

Sympy [A]  time = 3.08808, size = 32, normalized size = 0.15 \begin{align*} 6 \operatorname{RootSum}{\left (20736 t^{4} + 576 t^{2} + 13, \left ( t \mapsto t \log{\left (576 t^{3} + 8 t + \sqrt{3 x + 2} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*RootSum(20736*_t**4 + 576*_t**2 + 13, Lambda(_t, _t*log(576*_t**3 + 8*_t + sqrt(3*x + 2))))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{3 \, x + 2}}{x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(3*x + 2)/(x^2 + 1), x)