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

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

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Rubi [A]  time = 0.23, antiderivative size = 214, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 )}} \end {gather*}

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 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 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 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]

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 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]

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*}

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Mathematica [C]  time = 0.03, size = 59, normalized size = 0.28 \begin {gather*} 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 ) \end {gather*}

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]]

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IntegrateAlgebraic [C]  time = 0.11, size = 61, normalized size = 0.29 \begin {gather*} \sqrt {2-3 i} \tan ^{-1}\left (\sqrt {-\frac {2}{13}-\frac {3 i}{13}} \sqrt {3 x+2}\right )+\sqrt {2+3 i} \tan ^{-1}\left (\sqrt {-\frac {2}{13}+\frac {3 i}{13}} \sqrt {3 x+2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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]]

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fricas [A]  time = 0.42, size = 282, normalized size = 1.32 \begin {gather*} \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 {gather*}

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)

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giac [A]  time = 1.10, size = 179, normalized size = 0.84 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.70, size = 360, normalized size = 1.68 \begin {gather*} -\frac {\left (4+2 \sqrt {13}\right ) \arctan \left (\frac {2 \sqrt {3 x +2}-\sqrt {4+2 \sqrt {13}}}{\sqrt {-4+2 \sqrt {13}}}\right )}{3 \sqrt {-4+2 \sqrt {13}}}+\frac {\sqrt {13}\, \left (4+2 \sqrt {13}\right ) \arctan \left (\frac {2 \sqrt {3 x +2}-\sqrt {4+2 \sqrt {13}}}{\sqrt {-4+2 \sqrt {13}}}\right )}{6 \sqrt {-4+2 \sqrt {13}}}-\frac {\left (4+2 \sqrt {13}\right ) \arctan \left (\frac {2 \sqrt {3 x +2}+\sqrt {4+2 \sqrt {13}}}{\sqrt {-4+2 \sqrt {13}}}\right )}{3 \sqrt {-4+2 \sqrt {13}}}+\frac {\sqrt {13}\, \left (4+2 \sqrt {13}\right ) \arctan \left (\frac {2 \sqrt {3 x +2}+\sqrt {4+2 \sqrt {13}}}{\sqrt {-4+2 \sqrt {13}}}\right )}{6 \sqrt {-4+2 \sqrt {13}}}-\frac {\sqrt {4+2 \sqrt {13}}\, \ln \left (3 x +2+\sqrt {13}-\sqrt {3 x +2}\, \sqrt {4+2 \sqrt {13}}\right )}{6}+\frac {\sqrt {4+2 \sqrt {13}}\, \sqrt {13}\, \ln \left (3 x +2+\sqrt {13}-\sqrt {3 x +2}\, \sqrt {4+2 \sqrt {13}}\right )}{12}+\frac {\sqrt {4+2 \sqrt {13}}\, \ln \left (3 x +2+\sqrt {13}+\sqrt {3 x +2}\, \sqrt {4+2 \sqrt {13}}\right )}{6}-\frac {\sqrt {4+2 \sqrt {13}}\, \sqrt {13}\, \ln \left (3 x +2+\sqrt {13}+\sqrt {3 x +2}\, \sqrt {4+2 \sqrt {13}}\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {3 \, x + 2}}{x^{2} + 1}\,{d x} \end {gather*}

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)

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mupad [B]  time = 0.13, size = 179, normalized size = 0.84 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 4.36, size = 32, normalized size = 0.15 \begin {gather*} 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 {gather*}

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))))

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