\(\int \frac {1+a x^2}{(-1+a x^2) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\) [2244]

Optimal result

Integrand size = 38, antiderivative size = 167 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=-\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}+\text {RootSum}\left [b^2-4 a \text {$\#$1}^4-2 b \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-b \log \left (\sqrt {a x+\sqrt {b+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 a \text {$\#$1}^3+b \text {$\#$1}^3-\text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.91, number of steps used = 23, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6857, 2142, 14, 2144, 1642, 842, 840, 1180, 214, 211} \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {\sqrt {a^2 x^2+b}+a x}}{\sqrt {\sqrt {a}-\sqrt {a+b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a}-\sqrt {a+b}}}-\frac {2 \arctan \left (\frac {\sqrt {\sqrt {a^2 x^2+b}+a x}}{\sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {a} \sqrt {\sqrt {a+b}+\sqrt {a}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2 x^2+b}+a x}}{\sqrt {\sqrt {a}-\sqrt {a+b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a}-\sqrt {a+b}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2 x^2+b}+a x}}{\sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {a} \sqrt {\sqrt {a+b}+\sqrt {a}}}-\frac {b}{3 a \left (\sqrt {a^2 x^2+b}+a x\right )^{3/2}}+\frac {\sqrt {\sqrt {a^2 x^2+b}+a x}}{a} \]

[In]

Int[(1 + a*x^2)/((-1 + a*x^2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]),x]

[Out]

-1/3*b/(a*(a*x + Sqrt[b + a^2*x^2])^(3/2)) + Sqrt[a*x + Sqrt[b + a^2*x^2]]/a - (2*ArcTan[Sqrt[a*x + Sqrt[b + a
^2*x^2]]/Sqrt[Sqrt[a] - Sqrt[a + b]]])/(Sqrt[a]*Sqrt[Sqrt[a] - Sqrt[a + b]]) - (2*ArcTan[Sqrt[a*x + Sqrt[b + a
^2*x^2]]/Sqrt[Sqrt[a] + Sqrt[a + b]]])/(Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[a + b]]) - (2*ArcTanh[Sqrt[a*x + Sqrt[b +
a^2*x^2]]/Sqrt[Sqrt[a] - Sqrt[a + b]]])/(Sqrt[a]*Sqrt[Sqrt[a] - Sqrt[a + b]]) - (2*ArcTanh[Sqrt[a*x + Sqrt[b +
a^2*x^2]]/Sqrt[Sqrt[a] + Sqrt[a + b]]])/(Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[a + b]])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 211

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

Rule 214

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

Rule 840

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

Rule 842

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e
*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d +
e*x)^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1180

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

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2142

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[(g + h*x^n)^p*((d^2 + a*f^2 - 2*d*x + x^2)/(d - x)^2), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rule 2144

Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dist[1/(2^(
m + 1)*e^(m + 1)), Subst[Int[x^(n - m - 2)*(a*f^2 + x^2)*((-a)*f^2*h + 2*e*g*x + h*x^2)^m, x], x, e*x + f*Sqrt
[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[m]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {a x+\sqrt {b+a^2 x^2}}}+\frac {2}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}}\right ) \, dx \\ & = 2 \int \frac {1}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx+\int \frac {1}{\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \\ & = 2 \int \left (-\frac {1}{2 \left (1-\sqrt {a} x\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}}-\frac {1}{2 \left (1+\sqrt {a} x\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}}\right ) \, dx+\frac {\text {Subst}\left (\int \frac {b+x^2}{x^{5/2}} \, dx,x,a x+\sqrt {b+a^2 x^2}\right )}{2 a} \\ & = \frac {\text {Subst}\left (\int \left (\frac {b}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,a x+\sqrt {b+a^2 x^2}\right )}{2 a}-\int \frac {1}{\left (1-\sqrt {a} x\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx-\int \frac {1}{\left (1+\sqrt {a} x\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \\ & = -\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-\text {Subst}\left (\int \frac {b+x^2}{x^{3/2} \left (\sqrt {a} b+2 a x-\sqrt {a} x^2\right )} \, dx,x,a x+\sqrt {b+a^2 x^2}\right )-\text {Subst}\left (\int \frac {b+x^2}{x^{3/2} \left (-\sqrt {a} b+2 a x+\sqrt {a} x^2\right )} \, dx,x,a x+\sqrt {b+a^2 x^2}\right ) \\ & = -\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-\text {Subst}\left (\int \left (-\frac {1}{\sqrt {a} x^{3/2}}+\frac {2 \left (b+\sqrt {a} x\right )}{x^{3/2} \left (\sqrt {a} b+2 a x-\sqrt {a} x^2\right )}\right ) \, dx,x,a x+\sqrt {b+a^2 x^2}\right )-\text {Subst}\left (\int \left (\frac {1}{\sqrt {a} x^{3/2}}+\frac {2 \left (b-\sqrt {a} x\right )}{x^{3/2} \left (-\sqrt {a} b+2 a x+\sqrt {a} x^2\right )}\right ) \, dx,x,a x+\sqrt {b+a^2 x^2}\right ) \\ & = -\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-2 \text {Subst}\left (\int \frac {b+\sqrt {a} x}{x^{3/2} \left (\sqrt {a} b+2 a x-\sqrt {a} x^2\right )} \, dx,x,a x+\sqrt {b+a^2 x^2}\right )-2 \text {Subst}\left (\int \frac {b-\sqrt {a} x}{x^{3/2} \left (-\sqrt {a} b+2 a x+\sqrt {a} x^2\right )} \, dx,x,a x+\sqrt {b+a^2 x^2}\right ) \\ & = -\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-\frac {2 \text {Subst}\left (\int \frac {-a b+\sqrt {a} b x}{\sqrt {x} \left (\sqrt {a} b+2 a x-\sqrt {a} x^2\right )} \, dx,x,a x+\sqrt {b+a^2 x^2}\right )}{\sqrt {a} b}+\frac {2 \text {Subst}\left (\int \frac {-a b-\sqrt {a} b x}{\sqrt {x} \left (-\sqrt {a} b+2 a x+\sqrt {a} x^2\right )} \, dx,x,a x+\sqrt {b+a^2 x^2}\right )}{\sqrt {a} b} \\ & = -\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-\frac {4 \text {Subst}\left (\int \frac {-a b+\sqrt {a} b x^2}{\sqrt {a} b+2 a x^2-\sqrt {a} x^4} \, dx,x,\sqrt {a x+\sqrt {b+a^2 x^2}}\right )}{\sqrt {a} b}+\frac {4 \text {Subst}\left (\int \frac {-a b-\sqrt {a} b x^2}{-\sqrt {a} b+2 a x^2+\sqrt {a} x^4} \, dx,x,\sqrt {a x+\sqrt {b+a^2 x^2}}\right )}{\sqrt {a} b} \\ & = -\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-2 \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {a+b}-\sqrt {a} x^2} \, dx,x,\sqrt {a x+\sqrt {b+a^2 x^2}}\right )-2 \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {a+b}-\sqrt {a} x^2} \, dx,x,\sqrt {a x+\sqrt {b+a^2 x^2}}\right )-2 \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {a+b}+\sqrt {a} x^2} \, dx,x,\sqrt {a x+\sqrt {b+a^2 x^2}}\right )-2 \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {a+b}+\sqrt {a} x^2} \, dx,x,\sqrt {a x+\sqrt {b+a^2 x^2}}\right ) \\ & = -\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-\frac {2 \arctan \left (\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {a}-\sqrt {a+b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a}-\sqrt {a+b}}}-\frac {2 \arctan \left (\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a}+\sqrt {a+b}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {a}-\sqrt {a+b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a}-\sqrt {a+b}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a}+\sqrt {a+b}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.96 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\frac {2 \left (b+3 a x \left (a x+\sqrt {b+a^2 x^2}\right )\right )}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\text {RootSum}\left [b^2-4 a \text {$\#$1}^4-2 b \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {b \log \left (\sqrt {a x+\sqrt {b+a^2 x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {a x+\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 a \text {$\#$1}^3-b \text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \]

[In]

Integrate[(1 + a*x^2)/((-1 + a*x^2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]),x]

[Out]

(2*(b + 3*a*x*(a*x + Sqrt[b + a^2*x^2])))/(3*a*(a*x + Sqrt[b + a^2*x^2])^(3/2)) + RootSum[b^2 - 4*a*#1^4 - 2*b
*#1^4 + #1^8 & , (b*Log[Sqrt[a*x + Sqrt[b + a^2*x^2]] - #1] + Log[Sqrt[a*x + Sqrt[b + a^2*x^2]] - #1]*#1^4)/(-
2*a*#1^3 - b*#1^3 + #1^7) & ]

Maple [N/A] (verified)

Not integrable

Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.20

\[\int \frac {a \,x^{2}+1}{\left (a \,x^{2}-1\right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b}}}d x\]

[In]

int((a*x^2+1)/(a*x^2-1)/(a*x+(a^2*x^2+b)^(1/2))^(1/2),x)

[Out]

int((a*x^2+1)/(a*x^2-1)/(a*x+(a^2*x^2+b)^(1/2))^(1/2),x)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.27 (sec) , antiderivative size = 1046, normalized size of antiderivative = 6.26 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\frac {3 \, a b \sqrt {-\sqrt {\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} + 2 \, a + b}{a^{2} b^{2}}}} \log \left (8 \, {\left (a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} - a\right )} \sqrt {-\sqrt {\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} + 2 \, a + b}{a^{2} b^{2}}}} + 8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right ) - 3 \, a b \sqrt {-\sqrt {\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} + 2 \, a + b}{a^{2} b^{2}}}} \log \left (-8 \, {\left (a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} - a\right )} \sqrt {-\sqrt {\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} + 2 \, a + b}{a^{2} b^{2}}}} + 8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right ) - 3 \, a b \sqrt {-\sqrt {-\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} - 2 \, a - b}{a^{2} b^{2}}}} \log \left (8 \, {\left (a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} + a\right )} \sqrt {-\sqrt {-\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} - 2 \, a - b}{a^{2} b^{2}}}} + 8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right ) + 3 \, a b \sqrt {-\sqrt {-\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} - 2 \, a - b}{a^{2} b^{2}}}} \log \left (-8 \, {\left (a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} + a\right )} \sqrt {-\sqrt {-\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} - 2 \, a - b}{a^{2} b^{2}}}} + 8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right ) + 3 \, a b \left (\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} + 2 \, a + b}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (8 \, {\left (a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} - a\right )} \left (\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} + 2 \, a + b}{a^{2} b^{2}}\right )^{\frac {1}{4}} + 8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right ) - 3 \, a b \left (\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} + 2 \, a + b}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (-8 \, {\left (a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} - a\right )} \left (\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} + 2 \, a + b}{a^{2} b^{2}}\right )^{\frac {1}{4}} + 8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right ) - 3 \, a b \left (-\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} - 2 \, a - b}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (8 \, {\left (a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} + a\right )} \left (-\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} - 2 \, a - b}{a^{2} b^{2}}\right )^{\frac {1}{4}} + 8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right ) + 3 \, a b \left (-\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} - 2 \, a - b}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (-8 \, {\left (a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} + a\right )} \left (-\frac {2 \, a^{2} b^{2} \sqrt {\frac {a + b}{a^{3} b^{4}}} - 2 \, a - b}{a^{2} b^{2}}\right )^{\frac {1}{4}} + 8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right ) - 2 \, {\left (a^{2} x^{2} - \sqrt {a^{2} x^{2} + b} a x - b\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}{3 \, a b} \]

[In]

integrate((a*x^2+1)/(a*x^2-1)/(a*x+(a^2*x^2+b)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

1/3*(3*a*b*sqrt(-sqrt((2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) + 2*a + b)/(a^2*b^2)))*log(8*(a^2*b^2*sqrt((a + b)/(a
^3*b^4)) - a)*sqrt(-sqrt((2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) + 2*a + b)/(a^2*b^2))) + 8*sqrt(a*x + sqrt(a^2*x^2
+ b))) - 3*a*b*sqrt(-sqrt((2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) + 2*a + b)/(a^2*b^2)))*log(-8*(a^2*b^2*sqrt((a +
b)/(a^3*b^4)) - a)*sqrt(-sqrt((2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) + 2*a + b)/(a^2*b^2))) + 8*sqrt(a*x + sqrt(a
^2*x^2 + b))) - 3*a*b*sqrt(-sqrt(-(2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) - 2*a - b)/(a^2*b^2)))*log(8*(a^2*b^2*sqr
t((a + b)/(a^3*b^4)) + a)*sqrt(-sqrt(-(2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) - 2*a - b)/(a^2*b^2))) + 8*sqrt(a*x +
sqrt(a^2*x^2 + b))) + 3*a*b*sqrt(-sqrt(-(2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) - 2*a - b)/(a^2*b^2)))*log(-8*(a^2
*b^2*sqrt((a + b)/(a^3*b^4)) + a)*sqrt(-sqrt(-(2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) - 2*a - b)/(a^2*b^2))) + 8*sq
rt(a*x + sqrt(a^2*x^2 + b))) + 3*a*b*((2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) + 2*a + b)/(a^2*b^2))^(1/4)*log(8*(a^
2*b^2*sqrt((a + b)/(a^3*b^4)) - a)*((2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) + 2*a + b)/(a^2*b^2))^(1/4) + 8*sqrt(a*
x + sqrt(a^2*x^2 + b))) - 3*a*b*((2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) + 2*a + b)/(a^2*b^2))^(1/4)*log(-8*(a^2*b^
2*sqrt((a + b)/(a^3*b^4)) - a)*((2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) + 2*a + b)/(a^2*b^2))^(1/4) + 8*sqrt(a*x +
sqrt(a^2*x^2 + b))) - 3*a*b*(-(2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) - 2*a - b)/(a^2*b^2))^(1/4)*log(8*(a^2*b^2*sq
rt((a + b)/(a^3*b^4)) + a)*(-(2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) - 2*a - b)/(a^2*b^2))^(1/4) + 8*sqrt(a*x + sqr
t(a^2*x^2 + b))) + 3*a*b*(-(2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) - 2*a - b)/(a^2*b^2))^(1/4)*log(-8*(a^2*b^2*sqrt
((a + b)/(a^3*b^4)) + a)*(-(2*a^2*b^2*sqrt((a + b)/(a^3*b^4)) - 2*a - b)/(a^2*b^2))^(1/4) + 8*sqrt(a*x + sqrt(
a^2*x^2 + b))) - 2*(a^2*x^2 - sqrt(a^2*x^2 + b)*a*x - b)*sqrt(a*x + sqrt(a^2*x^2 + b)))/(a*b)

Sympy [N/A]

Not integrable

Time = 2.78 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.19 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\int \frac {a x^{2} + 1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b}} \left (a x^{2} - 1\right )}\, dx \]

[In]

integrate((a*x**2+1)/(a*x**2-1)/(a*x+(a**2*x**2+b)**(1/2))**(1/2),x)

[Out]

Integral((a*x**2 + 1)/(sqrt(a*x + sqrt(a**2*x**2 + b))*(a*x**2 - 1)), x)

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.22 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\int { \frac {a x^{2} + 1}{{\left (a x^{2} - 1\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} \,d x } \]

[In]

integrate((a*x^2+1)/(a*x^2-1)/(a*x+(a^2*x^2+b)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate((a*x^2 + 1)/((a*x^2 - 1)*sqrt(a*x + sqrt(a^2*x^2 + b))), x)

Giac [N/A]

Not integrable

Time = 1.69 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.22 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\int { \frac {a x^{2} + 1}{{\left (a x^{2} - 1\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} \,d x } \]

[In]

integrate((a*x^2+1)/(a*x^2-1)/(a*x+(a^2*x^2+b)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate((a*x^2 + 1)/((a*x^2 - 1)*sqrt(a*x + sqrt(a^2*x^2 + b))), x)

Mupad [N/A]

Not integrable

Time = 6.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.22 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\int \frac {a\,x^2+1}{\sqrt {\sqrt {a^2\,x^2+b}+a\,x}\,\left (a\,x^2-1\right )} \,d x \]

[In]

int((a*x^2 + 1)/(((b + a^2*x^2)^(1/2) + a*x)^(1/2)*(a*x^2 - 1)),x)

[Out]

int((a*x^2 + 1)/(((b + a^2*x^2)^(1/2) + a*x)^(1/2)*(a*x^2 - 1)), x)