3.20.87 \(\int \frac {1}{(d+c x^2) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [1987]

Optimal. Leaf size=140 \[ \frac {1}{2} a \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\& ,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}^3-2 a^2 d \text {$\#$1}^3-c \text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1309\) vs. \(2(140)=280\).
time = 3.37, antiderivative size = 1309, normalized size of antiderivative = 9.35, number of steps used = 30, number of rules used = 10, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {6857, 2144, 1642, 842, 840, 1183, 648, 632, 210, 642} \begin {gather*} \frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \text {ArcTan}\left (\frac {\sqrt {-c} \left (\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}}-\sqrt {2} \sqrt [4]{-c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \text {ArcTan}\left (\frac {\sqrt {-c} \left (\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}}+\sqrt {2} \sqrt [4]{-c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \text {ArcTan}\left (\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}-\sqrt {2} (-c)^{3/4} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {-c} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}} (-c)^{3/4}+\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{\sqrt {-c} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}-\frac {\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \log \left (\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {2} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} \sqrt [4]{-c}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \log \left (\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )+\sqrt {2} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} \sqrt [4]{-c}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \log \left ((-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} (-c)^{3/4}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \log \left ((-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )+\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} (-c)^{3/4}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]]*ArcTan[(Sqrt[-c]*(Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]] - Sqrt[2]*(
-c)^(1/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]))/Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]]])/(Sqrt[2]*Sqrt[-b^2]*(-
c)^(5/4)*Sqrt[d]) - (Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]]*ArcTan[(Sqrt[-c]*(Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*
Sqrt[d]] + Sqrt[2]*(-c)^(1/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]))/Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]]])/(S
qrt[2]*Sqrt[-b^2]*(-c)^(5/4)*Sqrt[d]) - (Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]]*ArcTan[(Sqrt[Sqrt[-b^2]*(-c)^(3
/2) + a*c*Sqrt[d]] - Sqrt[2]*(-c)^(3/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(Sqrt[-c]*Sqrt[Sqrt[-b^2]*Sqrt[-c] +
a*Sqrt[d]])])/(Sqrt[2]*Sqrt[-b^2]*(-c)^(3/4)*Sqrt[d]) + (Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]]*ArcTan[(Sqrt[Sq
rt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]] + Sqrt[2]*(-c)^(3/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(Sqrt[-c]*Sqrt[Sqrt[-
b^2]*Sqrt[-c] + a*Sqrt[d]])])/(Sqrt[2]*Sqrt[-b^2]*(-c)^(3/4)*Sqrt[d]) - (Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]]
*Log[Sqrt[-b^2]*(-c)^(1/4) - Sqrt[2]*Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] + (
-c)^(1/4)*(a*x + Sqrt[b^2 + a^2*x^2])])/(2*Sqrt[2]*Sqrt[-b^2]*(-c)^(3/4)*Sqrt[d]) + (Sqrt[Sqrt[-b^2]*Sqrt[-c]
+ a*Sqrt[d]]*Log[Sqrt[-b^2]*(-c)^(1/4) + Sqrt[2]*Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]]*Sqrt[a*x + Sqrt[b^2 + a
^2*x^2]] + (-c)^(1/4)*(a*x + Sqrt[b^2 + a^2*x^2])])/(2*Sqrt[2]*Sqrt[-b^2]*(-c)^(3/4)*Sqrt[d]) + (Sqrt[Sqrt[-b^
2]*(-c)^(3/2) + a*c*Sqrt[d]]*Log[Sqrt[-b^2]*(-c)^(3/4) - Sqrt[2]*Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]]*Sqr
t[a*x + Sqrt[b^2 + a^2*x^2]] + (-c)^(3/4)*(a*x + Sqrt[b^2 + a^2*x^2])])/(2*Sqrt[2]*Sqrt[-b^2]*(-c)^(5/4)*Sqrt[
d]) - (Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]]*Log[Sqrt[-b^2]*(-c)^(3/4) + Sqrt[2]*Sqrt[Sqrt[-b^2]*(-c)^(3/2
) + a*c*Sqrt[d]]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] + (-c)^(3/4)*(a*x + Sqrt[b^2 + a^2*x^2])])/(2*Sqrt[2]*Sqrt[-b
^2]*(-c)^(5/4)*Sqrt[d])

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 648

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

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 1183

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[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 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*} \int \frac {1}{\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\int \left (\frac {1}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {1}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx\\ &=\frac {\int \frac {1}{\left (\sqrt {d}-\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx}{2 \sqrt {d}}+\frac {\int \frac {1}{\left (\sqrt {d}+\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx}{2 \sqrt {d}}\\ &=\frac {\text {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (b^2 \sqrt {-c}+2 a \sqrt {d} x-\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 \sqrt {d}}+\frac {\text {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (-b^2 \sqrt {-c}+2 a \sqrt {d} x+\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 \sqrt {d}}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{\sqrt {-c} x^{3/2}}+\frac {2 \left (b^2 c-a \sqrt {-c} \sqrt {d} x\right )}{c x^{3/2} \left (b^2 \sqrt {-c}+2 a \sqrt {d} x-\sqrt {-c} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 \sqrt {d}}+\frac {\text {Subst}\left (\int \left (\frac {1}{\sqrt {-c} x^{3/2}}+\frac {2 \left (b^2 c+a \sqrt {-c} \sqrt {d} x\right )}{c x^{3/2} \left (-b^2 \sqrt {-c}+2 a \sqrt {d} x+\sqrt {-c} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 \sqrt {d}}\\ &=\frac {\text {Subst}\left (\int \frac {b^2 c-a \sqrt {-c} \sqrt {d} x}{x^{3/2} \left (b^2 \sqrt {-c}+2 a \sqrt {d} x-\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{c \sqrt {d}}+\frac {\text {Subst}\left (\int \frac {b^2 c+a \sqrt {-c} \sqrt {d} x}{x^{3/2} \left (-b^2 \sqrt {-c}+2 a \sqrt {d} x+\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{c \sqrt {d}}\\ &=-\frac {\text {Subst}\left (\int \frac {-a b^2 c \sqrt {d}-b^2 (-c)^{3/2} x}{\sqrt {x} \left (b^2 \sqrt {-c}+2 a \sqrt {d} x-\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{b^2 (-c)^{3/2} \sqrt {d}}+\frac {\text {Subst}\left (\int \frac {-a b^2 c \sqrt {d}+b^2 (-c)^{3/2} x}{\sqrt {x} \left (-b^2 \sqrt {-c}+2 a \sqrt {d} x+\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{b^2 (-c)^{3/2} \sqrt {d}}\\ &=-\frac {2 \text {Subst}\left (\int \frac {-a b^2 c \sqrt {d}-b^2 (-c)^{3/2} x^2}{b^2 \sqrt {-c}+2 a \sqrt {d} x^2-\sqrt {-c} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 (-c)^{3/2} \sqrt {d}}+\frac {2 \text {Subst}\left (\int \frac {-a b^2 c \sqrt {d}+b^2 (-c)^{3/2} x^2}{-b^2 \sqrt {-c}+2 a \sqrt {d} x^2+\sqrt {-c} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 (-c)^{3/2} \sqrt {d}}\\ &=-\frac {\text {Subst}\left (\int \frac {-\frac {\sqrt {2} a b^2 c \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}}{\sqrt [4]{-c}}-\left (b^2 \sqrt {-b^2} (-c)^{3/2}-a b^2 c \sqrt {d}\right ) x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \left (-b^2\right )^{3/2} (-c)^{7/4} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}}-\frac {\text {Subst}\left (\int \frac {-\frac {\sqrt {2} a b^2 c \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}}{\sqrt [4]{-c}}+\left (b^2 \sqrt {-b^2} (-c)^{3/2}-a b^2 c \sqrt {d}\right ) x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \left (-b^2\right )^{3/2} (-c)^{7/4} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}}-\frac {\text {Subst}\left (\int \frac {-\frac {\sqrt {2} a b^2 c \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}}{(-c)^{3/4}}-\left (-b^2 \sqrt {-b^2} (-c)^{3/2}-a b^2 c \sqrt {d}\right ) x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \left (-b^2\right )^{3/2} (-c)^{5/4} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}}-\frac {\text {Subst}\left (\int \frac {-\frac {\sqrt {2} a b^2 c \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}}{(-c)^{3/4}}+\left (-b^2 \sqrt {-b^2} (-c)^{3/2}-a b^2 c \sqrt {d}\right ) x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \left (-b^2\right )^{3/2} (-c)^{5/4} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}}\\ &=-\frac {\left (\frac {a}{\sqrt {-b^2}}-\frac {\sqrt {-c}}{\sqrt {d}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 c}-\frac {\left (\frac {a}{\sqrt {-b^2}}-\frac {\sqrt {-c}}{\sqrt {d}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 c}-\frac {\left (\frac {a}{\sqrt {-b^2}}+\frac {\sqrt {-c}}{\sqrt {d}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 c}-\frac {\left (\frac {a}{\sqrt {-b^2}}+\frac {\sqrt {-c}}{\sqrt {d}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 c}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+2 x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+2 x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+2 x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+2 x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}\\ &=-\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \log \left (\sqrt {-b^2} \sqrt [4]{-c}-\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \log \left (\sqrt {-b^2} \sqrt [4]{-c}+\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \log \left (\sqrt {-b^2} (-c)^{3/4}-\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \log \left (\sqrt {-b^2} (-c)^{3/4}+\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}+\frac {\left (\frac {a}{\sqrt {-b^2}}-\frac {\sqrt {-c}}{\sqrt {d}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 \left (\sqrt {-b^2} c+a \sqrt {-c} \sqrt {d}\right )}{c}-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{c}+\frac {\left (\frac {a}{\sqrt {-b^2}}-\frac {\sqrt {-c}}{\sqrt {d}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 \left (\sqrt {-b^2} c+a \sqrt {-c} \sqrt {d}\right )}{c}-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{c}+\frac {\left (\frac {a}{\sqrt {-b^2}}+\frac {\sqrt {-c}}{\sqrt {d}}\right ) \text {Subst}\left (\int \frac {1}{-2 \left (\sqrt {-b^2}+\frac {a \sqrt {d}}{\sqrt {-c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{c}+\frac {\left (\frac {a}{\sqrt {-b^2}}+\frac {\sqrt {-c}}{\sqrt {d}}\right ) \text {Subst}\left (\int \frac {1}{-2 \left (\sqrt {-b^2}+\frac {a \sqrt {d}}{\sqrt {-c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{c}\\ &=\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \tan ^{-1}\left (\frac {(-c)^{3/4} \left (\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}-\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \tan ^{-1}\left (\frac {\sqrt [4]{-c} \left (\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}-\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \tan ^{-1}\left (\frac {(-c)^{3/4} \left (\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \tan ^{-1}\left (\frac {\sqrt [4]{-c} \left (\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \log \left (\sqrt {-b^2} \sqrt [4]{-c}-\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \log \left (\sqrt {-b^2} \sqrt [4]{-c}+\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \log \left (\sqrt {-b^2} (-c)^{3/4}-\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \log \left (\sqrt {-b^2} (-c)^{3/4}+\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 140, normalized size = 1.00 \begin {gather*} \frac {1}{2} a \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}^3-2 a^2 d \text {$\#$1}^3-c \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c \,x^{2}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.65, size = 1800, normalized size = 12.86 \begin {gather*} -2 \, \left (-\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - b^{2} c + 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left ({\left (b^{4} c^{4} d^{3} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - a^{2} c d^{2}\right )} \sqrt {a^{3} x + \sqrt {a^{2} x^{2} + b^{2}} a^{2} - {\left (2 \, a^{2} b^{4} c^{3} d^{3} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + a^{2} b^{2} c d - 2 \, a^{4} d^{2}\right )} \sqrt {-\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - b^{2} c + 2 \, a^{2} d}{b^{4} c^{3} d^{2}}}} \sqrt {-\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - b^{2} c + 2 \, a^{2} d}{b^{4} c^{3} d^{2}}} - {\left (a b^{4} c^{4} d^{3} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - a^{3} c d^{2}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \sqrt {-\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - b^{2} c + 2 \, a^{2} d}{b^{4} c^{3} d^{2}}}\right )} \left (-\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - b^{2} c + 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {1}{4}}}{a^{2}}\right ) + 2 \, \left (\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + b^{2} c - 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (b^{4} c^{4} d^{3} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + a^{2} c d^{2}\right )} \sqrt {a^{3} x + \sqrt {a^{2} x^{2} + b^{2}} a^{2} + {\left (2 \, a^{2} b^{4} c^{3} d^{3} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - a^{2} b^{2} c d + 2 \, a^{4} d^{2}\right )} \sqrt {\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + b^{2} c - 2 \, a^{2} d}{b^{4} c^{3} d^{2}}}} \left (\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + b^{2} c - 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {3}{4}} - {\left (a b^{4} c^{4} d^{3} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + a^{3} c d^{2}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + b^{2} c - 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {3}{4}}}{a^{2}}\right ) + \frac {1}{2} \, \left (\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + b^{2} c - 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {1}{4}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a + {\left (b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + a^{2} d\right )} \left (\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + b^{2} c - 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {1}{4}}\right ) - \frac {1}{2} \, \left (\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + b^{2} c - 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {1}{4}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a - {\left (b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + a^{2} d\right )} \left (\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} + b^{2} c - 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {1}{4}}\right ) - \frac {1}{2} \, \left (-\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - b^{2} c + 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {1}{4}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a + {\left (b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - a^{2} d\right )} \left (-\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - b^{2} c + 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {1}{4}}\right ) + \frac {1}{2} \, \left (-\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - b^{2} c + 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {1}{4}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a - {\left (b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - a^{2} d\right )} \left (-\frac {2 \, b^{4} c^{3} d^{2} \sqrt {-\frac {a^{2} b^{2} c - a^{4} d}{b^{8} c^{6} d^{3}}} - b^{2} c + 2 \, a^{2} d}{b^{4} c^{3} d^{2}}\right )^{\frac {1}{4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(-(2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) - b^2*c + 2*a^2*d)/(b^4*c^3*d^2))^(1/4)*arctan(((
b^4*c^4*d^3*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) - a^2*c*d^2)*sqrt(a^3*x + sqrt(a^2*x^2 + b^2)*a^2 - (2*a^
2*b^4*c^3*d^3*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) + a^2*b^2*c*d - 2*a^4*d^2)*sqrt(-(2*b^4*c^3*d^2*sqrt(-(
a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) - b^2*c + 2*a^2*d)/(b^4*c^3*d^2)))*sqrt(-(2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a
^4*d)/(b^8*c^6*d^3)) - b^2*c + 2*a^2*d)/(b^4*c^3*d^2)) - (a*b^4*c^4*d^3*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3
)) - a^3*c*d^2)*sqrt(a*x + sqrt(a^2*x^2 + b^2))*sqrt(-(2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3))
- b^2*c + 2*a^2*d)/(b^4*c^3*d^2)))*(-(2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) - b^2*c + 2*a^2*d
)/(b^4*c^3*d^2))^(1/4)/a^2) + 2*((2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) + b^2*c - 2*a^2*d)/(b
^4*c^3*d^2))^(1/4)*arctan(((b^4*c^4*d^3*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) + a^2*c*d^2)*sqrt(a^3*x + sqr
t(a^2*x^2 + b^2)*a^2 + (2*a^2*b^4*c^3*d^3*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) - a^2*b^2*c*d + 2*a^4*d^2)*
sqrt((2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) + b^2*c - 2*a^2*d)/(b^4*c^3*d^2)))*((2*b^4*c^3*d^
2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) + b^2*c - 2*a^2*d)/(b^4*c^3*d^2))^(3/4) - (a*b^4*c^4*d^3*sqrt(-(a^2
*b^2*c - a^4*d)/(b^8*c^6*d^3)) + a^3*c*d^2)*sqrt(a*x + sqrt(a^2*x^2 + b^2))*((2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c -
a^4*d)/(b^8*c^6*d^3)) + b^2*c - 2*a^2*d)/(b^4*c^3*d^2))^(3/4))/a^2) + 1/2*((2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c -
a^4*d)/(b^8*c^6*d^3)) + b^2*c - 2*a^2*d)/(b^4*c^3*d^2))^(1/4)*log(sqrt(a*x + sqrt(a^2*x^2 + b^2))*a + (b^4*c^3
*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) + a^2*d)*((2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)
) + b^2*c - 2*a^2*d)/(b^4*c^3*d^2))^(1/4)) - 1/2*((2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) + b^
2*c - 2*a^2*d)/(b^4*c^3*d^2))^(1/4)*log(sqrt(a*x + sqrt(a^2*x^2 + b^2))*a - (b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^
4*d)/(b^8*c^6*d^3)) + a^2*d)*((2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) + b^2*c - 2*a^2*d)/(b^4*
c^3*d^2))^(1/4)) - 1/2*(-(2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) - b^2*c + 2*a^2*d)/(b^4*c^3*d
^2))^(1/4)*log(sqrt(a*x + sqrt(a^2*x^2 + b^2))*a + (b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) - a^2
*d)*(-(2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) - b^2*c + 2*a^2*d)/(b^4*c^3*d^2))^(1/4)) + 1/2*(
-(2*b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) - b^2*c + 2*a^2*d)/(b^4*c^3*d^2))^(1/4)*log(sqrt(a*x
+ sqrt(a^2*x^2 + b^2))*a - (b^4*c^3*d^2*sqrt(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) - a^2*d)*(-(2*b^4*c^3*d^2*sqr
t(-(a^2*b^2*c - a^4*d)/(b^8*c^6*d^3)) - b^2*c + 2*a^2*d)/(b^4*c^3*d^2))^(1/4))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (c x^{2} + d\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (c\,x^2+d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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