\(\int \frac {(c+d \sqrt {a+b x^2})^{3/2}}{x} \, dx\) [258]

Optimal result

Integrand size = 23, antiderivative size = 146 \[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx=2 c \sqrt {c+d \sqrt {a+b x^2}}+\frac {2}{3} \left (c+d \sqrt {a+b x^2}\right )^{3/2}-\left (c-\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \sqrt {a+b x^2}}}{\sqrt {c-\sqrt {a} d}}\right )-\left (c+\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \sqrt {a+b x^2}}}{\sqrt {c+\sqrt {a} d}}\right ) \] Output:

2*c*(c+d*(b*x^2+a)^(1/2))^(1/2)+2/3*(c+d*(b*x^2+a)^(1/2))^(3/2)-(c-a^(1/2) 
*d)^(3/2)*arctanh((c+d*(b*x^2+a)^(1/2))^(1/2)/(c-a^(1/2)*d)^(1/2))-(c+a^(1 
/2)*d)^(3/2)*arctanh((c+d*(b*x^2+a)^(1/2))^(1/2)/(c+a^(1/2)*d)^(1/2))
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01 \[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx=\frac {2}{3} \sqrt {c+d \sqrt {a+b x^2}} \left (4 c+d \sqrt {a+b x^2}\right )+\left (-c-\sqrt {a} d\right )^{3/2} \arctan \left (\frac {\sqrt {c+d \sqrt {a+b x^2}}}{\sqrt {-c-\sqrt {a} d}}\right )+\left (-c+\sqrt {a} d\right )^{3/2} \arctan \left (\frac {\sqrt {c+d \sqrt {a+b x^2}}}{\sqrt {-c+\sqrt {a} d}}\right ) \] Input:

Integrate[(c + d*Sqrt[a + b*x^2])^(3/2)/x,x]
 

Output:

(2*Sqrt[c + d*Sqrt[a + b*x^2]]*(4*c + d*Sqrt[a + b*x^2]))/3 + (-c - Sqrt[a 
]*d)^(3/2)*ArcTan[Sqrt[c + d*Sqrt[a + b*x^2]]/Sqrt[-c - Sqrt[a]*d]] + (-c 
+ Sqrt[a]*d)^(3/2)*ArcTan[Sqrt[c + d*Sqrt[a + b*x^2]]/Sqrt[-c + Sqrt[a]*d] 
]
 

Rubi [A] (warning: unable to verify)

Time = 1.02 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {7282, 896, 25, 1732, 561, 25, 27, 1602, 27, 25, 1602, 25, 25, 27, 1480, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (d \sqrt {a+b x^2}+c\right )^{3/2}}{x} \, dx\)

\(\Big \downarrow \) 7282

\(\displaystyle \frac {1}{2} \int \frac {\left (c+d \sqrt {b x^2+a}\right )^{3/2}}{x^2}dx^2\)

\(\Big \downarrow \) 896

\(\displaystyle \frac {1}{2} \int \frac {\left (c+d \sqrt {b x^2+a}\right )^{3/2}}{b x^2}d\left (b x^2+a\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{2} \int -\frac {\left (c+d \sqrt {b x^2+a}\right )^{3/2}}{b x^2}d\left (b x^2+a\right )\)

\(\Big \downarrow \) 1732

\(\displaystyle -\int \frac {\sqrt {b x^2+a} \left (c+d \sqrt {b x^2+a}\right )^{3/2}}{a-x^4}d\sqrt {b x^2+a}\)

\(\Big \downarrow \) 561

\(\displaystyle -\frac {2 \int -\frac {x^8 \left (c-x^4\right )}{d \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {b x^2+a}}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \int \frac {x^8 \left (c-x^4\right )}{d \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {b x^2+a}}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {x^8 \left (c-x^4\right )}{-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}}d\sqrt {c+d \sqrt {b x^2+a}}}{d^2}\)

\(\Big \downarrow \) 1602

\(\displaystyle \frac {2 \left (\frac {1}{3} d^2 \int -\frac {3 x^4 \left (c x^4+\left (a-\frac {c^2}{d^2}\right ) d^2\right )}{d^2 \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {b x^2+a}}+\frac {d^2 x^6}{3}\right )}{d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {d^2 x^6}{3}-\int -\frac {x^4 \left (-c x^4+c^2-a d^2\right )}{-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}}d\sqrt {c+d \sqrt {b x^2+a}}\right )}{d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (\int \frac {x^4 \left (-c x^4+c^2-a d^2\right )}{-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}}d\sqrt {c+d \sqrt {b x^2+a}}+\frac {d^2 x^6}{3}\right )}{d^2}\)

\(\Big \downarrow \) 1602

\(\displaystyle \frac {2 \left (d^2 \int -\frac {\left (c^2+a d^2\right ) x^4+c \left (a-\frac {c^2}{d^2}\right ) d^2}{d^2 \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {b x^2+a}}+c d^2 \sqrt {d \sqrt {a+b x^2}+c}+\frac {d^2 x^6}{3}\right )}{d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (-d^2 \int -\frac {c \left (c^2-a d^2\right )-\left (c^2+a d^2\right ) x^4}{d^2 \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {b x^2+a}}+c d^2 \sqrt {d \sqrt {a+b x^2}+c}+\frac {d^2 x^6}{3}\right )}{d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (d^2 \int \frac {c \left (c^2-a d^2\right )-\left (c^2+a d^2\right ) x^4}{d^2 \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {b x^2+a}}+c d^2 \sqrt {d \sqrt {a+b x^2}+c}+\frac {d^2 x^6}{3}\right )}{d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\int \frac {c \left (c^2-a d^2\right )-\left (c^2+a d^2\right ) x^4}{-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}}d\sqrt {c+d \sqrt {b x^2+a}}+c d^2 \sqrt {d \sqrt {a+b x^2}+c}+\frac {d^2 x^6}{3}\right )}{d^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {2 \left (-\frac {1}{2} \left (c-\sqrt {a} d\right )^2 \int \frac {1}{\frac {c-\sqrt {a} d}{d^2}-\frac {x^4}{d^2}}d\sqrt {c+d \sqrt {b x^2+a}}-\frac {1}{2} \left (\sqrt {a} d+c\right )^2 \int \frac {1}{\frac {c+\sqrt {a} d}{d^2}-\frac {x^4}{d^2}}d\sqrt {c+d \sqrt {b x^2+a}}+c d^2 \sqrt {d \sqrt {a+b x^2}+c}+\frac {d^2 x^6}{3}\right )}{d^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \left (-\frac {1}{2} d^2 \left (c-\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {d \sqrt {a+b x^2}+c}}{\sqrt {c-\sqrt {a} d}}\right )-\frac {1}{2} d^2 \left (\sqrt {a} d+c\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {d \sqrt {a+b x^2}+c}}{\sqrt {\sqrt {a} d+c}}\right )+c d^2 \sqrt {d \sqrt {a+b x^2}+c}+\frac {d^2 x^6}{3}\right )}{d^2}\)

Input:

Int[(c + d*Sqrt[a + b*x^2])^(3/2)/x,x]
 

Output:

(2*((d^2*x^6)/3 + c*d^2*Sqrt[c + d*Sqrt[a + b*x^2]] - (d^2*(c - Sqrt[a]*d) 
^(3/2)*ArcTanh[Sqrt[c + d*Sqrt[a + b*x^2]]/Sqrt[c - Sqrt[a]*d]])/2 - (d^2* 
(c + Sqrt[a]*d)^(3/2)*ArcTanh[Sqrt[c + d*Sqrt[a + b*x^2]]/Sqrt[c + Sqrt[a] 
*d]])/2))/d^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

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 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(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 1602

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 
1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3))   Int[(f*x)^(m - 2)* 
(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p 
+ 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c 
, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | 
| IntegerQ[m])
 

rule 1732

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb 
ol] :> With[{g = Denominator[n]}, Simp[g   Subst[Int[x^(g - 1)*(d + e*x^(g* 
n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p, q} 
, x] && EqQ[n2, 2*n] && FractionQ[n]
 

rule 7282

Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 
/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( 
lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ 
u] &&  !RationalFunctionQ[u, x]
 

Maple [F]

\[\int \frac {\left (c +d \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{x}d x\]

Input:

int((c+d*(b*x^2+a)^(1/2))^(3/2)/x,x)
 

Output:

int((c+d*(b*x^2+a)^(1/2))^(3/2)/x,x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx=\int \frac {\left (c + d \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{x}\, dx \] Input:

integrate((c+d*(b*x**2+a)**(1/2))**(3/2)/x,x)
 

Output:

Integral((c + d*sqrt(a + b*x**2))**(3/2)/x, x)
 

Maxima [F]

\[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx=\int { \frac {{\left (\sqrt {b x^{2} + a} d + c\right )}^{\frac {3}{2}}}{x} \,d x } \] Input:

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

Output:

integrate((sqrt(b*x^2 + a)*d + c)^(3/2)/x, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (112) = 224\).

Time = 0.16 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.88 \[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx=\frac {2 \, {\left (\sqrt {b x^{2} + a} d + c\right )}^{\frac {3}{2}} d \mathrm {sgn}\left ({\left (\sqrt {b x^{2} + a} d + c\right )} d - c d\right ) - 6 \, \sqrt {\sqrt {b x^{2} + a} d + c} c d \mathrm {sgn}\left ({\left (\sqrt {b x^{2} + a} d + c\right )} d - c d\right ) + 12 \, \sqrt {\sqrt {b x^{2} + a} d + c} c d - \frac {3 \, {\left (2 \, a c d^{3} {\left | d \right |} + 2 \, \sqrt {a} c^{2} d^{3} - {\left (a^{\frac {3}{2}} d^{3} + \sqrt {a} c^{2} d\right )} d^{2} \mathrm {sgn}\left ({\left (\sqrt {b x^{2} + a} d + c\right )} d - c d\right ) - {\left (a c d^{3} + c^{3} d\right )} {\left | d \right |} \mathrm {sgn}\left ({\left (\sqrt {b x^{2} + a} d + c\right )} d - c d\right )\right )} \arctan \left (\frac {\sqrt {\sqrt {b x^{2} + a} d + c}}{\sqrt {-c + \sqrt {a d^{2}}}}\right )}{{\left (\sqrt {a} d + c\right )} \sqrt {\sqrt {a} d - c} {\left | d \right |}} + \frac {3 \, {\left (2 \, a c d^{3} {\left | d \right |} - 2 \, \sqrt {a} c^{2} d^{3} + {\left (a^{\frac {3}{2}} d^{3} + \sqrt {a} c^{2} d\right )} d^{2} \mathrm {sgn}\left ({\left (\sqrt {b x^{2} + a} d + c\right )} d - c d\right ) - {\left (a c d^{3} + c^{3} d\right )} {\left | d \right |} \mathrm {sgn}\left ({\left (\sqrt {b x^{2} + a} d + c\right )} d - c d\right )\right )} \arctan \left (\frac {\sqrt {\sqrt {b x^{2} + a} d + c}}{\sqrt {-c - \sqrt {a d^{2}}}}\right )}{{\left (\sqrt {a} d - c\right )} \sqrt {-\sqrt {a} d - c} {\left | d \right |}}}{3 \, d} \] Input:

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

Output:

1/3*(2*(sqrt(b*x^2 + a)*d + c)^(3/2)*d*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d 
) - 6*sqrt(sqrt(b*x^2 + a)*d + c)*c*d*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) 
 + 12*sqrt(sqrt(b*x^2 + a)*d + c)*c*d - 3*(2*a*c*d^3*abs(d) + 2*sqrt(a)*c^ 
2*d^3 - (a^(3/2)*d^3 + sqrt(a)*c^2*d)*d^2*sgn((sqrt(b*x^2 + a)*d + c)*d - 
c*d) - (a*c*d^3 + c^3*d)*abs(d)*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d))*arct 
an(sqrt(sqrt(b*x^2 + a)*d + c)/sqrt(-c + sqrt(a*d^2)))/((sqrt(a)*d + c)*sq 
rt(sqrt(a)*d - c)*abs(d)) + 3*(2*a*c*d^3*abs(d) - 2*sqrt(a)*c^2*d^3 + (a^( 
3/2)*d^3 + sqrt(a)*c^2*d)*d^2*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) - (a*c* 
d^3 + c^3*d)*abs(d)*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d))*arctan(sqrt(sqrt 
(b*x^2 + a)*d + c)/sqrt(-c - sqrt(a*d^2)))/((sqrt(a)*d - c)*sqrt(-sqrt(a)* 
d - c)*abs(d)))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx=\int \frac {{\left (c+d\,\sqrt {b\,x^2+a}\right )}^{3/2}}{x} \,d x \] Input:

int((c + d*(a + b*x^2)^(1/2))^(3/2)/x,x)
 

Output:

int((c + d*(a + b*x^2)^(1/2))^(3/2)/x, x)
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 924, normalized size of antiderivative = 6.33 \[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx =\text {Too large to display} \] Input:

int((c+d*(b*x^2+a)^(1/2))^(3/2)/x,x)
 

Output:

( - 3*sqrt(a)*sqrt(sqrt(a)*d - c)*atan((sqrt(a + b*x**2)*sqrt(sqrt(a)*d - 
c)*sqrt(sqrt(a + b*x**2)*d + c)*a*d**3 - sqrt(a + b*x**2)*sqrt(sqrt(a)*d - 
 c)*sqrt(sqrt(a + b*x**2)*d + c)*c**2*d - sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt 
(sqrt(a + b*x**2)*d + c)*a*d**3 - sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a 
+ b*x**2)*d + c)*b*d**3*x**2 + sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b 
*x**2)*d + c)*c**2*d - 2*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)* 
a*c*d**2 - sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*b*c*d**2*x**2 
+ 2*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*c**3)/(2*a**2*d**4 + 
2*a*b*d**4*x**2 - 4*a*c**2*d**2 - 2*b*c**2*d**2*x**2 + 2*c**4))*a*d + 3*sq 
rt(sqrt(a)*d - c)*atan((sqrt(a + b*x**2)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + 
 b*x**2)*d + c)*a*d**3 - sqrt(a + b*x**2)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a 
+ b*x**2)*d + c)*c**2*d - sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2 
)*d + c)*a*d**3 - sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c) 
*b*d**3*x**2 + sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*c* 
*2*d - 2*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*a*c*d**2 - sqrt( 
sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*b*c*d**2*x**2 + 2*sqrt(sqrt(a) 
*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*c**3)/(2*a**2*d**4 + 2*a*b*d**4*x**2 
- 4*a*c**2*d**2 - 2*b*c**2*d**2*x**2 + 2*c**4))*a*c + 16*sqrt(a + b*x**2)* 
sqrt(sqrt(b)*sqrt(a + b*x**2)*d*x + sqrt(a + b*x**2)*c + sqrt(b)*c*x + a*d 
 + b*d*x**2)*sqrt(sqrt(a + b*x**2) + sqrt(b)*x)*c - 16*sqrt(b)*sqrt(sqr...