\(\int \frac {1}{\sqrt {a+b x^2} (c+d x^2) (e+f x^2)^2} \, dx\) [332]

Optimal result

Integrand size = 30, antiderivative size = 204 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {f^2 x \sqrt {a+b x^2}}{2 e (b e-a f) (d e-c f) \left (e+f x^2\right )}+\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f (2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{3/2} (d e-c f)^2} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(498\) vs. \(2(204)=408\).

Time = 12.79 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.44 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {-\frac {2 \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {c} \sqrt {-b c+a d}}-\frac {2 c^{3/2} f^2 \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {-b c+a d} (d e-c f)^2}+\frac {2 c \sqrt {e} f^2 \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {-b e+a f} (d e-c f)^2}+\frac {f \left (\frac {e \left (a f x \sqrt {a+b x^2}+a \sqrt {b} \left (e-f x^2\right )+2 b e x \left (\sqrt {b} x-\sqrt {a+b x^2}\right )\right )}{(-b e+a f) \left (e+f x^2\right ) \left (a+2 b x^2-2 \sqrt {b} x \sqrt {a+b x^2}\right )}+\frac {4 \sqrt {c} \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {-b c+a d}}+\frac {\sqrt {e} (4 b e-3 a f) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{(-b e+a f)^{3/2}}\right )}{-d e+c f}}{2 e^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {421, 291, 221, 402, 27, 291, 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.

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b e (2 d e-c f)-a f (3 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(d^2*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*Sqrt 
[b*c - a*d]*(d*e - c*f)^2) - (f*(-1/2*(f*(d*e - c*f)*x*Sqrt[a + b*x^2])/(e 
*(b*e - a*f)*(e + f*x^2)) + ((2*b*e*(2*d*e - c*f) - a*f*(3*d*e - c*f))*Arc 
Tanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f 
)^(3/2))))/(d*e - c*f)^2
 

Defintions of rubi rules used

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 291

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

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

rule 421

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2   Int[(c + d*x^2)^(q + 2)*((e + 
 f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2   Int[(c + d*x^2)^q*( 
e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} 
, x] && LtQ[q, -1]
 

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\frac {-\left (4 b d \,e^{2}+f \left (-3 a d -2 b c \right ) e +a c \,f^{2}\right ) \left (f \,x^{2}+e \right ) f \sqrt {\left (a d -b c \right ) c}\, \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\left (-2 d^{2} e \left (f \,x^{2}+e \right ) \left (a f -b e \right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\left (c f -d e \right ) \sqrt {b \,x^{2}+a}\, x \,f^{2} \sqrt {\left (a d -b c \right ) c}\right ) \sqrt {\left (a f -b e \right ) e}}{2 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (c f -d e \right )^{2} \left (a f -b e \right ) e \left (f \,x^{2}+e \right )}\)	\(231\)
default	\(\text {Expression too large to display}\)	\(1380\)

Input:

int(1/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
 

Output:

1/2/((a*d-b*c)*c)^(1/2)*(-(4*b*d*e^2+f*(-3*a*d-2*b*c)*e+a*c*f^2)*(f*x^2+e) 
*f*((a*d-b*c)*c)^(1/2)*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+(-2 
*d^2*e*(f*x^2+e)*(a*f-b*e)*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2)) 
+(c*f-d*e)*(b*x^2+a)^(1/2)*x*f^2*((a*d-b*c)*c)^(1/2))*((a*f-b*e)*e)^(1/2)) 
/((a*f-b*e)*e)^(1/2)/(c*f-d*e)^2/(a*f-b*e)/e/(f*x^2+e)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\int \frac {1}{\sqrt {a + b x^{2}} \left (c + d x^{2}\right ) \left (e + f x^{2}\right )^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (178) = 356\).

Time = 0.84 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.25 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=-\frac {1}{2} \, {\left (\frac {2 \, d^{2} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{{\left (b^{2} d^{2} e^{2} - 2 \, b^{2} c d e f + b^{2} c^{2} f^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} - \frac {{\left (4 \, b d e^{2} f - 2 \, b c e f^{2} - 3 \, a d e f^{2} + a c f^{3}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{{\left (b^{3} d^{2} e^{4} - 2 \, b^{3} c d e^{3} f - a b^{2} d^{2} e^{3} f + b^{3} c^{2} e^{2} f^{2} + 2 \, a b^{2} c d e^{2} f^{2} - a b^{2} c^{2} e f^{3}\right )} \sqrt {-b^{2} e^{2} + a b e f}} - \frac {2 \, {\left (2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b e f - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a f^{2} + a^{2} f^{2}\right )}}{{\left (b^{3} d e^{3} - b^{3} c e^{2} f - a b^{2} d e^{2} f + a b^{2} c e f^{2}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} f + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b e - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a f + a^{2} f\right )}}\right )} b^{\frac {5}{2}} \] Input:

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

Output:

-1/2*(2*d^2*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/s 
qrt(-b^2*c^2 + a*b*c*d))/((b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*sqrt 
(-b^2*c^2 + a*b*c*d)) - (4*b*d*e^2*f - 2*b*c*e*f^2 - 3*a*d*e*f^2 + a*c*f^3 
)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e 
^2 + a*b*e*f))/((b^3*d^2*e^4 - 2*b^3*c*d*e^3*f - a*b^2*d^2*e^3*f + b^3*c^2 
*e^2*f^2 + 2*a*b^2*c*d*e^2*f^2 - a*b^2*c^2*e*f^3)*sqrt(-b^2*e^2 + a*b*e*f) 
) - 2*(2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*e*f - (sqrt(b)*x - sqrt(b*x^2 + 
 a))^2*a*f^2 + a^2*f^2)/((b^3*d*e^3 - b^3*c*e^2*f - a*b^2*d*e^2*f + a*b^2* 
c*e*f^2)*((sqrt(b)*x - sqrt(b*x^2 + a))^4*f + 4*(sqrt(b)*x - sqrt(b*x^2 + 
a))^2*b*e - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*f + a^2*f)))*b^(5/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\int \frac {1}{\sqrt {b\,x^2+a}\,\left (d\,x^2+c\right )\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 1.49 (sec) , antiderivative size = 3033, normalized size of antiderivative = 14.87 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x 
**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**2*d**2*e**3*f**2 - 2*sqrt( 
c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt 
(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**2*d**2*e**2*f**3*x**2 + 4*sqrt(c)*sqr 
t(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sq 
rt(b)*x)/(sqrt(c)*sqrt(b)))*a*b*d**2*e**4*f + 4*sqrt(c)*sqrt(a*d - b*c)*at 
an((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt( 
c)*sqrt(b)))*a*b*d**2*e**3*f**2*x**2 - 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqr 
t(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt 
(b)))*b**2*d**2*e**5 - 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - s 
qrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*b**2*d**2* 
e**4*f*x**2 - 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sq 
rt(a + b*x**2) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**2*d**2*e**3*f**2 
 - 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x* 
*2) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**2*d**2*e**2*f**3*x**2 + 4*s 
qrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x**2) + 
sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*b*d**2*e**4*f + 4*sqrt(c)*sqrt(a*d 
 - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x**2) + sqrt(d)*sqrt(b) 
*x)/(sqrt(c)*sqrt(b)))*a*b*d**2*e**3*f**2*x**2 - 2*sqrt(c)*sqrt(a*d - b*c) 
*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x**2) + sqrt(d)*sqrt(b)*x)/...