\(\int \frac {A+B x^2}{(a+b x^2)^{3/2} (c+d x^2) (e+f x^2)^2} \, dx\) [2]

Optimal result

Integrand size = 37, antiderivative size = 356 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=-\frac {b \left (a B e (2 b d e-3 b c f+a d f)+A \left (a b c f^2-a^2 d f^2-2 b^2 e (d e-c f)\right )\right ) x}{2 a (b c-a d) e (b e-a f)^2 (d e-c f) \sqrt {a+b x^2}}-\frac {f (B e-A f) x}{2 e (b e-a f) (d e-c f) \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {d^2 (B c-A d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{3/2} (d e-c f)^2}+\frac {f (2 b e (A f (3 d e-2 c f)-B e (2 d e-c f))-a f (A f (3 d e-c f)-B e (d e+c f))) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{5/2} (d e-c f)^2} \] Output:

-1/2*b*(a*B*e*(a*d*f-3*b*c*f+2*b*d*e)+A*(a*b*c*f^2-a^2*d*f^2-2*b^2*e*(-c*f 
+d*e)))*x/a/(-a*d+b*c)/e/(-a*f+b*e)^2/(-c*f+d*e)/(b*x^2+a)^(1/2)-1/2*f*(-A 
*f+B*e)*x/e/(-a*f+b*e)/(-c*f+d*e)/(b*x^2+a)^(1/2)/(f*x^2+e)+d^2*(-A*d+B*c) 
*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2))/c^(1/2)/(-a*d+b*c)^(3 
/2)/(-c*f+d*e)^2+1/2*f*(2*b*e*(A*f*(-2*c*f+3*d*e)-B*e*(-c*f+2*d*e))-a*f*(A 
*f*(-c*f+3*d*e)-B*e*(c*f+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)^(5/2)/(-c*f+d*e)^2
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 18.03 (sec) , antiderivative size = 1481, normalized size of antiderivative = 4.16 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

Integrate[(A + B*x^2)/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)^2),x]
 

Output:

(d*(-(B*c) + A*d)*x*(-15*c*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 10*d* 
x^2*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 15*c*ArcTanh[Sqrt[((b*c - a* 
d)*x^2)/(c*(a + b*x^2))]] + 10*d*x^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a 
+ b*x^2))]] + 2*c*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2)*Hypergeometric 
2F1[2, 5/2, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 2*d*x^2*(((b*c - a*d 
)*x^2)/(c*(a + b*x^2)))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, ((b*c - a*d)* 
x^2)/(c*(a + b*x^2))]))/(5*a*c^2*(d*e - c*f)^2*(((b*c - a*d)*x^2)/(c*(a + 
b*x^2)))^(3/2)*Sqrt[a + b*x^2]*(1 + (b*x^2)/a)) + ((B*c - A*d)*f*x*(-15*e* 
Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))] - 10*f*x^2*Sqrt[((b*e - a*f)*x^2)/ 
(e*(a + b*x^2))] + 15*e*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]] + 
 10*f*x^2*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]] + 2*e*(((b*e - 
a*f)*x^2)/(e*(a + b*x^2)))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, ((b*e - a* 
f)*x^2)/(e*(a + b*x^2))] + 2*f*x^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(5/ 
2)*Hypergeometric2F1[2, 5/2, 7/2, ((b*e - a*f)*x^2)/(e*(a + b*x^2))]))/(5* 
a*e^2*(-(d*e) + c*f)^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2)*Sqrt[a + 
b*x^2]*(1 + (b*x^2)/a)) + ((B*e - A*f)*x*(-2625*Sqrt[((b*e - a*f)*x^2)/(e* 
(a + b*x^2))] - (5250*f*x^2*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))])/e - ( 
2310*f^2*x^4*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))])/e^2 + 70*(((b*e - a* 
f)*x^2)/(e*(a + b*x^2)))^(3/2) + (560*f*x^2*(((b*e - a*f)*x^2)/(e*(a + b*x 
^2)))^(3/2))/e + (280*f^2*x^4*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2)...
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {7276, 2009}

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.

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

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {d (A d-B c)}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (d e-c f)^2}+\frac {f (B c-A d)}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (c f-d e)^2}+\frac {B e-A f}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (d e-c f)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 (B c-A d) \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{3/2} (d e-c f)^2}-\frac {f (4 b e-a f) (B e-A f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{5/2} (d e-c f)}-\frac {f^2 (B c-A d) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2} (d e-c f)^2}-\frac {b d x (B c-A d)}{a \sqrt {a+b x^2} (b c-a d) (d e-c f)^2}+\frac {b x (a f+2 b e) (B e-A f)}{2 a e \sqrt {a+b x^2} (b e-a f)^2 (d e-c f)}+\frac {b f x (B c-A d)}{a \sqrt {a+b x^2} (b e-a f) (d e-c f)^2}-\frac {f x (B e-A f)}{2 e \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f) (d e-c f)}\)

Input:

Int[(A + B*x^2)/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)^2),x]
 

Output:

-((b*d*(B*c - A*d)*x)/(a*(b*c - a*d)*(d*e - c*f)^2*Sqrt[a + b*x^2])) + (b* 
(B*c - A*d)*f*x)/(a*(b*e - a*f)*(d*e - c*f)^2*Sqrt[a + b*x^2]) + (b*(2*b*e 
 + a*f)*(B*e - A*f)*x)/(2*a*e*(b*e - a*f)^2*(d*e - c*f)*Sqrt[a + b*x^2]) - 
 (f*(B*e - A*f)*x)/(2*e*(b*e - a*f)*(d*e - c*f)*Sqrt[a + b*x^2]*(e + f*x^2 
)) + (d^2*(B*c - A*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2] 
)])/(Sqrt[c]*(b*c - a*d)^(3/2)*(d*e - c*f)^2) - ((B*c - A*d)*f^2*ArcTanh[( 
Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*(b*e - a*f)^(3/2)* 
(d*e - c*f)^2) - (f*(4*b*e - a*f)*(B*e - A*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/ 
(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(5/2)*(d*e - c*f))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

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

Maple [A] (verified)

Time = 1.72 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.18

method	result	size
pseudoelliptic	\(\frac {-a \sqrt {\left (a d -b c \right ) c}\, \left (a d -b c \right ) \left (\left (\left (A \,f^{2}+B e f \right ) c -3 A d e f +B d \,e^{2}\right ) f a -4 b e \left (f \left (A f -\frac {B e}{2}\right ) c -\frac {3 A d e f}{2}+B d \,e^{2}\right )\right ) \sqrt {b \,x^{2}+a}\, f \left (f \,x^{2}+e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (-2 \sqrt {b \,x^{2}+a}\, a \,d^{2} e \left (f \,x^{2}+e \right ) \left (a f -b e \right )^{2} \left (A d -B c \right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+x \left (d \,f^{2} \left (A f -B e \right ) a^{3}-b \,f^{2} \left (-x^{2} d +c \right ) \left (A f -B e \right ) a^{2}-\left (f \left (f^{2} x^{2} A -3 f \,x^{2} B e -2 B \,e^{2}\right ) c +2 B \,e^{2} d \left (f \,x^{2}+e \right )\right ) b^{2} a -2 A \,b^{3} e \left (f \,x^{2}+e \right ) \left (c f -d e \right )\right ) \sqrt {\left (a d -b c \right ) c}\, \left (c f -d e \right )\right )}{2 \sqrt {b \,x^{2}+a}\, \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (c f -d e \right )^{2} \left (a d -b c \right ) e \left (f \,x^{2}+e \right ) \left (a f -b e \right )^{2} a}\)	\(421\)
default	\(\text {Expression too large to display}\)	\(2961\)

Input:

int((B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^2,x,method=_RETURNVERBOS 
E)
 

Output:

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

Fricas [F(-1)]

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

integrate((B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^2,x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((B*x**2+A)/(b*x**2+a)**(3/2)/(d*x**2+c)/(f*x**2+e)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^2,x, algorithm="ma 
xima")
 

Output:

integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(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. 785 vs. \(2 (326) = 652\).

Time = 2.14 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.21 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=-\frac {{\left (B a b^{2} - A b^{3}\right )} x}{{\left (a b^{3} c e^{2} - a^{2} b^{2} d e^{2} - 2 \, a^{2} b^{2} c e f + 2 \, a^{3} b d e f + a^{3} b c f^{2} - a^{4} d f^{2}\right )} \sqrt {b x^{2} + a}} - \frac {{\left (B \sqrt {b} c d^{2} - A \sqrt {b} d^{3}\right )} \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 c d^{2} e^{2} - a d^{3} e^{2} - 2 \, b c^{2} d e f + 2 \, a c d^{2} e f + b c^{3} f^{2} - a c^{2} d f^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} + \frac {{\left (4 \, B b^{\frac {3}{2}} d e^{3} f - 2 \, B b^{\frac {3}{2}} c e^{2} f^{2} - B a \sqrt {b} d e^{2} f^{2} - 6 \, A b^{\frac {3}{2}} d e^{2} f^{2} - B a \sqrt {b} c e f^{3} + 4 \, A b^{\frac {3}{2}} c e f^{3} + 3 \, A a \sqrt {b} d e f^{3} - A a \sqrt {b} c f^{4}\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 )}{2 \, {\left (b^{2} d^{2} e^{5} - 2 \, b^{2} c d e^{4} f - 2 \, a b d^{2} e^{4} f + b^{2} c^{2} e^{3} f^{2} + 4 \, a b c d e^{3} f^{2} + a^{2} d^{2} e^{3} f^{2} - 2 \, a b c^{2} e^{2} f^{3} - 2 \, a^{2} c d e^{2} f^{3} + a^{2} c^{2} e f^{4}\right )} \sqrt {-b^{2} e^{2} + a b e f}} + \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B b^{\frac {3}{2}} e^{2} f - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} e f^{2} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A b^{\frac {3}{2}} e f^{2} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a \sqrt {b} f^{3} + B a^{2} \sqrt {b} e f^{2} - A a^{2} \sqrt {b} f^{3}}{{\left (b^{2} d e^{4} - b^{2} c e^{3} f - 2 \, a b d e^{3} f + 2 \, a b c e^{2} f^{2} + a^{2} d e^{2} f^{2} - a^{2} c e f^{3}\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 )}} \] Input:

integrate((B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^2,x, algorithm="gi 
ac")
 

Output:

-(B*a*b^2 - A*b^3)*x/((a*b^3*c*e^2 - a^2*b^2*d*e^2 - 2*a^2*b^2*c*e*f + 2*a 
^3*b*d*e*f + a^3*b*c*f^2 - a^4*d*f^2)*sqrt(b*x^2 + a)) - (B*sqrt(b)*c*d^2 
- A*sqrt(b)*d^3)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a 
*d)/sqrt(-b^2*c^2 + a*b*c*d))/((b*c*d^2*e^2 - a*d^3*e^2 - 2*b*c^2*d*e*f + 
2*a*c*d^2*e*f + b*c^3*f^2 - a*c^2*d*f^2)*sqrt(-b^2*c^2 + a*b*c*d)) + 1/2*( 
4*B*b^(3/2)*d*e^3*f - 2*B*b^(3/2)*c*e^2*f^2 - B*a*sqrt(b)*d*e^2*f^2 - 6*A* 
b^(3/2)*d*e^2*f^2 - B*a*sqrt(b)*c*e*f^3 + 4*A*b^(3/2)*c*e*f^3 + 3*A*a*sqrt 
(b)*d*e*f^3 - A*a*sqrt(b)*c*f^4)*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^2*d^2*e^5 - 2*b^2*c*d*e^ 
4*f - 2*a*b*d^2*e^4*f + b^2*c^2*e^3*f^2 + 4*a*b*c*d*e^3*f^2 + a^2*d^2*e^3* 
f^2 - 2*a*b*c^2*e^2*f^3 - 2*a^2*c*d*e^2*f^3 + a^2*c^2*e*f^4)*sqrt(-b^2*e^2 
 + a*b*e*f)) + (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*b^(3/2)*e^2*f - (sqrt( 
b)*x - sqrt(b*x^2 + a))^2*B*a*sqrt(b)*e*f^2 - 2*(sqrt(b)*x - sqrt(b*x^2 + 
a))^2*A*b^(3/2)*e*f^2 + (sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a*sqrt(b)*f^3 + 
B*a^2*sqrt(b)*e*f^2 - A*a^2*sqrt(b)*f^3)/((b^2*d*e^4 - b^2*c*e^3*f - 2*a*b 
*d*e^3*f + 2*a*b*c*e^2*f^2 + a^2*d*e^2*f^2 - a^2*c*e*f^3)*((sqrt(b)*x - sq 
rt(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))
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((A + B*x^2)/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)^2),x)
 

Output:

int((A + B*x^2)/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)^2), x)
 

Reduce [B] (verification not implemented)

Time = 1.31 (sec) , antiderivative size = 3033, normalized size of antiderivative = 8.52 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((B*x^2+A)/(b*x^2+a)^(3/2)/(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)/...