\(\int \frac {a+b x^2}{(c+d x^2)^2 (e+f x^2)^2} \, dx\) [222]

Optimal result

Integrand size = 26, antiderivative size = 214 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=-\frac {f (2 b c e-a d e-a c f) x}{2 c e (d e-c f)^2 \left (e+f x^2\right )}-\frac {(b c-a d) x}{2 c (d e-c f) \left (c+d x^2\right ) \left (e+f x^2\right )}+\frac {\sqrt {d} (a d (d e-5 c f)+b c (d e+3 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (d e-c f)^3}+\frac {\sqrt {f} (a f (5 d e-c f)-b e (3 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} (d e-c f)^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {d (-b c+a d) x}{c (d e-c f)^2 \left (c+d x^2\right )}+\frac {f (-b e+a f) x}{e (d e-c f)^2 \left (e+f x^2\right )}-\frac {\sqrt {d} (a d (d e-5 c f)+b c (d e+3 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (-d e+c f)^3}-\frac {\sqrt {f} (a f (-5 d e+c f)+b e (3 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{3/2} (d e-c f)^3}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {402, 25, 402, 27, 397, 218}

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.

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {\frac {d e (a d (d e-5 c f)+b c (3 c f+d e)) \int \frac {1}{d x^2+c}dx}{d e-c f}+\frac {c f (a f (5 d e-c f)-b e (c f+3 d e)) \int \frac {1}{f x^2+e}dx}{d e-c f}}{e (d e-c f)}-\frac {f x (-a c f-a d e+2 b c e)}{e \left (e+f x^2\right ) (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right ) (d e-c f)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\sqrt {d} e \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) (a d (d e-5 c f)+b c (3 c f+d e))}{\sqrt {c} (d e-c f)}+\frac {c \sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (a f (5 d e-c f)-b e (c f+3 d e))}{\sqrt {e} (d e-c f)}}{e (d e-c f)}-\frac {f x (-a c f-a d e+2 b c e)}{e \left (e+f x^2\right ) (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right ) (d e-c f)}\)

Input:

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

Output:

-1/2*((b*c - a*d)*x)/(c*(d*e - c*f)*(c + d*x^2)*(e + f*x^2)) + (-((f*(2*b* 
c*e - a*d*e - a*c*f)*x)/(e*(d*e - c*f)*(e + f*x^2))) + ((Sqrt[d]*e*(a*d*(d 
*e - 5*c*f) + b*c*(d*e + 3*c*f))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(d* 
e - c*f)) + (c*Sqrt[f]*(a*f*(5*d*e - c*f) - b*e*(3*d*e + c*f))*ArcTan[(Sqr 
t[f]*x)/Sqrt[e]])/(Sqrt[e]*(d*e - c*f)))/(e*(d*e - c*f)))/(2*c*(d*e - c*f) 
)
 

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 218

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

rule 397

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

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]
 

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {f \left (\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) x}{2 e \left (f \,x^{2}+e \right )}+\frac {\left (a c \,f^{2}-5 a d e f +b c e f +3 b d \,e^{2}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 e \sqrt {e f}}\right )}{\left (c f -d e \right )^{3}}+\frac {d \left (\frac {\left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) x}{2 c \left (x^{2} d +c \right )}+\frac {\left (5 a c d f -a \,d^{2} e -3 b \,c^{2} f -b c d e \right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{2 c \sqrt {c d}}\right )}{\left (c f -d e \right )^{3}}\)	\(198\)
risch	\(\text {Expression too large to display}\)	\(11761\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (190) = 380\).

Time = 4.35 (sec) , antiderivative size = 2381, normalized size of antiderivative = 11.13 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(2*(2*b*c^2*d*e*f^2 - a*c^2*d*f^3 - (2*b*c*d^2 - a*d^3)*e^2*f)*x^3 + 
(((b*c*d^2 + a*d^3)*e^2*f + (3*b*c^2*d - 5*a*c*d^2)*e*f^2)*x^4 + (b*c^2*d 
+ a*c*d^2)*e^3 + (3*b*c^3 - 5*a*c^2*d)*e^2*f + ((b*c*d^2 + a*d^3)*e^3 + 4* 
(b*c^2*d - a*c*d^2)*e^2*f + (3*b*c^3 - 5*a*c^2*d)*e*f^2)*x^2)*sqrt(-d/c)*l 
og((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) - (3*b*c^2*d*e^3 + a*c^3*e* 
f^2 + (3*b*c*d^2*e^2*f + a*c^2*d*f^3 + (b*c^2*d - 5*a*c*d^2)*e*f^2)*x^4 + 
(b*c^3 - 5*a*c^2*d)*e^2*f + (3*b*c*d^2*e^3 + a*c^3*f^3 + (4*b*c^2*d - 5*a* 
c*d^2)*e^2*f + (b*c^3 - 4*a*c^2*d)*e*f^2)*x^2)*sqrt(-f/e)*log((f*x^2 + 2*e 
*x*sqrt(-f/e) - e)/(f*x^2 + e)) - 2*(a*c*d^2*e^2*f + a*c^3*f^3 + (b*c*d^2 
- a*d^3)*e^3 - (b*c^3 + a*c^2*d)*e*f^2)*x)/(c^2*d^3*e^5 - 3*c^3*d^2*e^4*f 
+ 3*c^4*d*e^3*f^2 - c^5*e^2*f^3 + (c*d^4*e^4*f - 3*c^2*d^3*e^3*f^2 + 3*c^3 
*d^2*e^2*f^3 - c^4*d*e*f^4)*x^4 + (c*d^4*e^5 - 2*c^2*d^3*e^4*f + 2*c^4*d*e 
^2*f^3 - c^5*e*f^4)*x^2), 1/4*(2*(2*b*c^2*d*e*f^2 - a*c^2*d*f^3 - (2*b*c*d 
^2 - a*d^3)*e^2*f)*x^3 - 2*(3*b*c^2*d*e^3 + a*c^3*e*f^2 + (3*b*c*d^2*e^2*f 
 + a*c^2*d*f^3 + (b*c^2*d - 5*a*c*d^2)*e*f^2)*x^4 + (b*c^3 - 5*a*c^2*d)*e^ 
2*f + (3*b*c*d^2*e^3 + a*c^3*f^3 + (4*b*c^2*d - 5*a*c*d^2)*e^2*f + (b*c^3 
- 4*a*c^2*d)*e*f^2)*x^2)*sqrt(f/e)*arctan(x*sqrt(f/e)) + (((b*c*d^2 + a*d^ 
3)*e^2*f + (3*b*c^2*d - 5*a*c*d^2)*e*f^2)*x^4 + (b*c^2*d + a*c*d^2)*e^3 + 
(3*b*c^3 - 5*a*c^2*d)*e^2*f + ((b*c*d^2 + a*d^3)*e^3 + 4*(b*c^2*d - a*c*d^ 
2)*e^2*f + (3*b*c^3 - 5*a*c^2*d)*e*f^2)*x^2)*sqrt(-d/c)*log((d*x^2 + 2*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.40 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {{\left (b c d^{2} e + a d^{3} e + 3 \, b c^{2} d f - 5 \, a c d^{2} f\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (c d^{3} e^{3} - 3 \, c^{2} d^{2} e^{2} f + 3 \, c^{3} d e f^{2} - c^{4} f^{3}\right )} \sqrt {c d}} - \frac {{\left (3 \, b d e^{2} f + b c e f^{2} - 5 \, a d e f^{2} + a c f^{3}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \, {\left (d^{3} e^{4} - 3 \, c d^{2} e^{3} f + 3 \, c^{2} d e^{2} f^{2} - c^{3} e f^{3}\right )} \sqrt {e f}} - \frac {2 \, b c d e f x^{3} - a d^{2} e f x^{3} - a c d f^{2} x^{3} + b c d e^{2} x - a d^{2} e^{2} x + b c^{2} e f x - a c^{2} f^{2} x}{2 \, {\left (c d^{2} e^{3} - 2 \, c^{2} d e^{2} f + c^{3} e f^{2}\right )} {\left (d f x^{4} + d e x^{2} + c f x^{2} + c e\right )}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 5.66 (sec) , antiderivative size = 9047, normalized size of antiderivative = 42.28 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((x*(a*c^2*f^2 + a*d^2*e^2 - b*c*d*e^2 - b*c^2*e*f))/(2*c*e*(c^2*f^2 + d^2 
*e^2 - 2*c*d*e*f)) + (d*f*x^3*(a*c*f + a*d*e - 2*b*c*e))/(2*c*e*(c^2*f^2 + 
 d^2*e^2 - 2*c*d*e*f)))/(c*e + x^2*(c*f + d*e) + d*f*x^4) + (atan(((((x*(a 
^2*c^4*d^3*f^7 + a^2*d^7*e^4*f^3 + 50*a^2*c^2*d^5*e^2*f^5 + 10*b^2*c^2*d^5 
*e^4*f^3 + 12*b^2*c^3*d^4*e^3*f^4 + 10*b^2*c^4*d^3*e^2*f^5 - 10*a^2*c*d^6* 
e^3*f^4 - 10*a^2*c^3*d^4*e*f^6 - 34*a*b*c^2*d^5*e^3*f^4 - 34*a*b*c^3*d^4*e 
^2*f^5 + 2*a*b*c*d^6*e^4*f^3 + 2*a*b*c^4*d^3*e*f^6))/(2*(c^2*d^4*e^6 + c^6 
*e^2*f^4 - 4*c^3*d^3*e^5*f - 4*c^5*d*e^3*f^3 + 6*c^4*d^2*e^4*f^2)) - (((2* 
a*c*d^10*e^9*f^2 + 2*a*c^9*d^2*e*f^10 - 20*a*c^2*d^9*e^8*f^3 + 80*a*c^3*d^ 
8*e^7*f^4 - 172*a*c^4*d^7*e^6*f^5 + 220*a*c^5*d^6*e^5*f^6 - 172*a*c^6*d^5* 
e^4*f^7 + 80*a*c^7*d^4*e^3*f^8 - 20*a*c^8*d^3*e^2*f^9 + 2*b*c^2*d^9*e^9*f^ 
2 - 10*b*c^3*d^8*e^8*f^3 + 18*b*c^4*d^7*e^7*f^4 - 10*b*c^5*d^6*e^6*f^5 - 1 
0*b*c^6*d^5*e^5*f^6 + 18*b*c^7*d^4*e^4*f^7 - 10*b*c^8*d^3*e^3*f^8 + 2*b*c^ 
9*d^2*e^2*f^9)/(c^2*d^6*e^8 + c^8*e^2*f^6 - 6*c^3*d^5*e^7*f - 6*c^7*d*e^3* 
f^5 + 15*c^4*d^4*e^6*f^2 - 20*c^5*d^3*e^5*f^3 + 15*c^6*d^2*e^4*f^4) - (x*( 
-e^3*f)^(1/2)*(a*c*f^2 + 3*b*d*e^2 - 5*a*d*e*f + b*c*e*f)*(16*c^2*d^9*e^9* 
f^2 - 80*c^3*d^8*e^8*f^3 + 144*c^4*d^7*e^7*f^4 - 80*c^5*d^6*e^6*f^5 - 80*c 
^6*d^5*e^5*f^6 + 144*c^7*d^4*e^4*f^7 - 80*c^8*d^3*e^3*f^8 + 16*c^9*d^2*e^2 
*f^9))/(8*(d^3*e^6 - c^3*e^3*f^3 + 3*c^2*d*e^4*f^2 - 3*c*d^2*e^5*f)*(c^2*d 
^4*e^6 + c^6*e^2*f^4 - 4*c^3*d^3*e^5*f - 4*c^5*d*e^3*f^3 + 6*c^4*d^2*e^...
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1095, normalized size of antiderivative = 5.12 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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