\(\int \frac {1}{(a+b x^2) (c+d x^2) (e+f x^2)^2} \, dx\) [254]

Optimal result

Integrand size = 28, antiderivative size = 203 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {f^2 x}{2 e (b e-a f) (d e-c f) \left (e+f x^2\right )}+\frac {b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d) (b e-a f)^2}-\frac {d^{5/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d) (d e-c f)^2}+\frac {f^{3/2} (b e (5 d e-3 c f)-a f (3 d e-c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} (b e-a f)^2 (d e-c f)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=-\frac {b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (-b c+a d) (b e-a f)^2}+\frac {\frac {f^2 (d e-c f) x}{e (b e-a f) \left (e+f x^2\right )}-\frac {2 d^{5/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)}+\frac {f^{3/2} (b e (5 d e-3 c f)+a f (-3 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{3/2} (b e-a f)^2}}{2 (d e-c f)^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.31, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {421, 303, 218, 402, 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.

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 303

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

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 303

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

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]
 

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 = 1.10 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.93

method	result	size
default	\(-\frac {b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a f -b e \right )^{2} \left (a d -b c \right ) \sqrt {a b}}+\frac {f^{2} \left (\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) x}{2 e \left (f \,x^{2}+e \right )}+\frac {\left (a c \,f^{2}-3 a d e f -3 b c e f +5 b d \,e^{2}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 e \sqrt {e f}}\right )}{\left (c f -d e \right )^{2} \left (a f -b e \right )^{2}}+\frac {d^{3} \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{\left (a d -b c \right ) \left (c f -d e \right )^{2} \sqrt {c d}}\)	\(189\)
risch	\(\text {Expression too large to display}\)	\(75155\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(1/(b*x^2+a)/(d*x^2+c)/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (175) = 350\).

Time = 0.13 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.76 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{3} c e^{2} - a b^{2} d e^{2} - 2 \, a b^{2} c e f + 2 \, a^{2} b d e f + a^{2} b c f^{2} - a^{3} d f^{2}\right )} \sqrt {a b}} - \frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c d^{2} e^{2} - a d^{3} e^{2} - 2 \, b c^{2} d e f + 2 \, a c d^{2} e f + b c^{3} f^{2} - a c^{2} d f^{2}\right )} \sqrt {c d}} + \frac {f^{2} x}{2 \, {\left (b d e^{3} - b c e^{2} f - a d e^{2} f + a c e f^{2}\right )} {\left (f x^{2} + e\right )}} + \frac {{\left (5 \, b d e^{2} f^{2} - 3 \, b c e f^{3} - 3 \, a d e f^{3} + a c f^{4}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \, {\left (b^{2} d^{2} e^{5} - 2 \, b^{2} c d e^{4} f - 2 \, a b d^{2} e^{4} f + b^{2} c^{2} e^{3} f^{2} + 4 \, a b c d e^{3} f^{2} + a^{2} d^{2} e^{3} f^{2} - 2 \, a b c^{2} e^{2} f^{3} - 2 \, a^{2} c d e^{2} f^{3} + a^{2} c^{2} e f^{4}\right )} \sqrt {e f}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 11.91 (sec) , antiderivative size = 120009, normalized size of antiderivative = 591.18 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan(((((x*(54*b^9*d^9*e^6*f^5 + a^2*b^7*c^4*d^5*f^11 + a^4*b^5*c^2*d^7*f 
^11 + 107*a^2*b^7*d^9*e^4*f^7 - 48*a^3*b^6*d^9*e^3*f^8 + 9*a^4*b^5*d^9*e^2 
*f^9 + 107*b^9*c^2*d^7*e^4*f^7 - 48*b^9*c^3*d^6*e^3*f^8 + 9*b^9*c^4*d^5*e^ 
2*f^9 - 118*a*b^8*d^9*e^5*f^6 - 118*b^9*c*d^8*e^5*f^6 + 192*a*b^8*c*d^8*e^ 
4*f^7 - 6*a*b^8*c^4*d^5*e*f^10 - 6*a^4*b^5*c*d^8*e*f^10 - 124*a*b^8*c^2*d^ 
7*e^3*f^8 + 40*a*b^8*c^3*d^6*e^2*f^9 - 124*a^2*b^7*c*d^8*e^3*f^8 - 8*a^2*b 
^7*c^3*d^6*e*f^10 + 40*a^3*b^6*c*d^8*e^2*f^9 - 8*a^3*b^6*c^2*d^7*e*f^10 + 
48*a^2*b^7*c^2*d^7*e^2*f^9))/(2*(b^4*d^4*e^10 + a^4*c^4*e^2*f^8 + a^4*d^4* 
e^6*f^4 + b^4*c^4*e^6*f^4 + 6*a^2*b^2*c^4*e^4*f^6 + 6*a^2*b^2*d^4*e^8*f^2 
+ 6*a^4*c^2*d^2*e^4*f^6 + 6*b^4*c^2*d^2*e^8*f^2 - 4*a*b^3*d^4*e^9*f - 4*b^ 
4*c*d^3*e^9*f - 4*a*b^3*c^4*e^5*f^5 - 4*a^3*b*c^4*e^3*f^7 - 4*a^3*b*d^4*e^ 
7*f^3 - 4*a^4*c*d^3*e^5*f^5 - 4*a^4*c^3*d*e^3*f^7 - 4*b^4*c^3*d*e^7*f^3 + 
16*a*b^3*c*d^3*e^8*f^2 + 16*a*b^3*c^3*d*e^6*f^4 + 16*a^3*b*c*d^3*e^6*f^4 + 
 16*a^3*b*c^3*d*e^4*f^6 - 24*a*b^3*c^2*d^2*e^7*f^3 - 24*a^2*b^2*c*d^3*e^7* 
f^3 - 24*a^2*b^2*c^3*d*e^5*f^5 - 24*a^3*b*c^2*d^2*e^5*f^5 + 36*a^2*b^2*c^2 
*d^2*e^6*f^4)) - (((2*a^2*b^8*c^8*d^2*f^13 - 2*a^3*b^7*c^7*d^3*f^13 - 2*a^ 
7*b^3*c^3*d^7*f^13 + 2*a^8*b^2*c^2*d^8*f^13 + 50*a^2*b^8*d^10*e^8*f^5 - 24 
0*a^3*b^7*d^10*e^7*f^6 + 466*a^4*b^6*d^10*e^6*f^7 - 464*a^5*b^5*d^10*e^5*f 
^8 + 246*a^6*b^4*d^10*e^4*f^9 - 64*a^7*b^3*d^10*e^3*f^10 + 6*a^8*b^2*d^10* 
e^2*f^11 + 50*b^10*c^2*d^8*e^8*f^5 - 240*b^10*c^3*d^7*e^7*f^6 + 466*b^1...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1214, normalized size of antiderivative = 5.98 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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