\(\int \frac {(a+b x^2)^2 (c+d x^2)^2}{(e+f x^2)^3} \, dx\) [229]

Optimal result

Integrand size = 28, antiderivative size = 262 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=-\frac {b d (3 b d e-2 b c f-2 a d f) x}{f^4}+\frac {b^2 d^2 x^3}{3 f^3}+\frac {(b e-a f)^2 (d e-c f)^2 x}{4 e f^4 \left (e+f x^2\right )^2}-\frac {(b e-a f) (d e-c f) (b e (13 d e-5 c f)-a f (5 d e+3 c f)) x}{8 e^2 f^4 \left (e+f x^2\right )}-\frac {\left (2 a b e f \left (15 d^2 e^2-6 c d e f-c^2 f^2\right )-b^2 e^2 \left (35 d^2 e^2-30 c d e f+3 c^2 f^2\right )-a^2 f^2 \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} f^{9/2}} \] Output:

-b*d*(-2*a*d*f-2*b*c*f+3*b*d*e)*x/f^4+1/3*b^2*d^2*x^3/f^3+1/4*(-a*f+b*e)^2 
*(-c*f+d*e)^2*x/e/f^4/(f*x^2+e)^2-1/8*(-a*f+b*e)*(-c*f+d*e)*(b*e*(-5*c*f+1 
3*d*e)-a*f*(3*c*f+5*d*e))*x/e^2/f^4/(f*x^2+e)-1/8*(2*a*b*e*f*(-c^2*f^2-6*c 
*d*e*f+15*d^2*e^2)-b^2*e^2*(3*c^2*f^2-30*c*d*e*f+35*d^2*e^2)-a^2*f^2*(3*c^ 
2*f^2+2*c*d*e*f+3*d^2*e^2))*arctan(f^(1/2)*x/e^(1/2))/e^(5/2)/f^(9/2)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=-\frac {b d (3 b d e-2 b c f-2 a d f) x}{f^4}+\frac {b^2 d^2 x^3}{3 f^3}+\frac {(b e-a f)^2 (d e-c f)^2 x}{4 e f^4 \left (e+f x^2\right )^2}-\frac {(b e-a f) (d e-c f) (b e (13 d e-5 c f)-a f (5 d e+3 c f)) x}{8 e^2 f^4 \left (e+f x^2\right )}+\frac {\left (2 a b e f \left (-15 d^2 e^2+6 c d e f+c^2 f^2\right )+b^2 e^2 \left (35 d^2 e^2-30 c d e f+3 c^2 f^2\right )+a^2 f^2 \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} f^{9/2}} \] Input:

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

Output:

-((b*d*(3*b*d*e - 2*b*c*f - 2*a*d*f)*x)/f^4) + (b^2*d^2*x^3)/(3*f^3) + ((b 
*e - a*f)^2*(d*e - c*f)^2*x)/(4*e*f^4*(e + f*x^2)^2) - ((b*e - a*f)*(d*e - 
 c*f)*(b*e*(13*d*e - 5*c*f) - a*f*(5*d*e + 3*c*f))*x)/(8*e^2*f^4*(e + f*x^ 
2)) + ((2*a*b*e*f*(-15*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + b^2*e^2*(35*d^2*e^ 
2 - 30*c*d*e*f + 3*c^2*f^2) + a^2*f^2*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2)) 
*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(8*e^(5/2)*f^(9/2))
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.57, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {425, 401, 25, 401, 299, 218, 403, 25, 299, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 401

\(\displaystyle \frac {b \left (-\frac {\int -\frac {\left (d x^2+c\right ) \left (d (5 b e-3 a f) x^2+c (b e+a f)\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}-\frac {(b e-a f) \left (-\frac {\int -\frac {\left (d x^2+c\right ) \left (d (5 b e-a f) x^2+c (b e+3 a f)\right )}{\left (f x^2+e\right )^2}dx}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {\int \frac {\left (d x^2+c\right ) \left (d (5 b e-3 a f) x^2+c (b e+a f)\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {\int \frac {\left (d x^2+c\right ) \left (d (5 b e-a f) x^2+c (b e+3 a f)\right )}{\left (f x^2+e\right )^2}dx}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {b \left (\frac {\int \frac {\left (d x^2+c\right ) \left (d (5 b e-3 a f) x^2+c (b e+a f)\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {\int \frac {c (a f (d e-3 c f)-b e (5 d e+c f))-d (b e (15 d e-c f)-3 a f (d e+c f)) x^2}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b \left (\frac {\int \frac {\left (d x^2+c\right ) \left (d (5 b e-3 a f) x^2+c (b e+a f)\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {\frac {\left (b e \left (-c^2 f^2-6 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right ) \int \frac {1}{f x^2+e}dx}{f}-\frac {d x (b e (15 d e-c f)-3 a f (c f+d e))}{f}}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {\int \frac {\left (d x^2+c\right ) \left (d (5 b e-3 a f) x^2+c (b e+a f)\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2-6 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{\sqrt {e} f^{3/2}}-\frac {d x (b e (15 d e-c f)-3 a f (c f+d e))}{f}}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b \left (\frac {\frac {\int -\frac {d (b e (15 d e-13 c f)-3 a f (3 d e-c f)) x^2+c (b e (5 d e-3 c f)-3 a f (d e+c f))}{f x^2+e}dx}{3 f}+\frac {d x \left (c+d x^2\right ) (5 b e-3 a f)}{3 f}}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2-6 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{\sqrt {e} f^{3/2}}-\frac {d x (b e (15 d e-c f)-3 a f (c f+d e))}{f}}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {\frac {d x \left (c+d x^2\right ) (5 b e-3 a f)}{3 f}-\frac {\int \frac {d (b e (15 d e-13 c f)-3 a f (3 d e-c f)) x^2+c (b e (5 d e-3 c f)-3 a f (d e+c f))}{f x^2+e}dx}{3 f}}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2-6 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{\sqrt {e} f^{3/2}}-\frac {d x (b e (15 d e-c f)-3 a f (c f+d e))}{f}}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b \left (\frac {\frac {d x \left (c+d x^2\right ) (5 b e-3 a f)}{3 f}-\frac {\frac {d x (b e (15 d e-13 c f)-3 a f (3 d e-c f))}{f}-\frac {3 (d e-c f) (b e (5 d e-c f)-a f (c f+3 d e)) \int \frac {1}{f x^2+e}dx}{f}}{3 f}}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2-6 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{\sqrt {e} f^{3/2}}-\frac {d x (b e (15 d e-c f)-3 a f (c f+d e))}{f}}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {\frac {d x \left (c+d x^2\right ) (5 b e-3 a f)}{3 f}-\frac {\frac {d x (b e (15 d e-13 c f)-3 a f (3 d e-c f))}{f}-\frac {3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f) (b e (5 d e-c f)-a f (c f+3 d e))}{\sqrt {e} f^{3/2}}}{3 f}}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2-6 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{\sqrt {e} f^{3/2}}-\frac {d x (b e (15 d e-c f)-3 a f (c f+d e))}{f}}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}\)

Input:

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

Output:

(b*(-1/2*((b*e - a*f)*x*(c + d*x^2)^2)/(e*f*(e + f*x^2)) + ((d*(5*b*e - 3* 
a*f)*x*(c + d*x^2))/(3*f) - ((d*(b*e*(15*d*e - 13*c*f) - 3*a*f*(3*d*e - c* 
f))*x)/f - (3*(d*e - c*f)*(b*e*(5*d*e - c*f) - a*f*(3*d*e + c*f))*ArcTan[( 
Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(3/2)))/(3*f))/(2*e*f)))/f - ((b*e - a*f)* 
(-1/4*((b*e - a*f)*x*(c + d*x^2)^2)/(e*f*(e + f*x^2)^2) + (-1/2*((b*e*(5*d 
*e - c*f) - a*f*(d*e + 3*c*f))*x*(c + d*x^2))/(e*f*(e + f*x^2)) - (-((d*(b 
*e*(15*d*e - c*f) - 3*a*f*(d*e + c*f))*x)/f) + ((b*e*(15*d^2*e^2 - 6*c*d*e 
*f - c^2*f^2) - a*f*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))*ArcTan[(Sqrt[f]*x 
)/Sqrt[e]])/(Sqrt[e]*f^(3/2)))/(2*e*f))/(4*e*f)))/f
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 401

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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

rule 403

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

rule 425

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ 
)^2)^(r_), x_Symbol] :> Simp[d/b   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 
 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b   Int[(a + b*x^2)^p*(c + d*x 
^2)^(q - 1)*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && ILt 
Q[p, 0] && GtQ[q, 0]
 

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.55

method	result	size
default	\(\frac {b d \left (\frac {1}{3} b d f \,x^{3}+2 a d f x +2 b c f x -3 b d e x \right )}{f^{4}}+\frac {\frac {\frac {f \left (3 a^{2} c^{2} f^{4}+2 a^{2} c d e \,f^{3}-5 a^{2} d^{2} e^{2} f^{2}+2 a b \,c^{2} e \,f^{3}-20 a b c d \,e^{2} f^{2}+18 a b \,d^{2} e^{3} f -5 b^{2} c^{2} e^{2} f^{2}+18 b^{2} c d \,e^{3} f -13 b^{2} d^{2} e^{4}\right ) x^{3}}{8 e^{2}}+\frac {\left (5 a^{2} c^{2} f^{4}-2 a^{2} c d e \,f^{3}-3 a^{2} d^{2} e^{2} f^{2}-2 a b \,c^{2} e \,f^{3}-12 a b c d \,e^{2} f^{2}+14 a b \,d^{2} e^{3} f -3 b^{2} c^{2} e^{2} f^{2}+14 b^{2} c d \,e^{3} f -11 b^{2} d^{2} e^{4}\right ) x}{8 e}}{\left (f \,x^{2}+e \right )^{2}}+\frac {\left (3 a^{2} c^{2} f^{4}+2 a^{2} c d e \,f^{3}+3 a^{2} d^{2} e^{2} f^{2}+2 a b \,c^{2} e \,f^{3}+12 a b c d \,e^{2} f^{2}-30 a b \,d^{2} e^{3} f +3 b^{2} c^{2} e^{2} f^{2}-30 b^{2} c d \,e^{3} f +35 b^{2} d^{2} e^{4}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 e^{2} \sqrt {e f}}}{f^{4}}\)	\(406\)
risch	\(\frac {b^{2} d^{2} x^{3}}{3 f^{3}}+\frac {2 b \,d^{2} a x}{f^{3}}+\frac {2 b^{2} d c x}{f^{3}}-\frac {3 b^{2} d^{2} e x}{f^{4}}+\frac {\frac {f \left (3 a^{2} c^{2} f^{4}+2 a^{2} c d e \,f^{3}-5 a^{2} d^{2} e^{2} f^{2}+2 a b \,c^{2} e \,f^{3}-20 a b c d \,e^{2} f^{2}+18 a b \,d^{2} e^{3} f -5 b^{2} c^{2} e^{2} f^{2}+18 b^{2} c d \,e^{3} f -13 b^{2} d^{2} e^{4}\right ) x^{3}}{8 e^{2}}+\frac {\left (5 a^{2} c^{2} f^{4}-2 a^{2} c d e \,f^{3}-3 a^{2} d^{2} e^{2} f^{2}-2 a b \,c^{2} e \,f^{3}-12 a b c d \,e^{2} f^{2}+14 a b \,d^{2} e^{3} f -3 b^{2} c^{2} e^{2} f^{2}+14 b^{2} c d \,e^{3} f -11 b^{2} d^{2} e^{4}\right ) x}{8 e}}{f^{4} \left (f \,x^{2}+e \right )^{2}}-\frac {3 \ln \left (f x +\sqrt {-e f}\right ) a^{2} c^{2}}{16 \sqrt {-e f}\, e^{2}}-\frac {\ln \left (f x +\sqrt {-e f}\right ) a^{2} c d}{8 f \sqrt {-e f}\, e}-\frac {3 \ln \left (f x +\sqrt {-e f}\right ) a^{2} d^{2}}{16 f^{2} \sqrt {-e f}}-\frac {\ln \left (f x +\sqrt {-e f}\right ) a b \,c^{2}}{8 f \sqrt {-e f}\, e}-\frac {3 \ln \left (f x +\sqrt {-e f}\right ) a b c d}{4 f^{2} \sqrt {-e f}}+\frac {15 e \ln \left (f x +\sqrt {-e f}\right ) a b \,d^{2}}{8 f^{3} \sqrt {-e f}}-\frac {3 \ln \left (f x +\sqrt {-e f}\right ) b^{2} c^{2}}{16 f^{2} \sqrt {-e f}}+\frac {15 e \ln \left (f x +\sqrt {-e f}\right ) b^{2} c d}{8 f^{3} \sqrt {-e f}}-\frac {35 e^{2} \ln \left (f x +\sqrt {-e f}\right ) b^{2} d^{2}}{16 f^{4} \sqrt {-e f}}+\frac {3 \ln \left (-f x +\sqrt {-e f}\right ) a^{2} c^{2}}{16 \sqrt {-e f}\, e^{2}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) a^{2} c d}{8 f \sqrt {-e f}\, e}+\frac {3 \ln \left (-f x +\sqrt {-e f}\right ) a^{2} d^{2}}{16 f^{2} \sqrt {-e f}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) a b \,c^{2}}{8 f \sqrt {-e f}\, e}+\frac {3 \ln \left (-f x +\sqrt {-e f}\right ) a b c d}{4 f^{2} \sqrt {-e f}}-\frac {15 e \ln \left (-f x +\sqrt {-e f}\right ) a b \,d^{2}}{8 f^{3} \sqrt {-e f}}+\frac {3 \ln \left (-f x +\sqrt {-e f}\right ) b^{2} c^{2}}{16 f^{2} \sqrt {-e f}}-\frac {15 e \ln \left (-f x +\sqrt {-e f}\right ) b^{2} c d}{8 f^{3} \sqrt {-e f}}+\frac {35 e^{2} \ln \left (-f x +\sqrt {-e f}\right ) b^{2} d^{2}}{16 f^{4} \sqrt {-e f}}\)	\(817\)

Input:

int((b*x^2+a)^2*(d*x^2+c)^2/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
 

Output:

b*d/f^4*(1/3*b*d*f*x^3+2*a*d*f*x+2*b*c*f*x-3*b*d*e*x)+1/f^4*((1/8*f*(3*a^2 
*c^2*f^4+2*a^2*c*d*e*f^3-5*a^2*d^2*e^2*f^2+2*a*b*c^2*e*f^3-20*a*b*c*d*e^2* 
f^2+18*a*b*d^2*e^3*f-5*b^2*c^2*e^2*f^2+18*b^2*c*d*e^3*f-13*b^2*d^2*e^4)/e^ 
2*x^3+1/8*(5*a^2*c^2*f^4-2*a^2*c*d*e*f^3-3*a^2*d^2*e^2*f^2-2*a*b*c^2*e*f^3 
-12*a*b*c*d*e^2*f^2+14*a*b*d^2*e^3*f-3*b^2*c^2*e^2*f^2+14*b^2*c*d*e^3*f-11 
*b^2*d^2*e^4)/e*x)/(f*x^2+e)^2+1/8*(3*a^2*c^2*f^4+2*a^2*c*d*e*f^3+3*a^2*d^ 
2*e^2*f^2+2*a*b*c^2*e*f^3+12*a*b*c*d*e^2*f^2-30*a*b*d^2*e^3*f+3*b^2*c^2*e^ 
2*f^2-30*b^2*c*d*e^3*f+35*b^2*d^2*e^4)/e^2/(e*f)^(1/2)*arctan(f*x/(e*f)^(1 
/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (244) = 488\).

Time = 0.11 (sec) , antiderivative size = 1238, normalized size of antiderivative = 4.73 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/48*(16*b^2*d^2*e^3*f^4*x^7 - 16*(7*b^2*d^2*e^4*f^3 - 6*(b^2*c*d + a*b*d 
^2)*e^3*f^4)*x^5 - 2*(175*b^2*d^2*e^5*f^2 - 9*a^2*c^2*e*f^6 - 150*(b^2*c*d 
 + a*b*d^2)*e^4*f^3 + 15*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3*f^4 - 6*(a*b* 
c^2 + a^2*c*d)*e^2*f^5)*x^3 - 3*(35*b^2*d^2*e^6 + 3*a^2*c^2*e^2*f^4 - 30*( 
b^2*c*d + a*b*d^2)*e^5*f + 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^4*f^2 + 2*( 
a*b*c^2 + a^2*c*d)*e^3*f^3 + (35*b^2*d^2*e^4*f^2 + 3*a^2*c^2*f^6 - 30*(b^2 
*c*d + a*b*d^2)*e^3*f^3 + 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2*f^4 + 2*(a 
*b*c^2 + a^2*c*d)*e*f^5)*x^4 + 2*(35*b^2*d^2*e^5*f + 3*a^2*c^2*e*f^5 - 30* 
(b^2*c*d + a*b*d^2)*e^4*f^2 + 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3*f^3 + 
2*(a*b*c^2 + a^2*c*d)*e^2*f^4)*x^2)*sqrt(-e*f)*log((f*x^2 - 2*sqrt(-e*f)*x 
 - e)/(f*x^2 + e)) - 6*(35*b^2*d^2*e^6*f - 5*a^2*c^2*e^2*f^5 - 30*(b^2*c*d 
 + a*b*d^2)*e^5*f^2 + 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^4*f^3 + 2*(a*b*c 
^2 + a^2*c*d)*e^3*f^4)*x)/(e^3*f^7*x^4 + 2*e^4*f^6*x^2 + e^5*f^5), 1/24*(8 
*b^2*d^2*e^3*f^4*x^7 - 8*(7*b^2*d^2*e^4*f^3 - 6*(b^2*c*d + a*b*d^2)*e^3*f^ 
4)*x^5 - (175*b^2*d^2*e^5*f^2 - 9*a^2*c^2*e*f^6 - 150*(b^2*c*d + a*b*d^2)* 
e^4*f^3 + 15*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3*f^4 - 6*(a*b*c^2 + a^2*c* 
d)*e^2*f^5)*x^3 + 3*(35*b^2*d^2*e^6 + 3*a^2*c^2*e^2*f^4 - 30*(b^2*c*d + a* 
b*d^2)*e^5*f + 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^4*f^2 + 2*(a*b*c^2 + a^ 
2*c*d)*e^3*f^3 + (35*b^2*d^2*e^4*f^2 + 3*a^2*c^2*f^6 - 30*(b^2*c*d + a*b*d 
^2)*e^3*f^3 + 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2*f^4 + 2*(a*b*c^2 + ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 675 vs. \(2 (260) = 520\).

Time = 19.17 (sec) , antiderivative size = 675, normalized size of antiderivative = 2.58 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=\frac {b^{2} d^{2} x^{3}}{3 f^{3}} + x \left (\frac {2 a b d^{2}}{f^{3}} + \frac {2 b^{2} c d}{f^{3}} - \frac {3 b^{2} d^{2} e}{f^{4}}\right ) - \frac {\sqrt {- \frac {1}{e^{5} f^{9}}} \cdot \left (3 a^{2} c^{2} f^{4} + 2 a^{2} c d e f^{3} + 3 a^{2} d^{2} e^{2} f^{2} + 2 a b c^{2} e f^{3} + 12 a b c d e^{2} f^{2} - 30 a b d^{2} e^{3} f + 3 b^{2} c^{2} e^{2} f^{2} - 30 b^{2} c d e^{3} f + 35 b^{2} d^{2} e^{4}\right ) \log {\left (- e^{3} f^{4} \sqrt {- \frac {1}{e^{5} f^{9}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{e^{5} f^{9}}} \cdot \left (3 a^{2} c^{2} f^{4} + 2 a^{2} c d e f^{3} + 3 a^{2} d^{2} e^{2} f^{2} + 2 a b c^{2} e f^{3} + 12 a b c d e^{2} f^{2} - 30 a b d^{2} e^{3} f + 3 b^{2} c^{2} e^{2} f^{2} - 30 b^{2} c d e^{3} f + 35 b^{2} d^{2} e^{4}\right ) \log {\left (e^{3} f^{4} \sqrt {- \frac {1}{e^{5} f^{9}}} + x \right )}}{16} + \frac {x^{3} \cdot \left (3 a^{2} c^{2} f^{5} + 2 a^{2} c d e f^{4} - 5 a^{2} d^{2} e^{2} f^{3} + 2 a b c^{2} e f^{4} - 20 a b c d e^{2} f^{3} + 18 a b d^{2} e^{3} f^{2} - 5 b^{2} c^{2} e^{2} f^{3} + 18 b^{2} c d e^{3} f^{2} - 13 b^{2} d^{2} e^{4} f\right ) + x \left (5 a^{2} c^{2} e f^{4} - 2 a^{2} c d e^{2} f^{3} - 3 a^{2} d^{2} e^{3} f^{2} - 2 a b c^{2} e^{2} f^{3} - 12 a b c d e^{3} f^{2} + 14 a b d^{2} e^{4} f - 3 b^{2} c^{2} e^{3} f^{2} + 14 b^{2} c d e^{4} f - 11 b^{2} d^{2} e^{5}\right )}{8 e^{4} f^{4} + 16 e^{3} f^{5} x^{2} + 8 e^{2} f^{6} x^{4}} \] Input:

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

Output:

b**2*d**2*x**3/(3*f**3) + x*(2*a*b*d**2/f**3 + 2*b**2*c*d/f**3 - 3*b**2*d* 
*2*e/f**4) - sqrt(-1/(e**5*f**9))*(3*a**2*c**2*f**4 + 2*a**2*c*d*e*f**3 + 
3*a**2*d**2*e**2*f**2 + 2*a*b*c**2*e*f**3 + 12*a*b*c*d*e**2*f**2 - 30*a*b* 
d**2*e**3*f + 3*b**2*c**2*e**2*f**2 - 30*b**2*c*d*e**3*f + 35*b**2*d**2*e* 
*4)*log(-e**3*f**4*sqrt(-1/(e**5*f**9)) + x)/16 + sqrt(-1/(e**5*f**9))*(3* 
a**2*c**2*f**4 + 2*a**2*c*d*e*f**3 + 3*a**2*d**2*e**2*f**2 + 2*a*b*c**2*e* 
f**3 + 12*a*b*c*d*e**2*f**2 - 30*a*b*d**2*e**3*f + 3*b**2*c**2*e**2*f**2 - 
 30*b**2*c*d*e**3*f + 35*b**2*d**2*e**4)*log(e**3*f**4*sqrt(-1/(e**5*f**9) 
) + x)/16 + (x**3*(3*a**2*c**2*f**5 + 2*a**2*c*d*e*f**4 - 5*a**2*d**2*e**2 
*f**3 + 2*a*b*c**2*e*f**4 - 20*a*b*c*d*e**2*f**3 + 18*a*b*d**2*e**3*f**2 - 
 5*b**2*c**2*e**2*f**3 + 18*b**2*c*d*e**3*f**2 - 13*b**2*d**2*e**4*f) + x* 
(5*a**2*c**2*e*f**4 - 2*a**2*c*d*e**2*f**3 - 3*a**2*d**2*e**3*f**2 - 2*a*b 
*c**2*e**2*f**3 - 12*a*b*c*d*e**3*f**2 + 14*a*b*d**2*e**4*f - 3*b**2*c**2* 
e**3*f**2 + 14*b**2*c*d*e**4*f - 11*b**2*d**2*e**5))/(8*e**4*f**4 + 16*e** 
3*f**5*x**2 + 8*e**2*f**6*x**4)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^2/(f*x^2+e)^3,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 = 0.12 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=\frac {{\left (35 \, b^{2} d^{2} e^{4} - 30 \, b^{2} c d e^{3} f - 30 \, a b d^{2} e^{3} f + 3 \, b^{2} c^{2} e^{2} f^{2} + 12 \, a b c d e^{2} f^{2} + 3 \, a^{2} d^{2} e^{2} f^{2} + 2 \, a b c^{2} e f^{3} + 2 \, a^{2} c d e f^{3} + 3 \, a^{2} c^{2} f^{4}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \, \sqrt {e f} e^{2} f^{4}} - \frac {13 \, b^{2} d^{2} e^{4} f x^{3} - 18 \, b^{2} c d e^{3} f^{2} x^{3} - 18 \, a b d^{2} e^{3} f^{2} x^{3} + 5 \, b^{2} c^{2} e^{2} f^{3} x^{3} + 20 \, a b c d e^{2} f^{3} x^{3} + 5 \, a^{2} d^{2} e^{2} f^{3} x^{3} - 2 \, a b c^{2} e f^{4} x^{3} - 2 \, a^{2} c d e f^{4} x^{3} - 3 \, a^{2} c^{2} f^{5} x^{3} + 11 \, b^{2} d^{2} e^{5} x - 14 \, b^{2} c d e^{4} f x - 14 \, a b d^{2} e^{4} f x + 3 \, b^{2} c^{2} e^{3} f^{2} x + 12 \, a b c d e^{3} f^{2} x + 3 \, a^{2} d^{2} e^{3} f^{2} x + 2 \, a b c^{2} e^{2} f^{3} x + 2 \, a^{2} c d e^{2} f^{3} x - 5 \, a^{2} c^{2} e f^{4} x}{8 \, {\left (f x^{2} + e\right )}^{2} e^{2} f^{4}} + \frac {b^{2} d^{2} f^{6} x^{3} - 9 \, b^{2} d^{2} e f^{5} x + 6 \, b^{2} c d f^{6} x + 6 \, a b d^{2} f^{6} x}{3 \, f^{9}} \] Input:

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

Output:

1/8*(35*b^2*d^2*e^4 - 30*b^2*c*d*e^3*f - 30*a*b*d^2*e^3*f + 3*b^2*c^2*e^2* 
f^2 + 12*a*b*c*d*e^2*f^2 + 3*a^2*d^2*e^2*f^2 + 2*a*b*c^2*e*f^3 + 2*a^2*c*d 
*e*f^3 + 3*a^2*c^2*f^4)*arctan(f*x/sqrt(e*f))/(sqrt(e*f)*e^2*f^4) - 1/8*(1 
3*b^2*d^2*e^4*f*x^3 - 18*b^2*c*d*e^3*f^2*x^3 - 18*a*b*d^2*e^3*f^2*x^3 + 5* 
b^2*c^2*e^2*f^3*x^3 + 20*a*b*c*d*e^2*f^3*x^3 + 5*a^2*d^2*e^2*f^3*x^3 - 2*a 
*b*c^2*e*f^4*x^3 - 2*a^2*c*d*e*f^4*x^3 - 3*a^2*c^2*f^5*x^3 + 11*b^2*d^2*e^ 
5*x - 14*b^2*c*d*e^4*f*x - 14*a*b*d^2*e^4*f*x + 3*b^2*c^2*e^3*f^2*x + 12*a 
*b*c*d*e^3*f^2*x + 3*a^2*d^2*e^3*f^2*x + 2*a*b*c^2*e^2*f^3*x + 2*a^2*c*d*e 
^2*f^3*x - 5*a^2*c^2*e*f^4*x)/((f*x^2 + e)^2*e^2*f^4) + 1/3*(b^2*d^2*f^6*x 
^3 - 9*b^2*d^2*e*f^5*x + 6*b^2*c*d*f^6*x + 6*a*b*d^2*f^6*x)/f^9
 

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=\frac {\frac {x^3\,\left (3\,a^2\,c^2\,f^5+2\,a^2\,c\,d\,e\,f^4-5\,a^2\,d^2\,e^2\,f^3+2\,a\,b\,c^2\,e\,f^4-20\,a\,b\,c\,d\,e^2\,f^3+18\,a\,b\,d^2\,e^3\,f^2-5\,b^2\,c^2\,e^2\,f^3+18\,b^2\,c\,d\,e^3\,f^2-13\,b^2\,d^2\,e^4\,f\right )}{8\,e^2}-\frac {x\,\left (-5\,a^2\,c^2\,f^4+2\,a^2\,c\,d\,e\,f^3+3\,a^2\,d^2\,e^2\,f^2+2\,a\,b\,c^2\,e\,f^3+12\,a\,b\,c\,d\,e^2\,f^2-14\,a\,b\,d^2\,e^3\,f+3\,b^2\,c^2\,e^2\,f^2-14\,b^2\,c\,d\,e^3\,f+11\,b^2\,d^2\,e^4\right )}{8\,e}}{e^2\,f^4+2\,e\,f^5\,x^2+f^6\,x^4}-x\,\left (\frac {3\,b^2\,d^2\,e}{f^4}-\frac {2\,b\,d\,\left (a\,d+b\,c\right )}{f^3}\right )+\frac {b^2\,d^2\,x^3}{3\,f^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {e}}\right )\,\left (3\,a^2\,c^2\,f^4+2\,a^2\,c\,d\,e\,f^3+3\,a^2\,d^2\,e^2\,f^2+2\,a\,b\,c^2\,e\,f^3+12\,a\,b\,c\,d\,e^2\,f^2-30\,a\,b\,d^2\,e^3\,f+3\,b^2\,c^2\,e^2\,f^2-30\,b^2\,c\,d\,e^3\,f+35\,b^2\,d^2\,e^4\right )}{8\,e^{5/2}\,f^{9/2}} \] Input:

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

Output:

((x^3*(3*a^2*c^2*f^5 - 13*b^2*d^2*e^4*f - 5*a^2*d^2*e^2*f^3 - 5*b^2*c^2*e^ 
2*f^3 + 2*a*b*c^2*e*f^4 + 2*a^2*c*d*e*f^4 + 18*a*b*d^2*e^3*f^2 + 18*b^2*c* 
d*e^3*f^2 - 20*a*b*c*d*e^2*f^3))/(8*e^2) - (x*(11*b^2*d^2*e^4 - 5*a^2*c^2* 
f^4 + 3*a^2*d^2*e^2*f^2 + 3*b^2*c^2*e^2*f^2 + 2*a*b*c^2*e*f^3 - 14*a*b*d^2 
*e^3*f + 2*a^2*c*d*e*f^3 - 14*b^2*c*d*e^3*f + 12*a*b*c*d*e^2*f^2))/(8*e))/ 
(e^2*f^4 + f^6*x^4 + 2*e*f^5*x^2) - x*((3*b^2*d^2*e)/f^4 - (2*b*d*(a*d + b 
*c))/f^3) + (b^2*d^2*x^3)/(3*f^3) + (atan((f^(1/2)*x)/e^(1/2))*(3*a^2*c^2* 
f^4 + 35*b^2*d^2*e^4 + 3*a^2*d^2*e^2*f^2 + 3*b^2*c^2*e^2*f^2 + 2*a*b*c^2*e 
*f^3 - 30*a*b*d^2*e^3*f + 2*a^2*c*d*e*f^3 - 30*b^2*c*d*e^3*f + 12*a*b*c*d* 
e^2*f^2))/(8*e^(5/2)*f^(9/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1195, normalized size of antiderivative = 4.56 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(9*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*e**2*f**4 + 18* 
sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*e*f**5*x**2 + 9*sq 
rt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*f**6*x**4 + 6*sqrt(f 
)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d*e**3*f**3 + 12*sqrt(f)*sq 
rt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d*e**2*f**4*x**2 + 6*sqrt(f)*sq 
rt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d*e*f**5*x**4 + 9*sqrt(f)*sqrt( 
e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**2*e**4*f**2 + 18*sqrt(f)*sqrt(e)* 
atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**2*e**3*f**3*x**2 + 9*sqrt(f)*sqrt(e) 
*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**2*e**2*f**4*x**4 + 6*sqrt(f)*sqrt(e 
)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*e**3*f**3 + 12*sqrt(f)*sqrt(e)*at 
an((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*e**2*f**4*x**2 + 6*sqrt(f)*sqrt(e)*at 
an((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*e*f**5*x**4 + 36*sqrt(f)*sqrt(e)*atan 
((f*x)/(sqrt(f)*sqrt(e)))*a*b*c*d*e**4*f**2 + 72*sqrt(f)*sqrt(e)*atan((f*x 
)/(sqrt(f)*sqrt(e)))*a*b*c*d*e**3*f**3*x**2 + 36*sqrt(f)*sqrt(e)*atan((f*x 
)/(sqrt(f)*sqrt(e)))*a*b*c*d*e**2*f**4*x**4 - 90*sqrt(f)*sqrt(e)*atan((f*x 
)/(sqrt(f)*sqrt(e)))*a*b*d**2*e**5*f - 180*sqrt(f)*sqrt(e)*atan((f*x)/(sqr 
t(f)*sqrt(e)))*a*b*d**2*e**4*f**2*x**2 - 90*sqrt(f)*sqrt(e)*atan((f*x)/(sq 
rt(f)*sqrt(e)))*a*b*d**2*e**3*f**3*x**4 + 9*sqrt(f)*sqrt(e)*atan((f*x)/(sq 
rt(f)*sqrt(e)))*b**2*c**2*e**4*f**2 + 18*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt( 
f)*sqrt(e)))*b**2*c**2*e**3*f**3*x**2 + 9*sqrt(f)*sqrt(e)*atan((f*x)/(s...