\(\int \frac {\sqrt {a+b x^2} (c+d x^2)}{(e+f x^2)^3} \, dx\) [265]

Optimal result

Integrand size = 28, antiderivative size = 170 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx=\frac {(d e-c f) x \left (a+b x^2\right )^{3/2}}{4 e (b e-a f) \left (e+f x^2\right )^2}+\frac {(4 b c e-a d e-3 a c f) x \sqrt {a+b x^2}}{8 e^2 (b e-a f) \left (e+f x^2\right )}+\frac {a (4 b c e-a d e-3 a c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{3/2}} \] Output:

1/4*(-c*f+d*e)*x*(b*x^2+a)^(3/2)/e/(-a*f+b*e)/(f*x^2+e)^2+1/8*(-3*a*c*f-a* 
d*e+4*b*c*e)*x*(b*x^2+a)^(1/2)/e^2/(-a*f+b*e)/(f*x^2+e)+1/8*a*(-3*a*c*f-a* 
d*e+4*b*c*e)*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(5/2)/( 
-a*f+b*e)^(3/2)
 

Mathematica [A] (warning: unable to verify)

Time = 11.75 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.76 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx=\frac {x \sqrt {a+b x^2} \left (\frac {4 d e \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}{\sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {f x^2}{e}}}+\frac {4 d e \arcsin \left (\frac {\sqrt {\left (-\frac {b}{a}+\frac {f}{e}\right ) x^2}}{\sqrt {1+\frac {f x^2}{e}}}\right )}{\sqrt {\left (-\frac {b}{a}+\frac {f}{e}\right ) x^2} \sqrt {1+\frac {b x^2}{a}}}-\frac {(d e-c f) \left (e \left (2 b e \left (2 e+f x^2\right )-a f \left (5 e+3 f x^2\right )\right )+\frac {a (4 b e-3 a f) \left (e+f x^2\right )^2 \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}} \left (a+b x^2\right )}\right )}{(b e-a f) \left (e+f x^2\right )^2}\right )}{8 e^3 f} \] Input:

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

Output:

(x*Sqrt[a + b*x^2]*((4*d*e*Sqrt[(e*(a + b*x^2))/(a*(e + f*x^2))])/(Sqrt[1 
+ (b*x^2)/a]*Sqrt[1 + (f*x^2)/e]) + (4*d*e*ArcSin[Sqrt[(-(b/a) + f/e)*x^2] 
/Sqrt[1 + (f*x^2)/e]])/(Sqrt[(-(b/a) + f/e)*x^2]*Sqrt[1 + (b*x^2)/a]) - (( 
d*e - c*f)*(e*(2*b*e*(2*e + f*x^2) - a*f*(5*e + 3*f*x^2)) + (a*(4*b*e - 3* 
a*f)*(e + f*x^2)^2*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/(Sqrt 
[((b*e - a*f)*x^2)/(e*(a + b*x^2))]*(a + b*x^2))))/((b*e - a*f)*(e + f*x^2 
)^2)))/(8*e^3*f)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {401, 25, 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.

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/4*((d*e - c*f)*x*Sqrt[a + b*x^2])/(e*f*(e + f*x^2)^2) + (((2*b*e*(d*e + 
 c*f) - a*f*(d*e + 3*c*f))*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2) 
) + (a*f*(4*b*c*e - a*d*e - 3*a*c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]* 
Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2)))/(4*e*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 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 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 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 = 0.92 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.91

method	result	size
pseudoelliptic	\(-\frac {a \left (-\frac {\sqrt {b \,x^{2}+a}\, \left (3 a c \,f^{2} x^{2}+a d \,x^{2} e f -2 b c e f \,x^{2}-2 b d \,e^{2} x^{2}+5 a c e f -a d \,e^{2}-4 b c \,e^{2}\right ) x}{a \left (f \,x^{2}+e \right )^{2}}+\frac {\left (3 a c f +a d e -4 b c e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{\sqrt {\left (a f -b e \right ) e}}\right )}{8 \left (a f -b e \right ) e^{2}}\)	\(155\)
default	\(\text {Expression too large to display}\)	\(4167\)

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (150) = 300\).

Time = 0.90 (sec) , antiderivative size = 854, normalized size of antiderivative = 5.02 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)/(f*x^2+e)^3,x, algorithm="fricas")
 

Output:

[-1/32*((3*a^2*c*e^2*f + (3*a^2*c*f^3 - (4*a*b*c - a^2*d)*e*f^2)*x^4 - (4* 
a*b*c - a^2*d)*e^3 + 2*(3*a^2*c*e*f^2 - (4*a*b*c - a^2*d)*e^2*f)*x^2)*sqrt 
(b*e^2 - a*e*f)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*( 
4*a*b*e^2 - 3*a^2*e*f)*x^2 + 4*((2*b*e - a*f)*x^3 + a*e*x)*sqrt(b*e^2 - a* 
e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2)) - 4*((2*b^2*d*e^4 + 3*a 
^2*c*e*f^3 + (2*b^2*c - 3*a*b*d)*e^3*f - (5*a*b*c - a^2*d)*e^2*f^2)*x^3 + 
(5*a^2*c*e^2*f^2 + (4*b^2*c + a*b*d)*e^4 - (9*a*b*c + a^2*d)*e^3*f)*x)*sqr 
t(b*x^2 + a))/(b^2*e^7 - 2*a*b*e^6*f + a^2*e^5*f^2 + (b^2*e^5*f^2 - 2*a*b* 
e^4*f^3 + a^2*e^3*f^4)*x^4 + 2*(b^2*e^6*f - 2*a*b*e^5*f^2 + a^2*e^4*f^3)*x 
^2), 1/16*((3*a^2*c*e^2*f + (3*a^2*c*f^3 - (4*a*b*c - a^2*d)*e*f^2)*x^4 - 
(4*a*b*c - a^2*d)*e^3 + 2*(3*a^2*c*e*f^2 - (4*a*b*c - a^2*d)*e^2*f)*x^2)*s 
qrt(-b*e^2 + a*e*f)*arctan(1/2*sqrt(-b*e^2 + a*e*f)*((2*b*e - a*f)*x^2 + a 
*e)*sqrt(b*x^2 + a)/((b^2*e^2 - a*b*e*f)*x^3 + (a*b*e^2 - a^2*e*f)*x)) + 2 
*((2*b^2*d*e^4 + 3*a^2*c*e*f^3 + (2*b^2*c - 3*a*b*d)*e^3*f - (5*a*b*c - a^ 
2*d)*e^2*f^2)*x^3 + (5*a^2*c*e^2*f^2 + (4*b^2*c + a*b*d)*e^4 - (9*a*b*c + 
a^2*d)*e^3*f)*x)*sqrt(b*x^2 + a))/(b^2*e^7 - 2*a*b*e^6*f + a^2*e^5*f^2 + ( 
b^2*e^5*f^2 - 2*a*b*e^4*f^3 + a^2*e^3*f^4)*x^4 + 2*(b^2*e^6*f - 2*a*b*e^5* 
f^2 + a^2*e^4*f^3)*x^2)]
 

Sympy [F(-1)]

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

integrate((b*x**2+a)**(1/2)*(d*x**2+c)/(f*x**2+e)**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)/(f*x^2+e)^3,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (150) = 300\).

Time = 0.35 (sec) , antiderivative size = 859, normalized size of antiderivative = 5.05 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)/(f*x^2+e)^3,x, algorithm="giac")
 

Output:

-1/8*(4*a*b^(3/2)*c*e - a^2*sqrt(b)*d*e - 3*a^2*sqrt(b)*c*f)*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))/ 
(sqrt(-b^2*e^2 + a*b*e*f)*(b*e^3 - a*e^2*f)) + 1/4*(8*(sqrt(b)*x - sqrt(b* 
x^2 + a))^6*b^(5/2)*d*e^3*f - 8*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)* 
d*e^2*f^2 - 4*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c*e*f^3 + (sqrt(b) 
*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d*e*f^3 + 3*(sqrt(b)*x - sqrt(b*x^2 + 
a))^6*a^2*sqrt(b)*c*f^4 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*d*e^4 
 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c*e^3*f - 24*(sqrt(b)*x - sq 
rt(b*x^2 + a))^4*a*b^(5/2)*d*e^3*f - 40*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a* 
b^(5/2)*c*e^2*f^2 + 14*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*d*e^2*f 
^2 + 30*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c*e*f^3 - 3*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^4*a^3*sqrt(b)*d*e*f^3 - 9*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^4*a^3*sqrt(b)*c*f^4 + 8*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*d*e^ 
3*f + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c*e^2*f^2 - 8*(sqrt(b 
)*x - sqrt(b*x^2 + a))^2*a^3*b^(3/2)*d*e^2*f^2 - 28*(sqrt(b)*x - sqrt(b*x^ 
2 + a))^2*a^3*b^(3/2)*c*e*f^3 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*sqrt 
(b)*d*e*f^3 + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*sqrt(b)*c*f^4 + 2*a^4* 
b^(3/2)*d*e^2*f^2 + 2*a^4*b^(3/2)*c*e*f^3 - a^5*sqrt(b)*d*e*f^3 - 3*a^5*sq 
rt(b)*c*f^4)/((b*e^3*f^2 - a*e^2*f^3)*((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)...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 2655, normalized size of antiderivative = 15.62 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x 
**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**3*c*e**2*f**3 - 12*sqrt(e) 
*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f 
)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**3*c*e*f**4*x**2 - 6*sqrt(e)*sqrt(a*f - 
b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x) 
/(sqrt(e)*sqrt(b)))*a**3*c*f**5*x**4 - 2*sqrt(e)*sqrt(a*f - b*e)*atan((sqr 
t(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt 
(b)))*a**3*d*e**3*f**2 - 4*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - 
 sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**3*d*e 
**2*f**3*x**2 - 2*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)* 
sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**3*d*e*f**4*x** 
4 + 20*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b* 
x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b*c*e**3*f**2 + 40*sqrt 
(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqr 
t(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b*c*e**2*f**3*x**2 + 20*sqrt(e)*sq 
rt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*s 
qrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b*c*e*f**4*x**4 + 4*sqrt(e)*sqrt(a*f - b 
*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/ 
(sqrt(e)*sqrt(b)))*a**2*b*d*e**4*f + 8*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt( 
a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqr...