\(\int \frac {A+B x^2+C x^4}{(d+e x^2)^{7/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx\) [152]

Optimal result

Integrand size = 49, antiderivative size = 464 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{7/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=-\frac {\left (C d^2-B d e+A e^2\right ) x \sqrt {a d+(b d+a e) x^2+b e x^4}}{6 d e (b d-a e) \left (d+e x^2\right )^{7/2}}+\frac {\left (2 b d \left (C d^2+e (2 B d-5 A e)\right )-a e \left (7 C d^2-e (B d+5 A e)\right )\right ) x \sqrt {a d+(b d+a e) x^2+b e x^4}}{24 d^2 e (b d-a e)^2 \left (d+e x^2\right )^{5/2}}+\frac {\left (4 b^2 d^2 \left (C d^2+e (2 B d-11 A e)\right )-3 a^2 e^2 \left (C d^2+e (B d+5 A e)\right )-2 a b d e \left (8 C d^2-e (5 B d+22 A e)\right )\right ) x \sqrt {a d+(b d+a e) x^2+b e x^4}}{48 d^3 e (b d-a e)^3 \left (d+e x^2\right )^{3/2}}+\frac {\left (A \left (16 b^3 d^3-24 a b^2 d^2 e+18 a^2 b d e^2-5 a^3 e^3\right )-a d \left (8 b^2 B d^2+a^2 e (C d+B e)-2 a b d (3 C d+2 B e)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b d-a e} x \sqrt {d+e x^2}}{\sqrt {d} \sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{16 d^{7/2} (b d-a e)^{7/2}} \] Output:

-1/6*(A*e^2-B*d*e+C*d^2)*x*(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)/d/e/(-a*e+b*d 
)/(e*x^2+d)^(7/2)+1/24*(2*b*d*(C*d^2+e*(-5*A*e+2*B*d))-a*e*(7*C*d^2-e*(5*A 
*e+B*d)))*x*(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)/d^2/e/(-a*e+b*d)^2/(e*x^2+d) 
^(5/2)+1/48*(4*b^2*d^2*(C*d^2+e*(-11*A*e+2*B*d))-3*a^2*e^2*(C*d^2+e*(5*A*e 
+B*d))-2*a*b*d*e*(8*C*d^2-e*(22*A*e+5*B*d)))*x*(a*d+(a*e+b*d)*x^2+b*e*x^4) 
^(1/2)/d^3/e/(-a*e+b*d)^3/(e*x^2+d)^(3/2)+1/16*(A*(-5*a^3*e^3+18*a^2*b*d*e 
^2-24*a*b^2*d^2*e+16*b^3*d^3)-a*d*(8*b^2*B*d^2+a^2*e*(B*e+C*d)-2*a*b*d*(2* 
B*e+3*C*d)))*arctanh((-a*e+b*d)^(1/2)*x*(e*x^2+d)^(1/2)/d^(1/2)/(a*d+(a*e+ 
b*d)*x^2+b*e*x^4)^(1/2))/d^(7/2)/(-a*e+b*d)^(7/2)
 

Mathematica [A] (verified)

Time = 13.08 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{7/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {x \left (-24 C d^3 (b d-a e)^2 \left (d+e x^2\right )^2 \left (e \left (a+b x^2\right )-\frac {(2 b d-a e) \left (d+e x^2\right ) \text {arctanh}\left (\sqrt {\frac {(b d-a e) x^2}{d \left (a+b x^2\right )}}\right )}{d \sqrt {\frac {(b d-a e) x^2}{d \left (a+b x^2\right )}}}\right )+6 d (b d-a e) (2 C d-B e) \left (d+e x^2\right ) \left (d e \left (a+b x^2\right ) \left (2 b d \left (4 d+3 e x^2\right )-a e \left (5 d+3 e x^2\right )\right )-\frac {\left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right ) \left (d+e x^2\right )^2 \text {arctanh}\left (\sqrt {\frac {(b d-a e) x^2}{d \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b d-a e) x^2}{d \left (a+b x^2\right )}}}\right )-\left (C d^2+e (-B d+A e)\right ) \left (d e \left (a+b x^2\right ) \left (4 b^2 d^2 \left (18 d^2+27 d e x^2+11 e^2 x^4\right )+a^2 e^2 \left (33 d^2+40 d e x^2+15 e^2 x^4\right )-2 a b d e \left (45 d^2+59 d e x^2+22 e^2 x^4\right )\right )-\frac {3 (2 b d-a e) \left (8 b^2 d^2-8 a b d e+5 a^2 e^2\right ) \left (d+e x^2\right )^3 \text {arctanh}\left (\sqrt {\frac {(b d-a e) x^2}{d \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b d-a e) x^2}{d \left (a+b x^2\right )}}}\right )\right )}{48 d^4 e^2 (b d-a e)^3 \left (d+e x^2\right )^{5/2} \sqrt {\left (a+b x^2\right ) \left (d+e x^2\right )}} \] Input:

Integrate[(A + B*x^2 + C*x^4)/((d + e*x^2)^(7/2)*Sqrt[a*d + (b*d + a*e)*x^ 
2 + b*e*x^4]),x]
 

Output:

(x*(-24*C*d^3*(b*d - a*e)^2*(d + e*x^2)^2*(e*(a + b*x^2) - ((2*b*d - a*e)* 
(d + e*x^2)*ArcTanh[Sqrt[((b*d - a*e)*x^2)/(d*(a + b*x^2))]])/(d*Sqrt[((b* 
d - a*e)*x^2)/(d*(a + b*x^2))])) + 6*d*(b*d - a*e)*(2*C*d - B*e)*(d + e*x^ 
2)*(d*e*(a + b*x^2)*(2*b*d*(4*d + 3*e*x^2) - a*e*(5*d + 3*e*x^2)) - ((8*b^ 
2*d^2 - 8*a*b*d*e + 3*a^2*e^2)*(d + e*x^2)^2*ArcTanh[Sqrt[((b*d - a*e)*x^2 
)/(d*(a + b*x^2))]])/Sqrt[((b*d - a*e)*x^2)/(d*(a + b*x^2))]) - (C*d^2 + e 
*(-(B*d) + A*e))*(d*e*(a + b*x^2)*(4*b^2*d^2*(18*d^2 + 27*d*e*x^2 + 11*e^2 
*x^4) + a^2*e^2*(33*d^2 + 40*d*e*x^2 + 15*e^2*x^4) - 2*a*b*d*e*(45*d^2 + 5 
9*d*e*x^2 + 22*e^2*x^4)) - (3*(2*b*d - a*e)*(8*b^2*d^2 - 8*a*b*d*e + 5*a^2 
*e^2)*(d + e*x^2)^3*ArcTanh[Sqrt[((b*d - a*e)*x^2)/(d*(a + b*x^2))]])/Sqrt 
[((b*d - a*e)*x^2)/(d*(a + b*x^2))])))/(48*d^4*e^2*(b*d - a*e)^3*(d + e*x^ 
2)^(5/2)*Sqrt[(a + b*x^2)*(d + e*x^2)])
 

Rubi [A] (verified)

Time = 1.30 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.42, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1395, 7293, 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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{7/2} \sqrt {x^2 (a e+b d)+a d+b e x^4}} \, dx\)

\(\Big \downarrow \) 1395

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \int \frac {C x^4+B x^2+A}{\sqrt {b x^2+a} \left (e x^2+d\right )^4}dx}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {(2 b d-a e) \left (5 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \left (A e^2-B d e+C d^2\right ) \text {arctanh}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{16 d^{7/2} e^2 (b d-a e)^{7/2}}-\frac {x \sqrt {a+b x^2} \left (15 a^2 e^2-44 a b d e+44 b^2 d^2\right ) \left (A e^2-B d e+C d^2\right )}{48 d^3 e \left (d+e x^2\right ) (b d-a e)^3}-\frac {\left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) (2 C d-B e) \text {arctanh}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{8 d^{5/2} e^2 (b d-a e)^{5/2}}-\frac {5 x \sqrt {a+b x^2} (2 b d-a e) \left (A e^2-B d e+C d^2\right )}{24 d^2 e \left (d+e x^2\right )^2 (b d-a e)^2}-\frac {x \sqrt {a+b x^2} \left (A e^2-B d e+C d^2\right )}{6 d e \left (d+e x^2\right )^3 (b d-a e)}+\frac {C (2 b d-a e) \text {arctanh}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{2 d^{3/2} e^2 (b d-a e)^{3/2}}+\frac {3 x \sqrt {a+b x^2} (2 b d-a e) (2 C d-B e)}{8 d^2 e \left (d+e x^2\right ) (b d-a e)^2}+\frac {x \sqrt {a+b x^2} (2 C d-B e)}{4 d e \left (d+e x^2\right )^2 (b d-a e)}-\frac {C x \sqrt {a+b x^2}}{2 d e \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

Input:

Int[(A + B*x^2 + C*x^4)/((d + e*x^2)^(7/2)*Sqrt[a*d + (b*d + a*e)*x^2 + b* 
e*x^4]),x]
 

Output:

(Sqrt[a + b*x^2]*Sqrt[d + e*x^2]*(-1/6*((C*d^2 - B*d*e + A*e^2)*x*Sqrt[a + 
 b*x^2])/(d*e*(b*d - a*e)*(d + e*x^2)^3) + ((2*C*d - B*e)*x*Sqrt[a + b*x^2 
])/(4*d*e*(b*d - a*e)*(d + e*x^2)^2) - (5*(2*b*d - a*e)*(C*d^2 - B*d*e + A 
*e^2)*x*Sqrt[a + b*x^2])/(24*d^2*e*(b*d - a*e)^2*(d + e*x^2)^2) - (C*x*Sqr 
t[a + b*x^2])/(2*d*e*(b*d - a*e)*(d + e*x^2)) + (3*(2*b*d - a*e)*(2*C*d - 
B*e)*x*Sqrt[a + b*x^2])/(8*d^2*e*(b*d - a*e)^2*(d + e*x^2)) - ((44*b^2*d^2 
 - 44*a*b*d*e + 15*a^2*e^2)*(C*d^2 - B*d*e + A*e^2)*x*Sqrt[a + b*x^2])/(48 
*d^3*e*(b*d - a*e)^3*(d + e*x^2)) + (C*(2*b*d - a*e)*ArcTanh[(Sqrt[b*d - a 
*e]*x)/(Sqrt[d]*Sqrt[a + b*x^2])])/(2*d^(3/2)*e^2*(b*d - a*e)^(3/2)) - ((2 
*C*d - B*e)*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)*ArcTanh[(Sqrt[b*d - a*e]*x 
)/(Sqrt[d]*Sqrt[a + b*x^2])])/(8*d^(5/2)*e^2*(b*d - a*e)^(5/2)) + ((2*b*d 
- a*e)*(8*b^2*d^2 - 8*a*b*d*e + 5*a^2*e^2)*(C*d^2 - B*d*e + A*e^2)*ArcTanh 
[(Sqrt[b*d - a*e]*x)/(Sqrt[d]*Sqrt[a + b*x^2])])/(16*d^(7/2)*e^2*(b*d - a* 
e)^(7/2))))/Sqrt[a*d + (b*d + a*e)*x^2 + b*e*x^4]
 

Defintions of rubi rules used

rule 1395

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( 
x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d 
+ e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p 
+ q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E 
qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 
1] && EqQ[n, 2])
 

rule 2009

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(10035\) vs. \(2(432)=864\).

Time = 0.46 (sec) , antiderivative size = 10036, normalized size of antiderivative = 21.63

method	result	size
default	\(\text {Expression too large to display}\)	\(10036\)

Input:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x,me 
thod=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1247 vs. \(2 (432) = 864\).

Time = 0.23 (sec) , antiderivative size = 2520, normalized size of antiderivative = 5.43 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{7/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2 
),x, algorithm="fricas")
 

Output:

[-1/96*(3*(5*A*a^3*d^4*e^3 + (5*A*a^3*e^7 - 2*(3*C*a^2*b - 4*B*a*b^2 + 8*A 
*b^3)*d^3*e^4 + (C*a^3 - 4*B*a^2*b + 24*A*a*b^2)*d^2*e^5 + (B*a^3 - 18*A*a 
^2*b)*d*e^6)*x^8 - 2*(3*C*a^2*b - 4*B*a*b^2 + 8*A*b^3)*d^7 + (C*a^3 - 4*B* 
a^2*b + 24*A*a*b^2)*d^6*e + (B*a^3 - 18*A*a^2*b)*d^5*e^2 + 4*(5*A*a^3*d*e^ 
6 - 2*(3*C*a^2*b - 4*B*a*b^2 + 8*A*b^3)*d^4*e^3 + (C*a^3 - 4*B*a^2*b + 24* 
A*a*b^2)*d^3*e^4 + (B*a^3 - 18*A*a^2*b)*d^2*e^5)*x^6 + 6*(5*A*a^3*d^2*e^5 
- 2*(3*C*a^2*b - 4*B*a*b^2 + 8*A*b^3)*d^5*e^2 + (C*a^3 - 4*B*a^2*b + 24*A* 
a*b^2)*d^4*e^3 + (B*a^3 - 18*A*a^2*b)*d^3*e^4)*x^4 + 4*(5*A*a^3*d^3*e^4 - 
2*(3*C*a^2*b - 4*B*a*b^2 + 8*A*b^3)*d^6*e + (C*a^3 - 4*B*a^2*b + 24*A*a*b^ 
2)*d^5*e^2 + (B*a^3 - 18*A*a^2*b)*d^4*e^3)*x^2)*sqrt(b*d^2 - a*d*e)*log((2 
*b*d^2*x^2 + (2*b*d*e - a*e^2)*x^4 + a*d^2 + 2*sqrt(b*e*x^4 + (b*d + a*e)* 
x^2 + a*d)*sqrt(b*d^2 - a*d*e)*sqrt(e*x^2 + d)*x)/(e^2*x^4 + 2*d*e*x^2 + d 
^2)) - 2*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*((4*C*b^3*d^6*e + 15*A*a^3* 
d*e^6 - 4*(5*C*a*b^2 - 2*B*b^3)*d^5*e^2 + (13*C*a^2*b + 2*B*a*b^2 - 44*A*b 
^3)*d^4*e^3 + (3*C*a^3 - 13*B*a^2*b + 88*A*a*b^2)*d^3*e^4 + (3*B*a^3 - 59* 
A*a^2*b)*d^2*e^5)*x^5 + 2*(6*C*b^3*d^7 + 20*A*a^3*d^2*e^5 - (31*C*a*b^2 - 
12*B*b^3)*d^6*e + (29*C*a^2*b - 5*B*a*b^2 - 54*A*b^3)*d^5*e^2 - (4*C*a^3 + 
 11*B*a^2*b - 113*A*a*b^2)*d^4*e^3 + (4*B*a^3 - 79*A*a^2*b)*d^3*e^4)*x^3 + 
 3*(11*A*a^3*d^3*e^4 - 2*(3*C*a*b^2 - 4*B*b^3)*d^7 + (7*C*a^2*b - 12*B*a*b 
^2 - 24*A*b^3)*d^6*e - (C*a^3 - 5*B*a^2*b - 54*A*a*b^2)*d^5*e^2 - (B*a^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{7/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\text {Timed out} \] Input:

integrate((C*x**4+B*x**2+A)/(e*x**2+d)**(7/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4 
)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{7/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} {\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2 
),x, algorithm="maxima")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*(e*x^ 
2 + d)^(7/2)), x)
 

Giac [F]

\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{7/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} {\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2 
),x, algorithm="giac")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*(e*x^ 
2 + d)^(7/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{7/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (e\,x^2+d\right )}^{7/2}\,\sqrt {b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d}} \,d x \] Input:

int((A + B*x^2 + C*x^4)/((d + e*x^2)^(7/2)*(a*d + x^2*(a*e + b*d) + b*e*x^ 
4)^(1/2)),x)
 

Output:

int((A + B*x^2 + C*x^4)/((d + e*x^2)^(7/2)*(a*d + x^2*(a*e + b*d) + b*e*x^ 
4)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 1.55 (sec) , antiderivative size = 5903, normalized size of antiderivative = 12.72 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{7/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx =\text {Too large to display} \] Input:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x)
 

Output:

( - 15*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b* 
x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**5*d**3*e**6 - 45*sqrt(d)* 
sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e) 
*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**5*d**2*e**7*x**2 - 45*sqrt(d)*sqrt(a*e - 
 b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x 
)/(sqrt(d)*sqrt(b)))*a**5*d*e**8*x**4 - 15*sqrt(d)*sqrt(a*e - b*d)*atan((s 
qrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sq 
rt(b)))*a**5*e**9*x**6 + 81*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) 
- sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**4*b* 
d**4*e**5 + 243*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sq 
rt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**4*b*d**3*e**6*x* 
*2 + 243*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + 
b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**4*b*d**2*e**7*x**4 + 81 
*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) 
- sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**4*b*d*e**8*x**6 - 3*sqrt(d)*sqr 
t(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sq 
rt(b)*x)/(sqrt(d)*sqrt(b)))*a**4*c*d**5*e**4 - 9*sqrt(d)*sqrt(a*e - b*d)*a 
tan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt 
(d)*sqrt(b)))*a**4*c*d**4*e**5*x**2 - 9*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt 
(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sq...