\(\int \frac {1}{(d+e x^2)^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx\) [7]

Optimal result

Integrand size = 37, antiderivative size = 220 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=-\frac {e x \sqrt {a d+(b d+a e) x^2+b e x^4}}{4 d (b d-a e) \left (d+e x^2\right )^{5/2}}-\frac {3 e (2 b d-a e) x \sqrt {a d+(b d+a e) x^2+b e x^4}}{8 d^2 (b d-a e)^2 \left (d+e x^2\right )^{3/2}}+\frac {\left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {b d-a e} x \sqrt {d+e x^2}}{\sqrt {d} \sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{8 d^{5/2} (b d-a e)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {-\frac {\sqrt {d} e x \left (a+b x^2\right ) \left (2 b d \left (4 d+3 e x^2\right )-a e \left (5 d+3 e x^2\right )\right )}{(b d-a e)^2}-\frac {\left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right ) \sqrt {a+b x^2} \left (d+e x^2\right )^2 \arctan \left (\frac {-e x \sqrt {a+b x^2}+\sqrt {b} \left (d+e x^2\right )}{\sqrt {d} \sqrt {-b d+a e}}\right )}{(-b d+a e)^{5/2}}}{8 d^{5/2} \left (d+e x^2\right )^{3/2} \sqrt {\left (a+b x^2\right ) \left (d+e x^2\right )}} \] Input:

Integrate[1/((d + e*x^2)^(5/2)*Sqrt[a*d + (b*d + a*e)*x^2 + b*e*x^4]),x]
 

Output:

(-((Sqrt[d]*e*x*(a + b*x^2)*(2*b*d*(4*d + 3*e*x^2) - a*e*(5*d + 3*e*x^2))) 
/(b*d - a*e)^2) - ((8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)*Sqrt[a + b*x^2]*(d 
+ e*x^2)^2*ArcTan[(-(e*x*Sqrt[a + b*x^2]) + Sqrt[b]*(d + e*x^2))/(Sqrt[d]* 
Sqrt[-(b*d) + a*e])])/(-(b*d) + a*e)^(5/2))/(8*d^(5/2)*(d + e*x^2)^(3/2)*S 
qrt[(a + b*x^2)*(d + e*x^2)])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1395, 316, 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.

\(\displaystyle \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {x^2 (a e+b d)+a d+b e x^4}} \, dx\)

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {8 b^2 d^2-8 a b e d+3 a^2 e^2}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{2 d (b d-a e)}-\frac {3 e x \sqrt {a+b x^2} (2 b d-a e)}{2 d \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}-\frac {e x \sqrt {a+b x^2}}{4 d \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{2 d (b d-a e)}-\frac {3 e x \sqrt {a+b x^2} (2 b d-a e)}{2 d \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}-\frac {e x \sqrt {a+b x^2}}{4 d \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \int \frac {1}{d-\frac {(b d-a e) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 d (b d-a e)}-\frac {3 e x \sqrt {a+b x^2} (2 b d-a e)}{2 d \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}-\frac {e x \sqrt {a+b x^2}}{4 d \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \text {arctanh}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{2 d^{3/2} (b d-a e)^{3/2}}-\frac {3 e x \sqrt {a+b x^2} (2 b d-a e)}{2 d \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}-\frac {e x \sqrt {a+b x^2}}{4 d \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

Input:

Int[1/((d + e*x^2)^(5/2)*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]*(-1/4*(e*x*Sqrt[a + b*x^2])/(d*(b*d - a*e 
)*(d + e*x^2)^2) + ((-3*e*(2*b*d - a*e)*x*Sqrt[a + b*x^2])/(2*d*(b*d - a*e 
)*(d + e*x^2)) + ((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])])/(2*d^(3/2)*(b*d - a*e)^(3/2)))/(4*d*(b 
*d - a*e))))/Sqrt[a*d + (b*d + a*e)*x^2 + b*e*x^4]
 

Defintions of rubi rules used

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 316

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2394\) vs. \(2(194)=388\).

Time = 0.45 (sec) , antiderivative size = 2395, normalized size of antiderivative = 10.89

method	result	size
default	\(\text {Expression too large to display}\)	\(2395\)

Input:

int(1/(e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x,method=_RETURNVE 
RBOSE)
 

Output:

-1/16*(-6*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e 
)/(e*x+(-d*e)^(1/2)))*a^3*d*e^4*x^2*b^(1/2)+6*ln(2*((b*x^2+a)^(1/2)*((a*e- 
b*d)/e)^(1/2)*e+(-d*e)^(1/2)*b*x+a*e)/(e*x-(-d*e)^(1/2)))*a^3*d*e^4*x^2*b^ 
(1/2)+8*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e)/ 
(e*x+(-d*e)^(1/2)))*b^(7/2)*d^3*e^2*x^4-8*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d) 
/e)^(1/2)*e+(-d*e)^(1/2)*b*x+a*e)/(e*x-(-d*e)^(1/2)))*b^(7/2)*d^3*e^2*x^4+ 
16*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e)/(e*x+ 
(-d*e)^(1/2)))*b^(7/2)*d^4*e*x^2-16*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1 
/2)*e+(-d*e)^(1/2)*b*x+a*e)/(e*x-(-d*e)^(1/2)))*b^(7/2)*d^4*e*x^2+11*ln(2* 
((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e)/(e*x+(-d*e)^( 
1/2)))*a^2*b^(3/2)*d^3*e^2-16*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e- 
(-d*e)^(1/2)*b*x+a*e)/(e*x+(-d*e)^(1/2)))*a*b^(5/2)*d^4*e-10*a^2*d*e^3*x*b 
^(1/2)*(-d*e)^(1/2)*((a*e-b*d)/e)^(1/2)*(b*x^2+a)^(1/2)+18*a*b^(3/2)*d*e^3 
*x^3*(-d*e)^(1/2)*((a*e-b*d)/e)^(1/2)*(b*x^2+a)^(1/2)+26*a*b^(3/2)*d^2*e^2 
*x*(-d*e)^(1/2)*((a*e-b*d)/e)^(1/2)*(b*x^2+a)^(1/2)+11*ln(2*((b*x^2+a)^(1/ 
2)*((a*e-b*d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e)/(e*x+(-d*e)^(1/2)))*a^2*b^( 
3/2)*d*e^4*x^4-16*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e-(-d*e)^(1/2) 
*b*x+a*e)/(e*x+(-d*e)^(1/2)))*a*b^(5/2)*d^2*e^3*x^4-11*ln(2*((b*x^2+a)^(1/ 
2)*((a*e-b*d)/e)^(1/2)*e+(-d*e)^(1/2)*b*x+a*e)/(e*x-(-d*e)^(1/2)))*a^2*b^( 
3/2)*d*e^4*x^4+16*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e+(-d*e)^(1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (194) = 388\).

Time = 0.10 (sec) , antiderivative size = 1068, normalized size of antiderivative = 4.85 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx =\text {Too large to display} \] Input:

integrate(1/(e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x, algorithm 
="fricas")
 

Output:

[1/16*((8*b^2*d^5 - 8*a*b*d^4*e + 3*a^2*d^3*e^2 + (8*b^2*d^2*e^3 - 8*a*b*d 
*e^4 + 3*a^2*e^5)*x^6 + 3*(8*b^2*d^3*e^2 - 8*a*b*d^2*e^3 + 3*a^2*d*e^4)*x^ 
4 + 3*(8*b^2*d^4*e - 8*a*b*d^3*e^2 + 3*a^2*d^2*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)*(3*(2*b^2*d^3*e^2 
- 3*a*b*d^2*e^3 + a^2*d*e^4)*x^3 + (8*b^2*d^4*e - 13*a*b*d^3*e^2 + 5*a^2*d 
^2*e^3)*x)*sqrt(e*x^2 + d))/(b^3*d^9 - 3*a*b^2*d^8*e + 3*a^2*b*d^7*e^2 - a 
^3*d^6*e^3 + (b^3*d^6*e^3 - 3*a*b^2*d^5*e^4 + 3*a^2*b*d^4*e^5 - a^3*d^3*e^ 
6)*x^6 + 3*(b^3*d^7*e^2 - 3*a*b^2*d^6*e^3 + 3*a^2*b*d^5*e^4 - a^3*d^4*e^5) 
*x^4 + 3*(b^3*d^8*e - 3*a*b^2*d^7*e^2 + 3*a^2*b*d^6*e^3 - a^3*d^5*e^4)*x^2 
), -1/8*((8*b^2*d^5 - 8*a*b*d^4*e + 3*a^2*d^3*e^2 + (8*b^2*d^2*e^3 - 8*a*b 
*d*e^4 + 3*a^2*e^5)*x^6 + 3*(8*b^2*d^3*e^2 - 8*a*b*d^2*e^3 + 3*a^2*d*e^4)* 
x^4 + 3*(8*b^2*d^4*e - 8*a*b*d^3*e^2 + 3*a^2*d^2*e^3)*x^2)*sqrt(-b*d^2 + a 
*d*e)*arctan(sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(-b*d^2 + a*d*e)*sq 
rt(e*x^2 + d)*x/(b*d*e*x^4 + a*d^2 + (b*d^2 + a*d*e)*x^2)) + sqrt(b*e*x^4 
+ (b*d + a*e)*x^2 + a*d)*(3*(2*b^2*d^3*e^2 - 3*a*b*d^2*e^3 + a^2*d*e^4)*x^ 
3 + (8*b^2*d^4*e - 13*a*b*d^3*e^2 + 5*a^2*d^2*e^3)*x)*sqrt(e*x^2 + d))/(b^ 
3*d^9 - 3*a*b^2*d^8*e + 3*a^2*b*d^7*e^2 - a^3*d^6*e^3 + (b^3*d^6*e^3 - 3*a 
*b^2*d^5*e^4 + 3*a^2*b*d^4*e^5 - a^3*d^3*e^6)*x^6 + 3*(b^3*d^7*e^2 - 3*...
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\int \frac {1}{\sqrt {\left (a + b x^{2}\right ) \left (d + e x^{2}\right )} \left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(e*x**2+d)**(5/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4)**(1/2),x)
 

Output:

Integral(1/(sqrt((a + b*x**2)*(d + e*x**2))*(d + e*x**2)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\int { \frac {1}{\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x, algorithm 
="maxima")
 

Output:

integrate(1/(sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*(e*x^2 + d)^(5/2)), x)
 

Giac [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\int { \frac {1}{\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x, algorithm 
="giac")
 

Output:

integrate(1/(sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*(e*x^2 + d)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\int \frac {1}{{\left (e\,x^2+d\right )}^{5/2}\,\sqrt {b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d}} \,d x \] Input:

int(1/((d + e*x^2)^(5/2)*(a*d + x^2*(a*e + b*d) + b*e*x^4)^(1/2)),x)
 

Output:

int(1/((d + e*x^2)^(5/2)*(a*d + x^2*(a*e + b*d) + b*e*x^4)^(1/2)), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 2020, normalized size of antiderivative = 9.18 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx =\text {Too large to display} \] Input:

int(1/(e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x)
 

Output:

( - 6*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**3*d**2*e**3 - 12*sqrt(d)*s 
qrt(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**3*d*e**4*x**2 - 6*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)/(sq 
rt(d)*sqrt(b)))*a**3*e**5*x**4 + 28*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**2*b*d**3*e**2 + 56*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqr 
t(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**2*b*d**2* 
e**3*x**2 + 28*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqr 
t(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**2*b*d*e**4*x**4 - 
 48*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*b**2*d**4*e - 96*sqrt(d)*sqrt 
(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqr 
t(b)*x)/(sqrt(d)*sqrt(b)))*a*b**2*d**3*e**2*x**2 - 48*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*b**2*d**2*e**3*x**4 + 32*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)))*b**3*d**5 + 64*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)))*b**3*d...