\(\int \frac {(a+b x^2)^{3/2} (c+d x^2)}{e+f x^2} \, dx\) [293]

Optimal result

Integrand size = 28, antiderivative size = 185 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{e+f x^2} \, dx=-\frac {(4 b d e-4 b c f-3 a d f) x \sqrt {a+b x^2}}{8 f^2}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 f}+\frac {\left (3 a^2 d f^2+8 b^2 e (d e-c f)-12 a b f (d e-c f)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b} f^3}-\frac {(b e-a f)^{3/2} (d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f^3} \] Output:

-1/8*(-3*a*d*f-4*b*c*f+4*b*d*e)*x*(b*x^2+a)^(1/2)/f^2+1/4*d*x*(b*x^2+a)^(3 
/2)/f+1/8*(3*a^2*d*f^2+8*b^2*e*(-c*f+d*e)-12*a*b*f*(-c*f+d*e))*arctanh(b^( 
1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/f^3-(-a*f+b*e)^(3/2)*(-c*f+d*e)*arctanh((- 
a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(1/2)/f^3
 

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{e+f x^2} \, dx=\frac {f x \sqrt {a+b x^2} \left (5 a d f+b \left (-4 d e+4 c f+2 d f x^2\right )\right )+\frac {8 (-b e+a f)^{3/2} (d e-c f) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e}}-\frac {\left (3 a^2 d f^2+8 b^2 e (d e-c f)+12 a b f (-d e+c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{8 f^3} \] Input:

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

Output:

(f*x*Sqrt[a + b*x^2]*(5*a*d*f + b*(-4*d*e + 4*c*f + 2*d*f*x^2)) + (8*(-(b* 
e) + a*f)^(3/2)*(d*e - c*f)*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + 
f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f])])/Sqrt[e] - ((3*a^2*d*f^2 + 8*b^2*e*( 
d*e - c*f) + 12*a*b*f*(-(d*e) + c*f))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]) 
/Sqrt[b])/(8*f^3)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {403, 25, 403, 398, 224, 219, 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.

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {d x \left (a+b x^2\right )^{3/2}}{4 f}-\frac {\frac {\int \frac {a (a f (5 d e-8 c f)-4 b e (d e-c f))-\left (8 e (d e-c f) b^2-12 a f (d e-c f) b+3 a^2 d f^2\right ) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 f}+\frac {x \sqrt {a+b x^2} (-3 a d f-4 b c f+4 b d e)}{2 f}}{4 f}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {d x \left (a+b x^2\right )^{3/2}}{4 f}-\frac {\frac {\frac {8 (b e-a f)^2 (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {\left (3 a^2 d f^2-12 a b f (d e-c f)+8 b^2 e (d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{f}}{2 f}+\frac {x \sqrt {a+b x^2} (-3 a d f-4 b c f+4 b d e)}{2 f}}{4 f}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d x \left (a+b x^2\right )^{3/2}}{4 f}-\frac {\frac {\frac {8 (b e-a f)^2 (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {\left (3 a^2 d f^2-12 a b f (d e-c f)+8 b^2 e (d e-c f)\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}}{2 f}+\frac {x \sqrt {a+b x^2} (-3 a d f-4 b c f+4 b d e)}{2 f}}{4 f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d x \left (a+b x^2\right )^{3/2}}{4 f}-\frac {\frac {\frac {8 (b e-a f)^2 (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d f^2-12 a b f (d e-c f)+8 b^2 e (d e-c f)\right )}{\sqrt {b} f}}{2 f}+\frac {x \sqrt {a+b x^2} (-3 a d f-4 b c f+4 b d e)}{2 f}}{4 f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d x \left (a+b x^2\right )^{3/2}}{4 f}-\frac {\frac {\frac {8 (b e-a f)^2 (d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d f^2-12 a b f (d e-c f)+8 b^2 e (d e-c f)\right )}{\sqrt {b} f}}{2 f}+\frac {x \sqrt {a+b x^2} (-3 a d f-4 b c f+4 b d e)}{2 f}}{4 f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d x \left (a+b x^2\right )^{3/2}}{4 f}-\frac {\frac {\frac {8 (b e-a f)^{3/2} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d f^2-12 a b f (d e-c f)+8 b^2 e (d e-c f)\right )}{\sqrt {b} f}}{2 f}+\frac {x \sqrt {a+b x^2} (-3 a d f-4 b c f+4 b d e)}{2 f}}{4 f}\)

Input:

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

Output:

(d*x*(a + b*x^2)^(3/2))/(4*f) - (((4*b*d*e - 4*b*c*f - 3*a*d*f)*x*Sqrt[a + 
 b*x^2])/(2*f) + (-(((3*a^2*d*f^2 + 8*b^2*e*(d*e - c*f) - 12*a*b*f*(d*e - 
c*f))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*f)) + (8*(b*e - a*f)^ 
(3/2)*(d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/ 
(Sqrt[e]*f))/(2*f))/(4*f)
 

Defintions of rubi rules used

rule 25

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

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 398

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

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]
 

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(-\frac {-\frac {f \sqrt {b \,x^{2}+a}\, \left (2 b d f \,x^{2}+5 a d f +4 b c f -4 b d e \right ) x}{4}-\frac {\left (3 a^{2} d \,f^{2}+12 a b c \,f^{2}-12 a b d e f -8 b^{2} c e f +8 b^{2} d \,e^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{4 \sqrt {b}}+\frac {2 \left (a f -b e \right )^{2} \left (c f -d e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{\sqrt {\left (a f -b e \right ) e}}}{2 f^{3}}\)	\(167\)
risch	\(\frac {x \left (2 b d f \,x^{2}+5 a d f +4 b c f -4 b d e \right ) \sqrt {b \,x^{2}+a}}{8 f^{2}}+\frac {\frac {\left (3 a^{2} d \,f^{2}+12 a b c \,f^{2}-12 a b d e f -8 b^{2} c e f +8 b^{2} d \,e^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{f \sqrt {b}}+\frac {4 \left (a^{2} c \,f^{3}-a^{2} d e \,f^{2}-2 a b c e \,f^{2}+2 a b d \,e^{2} f +b^{2} c \,e^{2} f -b^{2} d \,e^{3}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}-\frac {4 \left (a^{2} c \,f^{3}-a^{2} d e \,f^{2}-2 a b c e \,f^{2}+2 a b d \,e^{2} f +b^{2} c \,e^{2} f -b^{2} d \,e^{3}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}}{8 f^{2}}\)	\(526\)
default	\(\text {Expression too large to display}\)	\(1305\)

Input:

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

Output:

-1/2/f^3*(-1/4*f*(b*x^2+a)^(1/2)*(2*b*d*f*x^2+5*a*d*f+4*b*c*f-4*b*d*e)*x-1 
/4*(3*a^2*d*f^2+12*a*b*c*f^2-12*a*b*d*e*f-8*b^2*c*e*f+8*b^2*d*e^2)/b^(1/2) 
*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+2*(a*f-b*e)^2*(c*f-d*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 [A] (verification not implemented)

Time = 7.53 (sec) , antiderivative size = 1107, normalized size of antiderivative = 5.98 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{e+f x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/16*((8*b^2*d*e^2 - 4*(2*b^2*c + 3*a*b*d)*e*f + 3*(4*a*b*c + a^2*d)*f^2) 
*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 4*(b^2*d*e^2 + 
a*b*c*f^2 - (b^2*c + a*b*d)*e*f)*sqrt((b*e - a*f)/e)*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*(a*e^2 
*x + (2*b*e^2 - a*e*f)*x^3)*sqrt(b*x^2 + a)*sqrt((b*e - a*f)/e))/(f^2*x^4 
+ 2*e*f*x^2 + e^2)) + 2*(2*b^2*d*f^2*x^3 - (4*b^2*d*e*f - (4*b^2*c + 5*a*b 
*d)*f^2)*x)*sqrt(b*x^2 + a))/(b*f^3), -1/8*((8*b^2*d*e^2 - 4*(2*b^2*c + 3* 
a*b*d)*e*f + 3*(4*a*b*c + a^2*d)*f^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^ 
2 + a)) - 2*(b^2*d*e^2 + a*b*c*f^2 - (b^2*c + a*b*d)*e*f)*sqrt((b*e - a*f) 
/e)*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*(a*e^2*x + (2*b*e^2 - a*e*f)*x^3)*sqrt(b*x^2 + a)*sqrt( 
(b*e - a*f)/e))/(f^2*x^4 + 2*e*f*x^2 + e^2)) - (2*b^2*d*f^2*x^3 - (4*b^2*d 
*e*f - (4*b^2*c + 5*a*b*d)*f^2)*x)*sqrt(b*x^2 + a))/(b*f^3), 1/16*(8*(b^2* 
d*e^2 + a*b*c*f^2 - (b^2*c + a*b*d)*e*f)*sqrt(-(b*e - a*f)/e)*arctan(1/2*( 
(2*b*e - a*f)*x^2 + a*e)*sqrt(b*x^2 + a)*sqrt(-(b*e - a*f)/e)/((b^2*e - a* 
b*f)*x^3 + (a*b*e - a^2*f)*x)) + (8*b^2*d*e^2 - 4*(2*b^2*c + 3*a*b*d)*e*f 
+ 3*(4*a*b*c + a^2*d)*f^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b 
)*x - a) + 2*(2*b^2*d*f^2*x^3 - (4*b^2*d*e*f - (4*b^2*c + 5*a*b*d)*f^2)*x) 
*sqrt(b*x^2 + a))/(b*f^3), -1/8*((8*b^2*d*e^2 - 4*(2*b^2*c + 3*a*b*d)*e*f 
+ 3*(4*a*b*c + a^2*d)*f^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) ...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 697, normalized size of antiderivative = 3.77 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{e+f x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 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)*sqrt(b)))*a*b*c*f**2 + 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)*sqrt(b)))*a*b*d*e*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)*sqrt(b 
)))*b**2*c*e*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)*sqrt(b)))*b**2*d*e**2 - 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)*sqrt(b)))*a*b*c*f**2 + 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)*sqrt(b)))*a*b*d*e*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)*sqrt(b)))*b* 
*2*c*e*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)*sqrt(b)))*b**2*d*e**2 + 5*sqrt(a 
 + b*x**2)*a*b*d*e*f**2*x + 4*sqrt(a + b*x**2)*b**2*c*e*f**2*x - 4*sqrt(a 
+ b*x**2)*b**2*d*e**2*f*x + 2*sqrt(a + b*x**2)*b**2*d*e*f**2*x**3 + 3*sqrt 
(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*d*e*f**2 + 12*sqrt(b) 
*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*c*e*f**2 - 12*sqrt(b)*log 
((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*d*e**2*f - 8*sqrt(b)*log((sqr 
t(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b**2*c*e**2*f + 8*sqrt(b)*log((sqrt...