\(\int \frac {\sqrt {a+b x^2} (c+d x^2)}{e+f x^2} \, dx\) [263]

Optimal result

Integrand size = 28, antiderivative size = 128 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{e+f x^2} \, dx=\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {(2 b d e-2 b c f-a d f) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b} f^2}+\frac {\sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{e+f x^2} \, dx=\frac {d f x \sqrt {a+b x^2}+\frac {2 \sqrt {-b e+a f} (d e-c f) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e}}+\frac {(2 b d e-2 b c f-a d f) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{2 f^2} \] Input:

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

Output:

(d*f*x*Sqrt[a + b*x^2] + (2*Sqrt[-(b*e) + a*f]*(d*e - c*f)*ArcTan[(-(f*x*S 
qrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f])])/Sqrt 
[e] + ((2*b*d*e - 2*b*c*f - a*d*f)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/Sq 
rt[b])/(2*f^2)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {403, 25, 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.

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(d*x*Sqrt[a + b*x^2])/(2*f) - (((2*b*d*e - 2*b*c*f - a*d*f)*ArcTanh[(Sqrt[ 
b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*f) - (2*Sqrt[b*e - a*f]*(d*e - c*f)*ArcTa 
nh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f))/(2*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 = 1.40 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(-\frac {-\sqrt {b \,x^{2}+a}\, d f x -\frac {\left (a d f +2 b c f -2 b d e \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{\sqrt {b}}+\frac {2 \left (a f -b e \right ) \left (c f -d e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{\sqrt {\left (a f -b e \right ) e}}}{2 f^{2}}\)	\(115\)
risch	\(\frac {d x \sqrt {b \,x^{2}+a}}{2 f}+\frac {\frac {\left (a d f +2 b c f -2 b d e \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{f \sqrt {b}}+\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}-\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}}{2 f}\)	\(415\)
default	\(\frac {d \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{f}-\frac {\left (c f -d e \right ) \left (\sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}-\frac {\sqrt {b}\, \sqrt {-e f}\, \ln \left (\frac {-\frac {b \sqrt {-e f}}{f}+b \left (x +\frac {\sqrt {-e f}}{f}\right )}{\sqrt {b}}+\sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}\right )}{f}-\frac {\left (a f -b e \right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{f \sqrt {\frac {a f -b e}{f}}}\right )}{2 \sqrt {-e f}\, f}+\frac {\left (c f -d e \right ) \left (\sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}+\frac {\sqrt {b}\, \sqrt {-e f}\, \ln \left (\frac {\frac {b \sqrt {-e f}}{f}+b \left (x -\frac {\sqrt {-e f}}{f}\right )}{\sqrt {b}}+\sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}\right )}{f}-\frac {\left (a f -b e \right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{f \sqrt {\frac {a f -b e}{f}}}\right )}{2 \sqrt {-e f}\, f}\)	\(707\)

Input:

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

Output:

-1/2/f^2*(-(b*x^2+a)^(1/2)*d*f*x-(a*d*f+2*b*c*f-2*b*d*e)/b^(1/2)*arctanh(( 
b*x^2+a)^(1/2)/x/b^(1/2))+2*(a*f-b*e)*(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 = 0.73 (sec) , antiderivative size = 777, normalized size of antiderivative = 6.07 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{e+f x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/4*(2*sqrt(b*x^2 + a)*b*d*f*x - (2*b*d*e - (2*b*c + a*d)*f)*sqrt(b)*log( 
-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - (b*d*e - b*c*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)*s 
qrt((b*e - a*f)/e))/(f^2*x^4 + 2*e*f*x^2 + e^2)))/(b*f^2), 1/4*(2*sqrt(b*x 
^2 + a)*b*d*f*x + 2*(2*b*d*e - (2*b*c + a*d)*f)*sqrt(-b)*arctan(sqrt(-b)*x 
/sqrt(b*x^2 + a)) - (b*d*e - b*c*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)))/(b*f^2), 1/4*(2*sqrt(b*x^2 + a)*b*d*f*x - 2*(b*d*e 
 - b*c*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)) 
- (2*b*d*e - (2*b*c + a*d)*f)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqr 
t(b)*x - a))/(b*f^2), 1/2*(sqrt(b*x^2 + a)*b*d*f*x + (2*b*d*e - (2*b*c + a 
*d)*f)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (b*d*e - b*c*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)))/(b*f^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x^2+a)^(1/2)*(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. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{e+f x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((b*x^2+a)^(1/2)*(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. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{e+f x^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (d\,x^2+c\right )}{f\,x^2+e} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.52 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{e+f x^2} \, dx=\frac {-2 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b c f +2 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b d e -2 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b c f +2 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b d e +\sqrt {b \,x^{2}+a}\, b d e f x +\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a d e f +2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b c e f -2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b d \,e^{2}}{2 b e \,f^{2}} \] Input:

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

Output:

( - 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)))*b*c*f + 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)))*b*d*e - 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)))*b*c*f 
 + 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)))*b*d*e + sqrt(a + b*x**2)*b*d*e 
*f*x + sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*d*e*f + 2*sqr 
t(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b*c*e*f - 2*sqrt(b)*log(( 
sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b*d*e**2)/(2*b*e*f**2)