\(\int \frac {c+d x^2}{(a+b x^2)^{3/2} (e+f x^2)} \, dx\) [340]

Optimal result

Integrand size = 28, antiderivative size = 92 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {(b c-a d) x}{a (b e-a f) \sqrt {a+b x^2}}+\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.18 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {(-b c+a d) x}{a (-b e+a f) \sqrt {a+b x^2}}+\frac {(d e-c f) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} (-b e+a f)^{3/2}} \] Input:

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

Output:

((-(b*c) + a*d)*x)/(a*(-(b*e) + a*f)*Sqrt[a + b*x^2]) + ((d*e - c*f)*ArcTa 
n[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a* 
f])])/(Sqrt[e]*(-(b*e) + a*f)^(3/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {402, 25, 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.

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 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 = 0.82 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.92

method	result	size
pseudoelliptic	\(\frac {\frac {\left (a d -b c \right ) x}{\sqrt {b \,x^{2}+a}}-\frac {\left (c f -d e \right ) a \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{\sqrt {\left (a f -b e \right ) e}}}{\left (a f -b e \right ) a}\)	\(85\)
default	\(\frac {d x}{f a \sqrt {b \,x^{2}+a}}-\frac {\left (c f -d e \right ) \left (\frac {f}{\left (a f -b e \right ) \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 {2 b \sqrt {-e f}\, \left (2 b \left (x +\frac {\sqrt {-e f}}{f}\right )-\frac {2 b \sqrt {-e f}}{f}\right )}{\left (a f -b e \right ) \left (\frac {4 b \left (a f -b e \right )}{f}+\frac {4 b^{2} e}{f}\right ) \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 {f \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 )}{\left (a f -b e \right ) \sqrt {\frac {a f -b e}{f}}}\right )}{2 \sqrt {-e f}\, f}+\frac {\left (c f -d e \right ) \left (\frac {f}{\left (a f -b e \right ) \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 {2 b \sqrt {-e f}\, \left (2 b \left (x -\frac {\sqrt {-e f}}{f}\right )+\frac {2 b \sqrt {-e f}}{f}\right )}{\left (a f -b e \right ) \left (\frac {4 b \left (a f -b e \right )}{f}+\frac {4 b^{2} e}{f}\right ) \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 {f \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 )}{\left (a f -b e \right ) \sqrt {\frac {a f -b e}{f}}}\right )}{2 \sqrt {-e f}\, f}\)	\(763\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (80) = 160\).

Time = 0.99 (sec) , antiderivative size = 509, normalized size of antiderivative = 5.53 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\left [\frac {4 \, {\left ({\left (b^{2} c - a b d\right )} e^{2} - {\left (a b c - a^{2} d\right )} e f\right )} \sqrt {b x^{2} + a} x + {\left (a^{2} d e - a^{2} c f + {\left (a b d e - a b c f\right )} x^{2}\right )} \sqrt {b e^{2} - a e f} \log \left (\frac {{\left (8 \, b^{2} e^{2} - 8 \, a b e f + a^{2} f^{2}\right )} x^{4} + a^{2} e^{2} + 2 \, {\left (4 \, a b e^{2} - 3 \, a^{2} e f\right )} x^{2} + 4 \, {\left ({\left (2 \, b e - a f\right )} x^{3} + a e x\right )} \sqrt {b e^{2} - a e f} \sqrt {b x^{2} + a}}{f^{2} x^{4} + 2 \, e f x^{2} + e^{2}}\right )}{4 \, {\left (a^{2} b^{2} e^{3} - 2 \, a^{3} b e^{2} f + a^{4} e f^{2} + {\left (a b^{3} e^{3} - 2 \, a^{2} b^{2} e^{2} f + a^{3} b e f^{2}\right )} x^{2}\right )}}, \frac {2 \, {\left ({\left (b^{2} c - a b d\right )} e^{2} - {\left (a b c - a^{2} d\right )} e f\right )} \sqrt {b x^{2} + a} x - {\left (a^{2} d e - a^{2} c f + {\left (a b d e - a b c f\right )} x^{2}\right )} \sqrt {-b e^{2} + a e f} \arctan \left (\frac {\sqrt {-b e^{2} + a e f} {\left ({\left (2 \, b e - a f\right )} x^{2} + a e\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} e^{2} - a b e f\right )} x^{3} + {\left (a b e^{2} - a^{2} e f\right )} x\right )}}\right )}{2 \, {\left (a^{2} b^{2} e^{3} - 2 \, a^{3} b e^{2} f + a^{4} e f^{2} + {\left (a b^{3} e^{3} - 2 \, a^{2} b^{2} e^{2} f + a^{3} b e f^{2}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[1/4*(4*((b^2*c - a*b*d)*e^2 - (a*b*c - a^2*d)*e*f)*sqrt(b*x^2 + a)*x + (a 
^2*d*e - a^2*c*f + (a*b*d*e - a*b*c*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)))/(a^2*b^2*e^3 - 2*a^3*b*e^2*f + a^4*e*f^2 + (a*b^ 
3*e^3 - 2*a^2*b^2*e^2*f + a^3*b*e*f^2)*x^2), 1/2*(2*((b^2*c - a*b*d)*e^2 - 
 (a*b*c - a^2*d)*e*f)*sqrt(b*x^2 + a)*x - (a^2*d*e - a^2*c*f + (a*b*d*e - 
a*b*c*f)*x^2)*sqrt(-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)))/(a^2*b^2*e^3 - 2*a^3*b*e^2*f + a^4*e*f^2 + (a*b^3*e^3 - 2*a^2 
*b^2*e^2*f + a^3*b*e*f^2)*x^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((d*x^2+c)/(b*x^2+a)^(3/2)/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (80) = 160\).

Time = 0.14 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.43 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {{\left (b c - a d\right )} x}{{\left (a b e - a^{2} f\right )} \sqrt {b x^{2} + a}} + \frac {{\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \arctan \left (-\frac {2 \, b^{\frac {3}{2}} d e^{2} - 2 \, b^{\frac {3}{2}} c e f - a \sqrt {b} d e f + a \sqrt {b} c f^{2} + {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} \sqrt {b} d e - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} \sqrt {b} c f\right )} f}{2 \, {\left (\sqrt {-b e^{2} + a e f} b d e - \sqrt {-b e^{2} + a e f} b c f\right )}}\right )}{{\left (b d e - b c f\right )} \sqrt {-b e^{2} + a e f} {\left (b e - a f\right )}} \] Input:

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

Output:

(b*c - a*d)*x/((a*b*e - a^2*f)*sqrt(b*x^2 + a)) + (b*d^2*e^2 - 2*b*c*d*e*f 
 + b*c^2*f^2)*arctan(-1/2*(2*b^(3/2)*d*e^2 - 2*b^(3/2)*c*e*f - a*sqrt(b)*d 
*e*f + a*sqrt(b)*c*f^2 + ((sqrt(b)*x - sqrt(b*x^2 + a))^2*sqrt(b)*d*e - (s 
qrt(b)*x - sqrt(b*x^2 + a))^2*sqrt(b)*c*f)*f)/(sqrt(-b*e^2 + a*e*f)*b*d*e 
- sqrt(-b*e^2 + a*e*f)*b*c*f))/((b*d*e - b*c*f)*sqrt(-b*e^2 + a*e*f)*(b*e 
- a*f))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 702, normalized size of antiderivative = 7.63 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {-\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 ) a^{2} b c f +\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 ) a^{2} b d e -\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 ) a \,b^{2} c f \,x^{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 ) a \,b^{2} d e \,x^{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 ) a^{2} b c f +\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 ) a^{2} b d e -\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 ) a \,b^{2} c f \,x^{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 ) a \,b^{2} d e \,x^{2}+\sqrt {b \,x^{2}+a}\, a^{2} b d e f x -\sqrt {b \,x^{2}+a}\, a \,b^{2} c e f x -\sqrt {b \,x^{2}+a}\, a \,b^{2} d \,e^{2} x +\sqrt {b \,x^{2}+a}\, b^{3} c \,e^{2} x +\sqrt {b}\, a^{3} d e f -\sqrt {b}\, a^{2} b c e f -\sqrt {b}\, a^{2} b d \,e^{2}+\sqrt {b}\, a^{2} b d e f \,x^{2}+\sqrt {b}\, a \,b^{2} c \,e^{2}-\sqrt {b}\, a \,b^{2} c e f \,x^{2}-\sqrt {b}\, a \,b^{2} d \,e^{2} x^{2}+\sqrt {b}\, b^{3} c \,e^{2} x^{2}}{a b e \left (a^{2} b \,f^{2} x^{2}-2 a \,b^{2} e f \,x^{2}+b^{3} e^{2} x^{2}+a^{3} f^{2}-2 a^{2} b e f +a \,b^{2} e^{2}\right )} \] Input:

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

Output:

( - 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*f + 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 - 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**2*c*f*x**2 + sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*s 
qrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*b**2*d*e*x**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**2*b*c*f + 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)/(s 
qrt(e)*sqrt(b)))*a**2*b*d*e - 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** 
2*c*f*x**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*b**2*d*e*x**2 + sqrt 
(a + b*x**2)*a**2*b*d*e*f*x - sqrt(a + b*x**2)*a*b**2*c*e*f*x - sqrt(a + b 
*x**2)*a*b**2*d*e**2*x + sqrt(a + b*x**2)*b**3*c*e**2*x + sqrt(b)*a**3*d*e 
*f - sqrt(b)*a**2*b*c*e*f - sqrt(b)*a**2*b*d*e**2 + sqrt(b)*a**2*b*d*e*f*x 
**2 + sqrt(b)*a*b**2*c*e**2 - sqrt(b)*a*b**2*c*e*f*x**2 - sqrt(b)*a*b**2*d 
*e**2*x**2 + sqrt(b)*b**3*c*e**2*x**2)/(a*b*e*(a**3*f**2 - 2*a**2*b*e*f + 
a**2*b*f**2*x**2 + a*b**2*e**2 - 2*a*b**2*e*f*x**2 + b**3*e**2*x**2))