\(\int \frac {(a+b x^2)^2}{(c+d x^2) (e+f x^2)} \, dx\) [236]

Optimal result

Integrand size = 28, antiderivative size = 101 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {b^2 x}{d f}+\frac {(b c-a d)^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{3/2} (d e-c f)}-\frac {(b e-a f)^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2} (d e-c f)} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {b^2 x}{d f}-\frac {(b c-a d)^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{3/2} (-d e+c f)}-\frac {(b e-a f)^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2} (d e-c f)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.44, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {420, 299, 218, 397, 218}

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.

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 397

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

rule 420

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

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {b^{2} x}{d f}+\frac {\left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{f \left (c f -d e \right ) \sqrt {e f}}+\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{\left (c f -d e \right ) d \sqrt {c d}}\)	\(113\)
risch	\(\frac {b^{2} x}{d f}-\frac {f \ln \left (e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) a^{2}}{2 \sqrt {-e f}\, \left (c f -d e \right )}+\frac {\ln \left (e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) a b e}{\sqrt {-e f}\, \left (c f -d e \right )}-\frac {\ln \left (e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) b^{2} e^{2}}{2 f \sqrt {-e f}\, \left (c f -d e \right )}+\frac {f \ln \left (-e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) a^{2}}{2 \sqrt {-e f}\, \left (c f -d e \right )}-\frac {\ln \left (-e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) a b e}{\sqrt {-e f}\, \left (c f -d e \right )}+\frac {\ln \left (-e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) b^{2} e^{2}}{2 f \sqrt {-e f}\, \left (c f -d e \right )}-\frac {d \ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a^{2}}{2 \sqrt {-c d}\, \left (c f -d e \right )}+\frac {\ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a b c}{\sqrt {-c d}\, \left (c f -d e \right )}-\frac {\ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) b^{2} c^{2}}{2 d \sqrt {-c d}\, \left (c f -d e \right )}+\frac {d \ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a^{2}}{2 \sqrt {-c d}\, \left (c f -d e \right )}-\frac {\ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a b c}{\sqrt {-c d}\, \left (c f -d e \right )}+\frac {\ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) b^{2} c^{2}}{2 d \sqrt {-c d}\, \left (c f -d e \right )}\)	\(489\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 1.04 (sec) , antiderivative size = 678, normalized size of antiderivative = 6.71 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\left [\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-c d} e f^{2} \log \left (\frac {d x^{2} + 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + {\left (b^{2} c d^{2} e^{2} - 2 \, a b c d^{2} e f + a^{2} c d^{2} f^{2}\right )} \sqrt {-e f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-e f} x - e}{f x^{2} + e}\right ) + 2 \, {\left (b^{2} c d^{2} e^{2} f - b^{2} c^{2} d e f^{2}\right )} x}{2 \, {\left (c d^{3} e^{2} f^{2} - c^{2} d^{2} e f^{3}\right )}}, \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-c d} e f^{2} \log \left (\frac {d x^{2} + 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) - 2 \, {\left (b^{2} c d^{2} e^{2} - 2 \, a b c d^{2} e f + a^{2} c d^{2} f^{2}\right )} \sqrt {e f} \arctan \left (\frac {\sqrt {e f} x}{e}\right ) + 2 \, {\left (b^{2} c d^{2} e^{2} f - b^{2} c^{2} d e f^{2}\right )} x}{2 \, {\left (c d^{3} e^{2} f^{2} - c^{2} d^{2} e f^{3}\right )}}, \frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d} e f^{2} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + {\left (b^{2} c d^{2} e^{2} - 2 \, a b c d^{2} e f + a^{2} c d^{2} f^{2}\right )} \sqrt {-e f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-e f} x - e}{f x^{2} + e}\right ) + 2 \, {\left (b^{2} c d^{2} e^{2} f - b^{2} c^{2} d e f^{2}\right )} x}{2 \, {\left (c d^{3} e^{2} f^{2} - c^{2} d^{2} e f^{3}\right )}}, \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d} e f^{2} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - {\left (b^{2} c d^{2} e^{2} - 2 \, a b c d^{2} e f + a^{2} c d^{2} f^{2}\right )} \sqrt {e f} \arctan \left (\frac {\sqrt {e f} x}{e}\right ) + {\left (b^{2} c d^{2} e^{2} f - b^{2} c^{2} d e f^{2}\right )} x}{c d^{3} e^{2} f^{2} - c^{2} d^{2} e f^{3}}\right ] \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x^2+a)^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 [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {b^{2} x}{d f} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (d^{2} e - c d f\right )} \sqrt {c d}} - \frac {{\left (b^{2} e^{2} - 2 \, a b e f + a^{2} f^{2}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{{\left (d e f - c f^{2}\right )} \sqrt {e f}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 3.55 (sec) , antiderivative size = 3684, normalized size of antiderivative = 36.48 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan((((((-c*d^3)^(1/2)*(a*d - b*c)^2*((4*a^2*c^2*d^3*f^5 + 4*a^2*d^5*e^2 
*f^3 + 8*b^2*c^2*d^3*e^2*f^3 - 8*a^2*c*d^4*e*f^4 - 4*b^2*c*d^4*e^3*f^2 - 4 
*b^2*c^3*d^2*e*f^4)/(d*f) + (x*(-c*d^3)^(1/2)*(a*d - b*c)^2*(4*c^3*d^3*f^6 
 + 4*d^6*e^3*f^3 - 4*c*d^5*e^2*f^4 - 4*c^2*d^4*e*f^5))/(d*f*(c^2*d^3*f - c 
*d^4*e))))/(2*(c^2*d^3*f - c*d^4*e)) + (2*x*(2*a^4*d^4*f^4 + b^4*c^4*f^4 + 
 b^4*d^4*e^4 + 6*a^2*b^2*c^2*d^2*f^4 + 6*a^2*b^2*d^4*e^2*f^2 - 4*a*b^3*c^3 
*d*f^4 - 4*a^3*b*c*d^3*f^4 - 4*a*b^3*d^4*e^3*f - 4*a^3*b*d^4*e*f^3))/(d*f) 
)*(-c*d^3)^(1/2)*(a*d - b*c)^2*1i)/(2*(c^2*d^3*f - c*d^4*e)) - ((((-c*d^3) 
^(1/2)*(a*d - b*c)^2*((4*a^2*c^2*d^3*f^5 + 4*a^2*d^5*e^2*f^3 + 8*b^2*c^2*d 
^3*e^2*f^3 - 8*a^2*c*d^4*e*f^4 - 4*b^2*c*d^4*e^3*f^2 - 4*b^2*c^3*d^2*e*f^4 
)/(d*f) - (x*(-c*d^3)^(1/2)*(a*d - b*c)^2*(4*c^3*d^3*f^6 + 4*d^6*e^3*f^3 - 
 4*c*d^5*e^2*f^4 - 4*c^2*d^4*e*f^5))/(d*f*(c^2*d^3*f - c*d^4*e))))/(2*(c^2 
*d^3*f - c*d^4*e)) - (2*x*(2*a^4*d^4*f^4 + b^4*c^4*f^4 + b^4*d^4*e^4 + 6*a 
^2*b^2*c^2*d^2*f^4 + 6*a^2*b^2*d^4*e^2*f^2 - 4*a*b^3*c^3*d*f^4 - 4*a^3*b*c 
*d^3*f^4 - 4*a*b^3*d^4*e^3*f - 4*a^3*b*d^4*e*f^3))/(d*f))*(-c*d^3)^(1/2)*( 
a*d - b*c)^2*1i)/(2*(c^2*d^3*f - c*d^4*e)))/((2*(b^6*c^2*d*e^3 - 2*a^5*b*d 
^3*f^3 + b^6*c^3*e^2*f + a^2*b^4*c^3*f^3 + a^2*b^4*d^3*e^3 - 2*a*b^5*c*d^2 
*e^3 - 2*a*b^5*c^3*e*f^2 - 4*a^3*b^3*c^2*d*f^3 + 5*a^4*b^2*c*d^2*f^3 - 4*a 
^3*b^3*d^3*e^2*f + 5*a^4*b^2*d^3*e*f^2 + 9*a^2*b^4*c*d^2*e^2*f + 9*a^2*b^4 
*c^2*d*e*f^2 - 12*a^3*b^3*c*d^2*e*f^2 - 6*a*b^5*c^2*d*e^2*f))/(d*f) + (...
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.11 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {-\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} d^{2} e \,f^{2}+2 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a b c d e \,f^{2}-\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{2} e \,f^{2}+\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a^{2} c \,d^{2} f^{2}-2 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a b c \,d^{2} e f +\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) b^{2} c \,d^{2} e^{2}+b^{2} c^{2} d e \,f^{2} x -b^{2} c \,d^{2} e^{2} f x}{c \,d^{2} e \,f^{2} \left (c f -d e \right )} \] Input:

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

Output:

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