\(\int \frac {f+g x}{\sqrt {d+e x} (a+b x+c x^2)} \, dx\) [306]

Optimal result

Integrand size = 27, antiderivative size = 244 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=-\frac {\sqrt {2} \left (2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.03 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.03 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=\frac {\sqrt {2} \left (\frac {\left (-2 i c f+\left (i b+\sqrt {-b^2+4 a c}\right ) g\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (2 i c f+\left (-i b+\sqrt {-b^2+4 a c}\right ) g\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}\right )}{\sqrt {c} \sqrt {-b^2+4 a c}} \] Input:

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

Output:

(Sqrt[2]*((((-2*I)*c*f + (I*b + Sqrt[-b^2 + 4*a*c])*g)*ArcTan[(Sqrt[2]*Sqr 
t[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/Sqrt[-2* 
c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e] + (((2*I)*c*f + ((-I)*b + Sqrt[-b^2 + 
4*a*c])*g)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sq 
rt[-b^2 + 4*a*c]*e]])/Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e]))/(Sqrt[ 
c]*Sqrt[-b^2 + 4*a*c])
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1197, 1480, 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.

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 2.89 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (-\frac {\left (-b e g +2 f c e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, g \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (b e g -2 f c e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, g \right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}}\)	\(229\)
derivativedivides	\(8 c \left (-\frac {\left (-b e g +2 f c e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, g \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (b e g -2 f c e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, g \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\)	\(258\)
default	\(8 c \left (-\frac {\left (-b e g +2 f c e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, g \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (b e g -2 f c e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, g \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\)	\(258\)

Input:

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

Output:

2^(1/2)/(-e^2*(4*a*c-b^2))^(1/2)*(-(-b*e*g+2*f*c*e+(-e^2*(4*a*c-b^2))^(1/2 
)*g)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2) 
*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))+(b*e*g-2*f*c*e 
+(-e^2*(4*a*c-b^2))^(1/2)*g)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2 
)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^ 
(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5621 vs. \(2 (200) = 400\).

Time = 0.61 (sec) , antiderivative size = 5621, normalized size of antiderivative = 23.04 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (200) = 400\).

Time = 0.29 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.89 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*(2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*sqrt(b^2 - 4*a*c)* 
c*e*f*abs(e) - 2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*sqrt(b^2 
 - 4*a*c)*c*d*g*abs(e) + sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)* 
(b^2 - 4*a*c)*e^2*g - 2*(2*c^2*d*e - b*c*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqr 
t(b^2 - 4*a*c)*c)*e)*f + (2*b*c*d*e - b^2*e^2)*sqrt(-4*c^2*d + 2*(b*c - sq 
rt(b^2 - 4*a*c)*c)*e)*g)*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*c*d - b 
*e + sqrt((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c))/((sqrt(b^2 - 
 4*a*c)*c^2*d^2 - sqrt(b^2 - 4*a*c)*b*c*d*e + sqrt(b^2 - 4*a*c)*a*c*e^2)*a 
bs(c)*abs(e)) + 1/4*(2*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*sq 
rt(b^2 - 4*a*c)*c*e*f*abs(e) - 2*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c 
)*c)*e)*sqrt(b^2 - 4*a*c)*c*d*g*abs(e) - sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 
 - 4*a*c)*c)*e)*(b^2 - 4*a*c)*e^2*g + 2*(2*c^2*d*e - b*c*e^2)*sqrt(-4*c^2* 
d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*f - (2*b*c*d*e - b^2*e^2)*sqrt(-4*c^2 
*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*g)*arctan(2*sqrt(1/2)*sqrt(e*x + d)/ 
sqrt(-(2*c*d - b*e - sqrt((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/ 
c))/((sqrt(b^2 - 4*a*c)*c^2*d^2 - sqrt(b^2 - 4*a*c)*b*c*d*e + sqrt(b^2 - 4 
*a*c)*a*c*e^2)*abs(c)*abs(e))
 

Mupad [B] (verification not implemented)

Time = 20.29 (sec) , antiderivative size = 9944, normalized size of antiderivative = 40.75 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan(((((b^4*d*g^2 + 8*a^2*c^2*d*g^2 + 2*b^2*c^2*d*f^2 - 8*a*c^3*d*f^2 - 
 a*b^3*e*g^2 - a*e*g^2*(-(4*a*c - b^2)^3)^(1/2) + b*d*g^2*(-(4*a*c - b^2)^ 
3)^(1/2) - b^3*c*e*f^2 + c*e*f^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c^2*e*f^ 
2 - 6*a*b^2*c*d*g^2 + 4*a^2*b*c*e*g^2 - 16*a^2*c^2*e*f*g - 2*b^3*c*d*f*g - 
 2*c*d*f*g*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b*c^2*d*f*g + 4*a*b^2*c*e*f*g)/( 
2*(16*a^2*c^4*d^2 + 16*a^3*c^3*e^2 + b^4*c^2*d^2 - 8*a*b^2*c^3*d^2 - b^5*c 
*d*e - 8*a^2*b^2*c^2*e^2 + a*b^4*c*e^2 + 8*a*b^3*c^2*d*e - 16*a^2*b*c^3*d* 
e)))^(1/2)*((d + e*x)^(1/2)*((b^4*d*g^2 + 8*a^2*c^2*d*g^2 + 2*b^2*c^2*d*f^ 
2 - 8*a*c^3*d*f^2 - a*b^3*e*g^2 - a*e*g^2*(-(4*a*c - b^2)^3)^(1/2) + b*d*g 
^2*(-(4*a*c - b^2)^3)^(1/2) - b^3*c*e*f^2 + c*e*f^2*(-(4*a*c - b^2)^3)^(1/ 
2) + 4*a*b*c^2*e*f^2 - 6*a*b^2*c*d*g^2 + 4*a^2*b*c*e*g^2 - 16*a^2*c^2*e*f* 
g - 2*b^3*c*d*f*g - 2*c*d*f*g*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b*c^2*d*f*g + 
 4*a*b^2*c*e*f*g)/(2*(16*a^2*c^4*d^2 + 16*a^3*c^3*e^2 + b^4*c^2*d^2 - 8*a* 
b^2*c^3*d^2 - b^5*c*d*e - 8*a^2*b^2*c^2*e^2 + a*b^4*c*e^2 + 8*a*b^3*c^2*d* 
e - 16*a^2*b*c^3*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c 
^3*e^3 + 64*a*c^4*d*e^2) - 8*b^2*c^2*e^3*f + 32*a*c^3*e^3*f - 32*a*c^3*d*e 
^2*g + 8*b^2*c^2*d*e^2*g) + (d + e*x)^(1/2)*(16*c^3*e^2*f^2 - 16*a*c^2*e^2 
*g^2 + 8*b^2*c*e^2*g^2 - 16*b*c^2*e^2*f*g))*((b^4*d*g^2 + 8*a^2*c^2*d*g^2 
+ 2*b^2*c^2*d*f^2 - 8*a*c^3*d*f^2 - a*b^3*e*g^2 - a*e*g^2*(-(4*a*c - b^2)^ 
3)^(1/2) + b*d*g^2*(-(4*a*c - b^2)^3)^(1/2) - b^3*c*e*f^2 + c*e*f^2*(-(...
 

Reduce [B] (verification not implemented)

Time = 21.99 (sec) , antiderivative size = 4738, normalized size of antiderivative = 19.42 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a*e 
**2 - b*d*e + c*d**2)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - 
 b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d* 
e + c*d**2) + b*e - 2*c*d))*a*c*e*g + 2*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e 
 + c*d**2) + b*e - 2*c*d)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((sqrt(2*sqrt( 
c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x)) 
/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*b*c*d*g + 2* 
sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a*e**2 - 
b*d*e + c*d**2)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 
 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c* 
d**2) + b*e - 2*c*d))*b*c*e*f - 4*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d 
**2) + b*e - 2*c*d)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((sqrt(2*sqrt(c)*sqr 
t(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt( 
2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*c**2*d*f + 2*sqrt( 
c)*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*atan((sqrt( 
2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d 
+ e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*a*b*e 
**2*g - 4*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c 
*d)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2* 
sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b...