\(\int \frac {(d+e x)^2}{(f+g x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\) [279]

Optimal result

Integrand size = 46, antiderivative size = 241 \[ \int \frac {(d+e x)^2}{(f+g x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{g (c e f+c d g-b e g) (f+g x)}+\frac {2 e \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{\sqrt {c} g^2}-\frac {e \sqrt {e f-d g} (2 c e f+4 c d g-3 b e g) \arctan \left (\frac {\sqrt {c e f+c d g-b e g} (d+e x)}{\sqrt {e f-d g} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{g^2 (c e f+c d g-b e g)^{3/2}} \] Output:

(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/g/(-b*e*g+c*d*g+c*e*f)/( 
g*x+f)+2*e*arctan(c^(1/2)*(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/ 
c^(1/2)/g^2-e*(-d*g+e*f)^(1/2)*(-3*b*e*g+4*c*d*g+2*c*e*f)*arctan((-b*e*g+c 
*d*g+c*e*f)^(1/2)*(e*x+d)/(-d*g+e*f)^(1/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2 
)^(1/2))/g^2/(-b*e*g+c*d*g+c*e*f)^(3/2)
 

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^2}{(f+g x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {e \left (\frac {g (-e f+d g) (d+e x) (-c d+b e+c e x)}{e (c e f+c d g-b e g) (f+g x)}-\frac {2 \sqrt {d+e x} \sqrt {c d-b e-c e x} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{\sqrt {c}}+\frac {\sqrt {e f-d g} (2 c e f+4 c d g-3 b e g) \sqrt {d+e x} \sqrt {c d-b e-c e x} \arctan \left (\frac {\sqrt {e f-d g} \sqrt {c d-b e-c e x}}{\sqrt {c e f+c d g-b e g} \sqrt {d+e x}}\right )}{(c e f+c d g-b e g)^{3/2}}\right )}{g^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:

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

Output:

(e*((g*(-(e*f) + d*g)*(d + e*x)*(-(c*d) + b*e + c*e*x))/(e*(c*e*f + c*d*g 
- b*e*g)*(f + g*x)) - (2*Sqrt[d + e*x]*Sqrt[c*d - b*e - c*e*x]*ArcTan[Sqrt 
[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])])/Sqrt[c] + (Sqrt[e*f - d*g]*( 
2*c*e*f + 4*c*d*g - 3*b*e*g)*Sqrt[d + e*x]*Sqrt[c*d - b*e - c*e*x]*ArcTan[ 
(Sqrt[e*f - d*g]*Sqrt[c*d - b*e - c*e*x])/(Sqrt[c*e*f + c*d*g - b*e*g]*Sqr 
t[d + e*x])])/(c*e*f + c*d*g - b*e*g)^(3/2)))/(g^2*Sqrt[(d + e*x)*(-(b*e) 
+ c*(d - e*x))])
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.30, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1266, 27, 1269, 1092, 217, 1154, 217}

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.

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

\(\Big \downarrow \) 1266

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1154

\(\displaystyle \frac {e \left (\frac {2 (e f-d g) (-3 b e g+4 c d g+2 c e f) \int \frac {1}{-\frac {\left (2 c g d^2+b e (e f-2 d g)+e^2 (2 c f-b g) x\right )^2}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 (e f-d g) (c e f+c d g-b e g)}d\frac {2 c g d^2+b e (e f-2 d g)+e^2 (2 c f-b g) x}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}}{g}+\frac {2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-b e g+c d g+c e f)}{\sqrt {c} g}\right )}{2 g (-b e g+c d g+c e f)}+\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{g (f+g x) (-b e g+c d g+c e f)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {e \left (\frac {2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-b e g+c d g+c e f)}{\sqrt {c} g}-\frac {\sqrt {e f-d g} (-3 b e g+4 c d g+2 c e f) \arctan \left (\frac {e^2 x (2 c f-b g)+b e (e f-2 d g)+2 c d^2 g}{2 \sqrt {e f-d g} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \sqrt {-b e g+c d g+c e f}}\right )}{g \sqrt {-b e g+c d g+c e f}}\right )}{2 g (-b e g+c d g+c e f)}+\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{g (f+g x) (-b e g+c d g+c e f)}\)

Input:

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

Output:

((e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(g*(c*e*f + c*d*g 
- b*e*g)*(f + g*x)) + (e*((2*(c*e*f + c*d*g - b*e*g)*ArcTan[(e*(b + 2*c*x) 
)/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(Sqrt[c]*g) - (S 
qrt[e*f - d*g]*(2*c*e*f + 4*c*d*g - 3*b*e*g)*ArcTan[(2*c*d^2*g + b*e*(e*f 
- 2*d*g) + e^2*(2*c*f - b*g)*x)/(2*Sqrt[e*f - d*g]*Sqrt[c*e*f + c*d*g - b* 
e*g]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(g*Sqrt[c*e*f + c*d*g - 
b*e*g])))/(2*g*(c*e*f + c*d*g - b*e*g))
 

Defintions of rubi rules used

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 217

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

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1266

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x)^ 
n, d + e*x, x], R = PolynomialRemainder[(f + g*x)^n, d + e*x, x]}, Simp[(e* 
R*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a* 
e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1 
)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R 
*(m + 1) - b*e*R*(m + p + 2) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && IGtQ[n, 1] && ILtQ[m, -1] && NeQ[c*d^2 - b* 
d*e + a*e^2, 0] && (NeQ[m + n, 0] || EqQ[p, -2^(-1)])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(745\) vs. \(2(223)=446\).

Time = 2.17 (sec) , antiderivative size = 746, normalized size of antiderivative = 3.10

method	result	size
default	\(\frac {e^{2} \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{g^{2} \sqrt {c \,e^{2}}}+\frac {\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \left (\frac {g^{2} \sqrt {-\left (x +\frac {f}{g}\right )^{2} c \,e^{2}-\frac {e^{2} \left (b g -2 c f \right ) \left (x +\frac {f}{g}\right )}{g}-\frac {b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}}{g^{2}}}}{\left (b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}\right ) \left (x +\frac {f}{g}\right )}+\frac {e^{2} \left (b g -2 c f \right ) g \ln \left (\frac {-\frac {2 \left (b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}\right )}{g^{2}}-\frac {e^{2} \left (b g -2 c f \right ) \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c \,e^{2}-\frac {e^{2} \left (b g -2 c f \right ) \left (x +\frac {f}{g}\right )}{g}-\frac {b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}}{g^{2}}}}{x +\frac {f}{g}}\right )}{2 \left (b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}\right ) \sqrt {-\frac {b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}}{g^{2}}}}\right )}{g^{4}}-\frac {2 e \left (d g -e f \right ) \ln \left (\frac {-\frac {2 \left (b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}\right )}{g^{2}}-\frac {e^{2} \left (b g -2 c f \right ) \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c \,e^{2}-\frac {e^{2} \left (b g -2 c f \right ) \left (x +\frac {f}{g}\right )}{g}-\frac {b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}}{g^{2}}}}{x +\frac {f}{g}}\right )}{g^{3} \sqrt {-\frac {b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}}{g^{2}}}}\)	\(746\)

Input:

int((e*x+d)^2/(g*x+f)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x,method=_R 
ETURNVERBOSE)
 

Output:

e^2/g^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x 
-b*d*e+c*d^2)^(1/2))+1/g^4*(d^2*g^2-2*d*e*f*g+e^2*f^2)*(1/(b*d*e*g^2-b*e^2 
*f*g-c*d^2*g^2+c*e^2*f^2)*g^2/(x+f/g)*(-(x+f/g)^2*c*e^2-e^2*(b*g-2*c*f)/g* 
(x+f/g)-(b*d*e*g^2-b*e^2*f*g-c*d^2*g^2+c*e^2*f^2)/g^2)^(1/2)+1/2*e^2*(b*g- 
2*c*f)*g/(b*d*e*g^2-b*e^2*f*g-c*d^2*g^2+c*e^2*f^2)/(-(b*d*e*g^2-b*e^2*f*g- 
c*d^2*g^2+c*e^2*f^2)/g^2)^(1/2)*ln((-2*(b*d*e*g^2-b*e^2*f*g-c*d^2*g^2+c*e^ 
2*f^2)/g^2-e^2*(b*g-2*c*f)/g*(x+f/g)+2*(-(b*d*e*g^2-b*e^2*f*g-c*d^2*g^2+c* 
e^2*f^2)/g^2)^(1/2)*(-(x+f/g)^2*c*e^2-e^2*(b*g-2*c*f)/g*(x+f/g)-(b*d*e*g^2 
-b*e^2*f*g-c*d^2*g^2+c*e^2*f^2)/g^2)^(1/2))/(x+f/g)))-2*e/g^3*(d*g-e*f)/(- 
(b*d*e*g^2-b*e^2*f*g-c*d^2*g^2+c*e^2*f^2)/g^2)^(1/2)*ln((-2*(b*d*e*g^2-b*e 
^2*f*g-c*d^2*g^2+c*e^2*f^2)/g^2-e^2*(b*g-2*c*f)/g*(x+f/g)+2*(-(b*d*e*g^2-b 
*e^2*f*g-c*d^2*g^2+c*e^2*f^2)/g^2)^(1/2)*(-(x+f/g)^2*c*e^2-e^2*(b*g-2*c*f) 
/g*(x+f/g)-(b*d*e*g^2-b*e^2*f*g-c*d^2*g^2+c*e^2*f^2)/g^2)^(1/2))/(x+f/g))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)^2/(g*x+f)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, al 
gorithm="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((4*c*e^2>0)', see `assume?` for 
more detai
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 6230, normalized size of antiderivative = 25.85 \[ \int \frac {(d+e x)^2}{(f+g x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*(4*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b 
**3*e**4*f*g**2 + 4*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b 
*e + 2*c*d))*b**3*e**4*g**3*x - 16*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e* 
x)*i)/sqrt( - b*e + 2*c*d))*b**2*c*d*e**3*f*g**2 - 16*sqrt(c)*asinh((sqrt( 
 - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c*d*e**3*g**3*x - 8*sq 
rt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c*e* 
*4*f**2*g - 8*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2 
*c*d))*b**2*c*e**4*f*g**2*x + 20*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x) 
*i)/sqrt( - b*e + 2*c*d))*b*c**2*d**2*e**2*f*g**2 + 20*sqrt(c)*asinh((sqrt 
( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**2*d**2*e**2*g**3*x + 
24*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c* 
*2*d*e**3*f**2*g + 24*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - 
 b*e + 2*c*d))*b*c**2*d*e**3*f*g**2*x + 4*sqrt(c)*asinh((sqrt( - b*e + c*d 
 - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**2*e**4*f**3 + 4*sqrt(c)*asinh((sqr 
t( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**2*e**4*f**2*g*x - 8* 
sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**3*d* 
*3*e*f*g**2 - 8*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 
 2*c*d))*c**3*d**3*e*g**3*x - 16*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x) 
*i)/sqrt( - b*e + 2*c*d))*c**3*d**2*e**2*f**2*g - 16*sqrt(c)*asinh((sqrt( 
- b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**3*d**2*e**2*f*g**2*x -...