\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x) (f+g x)^2} \, dx\) [272]

Optimal result

Integrand size = 44, antiderivative size = 261 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx=\frac {(2 c d f-a e g+c d g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^2 (f+g x)}+\frac {\sqrt {c} \sqrt {d} \left (3 a e^2 g-c d (4 e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\sqrt {e} g^3}-\frac {\sqrt {c d f-a e g} \left (a e^2 g-c d (4 e f-3 d g)\right ) \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{g^3 \sqrt {e f-d g}} \] Output:

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

Mathematica [A] (verified)

Time = 1.67 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\frac {g (-a e g+c d (2 f+g x))}{f+g x}+\frac {\sqrt {c d f-a e g} \left (a e^2 g+c d (-4 e f+3 d g)\right ) \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {-e f+d g} \sqrt {a e+c d x}}\right )}{\sqrt {-e f+d g} \sqrt {a e+c d x} \sqrt {d+e x}}+\frac {\sqrt {c} \sqrt {d} \left (3 a e^2 g+c d (-4 e f+d g)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x}}\right )}{g^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.25, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1215, 1230, 1269, 1092, 219, 1154, 219}

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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

((2*c*d*f - a*e*g + c*d*g*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/ 
(g^2*(f + g*x)) - (-((Sqrt[c]*Sqrt[d]*(3*a*e^2*g - c*d*(4*e*f - d*g))*ArcT 
anh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c 
*d^2 + a*e^2)*x + c*d*e*x^2])])/(Sqrt[e]*g)) - (Sqrt[c*d*f - a*e*g]*(4*c*d 
*e*f - 3*c*d^2*g - a*e^2*g)*ArcTanh[(c*d^2*f + a*e*(e*f - 2*d*g) - (a*e^2* 
g - c*d*(2*e*f - d*g))*x)/(2*Sqrt[e*f - d*g]*Sqrt[c*d*f - a*e*g]*Sqrt[a*d* 
e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(g*Sqrt[e*f - d*g]))/(2*g^2)
 

Defintions of rubi rules used

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 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 1215

Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( 
(d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + 
 c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - 
 b*d*e + a*e^2, 0] && GtQ[p, 0]
 

rule 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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. \(3014\) vs. \(2(235)=470\).

Time = 2.67 (sec) , antiderivative size = 3015, normalized size of antiderivative = 11.55

method	result	size
default	\(\text {Expression too large to display}\)	\(3015\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d)/(g*x+f)^2,x,method=_RE 
TURNVERBOSE)
 

Output:

e/(d*g-e*f)^2*(1/3*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+1/2*(a*e^ 
2-c*d^2)*(1/4*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2- 
c*d^2)*(x+d/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^2+d* 
e*c*(x+d/e))/(d*e*c)^(1/2)+(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/ 
(d*e*c)^(1/2)))+1/g/(d*g-e*f)*(-1/(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2 
)*g^2/(x+f/g)*(c*d*(x+f/g)^2*e+(a*e^2*g+c*d^2*g-2*c*d*e*f)/g*(x+f/g)+(a*d* 
e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2)^(5/2)+3/2*(a*e^2*g+c*d^2*g-2*c*d 
*e*f)*g/(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)*(1/3*(c*d*(x+f/g)^2*e+(a 
*e^2*g+c*d^2*g-2*c*d*e*f)/g*(x+f/g)+(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f 
^2)/g^2)^(3/2)+1/2*(a*e^2*g+c*d^2*g-2*c*d*e*f)/g*(1/4*(2*d*e*c*(x+f/g)+(a* 
e^2*g+c*d^2*g-2*c*d*e*f)/g)/d/e/c*(c*d*(x+f/g)^2*e+(a*e^2*g+c*d^2*g-2*c*d* 
e*f)/g*(x+f/g)+(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2)^(1/2)+1/8*(4 
*d*e*c*(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2-(a*e^2*g+c*d^2*g-2*c* 
d*e*f)^2/g^2)/d/e/c*ln((1/2*(a*e^2*g+c*d^2*g-2*c*d*e*f)/g+d*e*c*(x+f/g))/( 
d*e*c)^(1/2)+(c*d*(x+f/g)^2*e+(a*e^2*g+c*d^2*g-2*c*d*e*f)/g*(x+f/g)+(a*d*e 
*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2)^(1/2))/(d*e*c)^(1/2))+(a*d*e*g^2- 
a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2*((c*d*(x+f/g)^2*e+(a*e^2*g+c*d^2*g-2*c* 
d*e*f)/g*(x+f/g)+(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2)^(1/2)+1/2* 
(a*e^2*g+c*d^2*g-2*c*d*e*f)/g*ln((1/2*(a*e^2*g+c*d^2*g-2*c*d*e*f)/g+d*e*c* 
(x+f/g))/(d*e*c)^(1/2)+(c*d*(x+f/g)^2*e+(a*e^2*g+c*d^2*g-2*c*d*e*f)/g*(...
 

Fricas [F(-1)]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)/(g*x+f)^2,x, alg 
orithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)/(g*x+f)^2,x, alg 
orithm="maxima")
 

Output:

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

Giac [F(-1)]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)/(g*x+f)^2,x, alg 
orithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 19.72 (sec) , antiderivative size = 2635, normalized size of antiderivative = 10.10 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) 
 - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + a* 
e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x))*a* 
e**3*f*g + sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a* 
e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - c 
*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sqrt(c)*sqrt(d 
+ e*x))*a*e**3*g**2*x + 3*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)* 
sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f) 
*sqrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)* 
sqrt(c)*sqrt(d + e*x))*c*d**2*e*f*g + 3*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f 
)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*s 
qrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sq 
rt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x))*c*d**2*e*g**2*x - 4*sqrt(d*g - e*f)*s 
qrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(e)* 
sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g 
- 2*c*d*e*f) + sqrt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x))*c*d*e**2*f**2 - 4*sq 
rt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) - 
sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + a*e** 
2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x))*c*d*e 
**2*f*g*x + sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqr...