\(\int \frac {(d+e x)^2}{(f+g x) (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [293]

Optimal result

Integrand size = 44, antiderivative size = 270 \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 \left (a^2 e^4 g-a c d e^2 (e f-4 d g)-c^2 d^3 (e f+3 d g)+c d e \left (5 a e^2 g-c d (2 e f+3 d g)\right ) x\right )}{3 \left (c d^2-a e^2\right )^2 (c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 g^2 \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\sqrt {e f-d g} (c d f-a e g)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \left (\frac {c d (d+e x) \left (-6 a^2 e^3 g+c^2 d^2 (-d f+2 e f x+3 d g x)+a c d e (3 e f+4 d g-5 e g x)\right )}{\left (c d^2-a e^2\right )^2 (c d f-a e g)^2 (a e+c d x)}-\frac {3 g^2 \sqrt {a e+c d x} \sqrt {d+e x} \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {-e f+d g} \sqrt {a e+c d x}}\right )}{\sqrt {-e f+d g} (c d f-a e g)^{5/2}}\right )}{3 \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[(d + e*x)^2/((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^( 
5/2)),x]
 

Output:

(2*((c*d*(d + e*x)*(-6*a^2*e^3*g + c^2*d^2*(-(d*f) + 2*e*f*x + 3*d*g*x) + 
a*c*d*e*(3*e*f + 4*d*g - 5*e*g*x)))/((c*d^2 - a*e^2)^2*(c*d*f - a*e*g)^2*( 
a*e + c*d*x)) - (3*g^2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*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]*(c*d*f - a*e*g)^(5/2))))/(3*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1264, 27, 1235, 27, 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.

\(\displaystyle \int \frac {(d+e x)^2}{(f+g x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1264

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {\frac {2 \left (a^2 e^4 g+c d e x \left (5 a e^2 g-c d (3 d g+2 e f)\right )-a c d e^2 (e f-4 d g)-c^2 d^3 (3 d g+e f)\right )}{\left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}-\frac {2 \int \frac {3 \left (c d^2-a e^2\right )^2 g^2 (e f-d g)}{2 (f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{\left (c d^2-a e^2\right )^2 (e f-d g) (c d f-a e g)}}{3 (c d f-a e g)}-\frac {2 (d+e x)}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {2 \left (a^2 e^4 g+c d e x \left (5 a e^2 g-c d (3 d g+2 e f)\right )-a c d e^2 (e f-4 d g)-c^2 d^3 (3 d g+e f)\right )}{\left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}-\frac {3 g^2 \int \frac {1}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d f-a e g}}{3 (c d f-a e g)}-\frac {2 (d+e x)}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {\frac {6 g^2 \int \frac {1}{4 (e f-d g) (c d f-a e g)-\frac {\left (c f d^2+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\left (-\frac {c f d^2+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right )}{c d f-a e g}+\frac {2 \left (a^2 e^4 g+c d e x \left (5 a e^2 g-c d (3 d g+2 e f)\right )-a c d e^2 (e f-4 d g)-c^2 d^3 (3 d g+e f)\right )}{\left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}}{3 (c d f-a e g)}-\frac {2 (d+e x)}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {2 \left (a^2 e^4 g+c d e x \left (5 a e^2 g-c d (3 d g+2 e f)\right )-a c d e^2 (e f-4 d g)-c^2 d^3 (3 d g+e f)\right )}{\left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}-\frac {3 g^2 \text {arctanh}\left (\frac {-x \left (a e^2 g-c d (2 e f-d g)\right )+a e (e f-2 d g)+c d^2 f}{2 \sqrt {e f-d g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \sqrt {c d f-a e g}}\right )}{\sqrt {e f-d g} (c d f-a e g)^{3/2}}}{3 (c d f-a e g)}-\frac {2 (d+e x)}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}\)

Input:

Int[(d + e*x)^2/((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)), 
x]
 

Output:

(-2*(d + e*x))/(3*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^ 
(3/2)) - ((2*(a^2*e^4*g - a*c*d*e^2*(e*f - 4*d*g) - c^2*d^3*(e*f + 3*d*g) 
+ c*d*e*(5*a*e^2*g - c*d*(2*e*f + 3*d*g))*x))/((c*d^2 - a*e^2)^2*(c*d*f - 
a*e*g)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (3*g^2*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])])/(Sq 
rt[e*f - d*g]*(c*d*f - a*e*g)^(3/2)))/(3*(c*d*f - a*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 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 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 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 1264

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d + e*x) 
^m*(f + g*x)^n, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[(d + e*x 
)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[ 
(d + e*x)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + ( 
2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + S 
imp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*E 
xpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*R - b*S) 
)/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 1] 
 && LtQ[p, -1] && ILtQ[m, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1782\) vs. \(2(255)=510\).

Time = 2.05 (sec) , antiderivative size = 1783, normalized size of antiderivative = 6.60

method	result	size
default	\(\text {Expression too large to display}\)	\(1783\)

Input:

int((e*x+d)^2/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_RE 
TURNVERBOSE)
 

Output:

e/g^2*(e*g*(-1/3/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)-1/2*(a*e^2+ 
c*d^2)/d/e/c*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/ 
(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+16/3*d*e*c/(4*a*c*d^2*e^2-(a*e^2+c 
*d^2)^2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2) 
))+2*d*g*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d 
*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+16/3*d*e*c/(4*a*c*d^2*e^2-(a*e^2+c*d^2 
)^2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))-e* 
f*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e 
^2+c*d^2)*x+c*d*x^2*e)^(3/2)+16/3*d*e*c/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)^2* 
(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)))+(d^2*g^2 
-2*d*e*f*g+e^2*f^2)/g^3*(1/3/(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)^(3/2)-1/2*(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)*(2/3*(2*d*e*c*(x+f/g)+(a*e^2*g+c*d^2* 
g-2*c*d*e*f)/g)/(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)/(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)+16/3*d*e 
*c/(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)^2*(2*d*e*c*(x+f/g)+(a*e^2*g+c*d^2*g-2*c*d*e*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-...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1501 vs. \(2 (252) = 504\).

Time = 12.55 (sec) , antiderivative size = 3059, normalized size of antiderivative = 11.33 \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^2/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, alg 
orithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**2/(g*x+f)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x 
)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^2/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, alg 
orithm="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(((a*e^2)/g>0)', see `assume?` fo 
r more det
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x+d)^2/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, alg 
orithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[2,2,10]%%%},[4,2,6,0]%%%}+%%%{%%%{-4,[3,4,8]%%%},[4 
,2,5,0]%%
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{\left (f+g\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \] Input:

int((d + e*x)^2/((f + g*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)), 
x)
 

Output:

int((d + e*x)^2/((f + g*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)), 
 x)
 

Reduce [B] (verification not implemented)

Time = 3.92 (sec) , antiderivative size = 3593, normalized size of antiderivative = 13.31 \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^2/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
 

Output:

(3*sqrt(d*g - e*f)*sqrt(a*e + c*d*x)*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**3*e**5*g**2 - 6*sqrt(d*g - e*f)*sqrt(a*e + c*d*x)*sqr 
t(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(e)*sq 
rt(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**2*c*d**2*e**3*g**2 
+ 3*sqrt(d*g - e*f)*sqrt(a*e + c*d*x)*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)*sqr 
t(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**2*c*d*e**4*g**2*x + 3*sqrt(d*g - e*f)*sqrt(a*e + c*d 
*x)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqr 
t(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*c**2*d**4*e*g 
**2 - 6*sqrt(d*g - e*f)*sqrt(a*e + c*d*x)*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*c**2*d**3*e**2*g**2*x + 3*sqrt(d*g - e*f)*sqrt(a* 
e + c*d*x)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) - sqr 
t(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + a*e**...