\(\int \frac {A+B x+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}} \, dx\) [80]

Optimal result

Integrand size = 36, antiderivative size = 254 \[ \int \frac {A+B x+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{b (b c-a d) (b e-a f) (a+b x)}+\frac {2 C \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{b^2 \sqrt {d} \sqrt {f}}+\frac {\left (2 a^3 C d f-3 a^2 b C (d e+c f)-b^3 (2 B c e-A d e-A c f)+a b^2 (4 c C e+B d e+B c f-2 A d f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{b^2 (b c-a d)^{3/2} (b e-a f)^{3/2}} \] Output:

-(A*b^2-a*(B*b-C*a))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b*e)/( 
b*x+a)+2*C*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+e)^(1/2))/b^2/d^(1/2 
)/f^(1/2)+(2*a^3*C*d*f-3*a^2*b*C*(c*f+d*e)-b^3*(-A*c*f-A*d*e+2*B*c*e)+a*b^ 
2*(-2*A*d*f+B*c*f+B*d*e+4*C*c*e))*arctanh((-a*f+b*e)^(1/2)*(d*x+c)^(1/2)/( 
-a*d+b*c)^(1/2)/(f*x+e)^(1/2))/b^2/(-a*d+b*c)^(3/2)/(-a*f+b*e)^(3/2)
 

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {\frac {b \left (A b^2+a (-b B+a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x)}+\frac {\left (-2 a^3 C d f+3 a^2 b C (d e+c f)-a b^2 (4 c C e+B d e+B c f-2 A d f)+b^3 (2 B c e-A (d e+c f))\right ) \arctan \left (\frac {\sqrt {b c-a d} \sqrt {e+f x}}{\sqrt {-b e+a f} \sqrt {c+d x}}\right )}{(b c-a d)^{3/2} (-b e+a f)^{3/2}}-\frac {2 C \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{\sqrt {d} \sqrt {f}}}{b^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2116, 27, 175, 66, 104, 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.

\(\displaystyle \int \frac {A+B x+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}} \, dx\)

\(\Big \downarrow \) 2116

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-(((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*(b*c - a*d)*(b* 
e - a*f)*(a + b*x))) + ((4*C*(b*c - a*d)*(b*e - a*f)*ArcTanh[(Sqrt[f]*Sqrt 
[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(b*Sqrt[d]*Sqrt[f]) + (2*(2*a^3*C*d*f 
 - 3*a^2*b*C*(d*e + c*f) - b^3*(2*B*c*e - A*d*e - A*c*f) + a*b^2*(4*c*C*e 
+ B*d*e + B*c*f - 2*A*d*f))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[ 
b*c - a*d]*Sqrt[e + f*x])])/(b*Sqrt[b*c - a*d]*Sqrt[b*e - a*f]))/(2*b*(b*c 
 - a*d)*(b*e - a*f))
 

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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

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 2116

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_ 
.)*(x_))^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px, a + b*x, x], 
 R = PolynomialRemainder[Px, a + b*x, x]}, Simp[b*R*(a + b*x)^(m + 1)*(c + 
d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Si 
mp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n* 
(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f*R*(m + 1 
) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x] 
, x], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolyQ[Px, x] && ILtQ[m, 
-1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2972\) vs. \(2(226)=452\).

Time = 0.97 (sec) , antiderivative size = 2973, normalized size of antiderivative = 11.70

method	result	size
default	\(\text {Expression too large to display}\)	\(2973\)

Input:

int((C*x^2+B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2),x,method=_RETURNVE 
RBOSE)
 

Output:

-1/2*(f*x+e)^(1/2)*(d*x+c)^(1/2)*(-A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((f* 
x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f- 
a*d*e+2*b*c*e)/(b*x+a))*b^4*c*f*x*(d*f)^(1/2)-A*ln((-2*a*d*f*x+b*c*f*x+b*d 
*e*x+2*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/ 
2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*b^4*d*e*x*(d*f)^(1/2)+2*B*ln((-2*a*d*f* 
x+b*c*f*x+b*d*e*x+2*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2* 
c*e)/b^2)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*b^4*c*e*x*(d*f)^(1/2)-2*C* 
ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2) 
)*b^4*c*e*x*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+2*A*ln((-2*a*d*f 
*x+b*c*f*x+b*d*e*x+2*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2 
*c*e)/b^2)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^2*b^2*d*f*(d*f)^(1/2)-A 
*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c* 
f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a*b^3*c*f*(d 
*f)^(1/2)-A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((f*x+e)*(d*x+c))^(1/2)*((a^2 
*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a 
*b^3*d*e*(d*f)^(1/2)-B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((f*x+e)*(d*x+c))^ 
(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*d*e+2*b*c*e) 
/(b*x+a))*a^2*b^2*c*f*(d*f)^(1/2)-B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((f*x 
+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a 
*d*e+2*b*c*e)/(b*x+a))*a^2*b^2*d*e*(d*f)^(1/2)+2*B*ln((-2*a*d*f*x+b*c*f...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((C*x^2+B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2),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(((-(2*a*d*f)/b^2)>0)', see `assu 
me?` for m
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1319 vs. \(2 (225) = 450\).

Time = 0.87 (sec) , antiderivative size = 1319, normalized size of antiderivative = 5.19 \[ \int \frac {A+B x+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

(4*sqrt(d*f)*C*a*b^2*c*d^2*e - 2*sqrt(d*f)*B*b^3*c*d^2*e - 3*sqrt(d*f)*C*a 
^2*b*d^3*e + sqrt(d*f)*B*a*b^2*d^3*e + sqrt(d*f)*A*b^3*d^3*e - 3*sqrt(d*f) 
*C*a^2*b*c*d^2*f + sqrt(d*f)*B*a*b^2*c*d^2*f + sqrt(d*f)*A*b^3*c*d^2*f + 2 
*sqrt(d*f)*C*a^3*d^3*f - 2*sqrt(d*f)*A*a*b^2*d^3*f)*arctan(-1/2*(b*d^2*e + 
 b*c*d*f - 2*a*d^2*f - (sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d 
*f - c*d*f))^2*b)/(sqrt(-b^2*c*d*e*f + a*b*d^2*e*f + a*b*c*d*f^2 - a^2*d^2 
*f^2)*d))/((b^4*c*e*abs(d) - a*b^3*d*e*abs(d) - a*b^3*c*f*abs(d) + a^2*b^2 
*d*f*abs(d))*sqrt(-b^2*c*d*e*f + a*b*d^2*e*f + a*b*c*d*f^2 - a^2*d^2*f^2)* 
d) - 2*(sqrt(d*f)*C*a^2*b*d^5*e^2 - sqrt(d*f)*B*a*b^2*d^5*e^2 + sqrt(d*f)* 
A*b^3*d^5*e^2 - 2*sqrt(d*f)*C*a^2*b*c*d^4*e*f + 2*sqrt(d*f)*B*a*b^2*c*d^4* 
e*f - 2*sqrt(d*f)*A*b^3*c*d^4*e*f + sqrt(d*f)*C*a^2*b*c^2*d^3*f^2 - sqrt(d 
*f)*B*a*b^2*c^2*d^3*f^2 + sqrt(d*f)*A*b^3*c^2*d^3*f^2 - sqrt(d*f)*(sqrt(d* 
f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^2*b*d^3*e + 
sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^ 
2*B*a*b^2*d^3*e - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + 
 c)*d*f - c*d*f))^2*A*b^3*d^3*e - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqr 
t(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^2*b*c*d^2*f + sqrt(d*f)*(sqrt(d*f) 
*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a*b^2*c*d^2*f - 
sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^ 
2*A*b^3*c*d^2*f + 2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 1.69 (sec) , antiderivative size = 5671, normalized size of antiderivative = 22.33 \[ \int \frac {A+B x+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

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

Output:

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