\(\int \frac {\sqrt {a+b x^2} (A+B x+C x^2)}{x^4 (c+d x)^2} \, dx\) [45]

Optimal result

Integrand size = 32, antiderivative size = 349 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4 (c+d x)^2} \, dx=-\frac {\left (3 c^2 C-3 B c d+4 A d^2\right ) \sqrt {a+b x^2}}{3 c^2 d^2 x^3}+\frac {\left (2 c^2 C-3 B c d+4 A d^2\right ) \sqrt {a+b x^2}}{2 c^3 d x^2}-\frac {\left (A b c^2+6 a c^2 C-9 a B c d+12 a A d^2\right ) \sqrt {a+b x^2}}{3 a c^4 x}+\frac {\left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2}}{c d^2 x^3 (c+d x)}-\frac {\left (b c^2 \left (c^2 C-2 B c d+3 A d^2\right )+a d^2 \left (2 c^2 C-3 B c d+4 A d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^5 \sqrt {b c^2+a d^2}}-\frac {\left (b c^2 (B c-2 A d)-2 a d \left (2 c^2 C-3 B c d+4 A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} c^5} \] Output:

-1/3*(4*A*d^2-3*B*c*d+3*C*c^2)*(b*x^2+a)^(1/2)/c^2/d^2/x^3+1/2*(4*A*d^2-3* 
B*c*d+2*C*c^2)*(b*x^2+a)^(1/2)/c^3/d/x^2-1/3*(12*A*a*d^2+A*b*c^2-9*B*a*c*d 
+6*C*a*c^2)*(b*x^2+a)^(1/2)/a/c^4/x+(A*d^2-B*c*d+C*c^2)*(b*x^2+a)^(1/2)/c/ 
d^2/x^3/(d*x+c)-(b*c^2*(3*A*d^2-2*B*c*d+C*c^2)+a*d^2*(4*A*d^2-3*B*c*d+2*C* 
c^2))*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/c^5/(a*d^2 
+b*c^2)^(1/2)-1/2*(b*c^2*(-2*A*d+B*c)-2*a*d*(4*A*d^2-3*B*c*d+2*C*c^2))*arc 
tanh((b*x^2+a)^(1/2)/a^(1/2))/a^(1/2)/c^5
 

Mathematica [A] (verified)

Time = 2.69 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4 (c+d x)^2} \, dx=-\frac {\frac {c \sqrt {a+b x^2} \left (2 A b c^2 x^2 (c+d x)+2 a A \left (c^3-2 c^2 d x+6 c d^2 x^2+12 d^3 x^3\right )+3 a c x \left (2 c C x (c+2 d x)+B \left (c^2-3 c d x-6 d^2 x^2\right )\right )\right )}{a x^3 (c+d x)}+\frac {12 \left (b c^2 \left (c^2 C-2 B c d+3 A d^2\right )+a d^2 \left (2 c^2 C-3 B c d+4 A d^2\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+48 \sqrt {a} A d^3 \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {6 c (b c (B c-2 A d)+2 a d (-2 c C+3 B d)) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}}{6 c^5} \] Input:

Integrate[(Sqrt[a + b*x^2]*(A + B*x + C*x^2))/(x^4*(c + d*x)^2),x]
 

Output:

-1/6*((c*Sqrt[a + b*x^2]*(2*A*b*c^2*x^2*(c + d*x) + 2*a*A*(c^3 - 2*c^2*d*x 
 + 6*c*d^2*x^2 + 12*d^3*x^3) + 3*a*c*x*(2*c*C*x*(c + 2*d*x) + B*(c^2 - 3*c 
*d*x - 6*d^2*x^2))))/(a*x^3*(c + d*x)) + (12*(b*c^2*(c^2*C - 2*B*c*d + 3*A 
*d^2) + a*d^2*(2*c^2*C - 3*B*c*d + 4*A*d^2))*ArcTan[(Sqrt[b]*(c + d*x) - d 
*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/Sqrt[-(b*c^2) - a*d^2] + 48*Sqr 
t[a]*A*d^3*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]] + (6*c*(b*c*(B*c 
 - 2*A*d) + 2*a*d*(-2*c*C + 3*B*d))*ArcTanh[(-(Sqrt[b]*x) + Sqrt[a + b*x^2 
])/Sqrt[a]])/Sqrt[a])/c^5
 

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2353, 2009}

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.

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

\(\Big \downarrow \) 2353

\(\displaystyle \int \left (\frac {\sqrt {a+b x^2} (B c-2 A d)}{c^3 x^3}+\frac {d^2 \sqrt {a+b x^2} \left (4 A d^2-3 B c d+2 c^2 C\right )}{c^5 (c+d x)}-\frac {d \sqrt {a+b x^2} \left (4 A d^2-3 B c d+2 c^2 C\right )}{c^5 x}+\frac {d^2 \sqrt {a+b x^2} \left (A d^2-B c d+c^2 C\right )}{c^4 (c+d x)^2}+\frac {\sqrt {a+b x^2} \left (3 A d^2-2 B c d+c^2 C\right )}{c^4 x^2}+\frac {A \sqrt {a+b x^2}}{c^2 x^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) (B c-2 A d)}{2 \sqrt {a} c^3}-\frac {\sqrt {a d^2+b c^2} \left (4 A d^2-3 B c d+2 c^2 C\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^5}+\frac {\sqrt {a} d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \left (4 A d^2-3 B c d+2 c^2 C\right )}{c^5}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (A d^2-B c d+c^2 C\right )}{c^4}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 A d^2-2 B c d+c^2 C\right )}{c^4}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (4 A d^2-3 B c d+2 c^2 C\right )}{c^4}+\frac {b \left (A d^2-B c d+c^2 C\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^3 \sqrt {a d^2+b c^2}}-\frac {\sqrt {a+b x^2} (B c-2 A d)}{2 c^3 x^2}-\frac {\sqrt {a+b x^2} \left (3 A d^2-2 B c d+c^2 C\right )}{c^4 x}-\frac {d \sqrt {a+b x^2} \left (A d^2-B c d+c^2 C\right )}{c^4 (c+d x)}-\frac {A \left (a+b x^2\right )^{3/2}}{3 a c^2 x^3}\)

Input:

Int[(Sqrt[a + b*x^2]*(A + B*x + C*x^2))/(x^4*(c + d*x)^2),x]
 

Output:

-1/2*((B*c - 2*A*d)*Sqrt[a + b*x^2])/(c^3*x^2) - ((c^2*C - 2*B*c*d + 3*A*d 
^2)*Sqrt[a + b*x^2])/(c^4*x) - (d*(c^2*C - B*c*d + A*d^2)*Sqrt[a + b*x^2]) 
/(c^4*(c + d*x)) - (A*(a + b*x^2)^(3/2))/(3*a*c^2*x^3) + (Sqrt[b]*(c^2*C - 
 B*c*d + A*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/c^4 + (Sqrt[b]*(c^2* 
C - 2*B*c*d + 3*A*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/c^4 - (Sqrt[b 
]*(2*c^2*C - 3*B*c*d + 4*A*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/c^4 
+ (b*(c^2*C - B*c*d + A*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sq 
rt[a + b*x^2])])/(c^3*Sqrt[b*c^2 + a*d^2]) - (Sqrt[b*c^2 + a*d^2]*(2*c^2*C 
 - 3*B*c*d + 4*A*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + 
b*x^2])])/c^5 - (b*(B*c - 2*A*d)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*Sqrt 
[a]*c^3) + (Sqrt[a]*d*(2*c^2*C - 3*B*c*d + 4*A*d^2)*ArcTanh[Sqrt[a + b*x^2 
]/Sqrt[a]])/c^5
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2353

Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) 
^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer 
Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
 

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.69

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (18 A a \,d^{2} x^{2}+2 A b \,c^{2} x^{2}-12 B a c d \,x^{2}+6 C a \,c^{2} x^{2}-6 A a c d x +3 B a \,c^{2} x +2 A \,c^{2} a \right )}{6 a \,c^{4} x^{3}}+\frac {\frac {\left (2 A a \,d^{4}+2 A b \,c^{2} d^{2}-2 B a c \,d^{3}-2 c^{3} B b d +2 C a \,c^{2} d^{2}+2 c^{4} C b \right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{2}}+\frac {\left (8 A a \,d^{3}+2 A b \,c^{2} d -6 B a c \,d^{2}-B b \,c^{3}+4 C a \,c^{2} d \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}-\frac {2 \left (4 A a \,d^{3}+2 A b \,c^{2} d -3 B a c \,d^{2}-B b \,c^{3}+2 C a \,c^{2} d \right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 c^{4}}\)	\(589\)
default	\(\text {Expression too large to display}\)	\(1071\)

Input:

int((b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^4/(d*x+c)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/6*(b*x^2+a)^(1/2)*(18*A*a*d^2*x^2+2*A*b*c^2*x^2-12*B*a*c*d*x^2+6*C*a*c^ 
2*x^2-6*A*a*c*d*x+3*B*a*c^2*x+2*A*a*c^2)/a/c^4/x^3+1/2/c^4*((2*A*a*d^4+2*A 
*b*c^2*d^2-2*B*a*c*d^3-2*B*b*c^3*d+2*C*a*c^2*d^2+2*C*b*c^4)/d^2*(-1/(a*d^2 
+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)- 
b*c*d/(a*d^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b* 
c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^ 
2+b*c^2)/d^2)^(1/2))/(x+c/d)))+1/c*(8*A*a*d^3+2*A*b*c^2*d-6*B*a*c*d^2-B*b* 
c^3+4*C*a*c^2*d)/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-2/c*(4*A*a* 
d^3+2*A*b*c^2*d-3*B*a*c*d^2-B*b*c^3+2*C*a*c^2*d)/((a*d^2+b*c^2)/d^2)^(1/2) 
*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x 
+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (322) = 644\).

Time = 10.01 (sec) , antiderivative size = 2786, normalized size of antiderivative = 7.98 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4 (c+d x)^2} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^4/(d*x+c)^2,x, algorithm="fricas 
")
 

Output:

[1/12*(6*((C*a*b*c^4*d - 2*B*a*b*c^3*d^2 - 3*B*a^2*c*d^4 + 4*A*a^2*d^5 + ( 
2*C*a^2 + 3*A*a*b)*c^2*d^3)*x^4 + (C*a*b*c^5 - 2*B*a*b*c^4*d - 3*B*a^2*c^2 
*d^3 + 4*A*a^2*c*d^4 + (2*C*a^2 + 3*A*a*b)*c^3*d^2)*x^3)*sqrt(b*c^2 + a*d^ 
2)*log((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 - 2* 
sqrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^ 
2)) - 3*((B*b^2*c^5*d + 7*B*a*b*c^3*d^3 + 6*B*a^2*c*d^5 - 8*A*a^2*d^6 - 2* 
(2*C*a*b + A*b^2)*c^4*d^2 - 2*(2*C*a^2 + 5*A*a*b)*c^2*d^4)*x^4 + (B*b^2*c^ 
6 + 7*B*a*b*c^4*d^2 + 6*B*a^2*c^2*d^4 - 8*A*a^2*c*d^5 - 2*(2*C*a*b + A*b^2 
)*c^5*d - 2*(2*C*a^2 + 5*A*a*b)*c^3*d^3)*x^3)*sqrt(a)*log(-(b*x^2 + 2*sqrt 
(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(2*A*a*b*c^6 + 2*A*a^2*c^4*d^2 - 2*(9* 
B*a*b*c^4*d^2 + 9*B*a^2*c^2*d^4 - 12*A*a^2*c*d^5 - (6*C*a*b + A*b^2)*c^5*d 
 - (6*C*a^2 + 13*A*a*b)*c^3*d^3)*x^3 - (9*B*a*b*c^5*d + 9*B*a^2*c^3*d^3 - 
12*A*a^2*c^2*d^4 - 2*(3*C*a*b + A*b^2)*c^6 - 2*(3*C*a^2 + 7*A*a*b)*c^4*d^2 
)*x^2 + (3*B*a*b*c^6 - 4*A*a*b*c^5*d + 3*B*a^2*c^4*d^2 - 4*A*a^2*c^3*d^3)* 
x)*sqrt(b*x^2 + a))/((a*b*c^7*d + a^2*c^5*d^3)*x^4 + (a*b*c^8 + a^2*c^6*d^ 
2)*x^3), -1/12*(12*((C*a*b*c^4*d - 2*B*a*b*c^3*d^2 - 3*B*a^2*c*d^4 + 4*A*a 
^2*d^5 + (2*C*a^2 + 3*A*a*b)*c^2*d^3)*x^4 + (C*a*b*c^5 - 2*B*a*b*c^4*d - 3 
*B*a^2*c^2*d^3 + 4*A*a^2*c*d^4 + (2*C*a^2 + 3*A*a*b)*c^3*d^2)*x^3)*sqrt(-b 
*c^2 - a*d^2)*arctan(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a 
*b*c^2 + a^2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)) + 3*((B*b^2*c^5*d + 7*B*a*...
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4 (c+d x)^2} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (A + B x + C x^{2}\right )}{x^{4} \left (c + d x\right )^{2}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(C*x**2+B*x+A)/x**4/(d*x+c)**2,x)
 

Output:

Integral(sqrt(a + b*x**2)*(A + B*x + C*x**2)/(x**4*(c + d*x)**2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4 (c+d x)^2} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x^{2} + a}}{{\left (d x + c\right )}^{2} x^{4}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^4/(d*x+c)^2,x, algorithm="maxima 
")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(b*x^2 + a)/((d*x + c)^2*x^4), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4 (c+d x)^2} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^4/(d*x+c)^2,x, algorithm="giac")
 

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4 (c+d x)^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (C\,x^2+B\,x+A\right )}{x^4\,{\left (c+d\,x\right )}^2} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(A + B*x + C*x^2))/(x^4*(c + d*x)^2),x)
 

Output:

int(((a + b*x^2)^(1/2)*(A + B*x + C*x^2))/(x^4*(c + d*x)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 2409, normalized size of antiderivative = 6.90 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4 (c+d x)^2} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^4/(d*x+c)^2,x)
 

Output:

(48*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d 
 + b*c*x)*a**3*c*d**4*x**3 + 48*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2) 
*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*d**5*x**4 + 36*sqrt(a*d**2 + b* 
c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*c** 
3*d**2*x**3 + 36*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + 
b*c**2) - a*d + b*c*x)*a**2*b*c**2*d**3*x**4 - 36*sqrt(a*d**2 + b*c**2)*lo 
g(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*c**2*d**3*x 
**3 - 36*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) 
- a*d + b*c*x)*a**2*b*c*d**4*x**4 + 24*sqrt(a*d**2 + b*c**2)*log(sqrt(a + 
b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*c**4*d**2*x**3 + 24*sqrt 
(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x 
)*a**2*c**3*d**3*x**4 - 24*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt 
(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**4*d*x**3 - 24*sqrt(a*d**2 + b*c 
**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**3 
*d**2*x**4 + 12*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b 
*c**2) - a*d + b*c*x)*a*b*c**6*x**3 + 12*sqrt(a*d**2 + b*c**2)*log(sqrt(a 
+ b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c**5*d*x**4 - 48*sqrt(a 
*d**2 + b*c**2)*log(c + d*x)*a**3*c*d**4*x**3 - 48*sqrt(a*d**2 + b*c**2)*l 
og(c + d*x)*a**3*d**5*x**4 - 36*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b* 
c**3*d**2*x**3 - 36*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c**2*d**3...