\(\int \frac {A+B x+C x^2}{x^3 (c+d x) (a+b x^2)^{3/2}} \, dx\) [156]

Optimal result

Integrand size = 32, antiderivative size = 293 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {A}{2 a c x^2 \sqrt {a+b x^2}}-\frac {B c-A d}{a c^2 x \sqrt {a+b x^2}}+\frac {b \left (c \left (2 a c (c C-B d)-A \left (3 b c^2+a d^2\right )\right )-2 \left (2 b c^2 (B c-A d)+a d \left (c^2 C+B c d-A d^2\right )\right ) x\right )}{2 a^2 c^2 \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}+\frac {d^3 \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^3 \left (b c^2+a d^2\right )^{3/2}}-\frac {\left (2 a c (c C-B d)-A \left (3 b c^2-2 a d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2} c^3} \] Output:

-1/2*A/a/c/x^2/(b*x^2+a)^(1/2)-(-A*d+B*c)/a/c^2/x/(b*x^2+a)^(1/2)+1/2*b*(c 
*(2*a*c*(-B*d+C*c)-A*(a*d^2+3*b*c^2))-2*(2*b*c^2*(-A*d+B*c)+a*d*(-A*d^2+B* 
c*d+C*c^2))*x)/a^2/c^2/(a*d^2+b*c^2)/(b*x^2+a)^(1/2)+d^3*(A*d^2-B*c*d+C*c^ 
2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/c^3/(a*d^2+b* 
c^2)^(3/2)-1/2*(2*a*c*(-B*d+C*c)-A*(-2*a*d^2+3*b*c^2))*arctanh((b*x^2+a)^( 
1/2)/a^(1/2))/a^(5/2)/c^3
 

Mathematica [A] (verified)

Time = 2.43 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {\frac {c \left (b^2 c^2 x^2 (3 A c+4 B c x-4 A d x)+a^2 d^2 (2 B c x+A (c-2 d x))+a b \left (A \left (c^3-2 c^2 d x+c d^2 x^2-2 d^3 x^3\right )+2 c x \left (c C x (-c+d x)+B \left (c^2+c d x+d^2 x^2\right )\right )\right )\right )}{a^2 \left (b c^2+a d^2\right ) x^2 \sqrt {a+b x^2}}+\frac {4 d^3 \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {6 A b c^2 \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {4 \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}}{2 c^3} \] Input:

Integrate[(A + B*x + C*x^2)/(x^3*(c + d*x)*(a + b*x^2)^(3/2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.19, 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.

\(\displaystyle \int \frac {A+B x+C x^2}{x^3 \left (a+b x^2\right )^{3/2} (c+d x)} \, dx\)

\(\Big \downarrow \) 2353

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*x + C*x^2)/(x^3*(c + d*x)*(a + b*x^2)^(3/2)),x]
 

Output:

(-3*A*b)/(2*a^2*c*Sqrt[a + b*x^2]) + (c^2*C - B*c*d + A*d^2)/(a*c^3*Sqrt[a 
 + b*x^2]) - A/(2*a*c*x^2*Sqrt[a + b*x^2]) - (B*c - A*d)/(a*c^2*x*Sqrt[a + 
 b*x^2]) - (2*b*(B*c - A*d)*x)/(a^2*c^2*Sqrt[a + b*x^2]) - (d*(c^2*C - B*c 
*d + A*d^2)*(a*d + b*c*x))/(a*c^3*(b*c^2 + a*d^2)*Sqrt[a + b*x^2]) + (d^3* 
(c^2*C - B*c*d + A*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a 
+ b*x^2])])/(c^3*(b*c^2 + a*d^2)^(3/2)) + (3*A*b*ArcTanh[Sqrt[a + b*x^2]/S 
qrt[a]])/(2*a^(5/2)*c) - ((c^2*C - B*c*d + A*d^2)*ArcTanh[Sqrt[a + b*x^2]/ 
Sqrt[a]])/(a^(3/2)*c^3)
 

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 = 0.30 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.62

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 A d x +2 B c x +A c \right )}{2 a^{2} c^{2} x^{2}}-\frac {\frac {\left (2 A a \,d^{2}-3 b A \,c^{2}-2 B a c d +2 C a \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}+\frac {b \,c^{2} \left (A b -B \sqrt {-a b}-a C \right ) \sqrt {b \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{\left (d \sqrt {-a b}-b c \right ) \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {b \,c^{2} \left (A b +B \sqrt {-a b}-a C \right ) \sqrt {b \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{\left (d \sqrt {-a b}+b c \right ) \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {2 b \,a^{2} \left (A \,d^{2}-B c d +C \,c^{2}\right ) d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (d \sqrt {-a b}+b c \right ) \left (d \sqrt {-a b}-b c \right ) c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 c^{2} a^{2}}\)	\(474\)
default	\(\frac {A \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{c}-\frac {\left (A d -B c \right ) \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{c^{2}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{c^{3}}-\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{c^{3}}\)	\(513\)

Input:

int((C*x^2+B*x+A)/x^3/(d*x+c)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(b*x^2+a)^(1/2)*(-2*A*d*x+2*B*c*x+A*c)/a^2/c^2/x^2-1/2/c^2/a^2*((2*A* 
a*d^2-3*A*b*c^2-2*B*a*c*d+2*C*a*c^2)/c/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a) 
^(1/2))/x)+b*c^2*(A*b-B*(-a*b)^(1/2)-a*C)/(d*(-a*b)^(1/2)-b*c)/(-a*b)^(1/2 
)/(x+(-a*b)^(1/2)/b)*(b*(x+(-a*b)^(1/2)/b)^2-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2 
)/b))^(1/2)+b*c^2*(A*b+B*(-a*b)^(1/2)-a*C)/(d*(-a*b)^(1/2)+b*c)/(-a*b)^(1/ 
2)/(x-(-a*b)^(1/2)/b)*(b*(x-(-a*b)^(1/2)/b)^2+2*(-a*b)^(1/2)*(x-(-a*b)^(1/ 
2)/b))^(1/2)+2*b*a^2*(A*d^2-B*c*d+C*c^2)*d^2/(d*(-a*b)^(1/2)+b*c)/(d*(-a*b 
)^(1/2)-b*c)/c/((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. 813 vs. \(2 (270) = 540\).

Time = 21.32 (sec) , antiderivative size = 3317, normalized size of antiderivative = 11.32 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/x^3/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {A+B x+C x^2}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {A + B x + C x^{2}}{x^{3} \left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )}\, dx \] Input:

integrate((C*x**2+B*x+A)/x**3/(d*x+c)/(b*x**2+a)**(3/2),x)
 

Output:

Integral((A + B*x + C*x**2)/(x**3*(a + b*x**2)**(3/2)*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )} x^{3}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/x^3/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)/((b*x^2 + a)^(3/2)*(d*x + c)*x^3), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.83 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {\frac {{\left (B a^{2} b^{3} c^{3} + C a^{3} b^{2} c^{2} d - A a^{2} b^{3} c^{2} d + B a^{3} b^{2} c d^{2} + C a^{4} b d^{3} - A a^{3} b^{2} d^{3}\right )} x}{a^{4} b^{2} c^{4} + 2 \, a^{5} b c^{2} d^{2} + a^{6} d^{4}} - \frac {C a^{3} b^{2} c^{3} - A a^{2} b^{3} c^{3} - B a^{3} b^{2} c^{2} d + C a^{4} b c d^{2} - A a^{3} b^{2} c d^{2} - B a^{4} b d^{3}}{a^{4} b^{2} c^{4} + 2 \, a^{5} b c^{2} d^{2} + a^{6} d^{4}}}{\sqrt {b x^{2} + a}} - \frac {2 \, {\left (C c^{2} d^{3} - B c d^{4} + A d^{5}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b c^{5} + a c^{3} d^{2}\right )} \sqrt {-b c^{2} - a d^{2}}} + \frac {{\left (2 \, C a c^{2} - 3 \, A b c^{2} - 2 \, B a c d + 2 \, A a d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} c^{3}} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b c + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a \sqrt {b} d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b c - 2 \, B a^{2} \sqrt {b} c + 2 \, A a^{2} \sqrt {b} d}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{2} c^{2}} \] Input:

integrate((C*x^2+B*x+A)/x^3/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

-((B*a^2*b^3*c^3 + C*a^3*b^2*c^2*d - A*a^2*b^3*c^2*d + B*a^3*b^2*c*d^2 + C 
*a^4*b*d^3 - A*a^3*b^2*d^3)*x/(a^4*b^2*c^4 + 2*a^5*b*c^2*d^2 + a^6*d^4) - 
(C*a^3*b^2*c^3 - A*a^2*b^3*c^3 - B*a^3*b^2*c^2*d + C*a^4*b*c*d^2 - A*a^3*b 
^2*c*d^2 - B*a^4*b*d^3)/(a^4*b^2*c^4 + 2*a^5*b*c^2*d^2 + a^6*d^4))/sqrt(b* 
x^2 + a) - 2*(C*c^2*d^3 - B*c*d^4 + A*d^5)*arctan(-((sqrt(b)*x - sqrt(b*x^ 
2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b*c^5 + a*c^3*d^2)*sqrt(-b* 
c^2 - a*d^2)) + (2*C*a*c^2 - 3*A*b*c^2 - 2*B*a*c*d + 2*A*a*d^2)*arctan(-(s 
qrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2*c^3) + ((sqrt(b)*x - s 
qrt(b*x^2 + a))^3*A*b*c + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a*sqrt(b)*c 
- 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a*sqrt(b)*d + (sqrt(b)*x - sqrt(b*x^ 
2 + a))*A*a*b*c - 2*B*a^2*sqrt(b)*c + 2*A*a^2*sqrt(b)*d)/(((sqrt(b)*x - sq 
rt(b*x^2 + a))^2 - a)^2*a^2*c^2)
 

Mupad [F(-1)]

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

int((A + B*x + C*x^2)/(x^3*(a + b*x^2)^(3/2)*(c + d*x)),x)
 

Output:

int((A + B*x + C*x^2)/(x^3*(a + b*x^2)^(3/2)*(c + d*x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 2277, normalized size of antiderivative = 7.77 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int((C*x^2+B*x+A)/x^3/(d*x+c)/(b*x^2+a)^(3/2),x)
 

Output:

(4*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**4*d**5*x**2 - 4*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*b*c*d**4*x**2 + 4*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*b*d**5*x**4 + 4*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**3*d**3*x**2 - 4*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**2*c*d 
**4*x**4 + 4*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**3*x**4 - 4*sqrt(a*d**2 + b*c**2)*log( 
c + d*x)*a**4*d**5*x**2 + 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*b*c*d* 
*4*x**2 - 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*b*d**5*x**4 - 4*sqrt(a 
*d**2 + b*c**2)*log(c + d*x)*a**3*c**3*d**3*x**2 + 4*sqrt(a*d**2 + b*c**2) 
*log(c + d*x)*a**2*b**2*c*d**4*x**4 - 4*sqrt(a*d**2 + b*c**2)*log(c + d*x) 
*a**2*b*c**3*d**3*x**4 - 2*sqrt(a + b*x**2)*a**4*c**2*d**4 + 4*sqrt(a + b* 
x**2)*a**4*c*d**5*x - 4*sqrt(a + b*x**2)*a**3*b*c**4*d**2 + 8*sqrt(a + b*x 
**2)*a**3*b*c**3*d**3*x - 2*sqrt(a + b*x**2)*a**3*b*c**2*d**4*x**2 - 4*sqr 
t(a + b*x**2)*a**3*b*c**2*d**4*x + 4*sqrt(a + b*x**2)*a**3*b*c*d**5*x**3 - 
 2*sqrt(a + b*x**2)*a**2*b**2*c**6 + 4*sqrt(a + b*x**2)*a**2*b**2*c**5*d*x 
 - 8*sqrt(a + b*x**2)*a**2*b**2*c**4*d**2*x**2 - 8*sqrt(a + b*x**2)*a**2*b 
**2*c**4*d**2*x + 12*sqrt(a + b*x**2)*a**2*b**2*c**3*d**3*x**3 - 4*sqrt...