\(\int \frac {(a+b x^2)^{3/2} (A+B x+C x^2)}{x^3 (c+d x)^2} \, dx\) [91]

Optimal result

Integrand size = 32, antiderivative size = 402 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^3 (c+d x)^2} \, dx=\frac {b C \sqrt {a+b x^2}}{d^2}-\frac {\left (2 b c^2 \left (c^2 C-B c d+A d^2\right )+a d^2 \left (2 c^2 C-2 B c d+3 A d^2\right )\right ) \sqrt {a+b x^2}}{2 c^2 d^4 x^2}+\frac {\left (b c^2 \left (c^2 C-B c d+A d^2\right )+a d^2 \left (c^2 C-2 B c d+3 A d^2\right )\right ) \sqrt {a+b x^2}}{c^3 d^3 x}+\frac {\left (b c^2+a d^2\right ) \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2}}{c d^4 x^2 (c+d x)}-\frac {b^{3/2} (2 c C-B d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^3}-\frac {\sqrt {b c^2+a d^2} \left (b c^3 (2 c C-B d)-a d^2 \left (c^2 C-2 B c d+3 A d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^4 d^3}-\frac {\sqrt {a} \left (2 a c (c C-2 B d)+3 A \left (b c^2+2 a d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 c^4} \] Output:

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

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^3 (c+d x)^2} \, dx=\frac {\sqrt {a+b x^2} \left (2 b c^2 x^2 \left (2 c^2 C-B c d+A d^2+c C d x\right )+a d^2 \left (A \left (-c^2+3 c d x+6 d^2 x^2\right )+2 c x (c C x-B (c+2 d x))\right )\right )}{2 c^3 d^2 x^2 (c+d x)}+\frac {2 \sqrt {-b c^2-a d^2} \left (b c^3 (2 c C-B d)-a d^2 \left (c^2 C-2 B c d+3 A d^2\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{c^4 d^3}+\frac {3 \sqrt {a} A b \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^2}-\frac {2 a^{3/2} \left (c^2 C-2 B c d+3 A d^2\right ) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^4}+\frac {b^{3/2} (2 c C-B d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{d^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.88 (sec) , antiderivative size = 669, normalized size of antiderivative = 1.66, 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.

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

\(\Big \downarrow \) 2353

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 0.36 (sec) , antiderivative size = 690, normalized size of antiderivative = 1.72

method	result	size
risch	\(-\frac {a \sqrt {b \,x^{2}+a}\, \left (-4 A d x +2 B c x +A c \right )}{2 c^{3} x^{2}}-\frac {\frac {2 \left (A \,a^{2} d^{6}+2 A a b \,c^{2} d^{4}+A \,b^{2} c^{4} d^{2}-B \,a^{2} c \,d^{5}-2 B a b \,c^{3} d^{3}-c^{5} B \,b^{2} d +C \,a^{2} c^{2} d^{4}+2 C a b \,c^{4} d^{2}+c^{6} C \,b^{2}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{5}}-\frac {2 \left (3 A \,a^{2} d^{6}+2 A a b \,c^{2} d^{4}-A \,b^{2} c^{4} d^{2}-2 B \,a^{2} c \,d^{5}+2 c^{5} B \,b^{2} d +C \,a^{2} c^{2} d^{4}-2 C a b \,c^{4} d^{2}-3 c^{6} C \,b^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{4} c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {\sqrt {a}\, \left (6 A a \,d^{2}+3 b A \,c^{2}-4 B a c d +2 C a \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c}-\frac {2 b^{2} c^{3} \left (\frac {B d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+\frac {C d \sqrt {b \,x^{2}+a}}{b}-\frac {2 C c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\right )}{d^{3}}}{2 c^{3}}\)	\(690\)
default	\(\text {Expression too large to display}\)	\(1649\)

Input:

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

Output:

-1/2*a*(b*x^2+a)^(1/2)*(-4*A*d*x+2*B*c*x+A*c)/c^3/x^2-1/2/c^3*(2*(A*a^2*d^ 
6+2*A*a*b*c^2*d^4+A*b^2*c^4*d^2-B*a^2*c*d^5-2*B*a*b*c^3*d^3-B*b^2*c^5*d+C* 
a^2*c^2*d^4+2*C*a*b*c^4*d^2+C*b^2*c^6)/d^5*(-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)))-2/d^4*(3*A*a^2*d^6+2*A*a*b*c^2*d^4-A*b^2*c^4*d^2-2*B*a^2*c*d^5+2 
*B*b^2*c^5*d+C*a^2*c^2*d^4-2*C*a*b*c^4*d^2-3*C*b^2*c^6)/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))+a^(1/2 
)/c*(6*A*a*d^2+3*A*b*c^2-4*B*a*c*d+2*C*a*c^2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^ 
(1/2))/x)-2*b^2*c^3/d^3*(B*d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)+C*d*(b* 
x^2+a)^(1/2)/b-2*C*c*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:16.7888 interp horner, loop index 0 
16.7889 interp resultant evaled at -3, 0% done22.132 interp dd 22.2036 int 
erp build
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1727, normalized size of antiderivative = 4.30 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^3 (c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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