\(\int \frac {\sqrt {a+b x^2}}{x^3 (c+d x)^3} \, dx\) [1022]

Optimal result

Integrand size = 22, antiderivative size = 253 \[ \int \frac {\sqrt {a+b x^2}}{x^3 (c+d x)^3} \, dx=-\frac {\sqrt {a+b x^2}}{2 c^3 x^2}+\frac {3 d \sqrt {a+b x^2}}{c^4 x}+\frac {d^2 \sqrt {a+b x^2}}{2 c^3 (c+d x)^2}+\frac {d^2 \left (5 b c^2+6 a d^2\right ) \sqrt {a+b x^2}}{2 c^4 \left (b c^2+a d^2\right ) (c+d x)}+\frac {d \left (6 b^2 c^4+19 a b c^2 d^2+12 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 c^5 \left (b c^2+a d^2\right )^{3/2}}-\frac {\left (b c^2+12 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} c^5} \] Output:

-1/2*(b*x^2+a)^(1/2)/c^3/x^2+3*d*(b*x^2+a)^(1/2)/c^4/x+1/2*d^2*(b*x^2+a)^( 
1/2)/c^3/(d*x+c)^2+1/2*d^2*(6*a*d^2+5*b*c^2)*(b*x^2+a)^(1/2)/c^4/(a*d^2+b* 
c^2)/(d*x+c)+1/2*d*(12*a^2*d^4+19*a*b*c^2*d^2+6*b^2*c^4)*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)^(3/2)-1/2*(12*a 
*d^2+b*c^2)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(1/2)/c^5
 

Mathematica [A] (verified)

Time = 1.80 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b x^2}}{x^3 (c+d x)^3} \, dx=\frac {\frac {c \sqrt {a+b x^2} \left (b c^2 \left (-c^3+4 c^2 d x+17 c d^2 x^2+11 d^3 x^3\right )+a d^2 \left (-c^3+4 c^2 d x+18 c d^2 x^2+12 d^3 x^3\right )\right )}{\left (b c^2+a d^2\right ) x^2 (c+d x)^2}-\frac {2 d \left (6 b^2 c^4+19 a b c^2 d^2+12 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {2 \left (b c^2+12 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 c^5} \] Input:

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

Output:

((c*Sqrt[a + b*x^2]*(b*c^2*(-c^3 + 4*c^2*d*x + 17*c*d^2*x^2 + 11*d^3*x^3) 
+ a*d^2*(-c^3 + 4*c^2*d*x + 18*c*d^2*x^2 + 12*d^3*x^3)))/((b*c^2 + a*d^2)* 
x^2*(c + d*x)^2) - (2*d*(6*b^2*c^4 + 19*a*b*c^2*d^2 + 12*a^2*d^4)*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) + (2*(b*c^2 + 12*a*d^2)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2 
])/Sqrt[a]])/Sqrt[a])/(2*c^5)
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2}}{x^3 (c+d x)^3} \, dx\)

\(\Big \downarrow \) 617

\(\displaystyle \int \left (-\frac {6 d^3 \sqrt {a+b x^2}}{c^5 (c+d x)}+\frac {6 d^2 \sqrt {a+b x^2}}{c^5 x}-\frac {3 d^3 \sqrt {a+b x^2}}{c^4 (c+d x)^2}-\frac {3 d \sqrt {a+b x^2}}{c^4 x^2}-\frac {d^3 \sqrt {a+b x^2}}{c^3 (c+d x)^3}+\frac {\sqrt {a+b x^2}}{c^3 x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 \sqrt {a} d^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^5}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} c^3}+\frac {6 d \sqrt {a d^2+b c^2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^5}-\frac {3 b d \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^3 \sqrt {a d^2+b c^2}}+\frac {a b d^3 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{2 c^3 \left (a d^2+b c^2\right )^{3/2}}+\frac {3 d^2 \sqrt {a+b x^2}}{c^4 (c+d x)}+\frac {3 d \sqrt {a+b x^2}}{c^4 x}-\frac {\sqrt {a+b x^2}}{2 c^3 x^2}+\frac {d^3 \sqrt {a+b x^2} (a d-b c x)}{2 c^3 (c+d x)^2 \left (a d^2+b c^2\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 617

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
 

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(893\) vs. \(2(223)=446\).

Time = 0.48 (sec) , antiderivative size = 894, normalized size of antiderivative = 3.53

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-6 d x +c \right )}{2 c^{4} x^{2}}-\frac {\frac {\left (12 a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}+\frac {2 \left (3 a \,d^{2}+b \,c^{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}-\frac {2 \left (6 a \,d^{2}+b \,c^{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 )}{c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 c \left (a \,d^{2}+b \,c^{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}}}}{2 \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 b c d \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 )}{2 \left (a \,d^{2}+b \,c^{2}\right )}+\frac {b \,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 )}{2 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{2}}}{2 c^{4}}\)	\(894\)
default	\(\text {Expression too large to display}\)	\(1880\)

Input:

int((b*x^2+a)^(1/2)/x^3/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*(b*x^2+a)^(1/2)*(-6*d*x+c)/c^4/x^2-1/2/c^4*(1/c*(12*a*d^2+b*c^2)/a^(1 
/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+2/d*(3*a*d^2+b*c^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)))-2/c*(6*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^2*c*(a*d^2+ 
b*c^2)*(-1/2/(a*d^2+b*c^2)*d^2/(x+c/d)^2*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d 
^2+b*c^2)/d^2)^(1/2)+3/2*b*c*d/(a*d^2+b*c^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)))+1/2*b/(a*d^2+b*c^2)*d^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))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (224) = 448\).

Time = 1.09 (sec) , antiderivative size = 2629, normalized size of antiderivative = 10.39 \[ \int \frac {\sqrt {a+b x^2}}{x^3 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)/x^3/(d*x+c)^3,x, algorithm="fricas")
 

Output:

[1/4*(((6*a*b^2*c^4*d^3 + 19*a^2*b*c^2*d^5 + 12*a^3*d^7)*x^4 + 2*(6*a*b^2* 
c^5*d^2 + 19*a^2*b*c^3*d^4 + 12*a^3*c*d^6)*x^3 + (6*a*b^2*c^6*d + 19*a^2*b 
*c^4*d^3 + 12*a^3*c^2*d^5)*x^2)*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)) + ((b^3*c^6*d^2 + 14* 
a*b^2*c^4*d^4 + 25*a^2*b*c^2*d^6 + 12*a^3*d^8)*x^4 + 2*(b^3*c^7*d + 14*a*b 
^2*c^5*d^3 + 25*a^2*b*c^3*d^5 + 12*a^3*c*d^7)*x^3 + (b^3*c^8 + 14*a*b^2*c^ 
6*d^2 + 25*a^2*b*c^4*d^4 + 12*a^3*c^2*d^6)*x^2)*sqrt(a)*log(-(b*x^2 - 2*sq 
rt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(a*b^2*c^8 + 2*a^2*b*c^6*d^2 + a^3*c 
^4*d^4 - (11*a*b^2*c^5*d^3 + 23*a^2*b*c^3*d^5 + 12*a^3*c*d^7)*x^3 - (17*a* 
b^2*c^6*d^2 + 35*a^2*b*c^4*d^4 + 18*a^3*c^2*d^6)*x^2 - 4*(a*b^2*c^7*d + 2* 
a^2*b*c^5*d^3 + a^3*c^3*d^5)*x)*sqrt(b*x^2 + a))/((a*b^2*c^9*d^2 + 2*a^2*b 
*c^7*d^4 + a^3*c^5*d^6)*x^4 + 2*(a*b^2*c^10*d + 2*a^2*b*c^8*d^3 + a^3*c^6* 
d^5)*x^3 + (a*b^2*c^11 + 2*a^2*b*c^9*d^2 + a^3*c^7*d^4)*x^2), 1/4*(2*((6*a 
*b^2*c^4*d^3 + 19*a^2*b*c^2*d^5 + 12*a^3*d^7)*x^4 + 2*(6*a*b^2*c^5*d^2 + 1 
9*a^2*b*c^3*d^4 + 12*a^3*c*d^6)*x^3 + (6*a*b^2*c^6*d + 19*a^2*b*c^4*d^3 + 
12*a^3*c^2*d^5)*x^2)*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)) + 
 ((b^3*c^6*d^2 + 14*a*b^2*c^4*d^4 + 25*a^2*b*c^2*d^6 + 12*a^3*d^8)*x^4 + 2 
*(b^3*c^7*d + 14*a*b^2*c^5*d^3 + 25*a^2*b*c^3*d^5 + 12*a^3*c*d^7)*x^3 +...
 

Sympy [F]

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

integrate((b*x**2+a)**(1/2)/x**3/(d*x+c)**3,x)
                                                                                    
                                                                                    
 

Output:

Integral(sqrt(a + b*x**2)/(x**3*(c + d*x)**3), x)
 

Maxima [F]

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

integrate((b*x^2+a)^(1/2)/x^3/(d*x+c)^3,x, algorithm="maxima")
 

Output:

integrate(sqrt(b*x^2 + a)/((d*x + c)^3*x^3), x)
 

Giac [F(-2)]

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

integrate((b*x^2+a)^(1/2)/x^3/(d*x+c)^3,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

int((a + b*x^2)^(1/2)/(x^3*(c + d*x)^3),x)
 

Output:

int((a + b*x^2)^(1/2)/(x^3*(c + d*x)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1853, normalized size of antiderivative = 7.32 \[ \int \frac {\sqrt {a+b x^2}}{x^3 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^(1/2)/x^3/(d*x+c)^3,x)
 

Output:

(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**3*c**2*d**5*x**2 + 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**6*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**3*d**7*x**4 + 38*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**4*d**3*x**2 + 76*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**4*x**3 + 38*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**5*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**2* 
c**6*d*x**2 + 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**5*d**2*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**2*c**4* 
d**3*x**4 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*c**2*d**5*x**2 - 48 
*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*c*d**6*x**3 - 24*sqrt(a*d**2 + b* 
c**2)*log(c + d*x)*a**3*d**7*x**4 - 38*sqrt(a*d**2 + b*c**2)*log(c + d*x)* 
a**2*b*c**4*d**3*x**2 - 76*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c**3* 
d**4*x**3 - 38*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c**2*d**5*x**4 - 
12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**6*d*x**2 - 24*sqrt(a*d**2 
+ b*c**2)*log(c + d*x)*a*b**2*c**5*d**2*x**3 - 12*sqrt(a*d**2 + b*c**2)...