\(\int \frac {1}{x (c+d x) (a+b x^2)^3} \, dx\) [249]

Optimal result

Integrand size = 20, antiderivative size = 260 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^3} \, dx=\frac {b (c-d x)}{4 a \left (b c^2+a d^2\right ) \left (a+b x^2\right )^2}+\frac {b \left (4 c \left (b c^2+2 a d^2\right )-d \left (3 b c^2+7 a d^2\right ) x\right )}{8 a^2 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}-\frac {\sqrt {b} d \left (3 b^2 c^4+10 a b c^2 d^2+15 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} \left (b c^2+a d^2\right )^3}+\frac {\log (x)}{a^3 c}-\frac {d^6 \log (c+d x)}{c \left (b c^2+a d^2\right )^3}-\frac {b c \left (b^2 c^4+3 a b c^2 d^2+3 a^2 d^4\right ) \log \left (a+b x^2\right )}{2 a^3 \left (b c^2+a d^2\right )^3} \] Output:

1/4*b*(-d*x+c)/a/(a*d^2+b*c^2)/(b*x^2+a)^2+1/8*b*(4*c*(2*a*d^2+b*c^2)-d*(7 
*a*d^2+3*b*c^2)*x)/a^2/(a*d^2+b*c^2)^2/(b*x^2+a)-1/8*b^(1/2)*d*(15*a^2*d^4 
+10*a*b*c^2*d^2+3*b^2*c^4)*arctan(b^(1/2)*x/a^(1/2))/a^(5/2)/(a*d^2+b*c^2) 
^3+ln(x)/a^3/c-d^6*ln(d*x+c)/c/(a*d^2+b*c^2)^3-1/2*b*c*(3*a^2*d^4+3*a*b*c^ 
2*d^2+b^2*c^4)*ln(b*x^2+a)/a^3/(a*d^2+b*c^2)^3
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^3} \, dx=\frac {1}{8} \left (\frac {2 b (c-d x)}{a \left (b c^2+a d^2\right ) \left (a+b x^2\right )^2}+\frac {b \left (a d^2 (8 c-7 d x)+b c^2 (4 c-3 d x)\right )}{a^2 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}-\frac {\sqrt {b} d \left (3 b^2 c^4+10 a b c^2 d^2+15 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \left (b c^2+a d^2\right )^3}+\frac {8 \log (x)}{a^3 c}-\frac {8 d^6 \log (c+d x)}{c \left (b c^2+a d^2\right )^3}-\frac {4 b \left (b^2 c^5+3 a b c^3 d^2+3 a^2 c d^4\right ) \log \left (a+b x^2\right )}{a^3 \left (b c^2+a d^2\right )^3}\right ) \] Input:

Integrate[1/(x*(c + d*x)*(a + b*x^2)^3),x]
 

Output:

((2*b*(c - d*x))/(a*(b*c^2 + a*d^2)*(a + b*x^2)^2) + (b*(a*d^2*(8*c - 7*d* 
x) + b*c^2*(4*c - 3*d*x)))/(a^2*(b*c^2 + a*d^2)^2*(a + b*x^2)) - (Sqrt[b]* 
d*(3*b^2*c^4 + 10*a*b*c^2*d^2 + 15*a^2*d^4)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/( 
a^(5/2)*(b*c^2 + a*d^2)^3) + (8*Log[x])/(a^3*c) - (8*d^6*Log[c + d*x])/(c* 
(b*c^2 + a*d^2)^3) - (4*b*(b^2*c^5 + 3*a*b*c^3*d^2 + 3*a^2*c*d^4)*Log[a + 
b*x^2])/(a^3*(b*c^2 + a*d^2)^3))/8
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.30, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {615, 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.

\(\displaystyle \int \frac {1}{x \left (a+b x^2\right )^3 (c+d x)} \, dx\)

\(\Big \downarrow \) 615

\(\displaystyle \int \left (\frac {1}{a^3 c x}+\frac {b \left (-a^2 d^3-b c x \left (2 a d^2+b c^2\right )\right )}{a^2 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}+\frac {b \left (-a^3 d^5-b c x \left (3 a^2 d^4+3 a b c^2 d^2+b^2 c^4\right )\right )}{a^3 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}-\frac {b (a d+b c x)}{a \left (a+b x^2\right )^3 \left (a d^2+b c^2\right )}-\frac {d^7}{c (c+d x) \left (a d^2+b c^2\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt {b} d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} \left (a d^2+b c^2\right )}-\frac {\sqrt {b} d^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \left (a d^2+b c^2\right )^2}+\frac {\log (x)}{a^3 c}-\frac {3 b d x}{8 a^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )}+\frac {b \left (c \left (2 a d^2+b c^2\right )-a d^3 x\right )}{2 a^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^2}-\frac {b c \left (3 a^2 d^4+3 a b c^2 d^2+b^2 c^4\right ) \log \left (a+b x^2\right )}{2 a^3 \left (a d^2+b c^2\right )^3}-\frac {\sqrt {b} d^5 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \left (a d^2+b c^2\right )^3}+\frac {b (c-d x)}{4 a \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )}-\frac {d^6 \log (c+d x)}{c \left (a d^2+b c^2\right )^3}\)

Input:

Int[1/(x*(c + d*x)*(a + b*x^2)^3),x]
 

Output:

(b*(c - d*x))/(4*a*(b*c^2 + a*d^2)*(a + b*x^2)^2) - (3*b*d*x)/(8*a^2*(b*c^ 
2 + a*d^2)*(a + b*x^2)) + (b*(c*(b*c^2 + 2*a*d^2) - a*d^3*x))/(2*a^2*(b*c^ 
2 + a*d^2)^2*(a + b*x^2)) - (Sqrt[b]*d^5*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqr 
t[a]*(b*c^2 + a*d^2)^3) - (Sqrt[b]*d^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^( 
3/2)*(b*c^2 + a*d^2)^2) - (3*Sqrt[b]*d*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^( 
5/2)*(b*c^2 + a*d^2)) + Log[x]/(a^3*c) - (d^6*Log[c + d*x])/(c*(b*c^2 + a* 
d^2)^3) - (b*c*(b^2*c^4 + 3*a*b*c^2*d^2 + 3*a^2*d^4)*Log[a + b*x^2])/(2*a^ 
3*(b*c^2 + a*d^2)^3)
 

Defintions of rubi rules used

rule 615

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

rule 2009

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

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.16

method	result	size
default	\(\frac {\ln \left (x \right )}{a^{3} c}-\frac {b \left (\frac {\left (\frac {7}{8} a^{3} b \,d^{5}+\frac {5}{4} a^{2} b^{2} c^{2} d^{3}+\frac {3}{8} a \,b^{3} c^{4} d \right ) x^{3}+\left (-a^{3} b c \,d^{4}-\frac {3}{2} a^{2} b^{2} c^{3} d^{2}-\frac {1}{2} a \,b^{3} c^{5}\right ) x^{2}+\frac {a^{2} d \left (9 a^{2} d^{4}+14 b \,c^{2} d^{2} a +5 b^{2} c^{4}\right ) x}{8}-\frac {5 a^{4} c \,d^{4}}{4}-2 a^{3} c^{3} d^{2} b -\frac {3 a^{2} c^{5} b^{2}}{4}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (24 a^{2} c \,d^{4} b +24 a \,b^{2} c^{3} d^{2}+8 b^{3} c^{5}\right ) \ln \left (b \,x^{2}+a \right )}{16 b}+\frac {\left (15 a^{3} d^{5}+10 a^{2} b \,c^{2} d^{3}+3 a \,b^{2} c^{4} d \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{3} a^{3}}-\frac {d^{6} \ln \left (d x +c \right )}{c \left (a \,d^{2}+b \,c^{2}\right )^{3}}\)	\(302\)
risch	\(\text {Expression too large to display}\)	\(10761\)

Input:

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

Output:

ln(x)/a^3/c-b/(a*d^2+b*c^2)^3/a^3*(((7/8*a^3*b*d^5+5/4*a^2*b^2*c^2*d^3+3/8 
*a*b^3*c^4*d)*x^3+(-a^3*b*c*d^4-3/2*a^2*b^2*c^3*d^2-1/2*a*b^3*c^5)*x^2+1/8 
*a^2*d*(9*a^2*d^4+14*a*b*c^2*d^2+5*b^2*c^4)*x-5/4*a^4*c*d^4-2*a^3*c^3*d^2* 
b-3/4*a^2*c^5*b^2)/(b*x^2+a)^2+1/16*(24*a^2*b*c*d^4+24*a*b^2*c^3*d^2+8*b^3 
*c^5)/b*ln(b*x^2+a)+1/8*(15*a^3*d^5+10*a^2*b*c^2*d^3+3*a*b^2*c^4*d)/(a*b)^ 
(1/2)*arctan(b*x/(a*b)^(1/2)))-d^6*ln(d*x+c)/c/(a*d^2+b*c^2)^3
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.70 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^3} \, dx=-\frac {d^{6} \log \left (d x + c\right )}{b^{3} c^{7} + 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{3} d^{4} + a^{3} c d^{6}} - \frac {{\left (b^{3} c^{5} + 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c d^{4}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{3} b^{3} c^{6} + 3 \, a^{4} b^{2} c^{4} d^{2} + 3 \, a^{5} b c^{2} d^{4} + a^{6} d^{6}\right )}} - \frac {{\left (3 \, b^{3} c^{4} d + 10 \, a b^{2} c^{2} d^{3} + 15 \, a^{2} b d^{5}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a^{2} b^{3} c^{6} + 3 \, a^{3} b^{2} c^{4} d^{2} + 3 \, a^{4} b c^{2} d^{4} + a^{5} d^{6}\right )} \sqrt {a b}} + \frac {6 \, a b^{2} c^{3} + 10 \, a^{2} b c d^{2} - {\left (3 \, b^{3} c^{2} d + 7 \, a b^{2} d^{3}\right )} x^{3} + 4 \, {\left (b^{3} c^{3} + 2 \, a b^{2} c d^{2}\right )} x^{2} - {\left (5 \, a b^{2} c^{2} d + 9 \, a^{2} b d^{3}\right )} x}{8 \, {\left (a^{4} b^{2} c^{4} + 2 \, a^{5} b c^{2} d^{2} + a^{6} d^{4} + {\left (a^{2} b^{4} c^{4} + 2 \, a^{3} b^{3} c^{2} d^{2} + a^{4} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{3} b^{3} c^{4} + 2 \, a^{4} b^{2} c^{2} d^{2} + a^{5} b d^{4}\right )} x^{2}\right )}} + \frac {\log \left (x\right )}{a^{3} c} \] Input:

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

Output:

-d^6*log(d*x + c)/(b^3*c^7 + 3*a*b^2*c^5*d^2 + 3*a^2*b*c^3*d^4 + a^3*c*d^6 
) - 1/2*(b^3*c^5 + 3*a*b^2*c^3*d^2 + 3*a^2*b*c*d^4)*log(b*x^2 + a)/(a^3*b^ 
3*c^6 + 3*a^4*b^2*c^4*d^2 + 3*a^5*b*c^2*d^4 + a^6*d^6) - 1/8*(3*b^3*c^4*d 
+ 10*a*b^2*c^2*d^3 + 15*a^2*b*d^5)*arctan(b*x/sqrt(a*b))/((a^2*b^3*c^6 + 3 
*a^3*b^2*c^4*d^2 + 3*a^4*b*c^2*d^4 + a^5*d^6)*sqrt(a*b)) + 1/8*(6*a*b^2*c^ 
3 + 10*a^2*b*c*d^2 - (3*b^3*c^2*d + 7*a*b^2*d^3)*x^3 + 4*(b^3*c^3 + 2*a*b^ 
2*c*d^2)*x^2 - (5*a*b^2*c^2*d + 9*a^2*b*d^3)*x)/(a^4*b^2*c^4 + 2*a^5*b*c^2 
*d^2 + a^6*d^4 + (a^2*b^4*c^4 + 2*a^3*b^3*c^2*d^2 + a^4*b^2*d^4)*x^4 + 2*( 
a^3*b^3*c^4 + 2*a^4*b^2*c^2*d^2 + a^5*b*d^4)*x^2) + log(x)/(a^3*c)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.65 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^3} \, dx=-\frac {d^{7} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{7} d + 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{3} d^{5} + a^{3} c d^{7}} - \frac {{\left (b^{3} c^{5} + 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c d^{4}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{3} b^{3} c^{6} + 3 \, a^{4} b^{2} c^{4} d^{2} + 3 \, a^{5} b c^{2} d^{4} + a^{6} d^{6}\right )}} - \frac {{\left (3 \, b^{3} c^{4} d + 10 \, a b^{2} c^{2} d^{3} + 15 \, a^{2} b d^{5}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a^{2} b^{3} c^{6} + 3 \, a^{3} b^{2} c^{4} d^{2} + 3 \, a^{4} b c^{2} d^{4} + a^{5} d^{6}\right )} \sqrt {a b}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3} c} + \frac {6 \, a^{2} b^{3} c^{5} + 16 \, a^{3} b^{2} c^{3} d^{2} + 10 \, a^{4} b c d^{4} - {\left (3 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{2} d^{3} + 7 \, a^{3} b^{2} d^{5}\right )} x^{3} + 4 \, {\left (a b^{4} c^{5} + 3 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4}\right )} x^{2} - {\left (5 \, a^{2} b^{3} c^{4} d + 14 \, a^{3} b^{2} c^{2} d^{3} + 9 \, a^{4} b d^{5}\right )} x}{8 \, {\left (b c^{2} + a d^{2}\right )}^{3} {\left (b x^{2} + a\right )}^{2} a^{3}} \] Input:

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

Output:

-d^7*log(abs(d*x + c))/(b^3*c^7*d + 3*a*b^2*c^5*d^3 + 3*a^2*b*c^3*d^5 + a^ 
3*c*d^7) - 1/2*(b^3*c^5 + 3*a*b^2*c^3*d^2 + 3*a^2*b*c*d^4)*log(b*x^2 + a)/ 
(a^3*b^3*c^6 + 3*a^4*b^2*c^4*d^2 + 3*a^5*b*c^2*d^4 + a^6*d^6) - 1/8*(3*b^3 
*c^4*d + 10*a*b^2*c^2*d^3 + 15*a^2*b*d^5)*arctan(b*x/sqrt(a*b))/((a^2*b^3* 
c^6 + 3*a^3*b^2*c^4*d^2 + 3*a^4*b*c^2*d^4 + a^5*d^6)*sqrt(a*b)) + log(abs( 
x))/(a^3*c) + 1/8*(6*a^2*b^3*c^5 + 16*a^3*b^2*c^3*d^2 + 10*a^4*b*c*d^4 - ( 
3*a*b^4*c^4*d + 10*a^2*b^3*c^2*d^3 + 7*a^3*b^2*d^5)*x^3 + 4*(a*b^4*c^5 + 3 
*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c*d^4)*x^2 - (5*a^2*b^3*c^4*d + 14*a^3*b^2*c^ 
2*d^3 + 9*a^4*b*d^5)*x)/((b*c^2 + a*d^2)^3*(b*x^2 + a)^2*a^3)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 8.99 (sec) , antiderivative size = 1580, normalized size of antiderivative = 6.08 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((3*b^2*c^3 + 5*a*b*c*d^2)/(4*a*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) + (x^ 
2*(b^3*c^3 + 2*a*b^2*c*d^2))/(2*a^2*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) - 
 (d*x^3*(3*b^3*c^2 + 7*a*b^2*d^2))/(8*a^2*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d 
^2)) - (d*x*(5*b^2*c^2 + 9*a*b*d^2))/(8*a*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d 
^2)))/(a^2 + b^2*x^4 + 2*a*b*x^2) - log(36864*a^13*b^16*c^28*x - 65536*a^1 
7*d^28*(-a^7*b)^(3/2) + 65536*a^27*b^2*d^28*x - 1730497*a*c^4*d^24*(-a^7*b 
)^(7/2) - 1026394*b*c^6*d^22*(-a^7*b)^(7/2) - 36864*a^3*b^14*c^28*(-a^7*b) 
^(3/2) + 639488*a^9*c^2*d^26*(-a^7*b)^(5/2) + 36762413*b^9*c^20*d^8*(-a^7* 
b)^(5/2) + 86677810*a^2*b^7*c^16*d^12*(-a^7*b)^(5/2) + 73728092*a^3*b^6*c^ 
14*d^14*(-a^7*b)^(5/2) + 37433746*a^4*b^5*c^12*d^16*(-a^7*b)^(5/2) + 75976 
24*a^5*b^4*c^10*d^18*(-a^7*b)^(5/2) - 982531*a^6*b^3*c^8*d^20*(-a^7*b)^(5/ 
2) - 521856*a^4*b^13*c^26*d^2*(-a^7*b)^(3/2) - 3439185*a^5*b^12*c^24*d^4*( 
-a^7*b)^(3/2) - 13764058*a^6*b^11*c^22*d^6*(-a^7*b)^(3/2) + 521856*a^14*b^ 
15*c^26*d^2*x + 3439185*a^15*b^14*c^24*d^4*x + 13764058*a^16*b^13*c^22*d^6 
*x + 36762413*a^17*b^12*c^20*d^8*x + 67886008*a^18*b^11*c^18*d^10*x + 8667 
7810*a^19*b^10*c^16*d^12*x + 73728092*a^20*b^9*c^14*d^14*x + 37433746*a^21 
*b^8*c^12*d^16*x + 7597624*a^22*b^7*c^10*d^18*x - 982531*a^23*b^6*c^8*d^20 
*x + 1026394*a^24*b^5*c^6*d^22*x + 1730497*a^25*b^4*c^4*d^24*x + 639488*a^ 
26*b^3*c^2*d^26*x + 67886008*a*b^8*c^18*d^10*(-a^7*b)^(5/2))*(((15*a^2*d^5 
*(-a^7*b)^(1/2))/16 - (a^6*d^6)/(2*c) + (3*b^2*c^4*d*(-a^7*b)^(1/2))/16...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1028, normalized size of antiderivative = 3.95 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*c*d**5 - 10*sqrt 
(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c**3*d**3 - 30*sqrt(b)*sq 
rt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c*d**5*x**2 - 3*sqrt(b)*sqrt(a) 
*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**2*c**5*d - 20*sqrt(b)*sqrt(a)*atan( 
(b*x)/(sqrt(b)*sqrt(a)))*a**2*b**2*c**3*d**3*x**2 - 15*sqrt(b)*sqrt(a)*ata 
n((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**2*c*d**5*x**4 - 6*sqrt(b)*sqrt(a)*atan( 
(b*x)/(sqrt(b)*sqrt(a)))*a*b**3*c**5*d*x**2 - 10*sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a*b**3*c**3*d**3*x**4 - 3*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*b**4*c**5*d*x**4 - 12*log(a + b*x**2)*a**4*b*c**2*d**4 
- 12*log(a + b*x**2)*a**3*b**2*c**4*d**2 - 24*log(a + b*x**2)*a**3*b**2*c* 
*2*d**4*x**2 - 4*log(a + b*x**2)*a**2*b**3*c**6 - 24*log(a + b*x**2)*a**2* 
b**3*c**4*d**2*x**2 - 12*log(a + b*x**2)*a**2*b**3*c**2*d**4*x**4 - 8*log( 
a + b*x**2)*a*b**4*c**6*x**2 - 12*log(a + b*x**2)*a*b**4*c**4*d**2*x**4 - 
4*log(a + b*x**2)*b**5*c**6*x**4 - 8*log(c + d*x)*a**5*d**6 - 16*log(c + d 
*x)*a**4*b*d**6*x**2 - 8*log(c + d*x)*a**3*b**2*d**6*x**4 + 8*log(x)*a**5* 
d**6 + 24*log(x)*a**4*b*c**2*d**4 + 16*log(x)*a**4*b*d**6*x**2 + 24*log(x) 
*a**3*b**2*c**4*d**2 + 48*log(x)*a**3*b**2*c**2*d**4*x**2 + 8*log(x)*a**3* 
b**2*d**6*x**4 + 8*log(x)*a**2*b**3*c**6 + 48*log(x)*a**2*b**3*c**4*d**2*x 
**2 + 24*log(x)*a**2*b**3*c**2*d**4*x**4 + 16*log(x)*a*b**4*c**6*x**2 + 24 
*log(x)*a*b**4*c**4*d**2*x**4 + 8*log(x)*b**5*c**6*x**4 + 6*a**4*b*c**2...