\(\int \frac {1}{x (c+d x)^2 (a+b x^2)^2} \, dx\) [205]

Optimal result

Integrand size = 20, antiderivative size = 231 \[ \int \frac {1}{x (c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {d^4}{c \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {b \left (b c^2-a d^2-2 b c d x\right )}{2 a \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}-\frac {b^{3/2} c d \left (b c^2+5 a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \left (b c^2+a d^2\right )^3}+\frac {\log (x)}{a^2 c^2}-\frac {d^4 \left (5 b c^2+a d^2\right ) \log (c+d x)}{c^2 \left (b c^2+a d^2\right )^3}-\frac {b \left (b^2 c^4+3 a b c^2 d^2-2 a^2 d^4\right ) \log \left (a+b x^2\right )}{2 a^2 \left (b c^2+a d^2\right )^3} \] Output:

d^4/c/(a*d^2+b*c^2)^2/(d*x+c)+1/2*b*(-2*b*c*d*x-a*d^2+b*c^2)/a/(a*d^2+b*c^ 
2)^2/(b*x^2+a)-b^(3/2)*c*d*(5*a*d^2+b*c^2)*arctan(b^(1/2)*x/a^(1/2))/a^(3/ 
2)/(a*d^2+b*c^2)^3+ln(x)/a^2/c^2-d^4*(a*d^2+5*b*c^2)*ln(d*x+c)/c^2/(a*d^2+ 
b*c^2)^3-1/2*b*(-2*a^2*d^4+3*a*b*c^2*d^2+b^2*c^4)*ln(b*x^2+a)/a^2/(a*d^2+b 
*c^2)^3
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x (c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {d^4}{c \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {b \left (-a d^2+b c (c-2 d x)\right )}{2 a \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}-\frac {b^{3/2} c d \left (b c^2+5 a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \left (b c^2+a d^2\right )^3}+\frac {\log (x)}{a^2 c^2}-\frac {\left (5 b c^2 d^4+a d^6\right ) \log (c+d x)}{c^2 \left (b c^2+a d^2\right )^3}-\frac {b \left (b^2 c^4+3 a b c^2 d^2-2 a^2 d^4\right ) \log \left (a+b x^2\right )}{2 a^2 \left (b c^2+a d^2\right )^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.13, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

d^4/(c*(b*c^2 + a*d^2)^2*(c + d*x)) + (b*(b*c^2 - a*d^2 - 2*b*c*d*x))/(2*a 
*(b*c^2 + a*d^2)^2*(a + b*x^2)) - (4*b^(3/2)*c*d^3*ArcTan[(Sqrt[b]*x)/Sqrt 
[a]])/(Sqrt[a]*(b*c^2 + a*d^2)^3) - (b^(3/2)*c*d*ArcTan[(Sqrt[b]*x)/Sqrt[a 
]])/(a^(3/2)*(b*c^2 + a*d^2)^2) + Log[x]/(a^2*c^2) - (d^4*(5*b*c^2 + a*d^2 
)*Log[c + d*x])/(c^2*(b*c^2 + a*d^2)^3) - (b*(b^2*c^4 + 3*a*b*c^2*d^2 - 2* 
a^2*d^4)*Log[a + b*x^2])/(2*a^2*(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 = 0.36 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {\ln \left (x \right )}{a^{2} c^{2}}-\frac {b^{2} \left (\frac {\left (d^{3} c \,a^{2}+a b \,c^{3} d \right ) x +\frac {a \left (a^{2} d^{4}-b^{2} c^{4}\right )}{2 b}}{b \,x^{2}+a}+\frac {\left (-2 a^{2} d^{4}+3 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (5 d^{3} c \,a^{2}+a b \,c^{3} d \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{3} a^{2}}+\frac {d^{4}}{c \left (a \,d^{2}+b \,c^{2}\right )^{2} \left (d x +c \right )}-\frac {d^{4} \left (a \,d^{2}+5 b \,c^{2}\right ) \ln \left (d x +c \right )}{c^{2} \left (a \,d^{2}+b \,c^{2}\right )^{3}}\)	\(221\)
risch	\(\frac {\frac {b \,d^{2} \left (a \,d^{2}-b \,c^{2}\right ) x^{2}}{a c \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}-\frac {b d x}{2 a \left (a \,d^{2}+b \,c^{2}\right )}+\frac {2 a^{2} d^{4}-b \,c^{2} d^{2} a +b^{2} c^{4}}{2 a c \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}}{\left (d x +c \right ) \left (b \,x^{2}+a \right )}-\frac {d^{6} \ln \left (d x +c \right ) a}{c^{2} \left (a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right )}-\frac {5 d^{4} \ln \left (d x +c \right ) b}{a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}}+\frac {\ln \left (-x \right )}{c^{2} a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{7} d^{6}+3 a^{6} b \,c^{2} d^{4}+3 a^{5} b^{2} c^{4} d^{2}+a^{4} b^{3} c^{6}\right ) \textit {\_Z}^{2}+\left (-4 a^{4} b \,d^{4}+6 a^{3} b^{2} c^{2} d^{2}+2 a^{2} b^{3} c^{4}\right ) \textit {\_Z} +4 a \,b^{2} d^{2}+b^{3} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a^{10} c^{2} d^{12}-19 a^{9} b \,c^{4} d^{10}-39 a^{8} b^{2} c^{6} d^{8}-46 a^{7} b^{3} c^{8} d^{6}-34 a^{6} b^{4} c^{10} d^{4}-15 a^{5} b^{5} c^{12} d^{2}-3 a^{4} b^{6} c^{14}\right ) \textit {\_R}^{3}+\left (-38 a^{6} b^{2} c^{4} d^{8}-93 a^{5} b^{3} c^{6} d^{6}-75 a^{4} b^{4} c^{8} d^{4}-23 a^{3} b^{5} c^{10} d^{2}-3 a^{2} b^{6} c^{12}\right ) \textit {\_R}^{2}+\left (-4 a^{5} b \,d^{10}+28 a^{4} b^{2} c^{2} d^{8}-136 a^{3} b^{3} c^{4} d^{6}-36 a^{2} b^{4} c^{6} d^{4}-4 a \,b^{5} c^{8} d^{2}\right ) \textit {\_R} +8 a^{2} b^{2} d^{8}-20 a \,b^{3} c^{2} d^{6}+4 b^{4} c^{4} d^{4}\right ) x +\left (-2 a^{10} c^{3} d^{11}-6 a^{9} b \,c^{5} d^{9}-4 a^{8} b^{2} c^{7} d^{7}+4 a^{7} b^{3} c^{9} d^{5}+6 a^{6} b^{4} c^{11} d^{3}+2 a^{5} b^{5} c^{13} d \right ) \textit {\_R}^{3}+\left (4 a^{8} c \,d^{11}+24 a^{7} b \,c^{3} d^{9}+29 a^{6} b^{2} c^{5} d^{7}-a^{5} b^{3} c^{7} d^{5}-13 a^{4} b^{4} c^{9} d^{3}-3 a^{3} b^{5} c^{11} d \right ) \textit {\_R}^{2}+\left (-24 a^{5} b c \,d^{9}-48 a^{4} b^{2} c^{3} d^{7}-14 a^{3} b^{3} c^{5} d^{5}-2 a \,b^{5} c^{9} d \right ) \textit {\_R} +12 a^{2} b^{2} c \,d^{7}+4 b^{4} c^{5} d^{3}\right )\right )}{2}\)	\(893\)

Input:

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

Output:

ln(x)/a^2/c^2-b^2/(a*d^2+b*c^2)^3/a^2*(((a^2*c*d^3+a*b*c^3*d)*x+1/2*a*(a^2 
*d^4-b^2*c^4)/b)/(b*x^2+a)+1/2*(-2*a^2*d^4+3*a*b*c^2*d^2+b^2*c^4)/b*ln(b*x 
^2+a)+(5*a^2*c*d^3+a*b*c^3*d)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))+d^4/c/( 
a*d^2+b*c^2)^2/(d*x+c)-d^4*(a*d^2+5*b*c^2)*ln(d*x+c)/c^2/(a*d^2+b*c^2)^3
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 455 vs. \(2 (219) = 438\).

Time = 0.12 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.97 \[ \int \frac {1}{x (c+d x)^2 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (b^{3} c^{4} + 3 \, a b^{2} c^{2} d^{2} - 2 \, a^{2} b d^{4}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{3} c^{6} + 3 \, a^{3} b^{2} c^{4} d^{2} + 3 \, a^{4} b c^{2} d^{4} + a^{5} d^{6}\right )}} - \frac {{\left (5 \, b c^{2} d^{4} + a d^{6}\right )} \log \left (d x + c\right )}{b^{3} c^{8} + 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{4} d^{4} + a^{3} c^{2} d^{6}} - \frac {{\left (b^{3} c^{3} d + 5 \, a b^{2} c d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (a b^{3} c^{6} + 3 \, a^{2} b^{2} c^{4} d^{2} + 3 \, a^{3} b c^{2} d^{4} + a^{4} d^{6}\right )} \sqrt {a b}} + \frac {b^{2} c^{4} - a b c^{2} d^{2} + 2 \, a^{2} d^{4} - 2 \, {\left (b^{2} c^{2} d^{2} - a b d^{4}\right )} x^{2} - {\left (b^{2} c^{3} d + a b c d^{3}\right )} x}{2 \, {\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{4} d^{2} + a^{4} c^{2} d^{4} + {\left (a b^{3} c^{5} d + 2 \, a^{2} b^{2} c^{3} d^{3} + a^{3} b c d^{5}\right )} x^{3} + {\left (a b^{3} c^{6} + 2 \, a^{2} b^{2} c^{4} d^{2} + a^{3} b c^{2} d^{4}\right )} x^{2} + {\left (a^{2} b^{2} c^{5} d + 2 \, a^{3} b c^{3} d^{3} + a^{4} c d^{5}\right )} x\right )}} + \frac {\log \left (x\right )}{a^{2} c^{2}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (219) = 438\).

Time = 0.13 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.95 \[ \int \frac {1}{x (c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {1}{2} \, {\left (\frac {2 \, d^{8}}{{\left (b^{2} c^{5} d^{5} + 2 \, a b c^{3} d^{7} + a^{2} c d^{9}\right )} {\left (d x + c\right )}} - \frac {{\left (b^{3} c^{4} + 3 \, a b^{2} c^{2} d^{2} - 2 \, a^{2} b d^{4}\right )} \log \left (b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d^{2}}{{\left (d x + c\right )}^{2}}\right )}{a^{2} b^{3} c^{6} d + 3 \, a^{3} b^{2} c^{4} d^{3} + 3 \, a^{4} b c^{2} d^{5} + a^{5} d^{7}} - \frac {2 \, {\left (b^{3} c^{3} d^{2} + 5 \, a b^{2} c d^{4}\right )} \arctan \left (\frac {b c - \frac {b c^{2}}{d x + c} - \frac {a d^{2}}{d x + c}}{\sqrt {a b} d}\right )}{{\left (a b^{3} c^{6} + 3 \, a^{2} b^{2} c^{4} d^{2} + 3 \, a^{3} b c^{2} d^{4} + a^{4} d^{6}\right )} \sqrt {a b} d^{2}} + \frac {2 \, \log \left ({\left | -\frac {c}{d x + c} + 1 \right |}\right )}{a^{2} c^{2} d} - \frac {\frac {3 \, a b^{3} c^{2} d^{2} - a^{2} b^{2} d^{4}}{b c^{2} + a d^{2}} - \frac {4 \, {\left (a b^{3} c^{3} d^{3} - a^{2} b^{2} c d^{5}\right )}}{{\left (b c^{2} + a d^{2}\right )} {\left (d x + c\right )} d}}{{\left (b c^{2} + a d^{2}\right )}^{2} a^{2} {\left (b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d^{2}}{{\left (d x + c\right )}^{2}}\right )} d}\right )} d \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.67 (sec) , antiderivative size = 1576, normalized size of antiderivative = 6.82 \[ \int \frac {1}{x (c+d x)^2 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((2*a^2*d^4 + b^2*c^4 - a*b*c^2*d^2)/(2*a*c*(a*d^2 + b*c^2)^2) - (b*d*x)/( 
2*a*(a*d^2 + b*c^2)) + (b*d^2*x^2*(a*d^2 - b*c^2))/(a*c*(a^2*d^4 + b^2*c^4 
 + 2*a*b*c^2*d^2)))/(a*c + a*d*x + b*c*x^2 + b*d*x^3) - (log(c + d*x)*(a*d 
^6 + 5*b*c^2*d^4))/(b^3*c^8 + a^3*c^2*d^6 + 3*a*b^2*c^6*d^2 + 3*a^2*b*c^4* 
d^4) - (log(9*a^9*b^18*c^28*x + 16*a^23*b^4*d^28*x - 9*a^2*b^13*c^28*(-a^5 
*b^3)^(3/2) + 16*a^21*b^2*d^28*(-a^5*b^3)^(1/2) + 22904*a^7*c^8*d^20*(-a^5 
*b^3)^(5/2) - 288*a^15*c^2*d^26*(-a^5*b^3)^(3/2) + 1208*b^7*c^22*d^6*(-a^5 
*b^3)^(5/2) - 562*a*b^6*c^20*d^8*(-a^5*b^3)^(5/2) + 45704*a^6*b*c^10*d^18* 
(-a^5*b^3)^(5/2) - 2032*a^14*b*c^4*d^24*(-a^5*b^3)^(3/2) - 912*a^2*b^5*c^1 
8*d^10*(-a^5*b^3)^(5/2) + 14100*a^3*b^4*c^16*d^12*(-a^5*b^3)^(5/2) + 43392 
*a^4*b^3*c^14*d^14*(-a^5*b^3)^(5/2) + 58721*a^5*b^2*c^12*d^16*(-a^5*b^3)^( 
5/2) - 136*a^3*b^12*c^26*d^2*(-a^5*b^3)^(3/2) - 700*a^4*b^11*c^24*d^4*(-a^ 
5*b^3)^(3/2) - 8104*a^13*b^2*c^6*d^22*(-a^5*b^3)^(3/2) + 136*a^10*b^17*c^2 
6*d^2*x + 700*a^11*b^16*c^24*d^4*x + 1208*a^12*b^15*c^22*d^6*x - 562*a^13* 
b^14*c^20*d^8*x - 912*a^14*b^13*c^18*d^10*x + 14100*a^15*b^12*c^16*d^12*x 
+ 43392*a^16*b^11*c^14*d^14*x + 58721*a^17*b^10*c^12*d^16*x + 45704*a^18*b 
^9*c^10*d^18*x + 22904*a^19*b^8*c^8*d^20*x + 8104*a^20*b^7*c^6*d^22*x + 20 
32*a^21*b^6*c^4*d^24*x + 288*a^22*b^5*c^2*d^26*x)*(a^2*b^3*c^4 - 2*a^4*b*d 
^4 + 5*a*c*d^3*(-a^5*b^3)^(1/2) + b*c^3*d*(-a^5*b^3)^(1/2) + 3*a^3*b^2*c^2 
*d^2))/(2*(a^7*d^6 + a^4*b^3*c^6 + 3*a^6*b*c^2*d^4 + 3*a^5*b^2*c^4*d^2)...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1176, normalized size of antiderivative = 5.09 \[ \int \frac {1}{x (c+d x)^2 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 10*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b*c**4*d**3 - 10 
*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b*c**3*d**4*x - 2*sqrt 
(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**2*c**6*d - 2*sqrt(b)*sqrt(a 
)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**2*c**5*d**2*x - 10*sqrt(b)*sqrt(a)*at 
an((b*x)/(sqrt(b)*sqrt(a)))*a*b**2*c**4*d**3*x**2 - 10*sqrt(b)*sqrt(a)*ata 
n((b*x)/(sqrt(b)*sqrt(a)))*a*b**2*c**3*d**4*x**3 - 2*sqrt(b)*sqrt(a)*atan( 
(b*x)/(sqrt(b)*sqrt(a)))*b**3*c**6*d*x**2 - 2*sqrt(b)*sqrt(a)*atan((b*x)/( 
sqrt(b)*sqrt(a)))*b**3*c**5*d**2*x**3 + 2*log(a + b*x**2)*a**3*b*c**3*d**4 
 + 2*log(a + b*x**2)*a**3*b*c**2*d**5*x - 3*log(a + b*x**2)*a**2*b**2*c**5 
*d**2 - 3*log(a + b*x**2)*a**2*b**2*c**4*d**3*x + 2*log(a + b*x**2)*a**2*b 
**2*c**3*d**4*x**2 + 2*log(a + b*x**2)*a**2*b**2*c**2*d**5*x**3 - log(a + 
b*x**2)*a*b**3*c**7 - log(a + b*x**2)*a*b**3*c**6*d*x - 3*log(a + b*x**2)* 
a*b**3*c**5*d**2*x**2 - 3*log(a + b*x**2)*a*b**3*c**4*d**3*x**3 - log(a + 
b*x**2)*b**4*c**7*x**2 - log(a + b*x**2)*b**4*c**6*d*x**3 - 2*log(c + d*x) 
*a**4*c*d**6 - 2*log(c + d*x)*a**4*d**7*x - 10*log(c + d*x)*a**3*b*c**3*d* 
*4 - 10*log(c + d*x)*a**3*b*c**2*d**5*x - 2*log(c + d*x)*a**3*b*c*d**6*x** 
2 - 2*log(c + d*x)*a**3*b*d**7*x**3 - 10*log(c + d*x)*a**2*b**2*c**3*d**4* 
x**2 - 10*log(c + d*x)*a**2*b**2*c**2*d**5*x**3 + 2*log(x)*a**4*c*d**6 + 2 
*log(x)*a**4*d**7*x + 6*log(x)*a**3*b*c**3*d**4 + 6*log(x)*a**3*b*c**2*d** 
5*x + 2*log(x)*a**3*b*c*d**6*x**2 + 2*log(x)*a**3*b*d**7*x**3 + 6*log(x...