Integrand size = 20, antiderivative size = 168 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^2} \, dx=\frac {b (c-d x)}{2 a \left (b c^2+a d^2\right ) \left (a+b x^2\right )}-\frac {\sqrt {b} d \left (b c^2+3 a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \left (b c^2+a d^2\right )^2}+\frac {\log (x)}{a^2 c}-\frac {d^4 \log (c+d x)}{c \left (b c^2+a d^2\right )^2}-\frac {b c \left (b c^2+2 a d^2\right ) \log \left (a+b x^2\right )}{2 a^2 \left (b c^2+a d^2\right )^2} \] Output:
1/2*b*(-d*x+c)/a/(a*d^2+b*c^2)/(b*x^2+a)-1/2*b^(1/2)*d*(3*a*d^2+b*c^2)*arc tan(b^(1/2)*x/a^(1/2))/a^(3/2)/(a*d^2+b*c^2)^2+ln(x)/a^2/c-d^4*ln(d*x+c)/c /(a*d^2+b*c^2)^2-1/2*b*c*(2*a*d^2+b*c^2)*ln(b*x^2+a)/a^2/(a*d^2+b*c^2)^2
Time = 0.20 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {b (c-d x)}{a \left (b c^2+a d^2\right ) \left (a+b x^2\right )}-\frac {\sqrt {b} d \left (b c^2+3 a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \left (b c^2+a d^2\right )^2}+\frac {2 \log (x)}{a^2 c}-\frac {2 d^4 \log (c+d x)}{c \left (b c^2+a d^2\right )^2}-\frac {b \left (b c^3+2 a c d^2\right ) \log \left (a+b x^2\right )}{a^2 \left (b c^2+a d^2\right )^2}\right ) \] Input:
Integrate[1/(x*(c + d*x)*(a + b*x^2)^2),x]
Output:
((b*(c - d*x))/(a*(b*c^2 + a*d^2)*(a + b*x^2)) - (Sqrt[b]*d*(b*c^2 + 3*a*d ^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(b*c^2 + a*d^2)^2) + (2*Log[x])/ (a^2*c) - (2*d^4*Log[c + d*x])/(c*(b*c^2 + a*d^2)^2) - (b*(b*c^3 + 2*a*c*d ^2)*Log[a + b*x^2])/(a^2*(b*c^2 + a*d^2)^2))/2
Time = 0.58 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.17, 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.
| \(\displaystyle \int \frac {1}{x \left (a+b x^2\right )^2 (c+d x)} \, dx\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \int \left (\frac {b \left (-a^2 d^3-b c x \left (2 a d^2+b c^2\right )\right )}{a^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^2}+\frac {1}{a^2 c x}-\frac {b (a d+b c x)}{a \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )}-\frac {d^5}{c (c+d x) \left (a d^2+b c^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {b} d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \left (a d^2+b c^2\right )}-\frac {b c \left (2 a d^2+b c^2\right ) \log \left (a+b x^2\right )}{2 a^2 \left (a d^2+b c^2\right )^2}+\frac {\log (x)}{a^2 c}-\frac {\sqrt {b} d^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \left (a d^2+b c^2\right )^2}+\frac {b (c-d x)}{2 a \left (a+b x^2\right ) \left (a d^2+b c^2\right )}-\frac {d^4 \log (c+d x)}{c \left (a d^2+b c^2\right )^2}\) |
Input:
Int[1/(x*(c + d*x)*(a + b*x^2)^2),x]
Output:
(b*(c - d*x))/(2*a*(b*c^2 + a*d^2)*(a + b*x^2)) - (Sqrt[b]*d^3*ArcTan[(Sqr t[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*c^2 + a*d^2)^2) - (Sqrt[b]*d*ArcTan[(Sqrt[b] *x)/Sqrt[a]])/(2*a^(3/2)*(b*c^2 + a*d^2)) + Log[x]/(a^2*c) - (d^4*Log[c + d*x])/(c*(b*c^2 + a*d^2)^2) - (b*c*(b*c^2 + 2*a*d^2)*Log[a + b*x^2])/(2*a^ 2*(b*c^2 + a*d^2)^2)
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]
Time = 0.32 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(\frac {\ln \left (x \right )}{a^{2} c}-\frac {b \left (\frac {\left (\frac {1}{2} a^{2} d^{3}+\frac {1}{2} a b \,c^{2} d \right ) x -\frac {a c \left (a \,d^{2}+b \,c^{2}\right )}{2}}{b \,x^{2}+a}+\frac {\left (4 a b c \,d^{2}+2 c^{3} b^{2}\right ) \ln \left (b \,x^{2}+a \right )}{4 b}+\frac {\left (3 a^{2} d^{3}+a b \,c^{2} d \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2} a^{2}}-\frac {d^{4} \ln \left (d x +c \right )}{c \left (a \,d^{2}+b \,c^{2}\right )^{2}}\) | \(165\) |
| risch | \(\frac {-\frac {b d x}{2 a \left (a \,d^{2}+b \,c^{2}\right )}+\frac {b c}{2 a \left (a \,d^{2}+b \,c^{2}\right )}}{b \,x^{2}+a}-\frac {d^{4} \ln \left (d x +c \right )}{c \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {\ln \left (-x \right )}{c \,a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{6} d^{4}+2 a^{5} b \,c^{2} d^{2}+b^{2} a^{4} c^{4}\right ) \textit {\_Z}^{2}+\left (8 a^{3} b c \,d^{2}+4 a^{2} b^{2} c^{3}\right ) \textit {\_Z} +9 a b \,d^{2}+4 b^{2} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a^{8} d^{8}-11 a^{7} b \,c^{2} d^{6}-13 a^{6} b^{2} c^{4} d^{4}-9 a^{5} b^{3} c^{6} d^{2}-3 a^{4} b^{4} c^{8}\right ) \textit {\_R}^{3}+\left (-32 a^{5} b c \,d^{6}-58 a^{4} b^{2} c^{3} d^{4}-32 a^{3} b^{3} c^{5} d^{2}-6 a^{2} b^{4} c^{7}\right ) \textit {\_R}^{2}+\left (-52 a^{3} b \,d^{6}-36 a^{2} b^{2} c^{2} d^{4}-18 a \,b^{3} c^{4} d^{2}\right ) \textit {\_R} +32 b^{2} c \,d^{4}\right ) x +\left (-2 a^{8} c \,d^{7}-2 a^{7} b \,c^{3} d^{5}+2 a^{6} b^{2} c^{5} d^{3}+2 a^{5} b^{3} c^{7} d \right ) \textit {\_R}^{3}+\left (8 a^{6} d^{7}+3 a^{5} b \,c^{2} d^{5}-10 a^{4} b^{2} c^{4} d^{3}-5 a^{3} b^{3} c^{6} d \right ) \textit {\_R}^{2}+\left (14 a^{3} b c \,d^{5}-12 a \,b^{3} c^{5} d \right ) \textit {\_R} +24 a b \,d^{5}+32 b^{2} c^{2} d^{3}\right )\right )}{4}\) | \(502\) |
Input:
int(1/x/(d*x+c)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
ln(x)/a^2/c-b/(a*d^2+b*c^2)^2/a^2*(((1/2*a^2*d^3+1/2*a*b*c^2*d)*x-1/2*a*c* (a*d^2+b*c^2))/(b*x^2+a)+1/4*(4*a*b*c*d^2+2*b^2*c^3)/b*ln(b*x^2+a)+1/2*(3* a^2*d^3+a*b*c^2*d)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-d^4*ln(d*x+c)/c/(a *d^2+b*c^2)^2
Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (155) = 310\).
Time = 18.66 (sec) , antiderivative size = 671, normalized size of antiderivative = 3.99 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^2} \, dx=\left [\frac {2 \, a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + {\left (a^{2} b c^{3} d + 3 \, a^{3} c d^{3} + {\left (a b^{2} c^{3} d + 3 \, a^{2} b c d^{3}\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2 \, {\left (a b^{2} c^{3} d + a^{2} b c d^{3}\right )} x - 2 \, {\left (a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + {\left (b^{3} c^{4} + 2 \, a b^{2} c^{2} d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 4 \, {\left (a^{2} b d^{4} x^{2} + a^{3} d^{4}\right )} \log \left (d x + c\right ) + 4 \, {\left (a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + a^{3} d^{4} + {\left (b^{3} c^{4} + 2 \, a b^{2} c^{2} d^{2} + a^{2} b d^{4}\right )} x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{3} b^{2} c^{5} + 2 \, a^{4} b c^{3} d^{2} + a^{5} c d^{4} + {\left (a^{2} b^{3} c^{5} + 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c d^{4}\right )} x^{2}\right )}}, \frac {a b^{2} c^{4} + a^{2} b c^{2} d^{2} - {\left (a^{2} b c^{3} d + 3 \, a^{3} c d^{3} + {\left (a b^{2} c^{3} d + 3 \, a^{2} b c d^{3}\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - {\left (a b^{2} c^{3} d + a^{2} b c d^{3}\right )} x - {\left (a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + {\left (b^{3} c^{4} + 2 \, a b^{2} c^{2} d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (a^{2} b d^{4} x^{2} + a^{3} d^{4}\right )} \log \left (d x + c\right ) + 2 \, {\left (a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + a^{3} d^{4} + {\left (b^{3} c^{4} + 2 \, a b^{2} c^{2} d^{2} + a^{2} b d^{4}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{2} c^{5} + 2 \, a^{4} b c^{3} d^{2} + a^{5} c d^{4} + {\left (a^{2} b^{3} c^{5} + 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c d^{4}\right )} x^{2}\right )}}\right ] \] Input:
integrate(1/x/(d*x+c)/(b*x^2+a)^2,x, algorithm="fricas")
Output:
[1/4*(2*a*b^2*c^4 + 2*a^2*b*c^2*d^2 + (a^2*b*c^3*d + 3*a^3*c*d^3 + (a*b^2* c^3*d + 3*a^2*b*c*d^3)*x^2)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/ (b*x^2 + a)) - 2*(a*b^2*c^3*d + a^2*b*c*d^3)*x - 2*(a*b^2*c^4 + 2*a^2*b*c^ 2*d^2 + (b^3*c^4 + 2*a*b^2*c^2*d^2)*x^2)*log(b*x^2 + a) - 4*(a^2*b*d^4*x^2 + a^3*d^4)*log(d*x + c) + 4*(a*b^2*c^4 + 2*a^2*b*c^2*d^2 + a^3*d^4 + (b^3 *c^4 + 2*a*b^2*c^2*d^2 + a^2*b*d^4)*x^2)*log(x))/(a^3*b^2*c^5 + 2*a^4*b*c^ 3*d^2 + a^5*c*d^4 + (a^2*b^3*c^5 + 2*a^3*b^2*c^3*d^2 + a^4*b*c*d^4)*x^2), 1/2*(a*b^2*c^4 + a^2*b*c^2*d^2 - (a^2*b*c^3*d + 3*a^3*c*d^3 + (a*b^2*c^3*d + 3*a^2*b*c*d^3)*x^2)*sqrt(b/a)*arctan(x*sqrt(b/a)) - (a*b^2*c^3*d + a^2* b*c*d^3)*x - (a*b^2*c^4 + 2*a^2*b*c^2*d^2 + (b^3*c^4 + 2*a*b^2*c^2*d^2)*x^ 2)*log(b*x^2 + a) - 2*(a^2*b*d^4*x^2 + a^3*d^4)*log(d*x + c) + 2*(a*b^2*c^ 4 + 2*a^2*b*c^2*d^2 + a^3*d^4 + (b^3*c^4 + 2*a*b^2*c^2*d^2 + a^2*b*d^4)*x^ 2)*log(x))/(a^3*b^2*c^5 + 2*a^4*b*c^3*d^2 + a^5*c*d^4 + (a^2*b^3*c^5 + 2*a ^3*b^2*c^3*d^2 + a^4*b*c*d^4)*x^2)]
Timed out. \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/x/(d*x+c)/(b*x**2+a)**2,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^2} \, dx=-\frac {d^{4} \log \left (d x + c\right )}{b^{2} c^{5} + 2 \, a b c^{3} d^{2} + a^{2} c d^{4}} - \frac {{\left (b^{2} c^{3} + 2 \, a b c d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{2} c^{4} + 2 \, a^{3} b c^{2} d^{2} + a^{4} d^{4}\right )}} - \frac {{\left (b^{2} c^{2} d + 3 \, a b d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + a^{3} d^{4}\right )} \sqrt {a b}} - \frac {b d x - b c}{2 \, {\left (a^{2} b c^{2} + a^{3} d^{2} + {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )} x^{2}\right )}} + \frac {\log \left (x\right )}{a^{2} c} \] Input:
integrate(1/x/(d*x+c)/(b*x^2+a)^2,x, algorithm="maxima")
Output:
-d^4*log(d*x + c)/(b^2*c^5 + 2*a*b*c^3*d^2 + a^2*c*d^4) - 1/2*(b^2*c^3 + 2 *a*b*c*d^2)*log(b*x^2 + a)/(a^2*b^2*c^4 + 2*a^3*b*c^2*d^2 + a^4*d^4) - 1/2 *(b^2*c^2*d + 3*a*b*d^3)*arctan(b*x/sqrt(a*b))/((a*b^2*c^4 + 2*a^2*b*c^2*d ^2 + a^3*d^4)*sqrt(a*b)) - 1/2*(b*d*x - b*c)/(a^2*b*c^2 + a^3*d^2 + (a*b^2 *c^2 + a^2*b*d^2)*x^2) + log(x)/(a^2*c)
Time = 0.13 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.42 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^2} \, dx=-\frac {d^{5} \log \left ({\left | d x + c \right |}\right )}{b^{2} c^{5} d + 2 \, a b c^{3} d^{3} + a^{2} c d^{5}} - \frac {{\left (b^{2} c^{3} + 2 \, a b c d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{2} c^{4} + 2 \, a^{3} b c^{2} d^{2} + a^{4} d^{4}\right )}} - \frac {{\left (b^{2} c^{2} d + 3 \, a b d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + a^{3} d^{4}\right )} \sqrt {a b}} + \frac {\log \left ({\left | x \right |}\right )}{a^{2} c} + \frac {a b^{2} c^{3} + a^{2} b c d^{2} - {\left (a b^{2} c^{2} d + a^{2} b d^{3}\right )} x}{2 \, {\left (b c^{2} + a d^{2}\right )}^{2} {\left (b x^{2} + a\right )} a^{2}} \] Input:
integrate(1/x/(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")
Output:
-d^5*log(abs(d*x + c))/(b^2*c^5*d + 2*a*b*c^3*d^3 + a^2*c*d^5) - 1/2*(b^2* c^3 + 2*a*b*c*d^2)*log(b*x^2 + a)/(a^2*b^2*c^4 + 2*a^3*b*c^2*d^2 + a^4*d^4 ) - 1/2*(b^2*c^2*d + 3*a*b*d^3)*arctan(b*x/sqrt(a*b))/((a*b^2*c^4 + 2*a^2* b*c^2*d^2 + a^3*d^4)*sqrt(a*b)) + log(abs(x))/(a^2*c) + 1/2*(a*b^2*c^3 + a ^2*b*c*d^2 - (a*b^2*c^2*d + a^2*b*d^3)*x)/((b*c^2 + a*d^2)^2*(b*x^2 + a)*a ^2)
Time = 9.14 (sec) , antiderivative size = 1032, normalized size of antiderivative = 6.14 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/(x*(a + b*x^2)^2*(c + d*x)),x)
Output:
((b*c)/(2*a*(a*d^2 + b*c^2)) - (b*d*x)/(2*a*(a*d^2 + b*c^2)))/(a + b*x^2) - (d^4*log(c + d*x))/(b^2*c^5 + a^2*c*d^4 + 2*a*b*c^3*d^2) + log(x)/(a^2*c ) - (log(144*a^9*b^12*c^20*x - 3729*c^4*d^16*(-a^5*b)^(7/2) - 256*a^12*d^2 0*(-a^5*b)^(3/2) + 256*a^19*b^2*d^20*x - 144*a^2*b^10*c^20*(-a^5*b)^(3/2) + 1824*a^6*c^2*d^18*(-a^5*b)^(5/2) + 11966*b^6*c^14*d^6*(-a^5*b)^(5/2) + 3 052*a^2*b^4*c^10*d^10*(-a^5*b)^(5/2) - 1537*a^3*b^3*c^8*d^12*(-a^5*b)^(5/2 ) + 1678*a^4*b^2*c^6*d^14*(-a^5*b)^(5/2) - 1480*a^3*b^9*c^18*d^2*(-a^5*b)^ (3/2) - 6001*a^4*b^8*c^16*d^4*(-a^5*b)^(3/2) + 1480*a^10*b^11*c^18*d^2*x + 6001*a^11*b^10*c^16*d^4*x + 11966*a^12*b^9*c^14*d^6*x + 11407*a^13*b^8*c^ 12*d^8*x + 3052*a^14*b^7*c^10*d^10*x - 1537*a^15*b^6*c^8*d^12*x + 1678*a^1 6*b^5*c^6*d^14*x + 3729*a^17*b^4*c^4*d^16*x + 1824*a^18*b^3*c^2*d^18*x + 1 1407*a*b^5*c^12*d^8*(-a^5*b)^(5/2))*(3*a*d^3*(-a^5*b)^(1/2) + 2*a^2*b^2*c^ 3 + 4*a^3*b*c*d^2 + b*c^2*d*(-a^5*b)^(1/2)))/(4*(a^6*d^4 + a^4*b^2*c^4 + 2 *a^5*b*c^2*d^2)) + (log(256*a^12*d^20*(-a^5*b)^(3/2) + 3729*c^4*d^16*(-a^5 *b)^(7/2) + 144*a^9*b^12*c^20*x + 256*a^19*b^2*d^20*x + 144*a^2*b^10*c^20* (-a^5*b)^(3/2) - 1824*a^6*c^2*d^18*(-a^5*b)^(5/2) - 11966*b^6*c^14*d^6*(-a ^5*b)^(5/2) - 3052*a^2*b^4*c^10*d^10*(-a^5*b)^(5/2) + 1537*a^3*b^3*c^8*d^1 2*(-a^5*b)^(5/2) - 1678*a^4*b^2*c^6*d^14*(-a^5*b)^(5/2) + 1480*a^3*b^9*c^1 8*d^2*(-a^5*b)^(3/2) + 6001*a^4*b^8*c^16*d^4*(-a^5*b)^(3/2) + 1480*a^10*b^ 11*c^18*d^2*x + 6001*a^11*b^10*c^16*d^4*x + 11966*a^12*b^9*c^14*d^6*x +...
Time = 0.20 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.48 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^2} \, dx=\frac {-3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} c \,d^{3}-\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b \,c^{3} d -3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b c \,d^{3} x^{2}-\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} c^{3} d \,x^{2}-2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b \,c^{2} d^{2}-\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{2} c^{4}-2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{2} c^{2} d^{2} x^{2}-\mathrm {log}\left (b \,x^{2}+a \right ) b^{3} c^{4} x^{2}-2 \,\mathrm {log}\left (d x +c \right ) a^{3} d^{4}-2 \,\mathrm {log}\left (d x +c \right ) a^{2} b \,d^{4} x^{2}+2 \,\mathrm {log}\left (x \right ) a^{3} d^{4}+4 \,\mathrm {log}\left (x \right ) a^{2} b \,c^{2} d^{2}+2 \,\mathrm {log}\left (x \right ) a^{2} b \,d^{4} x^{2}+2 \,\mathrm {log}\left (x \right ) a \,b^{2} c^{4}+4 \,\mathrm {log}\left (x \right ) a \,b^{2} c^{2} d^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) b^{3} c^{4} x^{2}-a^{2} b c \,d^{3} x -a \,b^{2} c^{3} d x -a \,b^{2} c^{2} d^{2} x^{2}-b^{3} c^{4} x^{2}}{2 a^{2} c \left (a^{2} b \,d^{4} x^{2}+2 a \,b^{2} c^{2} d^{2} x^{2}+b^{3} c^{4} x^{2}+a^{3} d^{4}+2 a^{2} b \,c^{2} d^{2}+a \,b^{2} c^{4}\right )} \] Input:
int(1/x/(d*x+c)/(b*x^2+a)^2,x)
Output:
( - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*c*d**3 - sqrt(b)* sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b*c**3*d - 3*sqrt(b)*sqrt(a)*atan( (b*x)/(sqrt(b)*sqrt(a)))*a*b*c*d**3*x**2 - sqrt(b)*sqrt(a)*atan((b*x)/(sqr t(b)*sqrt(a)))*b**2*c**3*d*x**2 - 2*log(a + b*x**2)*a**2*b*c**2*d**2 - log (a + b*x**2)*a*b**2*c**4 - 2*log(a + b*x**2)*a*b**2*c**2*d**2*x**2 - log(a + b*x**2)*b**3*c**4*x**2 - 2*log(c + d*x)*a**3*d**4 - 2*log(c + d*x)*a**2 *b*d**4*x**2 + 2*log(x)*a**3*d**4 + 4*log(x)*a**2*b*c**2*d**2 + 2*log(x)*a **2*b*d**4*x**2 + 2*log(x)*a*b**2*c**4 + 4*log(x)*a*b**2*c**2*d**2*x**2 + 2*log(x)*b**3*c**4*x**2 - a**2*b*c*d**3*x - a*b**2*c**3*d*x - a*b**2*c**2* d**2*x**2 - b**3*c**4*x**2)/(2*a**2*c*(a**3*d**4 + 2*a**2*b*c**2*d**2 + a* *2*b*d**4*x**2 + a*b**2*c**4 + 2*a*b**2*c**2*d**2*x**2 + b**3*c**4*x**2))