Integrand size = 20, antiderivative size = 183 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^2} \, dx=-\frac {1}{a^2 c x}-\frac {b (a d+b c x)}{2 a^2 \left (b c^2+a d^2\right ) \left (a+b x^2\right )}-\frac {b^{3/2} c \left (3 b c^2+5 a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \left (b c^2+a d^2\right )^2}-\frac {d \log (x)}{a^2 c^2}+\frac {d^5 \log (c+d x)}{c^2 \left (b c^2+a d^2\right )^2}+\frac {b d \left (b c^2+2 a d^2\right ) \log \left (a+b x^2\right )}{2 a^2 \left (b c^2+a d^2\right )^2} \] Output:
-1/a^2/c/x-1/2*b*(b*c*x+a*d)/a^2/(a*d^2+b*c^2)/(b*x^2+a)-1/2*b^(3/2)*c*(5* a*d^2+3*b*c^2)*arctan(b^(1/2)*x/a^(1/2))/a^(5/2)/(a*d^2+b*c^2)^2-d*ln(x)/a ^2/c^2+d^5*ln(d*x+c)/c^2/(a*d^2+b*c^2)^2+1/2*b*d*(2*a*d^2+b*c^2)*ln(b*x^2+ a)/a^2/(a*d^2+b*c^2)^2
Time = 0.21 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {2}{a^2 c x}-\frac {b (a d+b c x)}{a^2 \left (b c^2+a d^2\right ) \left (a+b x^2\right )}-\frac {b^{3/2} c \left (3 b c^2+5 a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \left (b c^2+a d^2\right )^2}-\frac {2 d \log (x)}{a^2 c^2}+\frac {2 d^5 \log (c+d x)}{\left (b c^3+a c d^2\right )^2}+\frac {b \left (b c^2 d+2 a d^3\right ) \log \left (a+b x^2\right )}{a^2 \left (b c^2+a d^2\right )^2}\right ) \] Input:
Integrate[1/(x^2*(c + d*x)*(a + b*x^2)^2),x]
Output:
(-2/(a^2*c*x) - (b*(a*d + b*c*x))/(a^2*(b*c^2 + a*d^2)*(a + b*x^2)) - (b^( 3/2)*c*(3*b*c^2 + 5*a*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(b*c^2 + a*d^2)^2) - (2*d*Log[x])/(a^2*c^2) + (2*d^5*Log[c + d*x])/(b*c^3 + a*c*d^2 )^2 + (b*(b*c^2*d + 2*a*d^3)*Log[a + b*x^2])/(a^2*(b*c^2 + a*d^2)^2))/2
Time = 0.62 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.21, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b x^2\right )^2 (c+d x)} \, dx\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \int \left (-\frac {b^2 (c-d x) \left (2 a d^2+b c^2\right )}{a^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^2}-\frac {d}{a^2 c^2 x}+\frac {1}{a^2 c x^2}-\frac {b^2 (c-d x)}{a \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )}+\frac {d^6}{c^2 (c+d x) \left (a d^2+b c^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^{3/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (2 a d^2+b c^2\right )}{a^{5/2} \left (a d^2+b c^2\right )^2}-\frac {b^{3/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \left (a d^2+b c^2\right )}-\frac {b (a d+b c x)}{2 a^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )}+\frac {b d \left (2 a d^2+b c^2\right ) \log \left (a+b x^2\right )}{2 a^2 \left (a d^2+b c^2\right )^2}-\frac {d \log (x)}{a^2 c^2}-\frac {1}{a^2 c x}+\frac {d^5 \log (c+d x)}{c^2 \left (a d^2+b c^2\right )^2}\) |
Input:
Int[1/(x^2*(c + d*x)*(a + b*x^2)^2),x]
Output:
-(1/(a^2*c*x)) - (b*(a*d + b*c*x))/(2*a^2*(b*c^2 + a*d^2)*(a + b*x^2)) - ( b^(3/2)*c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*(b*c^2 + a*d^2)) - (b^(3 /2)*c*(b*c^2 + 2*a*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(b*c^2 + a*d ^2)^2) - (d*Log[x])/(a^2*c^2) + (d^5*Log[c + d*x])/(c^2*(b*c^2 + a*d^2)^2) + (b*d*(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 = 0.33 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(-\frac {1}{a^{2} c x}-\frac {d \ln \left (x \right )}{a^{2} c^{2}}-\frac {b^{2} \left (\frac {\left (\frac {1}{2} a \,d^{2} c +\frac {1}{2} b \,c^{3}\right ) x +\frac {a d \left (a \,d^{2}+b \,c^{2}\right )}{2 b}}{b \,x^{2}+a}+\frac {\left (-4 a \,d^{3}-2 b \,c^{2} d \right ) \ln \left (b \,x^{2}+a \right )}{4 b}+\frac {\left (5 a \,d^{2} c +3 b \,c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2} a^{2}}+\frac {d^{5} \ln \left (d x +c \right )}{c^{2} \left (a \,d^{2}+b \,c^{2}\right )^{2}}\) | \(174\) |
| risch | \(\frac {-\frac {b \left (2 a \,d^{2}+3 b \,c^{2}\right ) x^{2}}{2 a^{2} c \left (a \,d^{2}+b \,c^{2}\right )}-\frac {b d x}{2 a \left (a \,d^{2}+b \,c^{2}\right )}-\frac {1}{a c}}{x \left (b \,x^{2}+a \right )}+\frac {d^{5} \ln \left (-d x -c \right )}{c^{2} \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}-\frac {d \ln \left (x \right )}{a^{2} c^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{7} d^{4}+2 a^{6} b \,c^{2} d^{2}+a^{5} b^{2} c^{4}\right ) \textit {\_Z}^{2}+\left (-8 a^{4} b \,d^{3}-4 a^{3} b^{2} c^{2} d \right ) \textit {\_Z} +16 a \,b^{2} d^{2}+9 b^{3} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a^{9} c^{2} d^{8}-11 a^{8} b \,c^{4} d^{6}-13 a^{7} b^{2} c^{6} d^{4}-9 a^{6} b^{3} c^{8} d^{2}-3 a^{5} b^{4} c^{10}\right ) \textit {\_R}^{3}+\left (32 a^{6} b \,c^{2} d^{7}+52 a^{5} b^{2} c^{4} d^{5}+20 a^{4} b^{3} c^{6} d^{3}\right ) \textit {\_R}^{2}+\left (-16 a^{4} b \,d^{8}-48 a^{3} b^{2} c^{2} d^{6}-132 a^{2} b^{3} c^{4} d^{4}-84 a \,b^{4} c^{6} d^{2}-18 b^{5} c^{8}\right ) \textit {\_R} +64 a \,b^{2} d^{7}\right ) x +\left (-2 a^{9} c^{3} d^{7}-2 a^{8} b \,c^{5} d^{5}+2 a^{7} b^{2} c^{7} d^{3}+2 a^{6} b^{3} c^{9} d \right ) \textit {\_R}^{3}+\left (-8 a^{7} c \,d^{8}+13 a^{5} b^{2} c^{5} d^{4}+2 a^{4} b^{3} c^{7} d^{2}-3 a^{3} b^{4} c^{9}\right ) \textit {\_R}^{2}+\left (32 a^{4} b c \,d^{7}-42 a^{2} b^{3} c^{5} d^{3}-12 a \,b^{4} c^{7} d \right ) \textit {\_R} -80 a \,b^{2} c \,d^{6}-72 b^{3} c^{3} d^{4}\right )\right )}{4}\) | \(572\) |
Input:
int(1/x^2/(d*x+c)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/a^2/c/x-d*ln(x)/a^2/c^2-b^2/(a*d^2+b*c^2)^2/a^2*(((1/2*a*d^2*c+1/2*b*c^ 3)*x+1/2*a*d*(a*d^2+b*c^2)/b)/(b*x^2+a)+1/4*(-4*a*d^3-2*b*c^2*d)/b*ln(b*x^ 2+a)+1/2*(5*a*c*d^2+3*b*c^3)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))+d^5*ln(d *x+c)/c^2/(a*d^2+b*c^2)^2
Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (169) = 338\).
Time = 51.86 (sec) , antiderivative size = 815, normalized size of antiderivative = 4.45 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(1/x^2/(d*x+c)/(b*x^2+a)^2,x, algorithm="fricas")
Output:
[-1/4*(4*a*b^2*c^5 + 8*a^2*b*c^3*d^2 + 4*a^3*c*d^4 + 2*(3*b^3*c^5 + 5*a*b^ 2*c^3*d^2 + 2*a^2*b*c*d^4)*x^2 - ((3*b^3*c^5 + 5*a*b^2*c^3*d^2)*x^3 + (3*a *b^2*c^5 + 5*a^2*b*c^3*d^2)*x)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 2*(a*b^2*c^4*d + a^2*b*c^2*d^3)*x - 2*((b^3*c^4*d + 2*a* b^2*c^2*d^3)*x^3 + (a*b^2*c^4*d + 2*a^2*b*c^2*d^3)*x)*log(b*x^2 + a) - 4*( a^2*b*d^5*x^3 + a^3*d^5*x)*log(d*x + c) + 4*((b^3*c^4*d + 2*a*b^2*c^2*d^3 + a^2*b*d^5)*x^3 + (a*b^2*c^4*d + 2*a^2*b*c^2*d^3 + a^3*d^5)*x)*log(x))/(( a^2*b^3*c^6 + 2*a^3*b^2*c^4*d^2 + a^4*b*c^2*d^4)*x^3 + (a^3*b^2*c^6 + 2*a^ 4*b*c^4*d^2 + a^5*c^2*d^4)*x), -1/2*(2*a*b^2*c^5 + 4*a^2*b*c^3*d^2 + 2*a^3 *c*d^4 + (3*b^3*c^5 + 5*a*b^2*c^3*d^2 + 2*a^2*b*c*d^4)*x^2 + ((3*b^3*c^5 + 5*a*b^2*c^3*d^2)*x^3 + (3*a*b^2*c^5 + 5*a^2*b*c^3*d^2)*x)*sqrt(b/a)*arcta n(x*sqrt(b/a)) + (a*b^2*c^4*d + a^2*b*c^2*d^3)*x - ((b^3*c^4*d + 2*a*b^2*c ^2*d^3)*x^3 + (a*b^2*c^4*d + 2*a^2*b*c^2*d^3)*x)*log(b*x^2 + a) - 2*(a^2*b *d^5*x^3 + a^3*d^5*x)*log(d*x + c) + 2*((b^3*c^4*d + 2*a*b^2*c^2*d^3 + a^2 *b*d^5)*x^3 + (a*b^2*c^4*d + 2*a^2*b*c^2*d^3 + a^3*d^5)*x)*log(x))/((a^2*b ^3*c^6 + 2*a^3*b^2*c^4*d^2 + a^4*b*c^2*d^4)*x^3 + (a^3*b^2*c^6 + 2*a^4*b*c ^4*d^2 + a^5*c^2*d^4)*x)]
Timed out. \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/x**2/(d*x+c)/(b*x**2+a)**2,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.46 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^2} \, dx=\frac {d^{5} \log \left (d x + c\right )}{b^{2} c^{6} + 2 \, a b c^{4} d^{2} + a^{2} c^{2} d^{4}} + \frac {{\left (b^{2} c^{2} d + 2 \, a b d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{2} c^{4} + 2 \, a^{3} b c^{2} d^{2} + a^{4} d^{4}\right )}} - \frac {{\left (3 \, b^{3} c^{3} + 5 \, a b^{2} c d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b^{2} c^{4} + 2 \, a^{3} b c^{2} d^{2} + a^{4} d^{4}\right )} \sqrt {a b}} - \frac {a b c d x + 2 \, a b c^{2} + 2 \, a^{2} d^{2} + {\left (3 \, b^{2} c^{2} + 2 \, a b d^{2}\right )} x^{2}}{2 \, {\left ({\left (a^{2} b^{2} c^{3} + a^{3} b c d^{2}\right )} x^{3} + {\left (a^{3} b c^{3} + a^{4} c d^{2}\right )} x\right )}} - \frac {d \log \left (x\right )}{a^{2} c^{2}} \] Input:
integrate(1/x^2/(d*x+c)/(b*x^2+a)^2,x, algorithm="maxima")
Output:
d^5*log(d*x + c)/(b^2*c^6 + 2*a*b*c^4*d^2 + a^2*c^2*d^4) + 1/2*(b^2*c^2*d + 2*a*b*d^3)*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*(3*b^3*c^3 + 5*a*b^2*c*d^2)*arctan(b*x/sqrt(a*b))/((a^2*b^2*c^4 + 2*a^3* b*c^2*d^2 + a^4*d^4)*sqrt(a*b)) - 1/2*(a*b*c*d*x + 2*a*b*c^2 + 2*a^2*d^2 + (3*b^2*c^2 + 2*a*b*d^2)*x^2)/((a^2*b^2*c^3 + a^3*b*c*d^2)*x^3 + (a^3*b*c^ 3 + a^4*c*d^2)*x) - d*log(x)/(a^2*c^2)
Time = 0.13 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.65 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^2} \, dx=\frac {d^{6} \log \left ({\left | d x + c \right |}\right )}{b^{2} c^{6} d + 2 \, a b c^{4} d^{3} + a^{2} c^{2} d^{5}} + \frac {{\left (b^{2} c^{2} d + 2 \, a b d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{2} c^{4} + 2 \, a^{3} b c^{2} d^{2} + a^{4} d^{4}\right )}} - \frac {{\left (3 \, b^{3} c^{3} + 5 \, a b^{2} c d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b^{2} c^{4} + 2 \, a^{3} b c^{2} d^{2} + a^{4} d^{4}\right )} \sqrt {a b}} - \frac {d \log \left ({\left | x \right |}\right )}{a^{2} c^{2}} - \frac {2 \, a b^{2} c^{5} + 4 \, a^{2} b c^{3} d^{2} + 2 \, a^{3} c d^{4} + {\left (3 \, b^{3} c^{5} + 5 \, a b^{2} c^{3} d^{2} + 2 \, a^{2} b c d^{4}\right )} x^{2} + {\left (a b^{2} c^{4} d + a^{2} b c^{2} d^{3}\right )} x}{2 \, {\left (b c^{2} + a d^{2}\right )}^{2} {\left (b x^{2} + a\right )} a^{2} c^{2} x} \] Input:
integrate(1/x^2/(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")
Output:
d^6*log(abs(d*x + c))/(b^2*c^6*d + 2*a*b*c^4*d^3 + a^2*c^2*d^5) + 1/2*(b^2 *c^2*d + 2*a*b*d^3)*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*(3*b^3*c^3 + 5*a*b^2*c*d^2)*arctan(b*x/sqrt(a*b))/((a^2*b^2*c^4 + 2*a^3*b*c^2*d^2 + a^4*d^4)*sqrt(a*b)) - d*log(abs(x))/(a^2*c^2) - 1/2*(2* a*b^2*c^5 + 4*a^2*b*c^3*d^2 + 2*a^3*c*d^4 + (3*b^3*c^5 + 5*a*b^2*c^3*d^2 + 2*a^2*b*c*d^4)*x^2 + (a*b^2*c^4*d + a^2*b*c^2*d^3)*x)/((b*c^2 + a*d^2)^2* (b*x^2 + a)*a^2*c^2*x)
Time = 9.58 (sec) , antiderivative size = 1276, normalized size of antiderivative = 6.97 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
int(1/(x^2*(a + b*x^2)^2*(c + d*x)),x)
Output:
(d^5*log(c + d*x))/(b^2*c^6 + a^2*c^2*d^4 + 2*a*b*c^4*d^2) - (1/(a*c) + (b *d*x)/(2*a*(a*d^2 + b*c^2)) + (b*x^2*(2*a*d^2 + 3*b*c^2))/(2*a^2*c*(a*d^2 + b*c^2)))/(a*x + b*x^3) + (log(81*a^10*b^16*c^24*x + 256*a^22*b^4*d^24*x - 81*a^3*b^11*c^24*(-a^5*b^3)^(3/2) + 256*a^20*b^2*d^24*(-a^5*b^3)^(1/2) - 5440*a^6*c^8*d^16*(-a^5*b^3)^(5/2) - 1536*a^14*c^2*d^22*(-a^5*b^3)^(3/2) + 6015*b^6*c^20*d^4*(-a^5*b^3)^(5/2) + 19348*a*b^5*c^18*d^6*(-a^5*b^3)^(5/ 2) + 6784*a^5*b*c^10*d^14*(-a^5*b^3)^(5/2) - 2304*a^13*b*c^4*d^20*(-a^5*b^ 3)^(3/2) + 38527*a^2*b^4*c^16*d^8*(-a^5*b^3)^(5/2) + 47558*a^3*b^3*c^14*d^ 10*(-a^5*b^3)^(5/2) + 32785*a^4*b^2*c^12*d^12*(-a^5*b^3)^(5/2) - 1062*a^4* b^10*c^22*d^2*(-a^5*b^3)^(3/2) + 1760*a^12*b^2*c^6*d^18*(-a^5*b^3)^(3/2) + 1062*a^11*b^15*c^22*d^2*x + 6015*a^12*b^14*c^20*d^4*x + 19348*a^13*b^13*c ^18*d^6*x + 38527*a^14*b^12*c^16*d^8*x + 47558*a^15*b^11*c^14*d^10*x + 327 85*a^16*b^10*c^12*d^12*x + 6784*a^17*b^9*c^10*d^14*x - 5440*a^18*b^8*c^8*d ^16*x - 1760*a^19*b^7*c^6*d^18*x + 2304*a^20*b^6*c^4*d^20*x + 1536*a^21*b^ 5*c^2*d^22*x)*(4*a^4*b*d^3 - 3*b*c^3*(-a^5*b^3)^(1/2) + 2*a^3*b^2*c^2*d - 5*a*c*d^2*(-a^5*b^3)^(1/2)))/(4*(a^7*d^4 + a^5*b^2*c^4 + 2*a^6*b*c^2*d^2)) + (log(81*a^10*b^16*c^24*x + 256*a^22*b^4*d^24*x + 81*a^3*b^11*c^24*(-a^5 *b^3)^(3/2) - 256*a^20*b^2*d^24*(-a^5*b^3)^(1/2) + 5440*a^6*c^8*d^16*(-a^5 *b^3)^(5/2) + 1536*a^14*c^2*d^22*(-a^5*b^3)^(3/2) - 6015*b^6*c^20*d^4*(-a^ 5*b^3)^(5/2) - 19348*a*b^5*c^18*d^6*(-a^5*b^3)^(5/2) - 6784*a^5*b*c^10*...
Time = 0.19 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.74 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^2} \, dx=\frac {-5 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b \,c^{3} d^{2} x -3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} c^{5} x -5 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} c^{3} d^{2} x^{3}-3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} c^{5} x^{3}+2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b \,c^{2} d^{3} x +\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{2} c^{4} d x +2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{2} c^{2} d^{3} x^{3}+\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{3} c^{4} d \,x^{3}+2 \,\mathrm {log}\left (d x +c \right ) a^{4} d^{5} x +2 \,\mathrm {log}\left (d x +c \right ) a^{3} b \,d^{5} x^{3}-2 \,\mathrm {log}\left (x \right ) a^{4} d^{5} x -4 \,\mathrm {log}\left (x \right ) a^{3} b \,c^{2} d^{3} x -2 \,\mathrm {log}\left (x \right ) a^{3} b \,d^{5} x^{3}-2 \,\mathrm {log}\left (x \right ) a^{2} b^{2} c^{4} d x -4 \,\mathrm {log}\left (x \right ) a^{2} b^{2} c^{2} d^{3} x^{3}-2 \,\mathrm {log}\left (x \right ) a \,b^{3} c^{4} d \,x^{3}-2 a^{4} c \,d^{4}-4 a^{3} b \,c^{3} d^{2}-2 a^{3} b c \,d^{4} x^{2}-2 a^{2} b^{2} c^{5}-5 a^{2} b^{2} c^{3} d^{2} x^{2}+a^{2} b^{2} c^{2} d^{3} x^{3}-3 a \,b^{3} c^{5} x^{2}+a \,b^{3} c^{4} d \,x^{3}}{2 a^{3} c^{2} x \left (a^{2} b \,d^{4} x^{2}+2 a \,b^{2} c^{2} d^{2} x^{2}+b^{3} c^{4} x^{2}+a^{3} d^{4}+2 a^{2} b \,c^{2} d^{2}+a \,b^{2} c^{4}\right )} \] Input:
int(1/x^2/(d*x+c)/(b*x^2+a)^2,x)
Output:
( - 5*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b*c**3*d**2*x - 3 *sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**2*c**5*x - 5*sqrt(b)*s qrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**2*c**3*d**2*x**3 - 3*sqrt(b)*sqr t(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**5*x**3 + 2*log(a + b*x**2)*a**3 *b*c**2*d**3*x + log(a + b*x**2)*a**2*b**2*c**4*d*x + 2*log(a + b*x**2)*a* *2*b**2*c**2*d**3*x**3 + log(a + b*x**2)*a*b**3*c**4*d*x**3 + 2*log(c + d* x)*a**4*d**5*x + 2*log(c + d*x)*a**3*b*d**5*x**3 - 2*log(x)*a**4*d**5*x - 4*log(x)*a**3*b*c**2*d**3*x - 2*log(x)*a**3*b*d**5*x**3 - 2*log(x)*a**2*b* *2*c**4*d*x - 4*log(x)*a**2*b**2*c**2*d**3*x**3 - 2*log(x)*a*b**3*c**4*d*x **3 - 2*a**4*c*d**4 - 4*a**3*b*c**3*d**2 - 2*a**3*b*c*d**4*x**2 - 2*a**2*b **2*c**5 - 5*a**2*b**2*c**3*d**2*x**2 + a**2*b**2*c**2*d**3*x**3 - 3*a*b** 3*c**5*x**2 + a*b**3*c**4*d*x**3)/(2*a**3*c**2*x*(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))