Integrand size = 22, antiderivative size = 156 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {1}{2 a^2 c^2 x^2}-\frac {b^3}{2 a^2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {d^3}{2 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {2 (b c+a d) \log (x)}{a^3 c^3}+\frac {b^3 (b c-2 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^3}+\frac {d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3} \] Output:
-1/2/a^2/c^2/x^2-1/2*b^3/a^2/(-a*d+b*c)^2/(b*x^2+a)-1/2*d^3/c^2/(-a*d+b*c) ^2/(d*x^2+c)-2*(a*d+b*c)*ln(x)/a^3/c^3+b^3*(-2*a*d+b*c)*ln(b*x^2+a)/a^3/(- a*d+b*c)^3+d^3*(-a*d+2*b*c)*ln(d*x^2+c)/c^3/(-a*d+b*c)^3
Time = 0.13 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {1}{a^2 c^2 x^2}-\frac {b^3}{a^2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {d^3}{c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {4 (b c+a d) \log (x)}{a^3 c^3}+\frac {2 b^3 (-b c+2 a d) \log \left (a+b x^2\right )}{a^3 (-b c+a d)^3}+\frac {2 d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3}\right ) \] Input:
Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^2),x]
Output:
(-(1/(a^2*c^2*x^2)) - b^3/(a^2*(b*c - a*d)^2*(a + b*x^2)) - d^3/(c^2*(b*c - a*d)^2*(c + d*x^2)) - (4*(b*c + a*d)*Log[x])/(a^3*c^3) + (2*b^3*(-(b*c) + 2*a*d)*Log[a + b*x^2])/(a^3*(-(b*c) + a*d)^3) + (2*d^3*(2*b*c - a*d)*Log [c + d*x^2])/(c^3*(b*c - a*d)^3))/2
Time = 0.38 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {354, 99, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^2+a\right )^2 \left (d x^2+c\right )^2}dx^2\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {2 (2 a d-b c) b^4}{a^3 (a d-b c)^3 \left (b x^2+a\right )}+\frac {b^4}{a^2 (a d-b c)^2 \left (b x^2+a\right )^2}+\frac {2 d^4 (2 b c-a d)}{c^3 (b c-a d)^3 \left (d x^2+c\right )}-\frac {2 (b c+a d)}{a^3 c^3 x^2}+\frac {d^4}{c^2 (b c-a d)^2 \left (d x^2+c\right )^2}+\frac {1}{a^2 c^2 x^4}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 b^3 (b c-2 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^3}-\frac {2 \log \left (x^2\right ) (a d+b c)}{a^3 c^3}-\frac {b^3}{a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac {1}{a^2 c^2 x^2}+\frac {2 d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3}-\frac {d^3}{c^2 \left (c+d x^2\right ) (b c-a d)^2}\right )\) |
Input:
Int[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^2),x]
Output:
(-(1/(a^2*c^2*x^2)) - b^3/(a^2*(b*c - a*d)^2*(a + b*x^2)) - d^3/(c^2*(b*c - a*d)^2*(c + d*x^2)) - (2*(b*c + a*d)*Log[x^2])/(a^3*c^3) + (2*b^3*(b*c - 2*a*d)*Log[a + b*x^2])/(a^3*(b*c - a*d)^3) + (2*d^3*(2*b*c - a*d)*Log[c + d*x^2])/(c^3*(b*c - a*d)^3))/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 0.61 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.01
| method | result | size |
| default | \(\frac {b^{4} \left (-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}+\frac {\left (4 a d -2 b c \right ) \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 a^{3} \left (a d -b c \right )^{3}}-\frac {1}{2 a^{2} c^{2} x^{2}}+\frac {\left (-2 a d -2 b c \right ) \ln \left (x \right )}{c^{3} a^{3}}+\frac {d^{4} \left (-\frac {\left (a d -b c \right ) c}{d \left (x^{2} d +c \right )}+\frac {\left (2 a d -4 b c \right ) \ln \left (x^{2} d +c \right )}{d}\right )}{2 c^{3} \left (a d -b c \right )^{3}}\) | \(157\) |
| norman | \(\frac {-\frac {1}{2 a c}+\frac {\left (2 d^{4} a^{4}-a^{3} b c \,d^{3}-a \,b^{3} c^{3} d +2 c^{4} b^{4}\right ) x^{4}}{2 c^{3} a^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (2 a^{3} d^{3}-a^{2} b c \,d^{2}-a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) b d \,x^{6}}{2 c^{3} a^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{x^{2} \left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}+\frac {b^{3} \left (2 a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d^{3} \left (a d -2 b c \right ) \ln \left (x^{2} d +c \right )}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {2 \left (a d +b c \right ) \ln \left (x \right )}{a^{3} c^{3}}\) | \(317\) |
| risch | \(\frac {-\frac {b d \left (a^{2} d^{2}-a b c d +b^{2} c^{2}\right ) x^{4}}{a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (2 a^{3} d^{3}-a^{2} b c \,d^{2}-a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) x^{2}}{2 a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {1}{2 a c}}{x^{2} \left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}-\frac {2 \ln \left (x \right ) d}{a^{2} c^{3}}-\frac {2 \ln \left (x \right ) b}{a^{3} c^{2}}+\frac {2 b^{3} \ln \left (b \,x^{2}+a \right ) d}{a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{4} \ln \left (b \,x^{2}+a \right ) c}{a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d^{4} \ln \left (-x^{2} d -c \right ) a}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {2 d^{3} \ln \left (-x^{2} d -c \right ) b}{c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(408\) |
| parallelrisch | \(-\frac {4 \ln \left (x \right ) x^{4} a^{5} d^{5}-4 \ln \left (x \right ) x^{4} b^{5} c^{5}+2 \ln \left (b \,x^{2}+a \right ) x^{4} b^{5} c^{5}-2 \ln \left (x^{2} d +c \right ) x^{4} a^{5} d^{5}-2 x^{6} a^{4} b \,d^{5}+2 x^{6} b^{5} c^{4} d -2 x^{4} a^{5} d^{5}-c^{5} b^{3} a^{2}-4 \ln \left (x \right ) x^{6} b^{5} c^{4} d +2 \ln \left (b \,x^{2}+a \right ) x^{6} b^{5} c^{4} d -2 \ln \left (x^{2} d +c \right ) x^{6} a^{4} b \,d^{5}+4 \ln \left (x \right ) x^{2} a^{5} c \,d^{4}-4 \ln \left (x \right ) x^{2} a \,b^{4} c^{5}+2 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{4} c^{5}-2 \ln \left (x^{2} d +c \right ) x^{2} a^{5} c \,d^{4}+3 x^{6} a^{3} b^{2} c \,d^{4}-3 x^{6} a \,b^{4} c^{3} d^{2}+3 x^{4} a^{4} b c \,d^{4}-x^{4} a^{3} b^{2} c^{2} d^{3}+x^{4} a^{2} b^{3} c^{3} d^{2}-3 x^{4} a \,b^{4} c^{4} d +4 \ln \left (x \right ) x^{6} a^{4} b \,d^{5}-8 \ln \left (x \right ) x^{6} a^{3} b^{2} c \,d^{4}+8 \ln \left (x \right ) x^{6} a \,b^{4} c^{3} d^{2}-4 \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{4} c^{3} d^{2}-3 a^{4} b \,c^{3} d^{2}+3 a^{3} b^{2} c^{4} d +4 \ln \left (x^{2} d +c \right ) x^{6} a^{3} b^{2} c \,d^{4}-4 \ln \left (x \right ) x^{4} a^{4} b c \,d^{4}-8 \ln \left (x \right ) x^{4} a^{3} b^{2} c^{2} d^{3}+8 \ln \left (x \right ) x^{4} a^{2} b^{3} c^{3} d^{2}+4 \ln \left (x \right ) x^{4} a \,b^{4} c^{4} d -4 \ln \left (b \,x^{2}+a \right ) x^{4} a^{2} b^{3} c^{3} d^{2}-2 \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{4} c^{4} d +2 \ln \left (x^{2} d +c \right ) x^{4} a^{4} b c \,d^{4}+4 \ln \left (x^{2} d +c \right ) x^{4} a^{3} b^{2} c^{2} d^{3}-8 \ln \left (x \right ) x^{2} a^{4} b \,c^{2} d^{3}+8 \ln \left (x \right ) x^{2} a^{2} b^{3} c^{4} d -4 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b^{3} c^{4} d +4 \ln \left (x^{2} d +c \right ) x^{2} a^{4} b \,c^{2} d^{3}+d^{3} c^{2} a^{5}+2 x^{4} b^{5} c^{5}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (x^{2} d +c \right ) \left (b \,x^{2}+a \right ) x^{2} c^{3} a^{3}}\) | \(769\) |
Input:
int(1/x^3/(b*x^2+a)^2/(d*x^2+c)^2,x,method=_RETURNVERBOSE)
Output:
1/2*b^4/a^3/(a*d-b*c)^3*(-(a*d-b*c)*a/b/(b*x^2+a)+(4*a*d-2*b*c)/b*ln(b*x^2 +a))-1/2/a^2/c^2/x^2+(-2*a*d-2*b*c)/c^3/a^3*ln(x)+1/2*d^4/c^3/(a*d-b*c)^3* (-(a*d-b*c)*c/d/(d*x^2+c)+(2*a*d-4*b*c)/d*ln(d*x^2+c))
Leaf count of result is larger than twice the leaf count of optimal. 667 vs. \(2 (150) = 300\).
Time = 4.97 (sec) , antiderivative size = 667, normalized size of antiderivative = 4.28 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + 2 \, {\left (a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{4} + {\left (2 \, a b^{4} c^{5} - 3 \, a^{2} b^{3} c^{4} d + 3 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}\right )} x^{2} - 2 \, {\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2}\right )} x^{6} + {\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2}\right )} x^{4} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left ({\left (2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{6} + {\left (2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{4} + {\left (2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 4 \, {\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{6} + {\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{4} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left ({\left (a^{3} b^{4} c^{6} d - 3 \, a^{4} b^{3} c^{5} d^{2} + 3 \, a^{5} b^{2} c^{4} d^{3} - a^{6} b c^{3} d^{4}\right )} x^{6} + {\left (a^{3} b^{4} c^{7} - 2 \, a^{4} b^{3} c^{6} d + 2 \, a^{6} b c^{4} d^{3} - a^{7} c^{3} d^{4}\right )} x^{4} + {\left (a^{4} b^{3} c^{7} - 3 \, a^{5} b^{2} c^{6} d + 3 \, a^{6} b c^{5} d^{2} - a^{7} c^{4} d^{3}\right )} x^{2}\right )}} \] Input:
integrate(1/x^3/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="fricas")
Output:
-1/2*(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3 + 2*(a *b^4*c^4*d - 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - a^4*b*c*d^4)*x^4 + (2 *a*b^4*c^5 - 3*a^2*b^3*c^4*d + 3*a^4*b*c^2*d^3 - 2*a^5*c*d^4)*x^2 - 2*((b^ 5*c^4*d - 2*a*b^4*c^3*d^2)*x^6 + (b^5*c^5 - a*b^4*c^4*d - 2*a^2*b^3*c^3*d^ 2)*x^4 + (a*b^4*c^5 - 2*a^2*b^3*c^4*d)*x^2)*log(b*x^2 + a) - 2*((2*a^3*b^2 *c*d^4 - a^4*b*d^5)*x^6 + (2*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 - a^5*d^5)*x^4 + (2*a^4*b*c^2*d^3 - a^5*c*d^4)*x^2)*log(d*x^2 + c) + 4*((b^5*c^4*d - 2*a* b^4*c^3*d^2 + 2*a^3*b^2*c*d^4 - a^4*b*d^5)*x^6 + (b^5*c^5 - a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 - a^5*d^5)*x^4 + (a*b^ 4*c^5 - 2*a^2*b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*d^4)*x^2)*log(x))/((a^3* b^4*c^6*d - 3*a^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4)*x^6 + ( a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a^7*c^3*d^4)*x^4 + (a^4* b^3*c^7 - 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x^2)
Timed out. \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/x**3/(b*x**2+a)**2/(d*x**2+c)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (150) = 300\).
Time = 0.05 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.44 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {{\left (b^{4} c - 2 \, a b^{3} d\right )} \log \left (b x^{2} + a\right )}{a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}} + \frac {{\left (2 \, b c d^{3} - a d^{4}\right )} \log \left (d x^{2} + c\right )}{b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}} - \frac {a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + 2 \, {\left (b^{3} c^{2} d - a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (2 \, b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} x^{2}}{2 \, {\left ({\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{6} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{4} + {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2}\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{2}\right )}{a^{3} c^{3}} \] Input:
integrate(1/x^3/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="maxima")
Output:
(b^4*c - 2*a*b^3*d)*log(b*x^2 + a)/(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5* b*c*d^2 - a^6*d^3) + (2*b*c*d^3 - a*d^4)*log(d*x^2 + c)/(b^3*c^6 - 3*a*b^2 *c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3) - 1/2*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + 2*(b^3*c^2*d - a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (2*b^3*c^3 - a* b^2*c^2*d - a^2*b*c*d^2 + 2*a^3*d^3)*x^2)/((a^2*b^3*c^4*d - 2*a^3*b^2*c^3* d^2 + a^4*b*c^2*d^3)*x^6 + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^4 + (a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*x^2) - (b*c + a*d)*log(x^2)/(a^3*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (150) = 300\).
Time = 0.13 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.13 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {{\left (b^{5} c - 2 \, a b^{4} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}} + \frac {{\left (2 \, b c d^{4} - a d^{5}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}} - \frac {2 \, b^{3} c^{2} d x^{4} - 2 \, a b^{2} c d^{2} x^{4} + 2 \, a^{2} b d^{3} x^{4} + 2 \, b^{3} c^{3} x^{2} - a b^{2} c^{2} d x^{2} - a^{2} b c d^{2} x^{2} + 2 \, a^{3} d^{3} x^{2} + a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}}{2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left (b d x^{6} + b c x^{4} + a d x^{4} + a c x^{2}\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{2}\right )}{a^{3} c^{3}} \] Input:
integrate(1/x^3/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="giac")
Output:
(b^5*c - 2*a*b^4*d)*log(abs(b*x^2 + a))/(a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3 *a^5*b^2*c*d^2 - a^6*b*d^3) + (2*b*c*d^4 - a*d^5)*log(abs(d*x^2 + c))/(b^3 *c^6*d - 3*a*b^2*c^5*d^2 + 3*a^2*b*c^4*d^3 - a^3*c^3*d^4) - 1/2*(2*b^3*c^2 *d*x^4 - 2*a*b^2*c*d^2*x^4 + 2*a^2*b*d^3*x^4 + 2*b^3*c^3*x^2 - a*b^2*c^2*d *x^2 - a^2*b*c*d^2*x^2 + 2*a^3*d^3*x^2 + a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c *d^2)/((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*(b*d*x^6 + b*c*x^4 + a* d*x^4 + a*c*x^2)) - (b*c + a*d)*log(x^2)/(a^3*c^3)
Time = 2.21 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.01 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {\frac {1}{2\,a\,c}+\frac {x^4\,\left (a^2\,b\,d^3-a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}{a^2\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^2\,\left (a\,d+b\,c\right )\,\left (2\,a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{2\,a^2\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^6+\left (a\,d+b\,c\right )\,x^4+a\,c\,x^2}-\frac {\ln \left (b\,x^2+a\right )\,\left (b^4\,c-2\,a\,b^3\,d\right )}{a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3}-\frac {\ln \left (d\,x^2+c\right )\,\left (a\,d^4-2\,b\,c\,d^3\right )}{-a^3\,c^3\,d^3+3\,a^2\,b\,c^4\,d^2-3\,a\,b^2\,c^5\,d+b^3\,c^6}-\frac {\ln \left (x\right )\,\left (2\,a\,d+2\,b\,c\right )}{a^3\,c^3} \] Input:
int(1/(x^3*(a + b*x^2)^2*(c + d*x^2)^2),x)
Output:
- (1/(2*a*c) + (x^4*(a^2*b*d^3 + b^3*c^2*d - a*b^2*c*d^2))/(a^2*c^2*(a^2*d ^2 + b^2*c^2 - 2*a*b*c*d)) + (x^2*(a*d + b*c)*(2*a^2*d^2 + 2*b^2*c^2 - 3*a *b*c*d))/(2*a^2*c^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x^4*(a*d + b*c) + a *c*x^2 + b*d*x^6) - (log(a + b*x^2)*(b^4*c - 2*a*b^3*d))/(a^6*d^3 - a^3*b^ 3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2) - (log(c + d*x^2)*(a*d^4 - 2*b*c* d^3))/(b^3*c^6 - a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*a*b^2*c^5*d) - (log(x)* (2*a*d + 2*b*c))/(a^3*c^3)
Time = 0.23 (sec) , antiderivative size = 1059, normalized size of antiderivative = 6.79 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/x^3/(b*x^2+a)^2/(d*x^2+c)^2,x)
Output:
(4*log(a + b*x**2)*a**3*b**3*c**4*d**2*x**2 + 4*log(a + b*x**2)*a**3*b**3* c**3*d**3*x**4 + 2*log(a + b*x**2)*a**2*b**4*c**5*d*x**2 + 6*log(a + b*x** 2)*a**2*b**4*c**4*d**2*x**4 + 4*log(a + b*x**2)*a**2*b**4*c**3*d**3*x**6 - 2*log(a + b*x**2)*a*b**5*c**6*x**2 + 2*log(a + b*x**2)*a*b**5*c**4*d**2*x **6 - 2*log(a + b*x**2)*b**6*c**6*x**4 - 2*log(a + b*x**2)*b**6*c**5*d*x** 6 + 2*log(c + d*x**2)*a**6*c*d**5*x**2 + 2*log(c + d*x**2)*a**6*d**6*x**4 - 2*log(c + d*x**2)*a**5*b*c**2*d**4*x**2 + 2*log(c + d*x**2)*a**5*b*d**6* x**6 - 4*log(c + d*x**2)*a**4*b**2*c**3*d**3*x**2 - 6*log(c + d*x**2)*a**4 *b**2*c**2*d**4*x**4 - 2*log(c + d*x**2)*a**4*b**2*c*d**5*x**6 - 4*log(c + d*x**2)*a**3*b**3*c**3*d**3*x**4 - 4*log(c + d*x**2)*a**3*b**3*c**2*d**4* x**6 - 4*log(x)*a**6*c*d**5*x**2 - 4*log(x)*a**6*d**6*x**4 + 4*log(x)*a**5 *b*c**2*d**4*x**2 - 4*log(x)*a**5*b*d**6*x**6 + 8*log(x)*a**4*b**2*c**3*d* *3*x**2 + 12*log(x)*a**4*b**2*c**2*d**4*x**4 + 4*log(x)*a**4*b**2*c*d**5*x **6 - 8*log(x)*a**3*b**3*c**4*d**2*x**2 + 8*log(x)*a**3*b**3*c**2*d**4*x** 6 - 4*log(x)*a**2*b**4*c**5*d*x**2 - 12*log(x)*a**2*b**4*c**4*d**2*x**4 - 8*log(x)*a**2*b**4*c**3*d**3*x**6 + 4*log(x)*a*b**5*c**6*x**2 - 4*log(x)*a *b**5*c**4*d**2*x**6 + 4*log(x)*b**6*c**6*x**4 + 4*log(x)*b**6*c**5*d*x**6 - a**6*c**2*d**4 - 2*a**6*c*d**5*x**2 + 2*a**5*b*c**3*d**3 + 3*a**5*b*c** 2*d**4*x**2 - a**4*b**2*c**3*d**3*x**2 + 2*a**4*b**2*c*d**5*x**6 - 2*a**3* b**3*c**5*d + a**3*b**3*c**4*d**2*x**2 - 4*a**3*b**3*c**2*d**4*x**6 + a...