Integrand size = 22, antiderivative size = 156 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {1}{2 a^2 c^2 x^2}-\frac {b^3}{2 a^2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {d^3}{2 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {2 (b c+a d) \log (x)}{a^3 c^3}+\frac {b^3 (b c-2 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^3}+\frac {d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 90} \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {b^3 (b c-2 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^3}-\frac {2 \log (x) (a d+b c)}{a^3 c^3}-\frac {b^3}{2 a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac {1}{2 a^2 c^2 x^2}+\frac {d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3}-\frac {d^3}{2 c^2 \left (c+d x^2\right ) (b c-a d)^2} \]
[In]
[Out]
Rule 90
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^2 (c+d x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a^2 c^2 x^2}-\frac {2 (b c+a d)}{a^3 c^3 x}+\frac {b^4}{a^2 (-b c+a d)^2 (a+b x)^2}+\frac {2 b^4 (-b c+2 a d)}{a^3 (-b c+a d)^3 (a+b x)}+\frac {d^4}{c^2 (b c-a d)^2 (c+d x)^2}+\frac {2 d^4 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = -\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} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {1}{a^2 c^2 x^2}-\frac {b^3}{a^2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {d^3}{c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {4 (b c+a d) \log (x)}{a^3 c^3}+\frac {2 b^3 (-b c+2 a d) \log \left (a+b x^2\right )}{a^3 (-b c+a d)^3}+\frac {2 d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3}\right ) \]
[In]
[Out]
Time = 2.76 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {1}{2 a^{2} c^{2} x^{2}}+\frac {\left (-2 a d -2 b c \right ) \ln \left (x \right )}{c^{3} a^{3}}+\frac {b^{4} \left (\frac {\left (4 a d -2 b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{3} \left (a d -b c \right )^{3}}+\frac {d^{4} \left (\frac {\left (2 a d -4 b c \right ) \ln \left (d \,x^{2}+c \right )}{d}-\frac {\left (a d -b c \right ) c}{d \left (d \,x^{2}+c \right )}\right )}{2 c^{3} \left (a d -b c \right )^{3}}\) | \(157\) |
norman | \(\frac {-\frac {1}{2 a c}+\frac {\left (2 a^{4} d^{4}-a^{3} b c \,d^{3}-a \,b^{3} c^{3} d +2 b^{4} c^{4}\right ) x^{4}}{2 c^{3} a^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (2 a^{3} d^{3}-a^{2} b c \,d^{2}-a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) b d \,x^{6}}{2 c^{3} a^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{x^{2} \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+\frac {b^{3} \left (2 a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d^{3} \left (a d -2 b c \right ) \ln \left (d \,x^{2}+c \right )}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {2 \left (a d +b c \right ) \ln \left (x \right )}{a^{3} c^{3}}\) | \(317\) |
risch | \(\frac {-\frac {b d \left (a^{2} d^{2}-a b c d +b^{2} c^{2}\right ) x^{4}}{a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (2 a^{3} d^{3}-a^{2} b c \,d^{2}-a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) x^{2}}{2 a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {1}{2 a c}}{x^{2} \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}-\frac {2 \ln \left (x \right ) d}{a^{2} c^{3}}-\frac {2 \ln \left (x \right ) b}{a^{3} c^{2}}+\frac {2 b^{3} \ln \left (b \,x^{2}+a \right ) d}{a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{4} \ln \left (b \,x^{2}+a \right ) c}{a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d^{4} \ln \left (-d \,x^{2}-c \right ) a}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {2 d^{3} \ln \left (-d \,x^{2}-c \right ) b}{c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(408\) |
parallelrisch | \(-\frac {4 \ln \left (d \,x^{2}+c \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 -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}+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}-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 (d \,x^{2}+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 (d \,x^{2}+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}-2 x^{4} a^{5} d^{5}-3 a^{4} b \,c^{3} d^{2}+3 a^{3} b^{2} c^{4} d +d^{3} c^{2} a^{5}-c^{5} b^{3} a^{2}+2 x^{4} b^{5} c^{5}-4 \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{4} c^{3} d^{2}+4 \ln \left (d \,x^{2}+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 (d \,x^{2}+c \right ) x^{4} a^{4} b c \,d^{4}+4 \ln \left (d \,x^{2}+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}+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 (d \,x^{2}+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 \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 (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) x^{2} c^{3} a^{3}}\) | \(769\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 667 vs. \(2 (150) = 300\).
Time = 5.16 (sec) , antiderivative size = 667, normalized size of antiderivative = 4.28 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + 2 \, {\left (a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{4} + {\left (2 \, a b^{4} c^{5} - 3 \, a^{2} b^{3} c^{4} d + 3 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}\right )} x^{2} - 2 \, {\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2}\right )} x^{6} + {\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2}\right )} x^{4} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left ({\left (2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{6} + {\left (2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{4} + {\left (2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 4 \, {\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{6} + {\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{4} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left ({\left (a^{3} b^{4} c^{6} d - 3 \, a^{4} b^{3} c^{5} d^{2} + 3 \, a^{5} b^{2} c^{4} d^{3} - a^{6} b c^{3} d^{4}\right )} x^{6} + {\left (a^{3} b^{4} c^{7} - 2 \, a^{4} b^{3} c^{6} d + 2 \, a^{6} b c^{4} d^{3} - a^{7} c^{3} d^{4}\right )} x^{4} + {\left (a^{4} b^{3} c^{7} - 3 \, a^{5} b^{2} c^{6} d + 3 \, a^{6} b c^{5} d^{2} - a^{7} c^{4} d^{3}\right )} x^{2}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (150) = 300\).
Time = 0.22 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.44 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {{\left (b^{4} c - 2 \, a b^{3} d\right )} \log \left (b x^{2} + a\right )}{a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}} + \frac {{\left (2 \, b c d^{3} - a d^{4}\right )} \log \left (d x^{2} + c\right )}{b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}} - \frac {a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + 2 \, {\left (b^{3} c^{2} d - a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (2 \, b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} x^{2}}{2 \, {\left ({\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{6} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{4} + {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2}\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{2}\right )}{a^{3} c^{3}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (150) = 300\).
Time = 0.29 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.13 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {{\left (b^{5} c - 2 \, a b^{4} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}} + \frac {{\left (2 \, b c d^{4} - a d^{5}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}} - \frac {2 \, b^{3} c^{2} d x^{4} - 2 \, a b^{2} c d^{2} x^{4} + 2 \, a^{2} b d^{3} x^{4} + 2 \, b^{3} c^{3} x^{2} - a b^{2} c^{2} d x^{2} - a^{2} b c d^{2} x^{2} + 2 \, a^{3} d^{3} x^{2} + a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}}{2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left (b d x^{6} + b c x^{4} + a d x^{4} + a c x^{2}\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{2}\right )}{a^{3} c^{3}} \]
[In]
[Out]
Time = 6.39 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.01 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {\frac {1}{2\,a\,c}+\frac {x^4\,\left (a^2\,b\,d^3-a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}{a^2\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^2\,\left (a\,d+b\,c\right )\,\left (2\,a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{2\,a^2\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^6+\left (a\,d+b\,c\right )\,x^4+a\,c\,x^2}-\frac {\ln \left (b\,x^2+a\right )\,\left (b^4\,c-2\,a\,b^3\,d\right )}{a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3}-\frac {\ln \left (d\,x^2+c\right )\,\left (a\,d^4-2\,b\,c\,d^3\right )}{-a^3\,c^3\,d^3+3\,a^2\,b\,c^4\,d^2-3\,a\,b^2\,c^5\,d+b^3\,c^6}-\frac {\ln \left (x\right )\,\left (2\,a\,d+2\,b\,c\right )}{a^3\,c^3} \]
[In]
[Out]