Integrand size = 22, antiderivative size = 215 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {1}{2 a^2 c^3 x^2}-\frac {b^4}{2 a^2 (b c-a d)^3 \left (a+b x^2\right )}-\frac {d^3}{4 c^2 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {d^3 (2 b c-a d)}{c^3 (b c-a d)^3 \left (c+d x^2\right )}-\frac {(2 b c+3 a d) \log (x)}{a^3 c^4}+\frac {b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4} \]
[Out]
Time = 0.22 (sec) , antiderivative size = 215, 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} \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}-\frac {\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac {b^4}{2 a^2 \left (a+b x^2\right ) (b c-a d)^3}+\frac {d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}-\frac {1}{2 a^2 c^3 x^2}-\frac {d^3 (2 b c-a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac {d^3}{4 c^2 \left (c+d x^2\right )^2 (b c-a d)^2} \]
[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)^3} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a^2 c^3 x^2}+\frac {-2 b c-3 a d}{a^3 c^4 x}-\frac {b^5}{a^2 (-b c+a d)^3 (a+b x)^2}-\frac {b^5 (-2 b c+5 a d)}{a^3 (-b c+a d)^4 (a+b x)}+\frac {d^4}{c^2 (b c-a d)^2 (c+d x)^3}+\frac {2 d^4 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)^2}+\frac {d^4 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )}{c^4 (b c-a d)^4 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {1}{2 a^2 c^3 x^2}-\frac {b^4}{2 a^2 (b c-a d)^3 \left (a+b x^2\right )}-\frac {d^3}{4 c^2 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {d^3 (2 b c-a d)}{c^3 (b c-a d)^3 \left (c+d x^2\right )}-\frac {(2 b c+3 a d) \log (x)}{a^3 c^4}+\frac {b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {1}{4} \left (-\frac {2}{a^2 c^3 x^2}+\frac {2 b^4}{a^2 (-b c+a d)^3 \left (a+b x^2\right )}-\frac {d^3}{c^2 (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {4 d^3 (-2 b c+a d)}{c^3 (b c-a d)^3 \left (c+d x^2\right )}-\frac {4 (2 b c+3 a d) \log (x)}{a^3 c^4}+\frac {2 b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^4}+\frac {2 d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{c^4 (b c-a d)^4}\right ) \]
[In]
[Out]
Time = 2.84 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.04
method | result | size |
default | \(-\frac {1}{2 a^{2} c^{3} x^{2}}+\frac {\left (-3 a d -2 b c \right ) \ln \left (x \right )}{a^{3} c^{4}}-\frac {b^{5} \left (\frac {\left (5 a d -2 b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{3} \left (a d -b c \right )^{4}}+\frac {d^{4} \left (-\frac {c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d \left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}-10 a b c d +10 b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{d}-\frac {2 c \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right )}{d \left (d \,x^{2}+c \right )}\right )}{2 c^{4} \left (a d -b c \right )^{4}}\) | \(223\) |
norman | \(\frac {-\frac {1}{2 a c}+\frac {\left (6 a^{5} d^{5}-12 a^{4} b c \,d^{4}+4 a^{3} b^{2} c^{2} d^{3}+a \,b^{4} c^{4} d -2 b^{5} c^{5}\right ) x^{4}}{2 c^{3} a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d \left (9 a^{5} d^{5}-7 a^{4} b c \,d^{4}-18 a^{3} b^{2} c^{2} d^{3}+8 a^{2} b^{3} c^{3} d^{2}+4 a \,b^{4} c^{4} d -8 b^{5} c^{5}\right ) x^{6}}{4 c^{4} a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d^{2} b \left (9 a^{4} d^{4}-19 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d -4 b^{4} c^{4}\right ) x^{8}}{4 c^{4} a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{x^{2} \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )^{2}}-\frac {\left (3 a d +2 b c \right ) \ln \left (x \right )}{a^{3} c^{4}}-\frac {b^{4} \left (5 a d -2 b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {d^{3} \left (3 a^{2} d^{2}-10 a b c d +10 b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 c^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(536\) |
risch | \(\frac {-\frac {b \,d^{2} \left (3 a^{3} d^{3}-7 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{6}}{2 a^{2} c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d \left (6 a^{4} d^{4}-5 a^{3} b c \,d^{3}-15 a^{2} b^{2} c^{2} d^{2}+10 a \,b^{3} c^{3} d -8 b^{4} c^{4}\right ) x^{4}}{4 a^{2} c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {\left (9 a^{4} d^{4}-19 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d -4 b^{4} c^{4}\right ) x^{2}}{4 a^{2} c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {1}{2 a c}}{x^{2} \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )^{2}}-\frac {3 \ln \left (x \right ) d}{a^{2} c^{4}}-\frac {2 \ln \left (x \right ) b}{a^{3} c^{3}}-\frac {5 b^{4} \ln \left (b \,x^{2}+a \right ) d}{2 a^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {b^{5} \ln \left (b \,x^{2}+a \right ) c}{a^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {3 d^{5} \ln \left (-d \,x^{2}-c \right ) a^{2}}{2 c^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {5 d^{4} \ln \left (-d \,x^{2}-c \right ) a b}{c^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {5 d^{3} \ln \left (-d \,x^{2}-c \right ) b^{2}}{c^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(699\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1376\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1227 vs. \(2 (205) = 410\).
Time = 16.97 (sec) , antiderivative size = 1227, normalized size of antiderivative = 5.71 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (205) = 410\).
Time = 0.23 (sec) , antiderivative size = 651, normalized size of antiderivative = 3.03 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {{\left (2 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )}} + \frac {{\left (10 \, b^{2} c^{2} d^{3} - 10 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )}} - \frac {2 \, a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 6 \, a^{3} b c^{3} d^{2} - 2 \, a^{4} c^{2} d^{3} + 2 \, {\left (2 \, b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 7 \, a^{2} b^{2} c d^{4} - 3 \, a^{3} b d^{5}\right )} x^{6} + {\left (8 \, b^{4} c^{4} d - 10 \, a b^{3} c^{3} d^{2} + 15 \, a^{2} b^{2} c^{2} d^{3} + 5 \, a^{3} b c d^{4} - 6 \, a^{4} d^{5}\right )} x^{4} + {\left (4 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d - 6 \, a^{2} b^{2} c^{3} d^{2} + 19 \, a^{3} b c^{2} d^{3} - 9 \, a^{4} c d^{4}\right )} x^{2}}{4 \, {\left ({\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{8} + {\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{6} + {\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x^{4} + {\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3}\right )} x^{2}\right )}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{4}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (205) = 410\).
Time = 0.37 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.97 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {{\left (2 \, b^{6} c - 5 \, a b^{5} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )}} + \frac {{\left (10 \, b^{2} c^{2} d^{4} - 10 \, a b c d^{5} + 3 \, a^{2} d^{6}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )}} + \frac {10 \, a^{2} b^{3} c^{2} d^{3} x^{4} - 10 \, a^{3} b^{2} c d^{4} x^{4} + 3 \, a^{4} b d^{5} x^{4} - 4 \, b^{5} c^{5} x^{2} + 10 \, a b^{4} c^{4} d x^{2} - 12 \, a^{2} b^{3} c^{3} d^{2} x^{2} + 18 \, a^{3} b^{2} c^{2} d^{3} x^{2} - 12 \, a^{4} b c d^{4} x^{2} + 3 \, a^{5} d^{5} x^{2} - 2 \, a b^{4} c^{5} + 8 \, a^{2} b^{3} c^{4} d - 12 \, a^{3} b^{2} c^{3} d^{2} + 8 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}}{4 \, {\left (a^{2} b^{4} c^{8} - 4 \, a^{3} b^{3} c^{7} d + 6 \, a^{4} b^{2} c^{6} d^{2} - 4 \, a^{5} b c^{5} d^{3} + a^{6} c^{4} d^{4}\right )} {\left (b x^{4} + a x^{2}\right )}} - \frac {30 \, b^{2} c^{2} d^{5} x^{4} - 30 \, a b c d^{6} x^{4} + 9 \, a^{2} d^{7} x^{4} + 68 \, b^{2} c^{3} d^{4} x^{2} - 72 \, a b c^{2} d^{5} x^{2} + 22 \, a^{2} c d^{6} x^{2} + 39 \, b^{2} c^{4} d^{3} - 44 \, a b c^{3} d^{4} + 14 \, a^{2} c^{2} d^{5}}{4 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} {\left (d x^{2} + c\right )}^{2}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{4}} \]
[In]
[Out]
Time = 7.22 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.55 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (2\,b^5\,c-5\,a\,b^4\,d\right )}{2\,a^7\,d^4-8\,a^6\,b\,c\,d^3+12\,a^5\,b^2\,c^2\,d^2-8\,a^4\,b^3\,c^3\,d+2\,a^3\,b^4\,c^4}-\frac {\frac {1}{2\,a\,c}-\frac {x^4\,\left (-6\,a^4\,d^5+5\,a^3\,b\,c\,d^4+15\,a^2\,b^2\,c^2\,d^3-10\,a\,b^3\,c^3\,d^2+8\,b^4\,c^4\,d\right )}{4\,a^2\,c^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x^2\,\left (9\,a^4\,d^4-19\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+2\,a\,b^3\,c^3\,d-4\,b^4\,c^4\right )}{4\,a^2\,c^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b\,d^2\,x^6\,\left (3\,a^3\,d^3-7\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{2\,a^2\,c^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x^4\,\left (b\,c^2+2\,a\,d\,c\right )+x^6\,\left (a\,d^2+2\,b\,c\,d\right )+a\,c^2\,x^2+b\,d^2\,x^8}+\frac {\ln \left (d\,x^2+c\right )\,\left (3\,a^2\,d^5-10\,a\,b\,c\,d^4+10\,b^2\,c^2\,d^3\right )}{2\,a^4\,c^4\,d^4-8\,a^3\,b\,c^5\,d^3+12\,a^2\,b^2\,c^6\,d^2-8\,a\,b^3\,c^7\,d+2\,b^4\,c^8}-\frac {\ln \left (x\right )\,\left (3\,a\,d+2\,b\,c\right )}{a^3\,c^4} \]
[In]
[Out]