Integrand size = 22, antiderivative size = 126 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {1}{2 a c^2 x^2}+\frac {d^2}{2 c^2 (b c-a d) \left (c+d x^2\right )}-\frac {(b c+2 a d) \log (x)}{a^2 c^3}+\frac {b^3 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^2}-\frac {d^2 (3 b c-2 a d) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2} \]
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Time = 0.11 (sec) , antiderivative size = 126, 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} \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b^3 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^2}-\frac {\log (x) (2 a d+b c)}{a^2 c^3}-\frac {d^2 (3 b c-2 a d) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2}+\frac {d^2}{2 c^2 \left (c+d x^2\right ) (b c-a d)}-\frac {1}{2 a c^2 x^2} \]
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Rule 90
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (a+b x) (c+d x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a c^2 x^2}+\frac {-b c-2 a d}{a^2 c^3 x}+\frac {b^4}{a^2 (-b c+a d)^2 (a+b x)}-\frac {d^3}{c^2 (b c-a d) (c+d x)^2}-\frac {d^3 (3 b c-2 a d)}{c^3 (b c-a d)^2 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {1}{2 a c^2 x^2}+\frac {d^2}{2 c^2 (b c-a d) \left (c+d x^2\right )}-\frac {(b c+2 a d) \log (x)}{a^2 c^3}+\frac {b^3 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^2}-\frac {d^2 (3 b c-2 a d) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {1}{a c^2 x^2}+\frac {d^2}{c^2 (b c-a d) \left (c+d x^2\right )}-\frac {2 (b c+2 a d) \log (x)}{a^2 c^3}+\frac {b^3 \log \left (a+b x^2\right )}{a^2 (b c-a d)^2}+\frac {d^2 (-3 b c+2 a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^2}\right ) \]
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Time = 2.75 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {1}{2 a \,c^{2} x^{2}}+\frac {\left (-2 a d -b c \right ) \ln \left (x \right )}{a^{2} c^{3}}+\frac {b^{3} \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a d -b c \right )^{2}}+\frac {d^{3} \left (\frac {\left (2 a d -3 b c \right ) \ln \left (d \,x^{2}+c \right )}{d}-\frac {\left (a d -b c \right ) c}{d \left (d \,x^{2}+c \right )}\right )}{2 c^{3} \left (a d -b c \right )^{2}}\) | \(120\) |
norman | \(\frac {-\frac {1}{2 a c}+\frac {\left (-2 a \,d^{3}+b c \,d^{2}\right ) x^{2}}{2 c^{2} d a \left (a d -b c \right )}}{\left (d \,x^{2}+c \right ) x^{2}}+\frac {b^{3} \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (2 a d +b c \right ) \ln \left (x \right )}{a^{2} c^{3}}+\frac {d^{2} \left (2 a d -3 b c \right ) \ln \left (d \,x^{2}+c \right )}{2 c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(166\) |
risch | \(\frac {-\frac {d \left (2 a d -b c \right ) x^{2}}{2 c^{2} a \left (a d -b c \right )}-\frac {1}{2 a c}}{\left (d \,x^{2}+c \right ) x^{2}}-\frac {2 \ln \left (x \right ) d}{a \,c^{3}}-\frac {\ln \left (x \right ) b}{a^{2} c^{2}}+\frac {b^{3} \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {d^{3} \ln \left (-d \,x^{2}-c \right ) a}{c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {3 d^{2} \ln \left (-d \,x^{2}-c \right ) b}{2 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(201\) |
parallelrisch | \(-\frac {4 \ln \left (x \right ) x^{4} a^{3} d^{5}-6 \ln \left (x \right ) x^{4} a^{2} b c \,d^{4}+2 \ln \left (x \right ) x^{4} b^{3} c^{3} d^{2}-\ln \left (b \,x^{2}+a \right ) x^{4} b^{3} c^{3} d^{2}-2 \ln \left (d \,x^{2}+c \right ) x^{4} a^{3} d^{5}+3 \ln \left (d \,x^{2}+c \right ) x^{4} a^{2} b c \,d^{4}+4 \ln \left (x \right ) x^{2} a^{3} c \,d^{4}-6 \ln \left (x \right ) x^{2} a^{2} b \,c^{2} d^{3}+2 \ln \left (x \right ) x^{2} b^{3} c^{4} d -\ln \left (b \,x^{2}+a \right ) x^{2} b^{3} c^{4} d -2 \ln \left (d \,x^{2}+c \right ) x^{2} a^{3} c \,d^{4}+3 \ln \left (d \,x^{2}+c \right ) x^{2} a^{2} b \,c^{2} d^{3}+2 x^{2} a^{3} c \,d^{4}-3 x^{2} a^{2} b \,c^{2} d^{3}+x^{2} a \,b^{2} c^{3} d^{2}+a^{3} c^{2} d^{3}-2 a^{2} b \,c^{3} d^{2}+a \,b^{2} c^{4} d}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (d \,x^{2}+c \right ) x^{2} a^{2} c^{3} d}\) | \(334\) |
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Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (118) = 236\).
Time = 1.47 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.40 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {a b^{2} c^{4} - 2 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2} + {\left (a b^{2} c^{3} d - 3 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x^{2} - {\left (b^{3} c^{3} d x^{4} + b^{3} c^{4} x^{2}\right )} \log \left (b x^{2} + a\right ) + {\left ({\left (3 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4}\right )} x^{4} + {\left (3 \, a^{2} b c^{2} d^{2} - 2 \, a^{3} c d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \, {\left ({\left (b^{3} c^{3} d - 3 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4}\right )} x^{4} + {\left (b^{3} c^{4} - 3 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left ({\left (a^{2} b^{2} c^{5} d - 2 \, a^{3} b c^{4} d^{2} + a^{4} c^{3} d^{3}\right )} x^{4} + {\left (a^{2} b^{2} c^{6} - 2 \, a^{3} b c^{5} d + a^{4} c^{4} d^{2}\right )} x^{2}\right )}} \]
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Timed out. \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
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none
Time = 0.22 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.49 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b^{3} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} - \frac {{\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}} - \frac {b c^{2} - a c d + {\left (b c d - 2 \, a d^{2}\right )} x^{2}}{2 \, {\left ({\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{4} + {\left (a b c^{4} - a^{2} c^{3} d\right )} x^{2}\right )}} - \frac {{\left (b c + 2 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{2} c^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (118) = 236\).
Time = 0.29 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.04 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )}} - \frac {{\left (3 \, b c d^{3} - 2 \, a d^{4}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )}} + \frac {b^{3} c^{2} d x^{4} + b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{2} d x^{2} + 6 \, a^{2} b c d^{2} x^{2} - 4 \, a^{3} d^{3} x^{2} - 2 \, a b^{2} c^{3} + 4 \, a^{2} b c^{2} d - 2 \, a^{3} c d^{2}}{4 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left (d x^{4} + c x^{2}\right )}} - \frac {{\left (b c + 2 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{2} c^{3}} \]
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Time = 5.93 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.36 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b^3\,\ln \left (b\,x^2+a\right )}{2\,\left (a^4\,d^2-2\,a^3\,b\,c\,d+a^2\,b^2\,c^2\right )}-\frac {\frac {1}{2\,a\,c}+\frac {x^2\,\left (2\,a\,d^2-b\,c\,d\right )}{2\,a\,c^2\,\left (a\,d-b\,c\right )}}{d\,x^4+c\,x^2}+\frac {\ln \left (d\,x^2+c\right )\,\left (2\,a\,d^3-3\,b\,c\,d^2\right )}{2\,a^2\,c^3\,d^2-4\,a\,b\,c^4\,d+2\,b^2\,c^5}-\frac {\ln \left (x\right )\,\left (2\,a\,d+b\,c\right )}{a^2\,c^3} \]
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