Integrand size = 17, antiderivative size = 66 \[ \int \frac {1}{x \left (b x^2+c x^4\right )^2} \, dx=-\frac {1}{4 b^2 x^4}+\frac {c}{b^3 x^2}+\frac {c^2}{2 b^3 \left (b+c x^2\right )}+\frac {3 c^2 \log (x)}{b^4}-\frac {3 c^2 \log \left (b+c x^2\right )}{2 b^4} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1598, 272, 46} \[ \int \frac {1}{x \left (b x^2+c x^4\right )^2} \, dx=-\frac {3 c^2 \log \left (b+c x^2\right )}{2 b^4}+\frac {3 c^2 \log (x)}{b^4}+\frac {c^2}{2 b^3 \left (b+c x^2\right )}+\frac {c}{b^3 x^2}-\frac {1}{4 b^2 x^4} \]
[In]
[Out]
Rule 46
Rule 272
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^5 \left (b+c x^2\right )^2} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 (b+c x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b^2 x^3}-\frac {2 c}{b^3 x^2}+\frac {3 c^2}{b^4 x}-\frac {c^3}{b^3 (b+c x)^2}-\frac {3 c^3}{b^4 (b+c x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {1}{4 b^2 x^4}+\frac {c}{b^3 x^2}+\frac {c^2}{2 b^3 \left (b+c x^2\right )}+\frac {3 c^2 \log (x)}{b^4}-\frac {3 c^2 \log \left (b+c x^2\right )}{2 b^4} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x \left (b x^2+c x^4\right )^2} \, dx=\frac {b \left (-\frac {b}{x^4}+\frac {4 c}{x^2}+\frac {2 c^2}{b+c x^2}\right )+12 c^2 \log (x)-6 c^2 \log \left (b+c x^2\right )}{4 b^4} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {1}{4 b^{2} x^{4}}+\frac {c}{b^{3} x^{2}}+\frac {3 c^{2} \ln \left (x \right )}{b^{4}}-\frac {c^{3} \left (\frac {3 \ln \left (c \,x^{2}+b \right )}{c}-\frac {b}{c \left (c \,x^{2}+b \right )}\right )}{2 b^{4}}\) | \(65\) |
norman | \(\frac {-\frac {1}{4 b}+\frac {3 c \,x^{2}}{4 b^{2}}-\frac {3 c^{3} x^{6}}{2 b^{4}}}{x^{4} \left (c \,x^{2}+b \right )}+\frac {3 c^{2} \ln \left (x \right )}{b^{4}}-\frac {3 c^{2} \ln \left (c \,x^{2}+b \right )}{2 b^{4}}\) | \(67\) |
risch | \(\frac {\frac {3 c^{2} x^{4}}{2 b^{3}}+\frac {3 c \,x^{2}}{4 b^{2}}-\frac {1}{4 b}}{x^{4} \left (c \,x^{2}+b \right )}+\frac {3 c^{2} \ln \left (x \right )}{b^{4}}-\frac {3 c^{2} \ln \left (c \,x^{2}+b \right )}{2 b^{4}}\) | \(67\) |
parallelrisch | \(\frac {12 c^{3} \ln \left (x \right ) x^{6}-6 c^{3} \ln \left (c \,x^{2}+b \right ) x^{6}-6 c^{3} x^{6}+12 b \,c^{2} \ln \left (x \right ) x^{4}-6 \ln \left (c \,x^{2}+b \right ) x^{4} b \,c^{2}+3 b^{2} c \,x^{2}-b^{3}}{4 b^{4} x^{4} \left (c \,x^{2}+b \right )}\) | \(95\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.36 \[ \int \frac {1}{x \left (b x^2+c x^4\right )^2} \, dx=\frac {6 \, b c^{2} x^{4} + 3 \, b^{2} c x^{2} - b^{3} - 6 \, {\left (c^{3} x^{6} + b c^{2} x^{4}\right )} \log \left (c x^{2} + b\right ) + 12 \, {\left (c^{3} x^{6} + b c^{2} x^{4}\right )} \log \left (x\right )}{4 \, {\left (b^{4} c x^{6} + b^{5} x^{4}\right )}} \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x \left (b x^2+c x^4\right )^2} \, dx=\frac {- b^{2} + 3 b c x^{2} + 6 c^{2} x^{4}}{4 b^{4} x^{4} + 4 b^{3} c x^{6}} + \frac {3 c^{2} \log {\left (x \right )}}{b^{4}} - \frac {3 c^{2} \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{4}} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x \left (b x^2+c x^4\right )^2} \, dx=\frac {6 \, c^{2} x^{4} + 3 \, b c x^{2} - b^{2}}{4 \, {\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} - \frac {3 \, c^{2} \log \left (c x^{2} + b\right )}{2 \, b^{4}} + \frac {3 \, c^{2} \log \left (x^{2}\right )}{2 \, b^{4}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x \left (b x^2+c x^4\right )^2} \, dx=\frac {3 \, c^{2} \log \left (x^{2}\right )}{2 \, b^{4}} - \frac {3 \, c^{2} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4}} + \frac {3 \, c^{3} x^{2} + 4 \, b c^{2}}{2 \, {\left (c x^{2} + b\right )} b^{4}} - \frac {9 \, c^{2} x^{4} - 4 \, b c x^{2} + b^{2}}{4 \, b^{4} x^{4}} \]
[In]
[Out]
Time = 12.92 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x \left (b x^2+c x^4\right )^2} \, dx=\frac {\frac {3\,c\,x^2}{4\,b^2}-\frac {1}{4\,b}+\frac {3\,c^2\,x^4}{2\,b^3}}{c\,x^6+b\,x^4}-\frac {3\,c^2\,\ln \left (c\,x^2+b\right )}{2\,b^4}+\frac {3\,c^2\,\ln \left (x\right )}{b^4} \]
[In]
[Out]