Integrand size = 15, antiderivative size = 86 \[ \int \frac {x}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {1}{4 b^3 x^4}+\frac {3 c}{2 b^4 x^2}+\frac {c^2}{4 b^3 \left (b+c x^2\right )^2}+\frac {3 c^2}{2 b^4 \left (b+c x^2\right )}+\frac {6 c^2 \log (x)}{b^5}-\frac {3 c^2 \log \left (b+c x^2\right )}{b^5} \]
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Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1598, 272, 46} \[ \int \frac {x}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {3 c^2 \log \left (b+c x^2\right )}{b^5}+\frac {6 c^2 \log (x)}{b^5}+\frac {3 c^2}{2 b^4 \left (b+c x^2\right )}+\frac {3 c}{2 b^4 x^2}+\frac {c^2}{4 b^3 \left (b+c x^2\right )^2}-\frac {1}{4 b^3 x^4} \]
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Rule 46
Rule 272
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^5 \left (b+c x^2\right )^3} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 (b+c x)^3} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b^3 x^3}-\frac {3 c}{b^4 x^2}+\frac {6 c^2}{b^5 x}-\frac {c^3}{b^3 (b+c x)^3}-\frac {3 c^3}{b^4 (b+c x)^2}-\frac {6 c^3}{b^5 (b+c x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {1}{4 b^3 x^4}+\frac {3 c}{2 b^4 x^2}+\frac {c^2}{4 b^3 \left (b+c x^2\right )^2}+\frac {3 c^2}{2 b^4 \left (b+c x^2\right )}+\frac {6 c^2 \log (x)}{b^5}-\frac {3 c^2 \log \left (b+c x^2\right )}{b^5} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {x}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {b \left (-b^3+4 b^2 c x^2+18 b c^2 x^4+12 c^3 x^6\right )}{x^4 \left (b+c x^2\right )^2}+24 c^2 \log (x)-12 c^2 \log \left (b+c x^2\right )}{4 b^5} \]
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Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {\frac {3 c^{3} x^{6}}{b^{4}}+\frac {9 c^{2} x^{4}}{2 b^{3}}+\frac {c \,x^{2}}{b^{2}}-\frac {1}{4 b}}{\left (c \,x^{2}+b \right )^{2} x^{4}}+\frac {6 c^{2} \ln \left (x \right )}{b^{5}}-\frac {3 c^{2} \ln \left (c \,x^{2}+b \right )}{b^{5}}\) | \(77\) |
norman | \(\frac {\frac {c \,x^{3}}{b^{2}}-\frac {x}{4 b}-\frac {6 c^{3} x^{7}}{b^{4}}-\frac {9 c^{4} x^{9}}{2 b^{5}}}{x^{5} \left (c \,x^{2}+b \right )^{2}}+\frac {6 c^{2} \ln \left (x \right )}{b^{5}}-\frac {3 c^{2} \ln \left (c \,x^{2}+b \right )}{b^{5}}\) | \(78\) |
default | \(-\frac {1}{4 b^{3} x^{4}}+\frac {3 c}{2 b^{4} x^{2}}+\frac {6 c^{2} \ln \left (x \right )}{b^{5}}-\frac {c^{3} \left (-\frac {b^{2}}{2 c \left (c \,x^{2}+b \right )^{2}}+\frac {6 \ln \left (c \,x^{2}+b \right )}{c}-\frac {3 b}{c \left (c \,x^{2}+b \right )}\right )}{2 b^{5}}\) | \(83\) |
parallelrisch | \(\frac {24 \ln \left (x \right ) x^{8} c^{4}-12 \ln \left (c \,x^{2}+b \right ) x^{8} c^{4}-18 c^{4} x^{8}+48 \ln \left (x \right ) x^{6} b \,c^{3}-24 \ln \left (c \,x^{2}+b \right ) x^{6} b \,c^{3}-24 c^{3} x^{6} b +24 \ln \left (x \right ) x^{4} b^{2} c^{2}-12 \ln \left (c \,x^{2}+b \right ) x^{4} b^{2} c^{2}+4 x^{2} c \,b^{3}-b^{4}}{4 b^{5} \left (c \,x^{2}+b \right )^{2} x^{4}}\) | \(136\) |
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Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.56 \[ \int \frac {x}{\left (b x^2+c x^4\right )^3} \, dx=\frac {12 \, b c^{3} x^{6} + 18 \, b^{2} c^{2} x^{4} + 4 \, b^{3} c x^{2} - b^{4} - 12 \, {\left (c^{4} x^{8} + 2 \, b c^{3} x^{6} + b^{2} c^{2} x^{4}\right )} \log \left (c x^{2} + b\right ) + 24 \, {\left (c^{4} x^{8} + 2 \, b c^{3} x^{6} + b^{2} c^{2} x^{4}\right )} \log \left (x\right )}{4 \, {\left (b^{5} c^{2} x^{8} + 2 \, b^{6} c x^{6} + b^{7} x^{4}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.05 \[ \int \frac {x}{\left (b x^2+c x^4\right )^3} \, dx=\frac {- b^{3} + 4 b^{2} c x^{2} + 18 b c^{2} x^{4} + 12 c^{3} x^{6}}{4 b^{6} x^{4} + 8 b^{5} c x^{6} + 4 b^{4} c^{2} x^{8}} + \frac {6 c^{2} \log {\left (x \right )}}{b^{5}} - \frac {3 c^{2} \log {\left (\frac {b}{c} + x^{2} \right )}}{b^{5}} \]
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Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.07 \[ \int \frac {x}{\left (b x^2+c x^4\right )^3} \, dx=\frac {12 \, c^{3} x^{6} + 18 \, b c^{2} x^{4} + 4 \, b^{2} c x^{2} - b^{3}}{4 \, {\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )}} - \frac {3 \, c^{2} \log \left (c x^{2} + b\right )}{b^{5}} + \frac {3 \, c^{2} \log \left (x^{2}\right )}{b^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92 \[ \int \frac {x}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {3 \, c^{2} \log \left ({\left | c x^{2} + b \right |}\right )}{b^{5}} + \frac {6 \, c^{2} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {12 \, c^{3} x^{6} + 18 \, b c^{2} x^{4} + 4 \, b^{2} c x^{2} - b^{3}}{4 \, {\left (c x^{4} + b x^{2}\right )}^{2} b^{4}} \]
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Time = 12.89 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02 \[ \int \frac {x}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {c\,x^2}{b^2}-\frac {1}{4\,b}+\frac {9\,c^2\,x^4}{2\,b^3}+\frac {3\,c^3\,x^6}{b^4}}{b^2\,x^4+2\,b\,c\,x^6+c^2\,x^8}-\frac {3\,c^2\,\ln \left (c\,x^2+b\right )}{b^5}+\frac {6\,c^2\,\ln \left (x\right )}{b^5} \]
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