Integrand size = 17, antiderivative size = 65 \[ \int \frac {x^8}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {x}{4 c \left (b+c x^2\right )^2}+\frac {x}{8 b c \left (b+c x^2\right )}+\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{3/2} c^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1598, 294, 205, 211} \[ \int \frac {x^8}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{3/2} c^{3/2}}+\frac {x}{8 b c \left (b+c x^2\right )}-\frac {x}{4 c \left (b+c x^2\right )^2} \]
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Rule 205
Rule 211
Rule 294
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2}{\left (b+c x^2\right )^3} \, dx \\ & = -\frac {x}{4 c \left (b+c x^2\right )^2}+\frac {\int \frac {1}{\left (b+c x^2\right )^2} \, dx}{4 c} \\ & = -\frac {x}{4 c \left (b+c x^2\right )^2}+\frac {x}{8 b c \left (b+c x^2\right )}+\frac {\int \frac {1}{b+c x^2} \, dx}{8 b c} \\ & = -\frac {x}{4 c \left (b+c x^2\right )^2}+\frac {x}{8 b c \left (b+c x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{3/2} c^{3/2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {x^8}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {\sqrt {b} \sqrt {c} x \left (-b+c x^2\right )}{\left (b+c x^2\right )^2}+\arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{3/2} c^{3/2}} \]
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Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {\frac {x^{3}}{8 b}-\frac {x}{8 c}}{\left (c \,x^{2}+b \right )^{2}}+\frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 c b \sqrt {b c}}\) | \(49\) |
risch | \(\frac {\frac {x^{3}}{8 b}-\frac {x}{8 c}}{\left (c \,x^{2}+b \right )^{2}}-\frac {\ln \left (c x +\sqrt {-b c}\right )}{16 \sqrt {-b c}\, c b}+\frac {\ln \left (-c x +\sqrt {-b c}\right )}{16 \sqrt {-b c}\, c b}\) | \(78\) |
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none
Time = 0.24 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.92 \[ \int \frac {x^8}{\left (b x^2+c x^4\right )^3} \, dx=\left [\frac {2 \, b c^{2} x^{3} - 2 \, b^{2} c x - {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {-b c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right )}{16 \, {\left (b^{2} c^{4} x^{4} + 2 \, b^{3} c^{3} x^{2} + b^{4} c^{2}\right )}}, \frac {b c^{2} x^{3} - b^{2} c x + {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right )}{8 \, {\left (b^{2} c^{4} x^{4} + 2 \, b^{3} c^{3} x^{2} + b^{4} c^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (51) = 102\).
Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.69 \[ \int \frac {x^8}{\left (b x^2+c x^4\right )^3} \, dx=- \frac {\sqrt {- \frac {1}{b^{3} c^{3}}} \log {\left (- b^{2} c \sqrt {- \frac {1}{b^{3} c^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{b^{3} c^{3}}} \log {\left (b^{2} c \sqrt {- \frac {1}{b^{3} c^{3}}} + x \right )}}{16} + \frac {- b x + c x^{3}}{8 b^{3} c + 16 b^{2} c^{2} x^{2} + 8 b c^{3} x^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95 \[ \int \frac {x^8}{\left (b x^2+c x^4\right )^3} \, dx=\frac {c x^{3} - b x}{8 \, {\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )}} + \frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b c} \]
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Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77 \[ \int \frac {x^8}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b c} + \frac {c x^{3} - b x}{8 \, {\left (c x^{2} + b\right )}^{2} b c} \]
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Time = 12.87 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85 \[ \int \frac {x^8}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{8\,b^{3/2}\,c^{3/2}}-\frac {\frac {x}{8\,c}-\frac {x^3}{8\,b}}{b^2+2\,b\,c\,x^2+c^2\,x^4} \]
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