Integrand size = 17, antiderivative size = 68 \[ \int \frac {x^{10}}{b x^2+c x^4} \, dx=-\frac {b^3 x}{c^4}+\frac {b^2 x^3}{3 c^3}-\frac {b x^5}{5 c^2}+\frac {x^7}{7 c}+\frac {b^{7/2} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{9/2}} \]
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Time = 0.03 (sec) , antiderivative size = 68, 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, 308, 211} \[ \int \frac {x^{10}}{b x^2+c x^4} \, dx=\frac {b^{7/2} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{9/2}}-\frac {b^3 x}{c^4}+\frac {b^2 x^3}{3 c^3}-\frac {b x^5}{5 c^2}+\frac {x^7}{7 c} \]
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Rule 211
Rule 308
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^8}{b+c x^2} \, dx \\ & = \int \left (-\frac {b^3}{c^4}+\frac {b^2 x^2}{c^3}-\frac {b x^4}{c^2}+\frac {x^6}{c}+\frac {b^4}{c^4 \left (b+c x^2\right )}\right ) \, dx \\ & = -\frac {b^3 x}{c^4}+\frac {b^2 x^3}{3 c^3}-\frac {b x^5}{5 c^2}+\frac {x^7}{7 c}+\frac {b^4 \int \frac {1}{b+c x^2} \, dx}{c^4} \\ & = -\frac {b^3 x}{c^4}+\frac {b^2 x^3}{3 c^3}-\frac {b x^5}{5 c^2}+\frac {x^7}{7 c}+\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{9/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {x^{10}}{b x^2+c x^4} \, dx=-\frac {b^3 x}{c^4}+\frac {b^2 x^3}{3 c^3}-\frac {b x^5}{5 c^2}+\frac {x^7}{7 c}+\frac {b^{7/2} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{9/2}} \]
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Time = 0.13 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {-\frac {1}{7} c^{3} x^{7}+\frac {1}{5} b \,c^{2} x^{5}-\frac {1}{3} b^{2} c \,x^{3}+b^{3} x}{c^{4}}+\frac {b^{4} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{c^{4} \sqrt {b c}}\) | \(60\) |
risch | \(\frac {x^{7}}{7 c}-\frac {b \,x^{5}}{5 c^{2}}+\frac {b^{2} x^{3}}{3 c^{3}}-\frac {b^{3} x}{c^{4}}+\frac {\sqrt {-b c}\, b^{3} \ln \left (-\sqrt {-b c}\, x +b \right )}{2 c^{5}}-\frac {\sqrt {-b c}\, b^{3} \ln \left (\sqrt {-b c}\, x +b \right )}{2 c^{5}}\) | \(90\) |
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Time = 0.26 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.18 \[ \int \frac {x^{10}}{b x^2+c x^4} \, dx=\left [\frac {30 \, c^{3} x^{7} - 42 \, b c^{2} x^{5} + 70 \, b^{2} c x^{3} + 105 \, b^{3} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} + 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right ) - 210 \, b^{3} x}{210 \, c^{4}}, \frac {15 \, c^{3} x^{7} - 21 \, b c^{2} x^{5} + 35 \, b^{2} c x^{3} + 105 \, b^{3} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) - 105 \, b^{3} x}{105 \, c^{4}}\right ] \]
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Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.57 \[ \int \frac {x^{10}}{b x^2+c x^4} \, dx=- \frac {b^{3} x}{c^{4}} + \frac {b^{2} x^{3}}{3 c^{3}} - \frac {b x^{5}}{5 c^{2}} - \frac {\sqrt {- \frac {b^{7}}{c^{9}}} \log {\left (x - \frac {c^{4} \sqrt {- \frac {b^{7}}{c^{9}}}}{b^{3}} \right )}}{2} + \frac {\sqrt {- \frac {b^{7}}{c^{9}}} \log {\left (x + \frac {c^{4} \sqrt {- \frac {b^{7}}{c^{9}}}}{b^{3}} \right )}}{2} + \frac {x^{7}}{7 c} \]
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.88 \[ \int \frac {x^{10}}{b x^2+c x^4} \, dx=\frac {b^{4} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{4}} + \frac {15 \, c^{3} x^{7} - 21 \, b c^{2} x^{5} + 35 \, b^{2} c x^{3} - 105 \, b^{3} x}{105 \, c^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.96 \[ \int \frac {x^{10}}{b x^2+c x^4} \, dx=\frac {b^{4} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{4}} + \frac {15 \, c^{6} x^{7} - 21 \, b c^{5} x^{5} + 35 \, b^{2} c^{4} x^{3} - 105 \, b^{3} c^{3} x}{105 \, c^{7}} \]
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Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.79 \[ \int \frac {x^{10}}{b x^2+c x^4} \, dx=\frac {x^7}{7\,c}-\frac {b\,x^5}{5\,c^2}-\frac {b^3\,x}{c^4}+\frac {b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{c^{9/2}}+\frac {b^2\,x^3}{3\,c^3} \]
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