Optimal. Leaf size=68 \[ \frac{b^2 x^3}{3 c^3}-\frac{b^3 x}{c^4}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{9/2}}-\frac{b x^5}{5 c^2}+\frac{x^7}{7 c} \]
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Rubi [A] time = 0.0382052, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1584, 302, 205} \[ \frac{b^2 x^3}{3 c^3}-\frac{b^3 x}{c^4}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{9/2}}-\frac{b x^5}{5 c^2}+\frac{x^7}{7 c} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{10}}{b x^2+c x^4} \, dx &=\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*}
Mathematica [A] time = 0.0264946, size = 68, normalized size = 1. \[ \frac{b^2 x^3}{3 c^3}-\frac{b^3 x}{c^4}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{9/2}}-\frac{b x^5}{5 c^2}+\frac{x^7}{7 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 60, normalized size = 0.9 \begin{align*}{\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{{b}^{4}}{{c}^{4}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45701, size = 336, normalized size = 4.94 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.38582, size = 107, normalized size = 1.57 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22828, size = 88, normalized size = 1.29 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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