Optimal. Leaf size=58 \[ -\frac{a^2}{b^3 x}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}+\frac{a}{3 b^2 x^3}-\frac{1}{5 b x^5} \]
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Rubi [A] time = 0.0261248, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 325, 205} \[ -\frac{a^2}{b^3 x}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}+\frac{a}{3 b^2 x^3}-\frac{1}{5 b x^5} \]
Antiderivative was successfully verified.
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Rule 263
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right ) x^8} \, dx &=\int \frac{1}{x^6 \left (b+a x^2\right )} \, dx\\ &=-\frac{1}{5 b x^5}-\frac{a \int \frac{1}{x^4 \left (b+a x^2\right )} \, dx}{b}\\ &=-\frac{1}{5 b x^5}+\frac{a}{3 b^2 x^3}+\frac{a^2 \int \frac{1}{x^2 \left (b+a x^2\right )} \, dx}{b^2}\\ &=-\frac{1}{5 b x^5}+\frac{a}{3 b^2 x^3}-\frac{a^2}{b^3 x}-\frac{a^3 \int \frac{1}{b+a x^2} \, dx}{b^3}\\ &=-\frac{1}{5 b x^5}+\frac{a}{3 b^2 x^3}-\frac{a^2}{b^3 x}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0234124, size = 58, normalized size = 1. \[ -\frac{a^2}{b^3 x}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}+\frac{a}{3 b^2 x^3}-\frac{1}{5 b x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 52, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{{b}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{5\,b{x}^{5}}}-{\frac{{a}^{2}}{{b}^{3}x}}+{\frac{a}{3\,{b}^{2}{x}^{3}}} \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 = 1.7193, size = 296, normalized size = 5.1 \begin{align*} \left [\frac{15 \, a^{2} x^{5} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) - 30 \, a^{2} x^{4} + 10 \, a b x^{2} - 6 \, b^{2}}{30 \, b^{3} x^{5}}, -\frac{15 \, a^{2} x^{5} \sqrt{\frac{a}{b}} \arctan \left (x \sqrt{\frac{a}{b}}\right ) + 15 \, a^{2} x^{4} - 5 \, a b x^{2} + 3 \, b^{2}}{15 \, b^{3} x^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.587015, size = 100, normalized size = 1.72 \begin{align*} \frac{\sqrt{- \frac{a^{5}}{b^{7}}} \log{\left (x - \frac{b^{4} \sqrt{- \frac{a^{5}}{b^{7}}}}{a^{3}} \right )}}{2} - \frac{\sqrt{- \frac{a^{5}}{b^{7}}} \log{\left (x + \frac{b^{4} \sqrt{- \frac{a^{5}}{b^{7}}}}{a^{3}} \right )}}{2} - \frac{15 a^{2} x^{4} - 5 a b x^{2} + 3 b^{2}}{15 b^{3} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16135, size = 70, normalized size = 1.21 \begin{align*} -\frac{a^{3} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{15 \, a^{2} x^{4} - 5 \, a b x^{2} + 3 \, b^{2}}{15 \, b^{3} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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