Optimal. Leaf size=69 \[ -\frac{b^2}{3 a^3 x^3}+\frac{b^3}{a^4 x}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}+\frac{b}{5 a^2 x^5}-\frac{1}{7 a x^7} \]
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Rubi [A] time = 0.0346157, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {325, 205} \[ -\frac{b^2}{3 a^3 x^3}+\frac{b^3}{a^4 x}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}+\frac{b}{5 a^2 x^5}-\frac{1}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^8 \left (a+b x^2\right )} \, dx &=-\frac{1}{7 a x^7}-\frac{b \int \frac{1}{x^6 \left (a+b x^2\right )} \, dx}{a}\\ &=-\frac{1}{7 a x^7}+\frac{b}{5 a^2 x^5}+\frac{b^2 \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx}{a^2}\\ &=-\frac{1}{7 a x^7}+\frac{b}{5 a^2 x^5}-\frac{b^2}{3 a^3 x^3}-\frac{b^3 \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{a^3}\\ &=-\frac{1}{7 a x^7}+\frac{b}{5 a^2 x^5}-\frac{b^2}{3 a^3 x^3}+\frac{b^3}{a^4 x}+\frac{b^4 \int \frac{1}{a+b x^2} \, dx}{a^4}\\ &=-\frac{1}{7 a x^7}+\frac{b}{5 a^2 x^5}-\frac{b^2}{3 a^3 x^3}+\frac{b^3}{a^4 x}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0246361, size = 69, normalized size = 1. \[ -\frac{b^2}{3 a^3 x^3}+\frac{b^3}{a^4 x}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}+\frac{b}{5 a^2 x^5}-\frac{1}{7 a x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 61, normalized size = 0.9 \begin{align*} -{\frac{1}{7\,a{x}^{7}}}-{\frac{{b}^{2}}{3\,{a}^{3}{x}^{3}}}+{\frac{b}{5\,{a}^{2}{x}^{5}}}+{\frac{{b}^{3}}{{a}^{4}x}}+{\frac{{b}^{4}}{{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.17482, size = 352, normalized size = 5.1 \begin{align*} \left [\frac{105 \, b^{3} x^{7} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 210 \, b^{3} x^{6} - 70 \, a b^{2} x^{4} + 42 \, a^{2} b x^{2} - 30 \, a^{3}}{210 \, a^{4} x^{7}}, \frac{105 \, b^{3} x^{7} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 105 \, b^{3} x^{6} - 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} - 15 \, a^{3}}{105 \, a^{4} x^{7}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.53454, size = 112, normalized size = 1.62 \begin{align*} - \frac{\sqrt{- \frac{b^{7}}{a^{9}}} \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{7}}{a^{9}}}}{b^{4}} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{7}}{a^{9}}} \log{\left (\frac{a^{5} \sqrt{- \frac{b^{7}}{a^{9}}}}{b^{4}} + x \right )}}{2} + \frac{- 15 a^{3} + 21 a^{2} b x^{2} - 35 a b^{2} x^{4} + 105 b^{3} x^{6}}{105 a^{4} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.99494, size = 84, normalized size = 1.22 \begin{align*} \frac{b^{4} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} + \frac{105 \, b^{3} x^{6} - 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} - 15 \, a^{3}}{105 \, a^{4} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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