Optimal. Leaf size=94 \[ -\frac{3 b^2}{2 a^4 x^3}+\frac{9 b^3}{2 a^5 x}+\frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{11/2}}+\frac{9 b}{10 a^3 x^5}-\frac{9}{14 a^2 x^7}+\frac{1}{2 a x^7 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0431358, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {290, 325, 205} \[ -\frac{3 b^2}{2 a^4 x^3}+\frac{9 b^3}{2 a^5 x}+\frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{11/2}}+\frac{9 b}{10 a^3 x^5}-\frac{9}{14 a^2 x^7}+\frac{1}{2 a x^7 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^8 \left (a+b x^2\right )^2} \, dx &=\frac{1}{2 a x^7 \left (a+b x^2\right )}+\frac{9 \int \frac{1}{x^8 \left (a+b x^2\right )} \, dx}{2 a}\\ &=-\frac{9}{14 a^2 x^7}+\frac{1}{2 a x^7 \left (a+b x^2\right )}-\frac{(9 b) \int \frac{1}{x^6 \left (a+b x^2\right )} \, dx}{2 a^2}\\ &=-\frac{9}{14 a^2 x^7}+\frac{9 b}{10 a^3 x^5}+\frac{1}{2 a x^7 \left (a+b x^2\right )}+\frac{\left (9 b^2\right ) \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx}{2 a^3}\\ &=-\frac{9}{14 a^2 x^7}+\frac{9 b}{10 a^3 x^5}-\frac{3 b^2}{2 a^4 x^3}+\frac{1}{2 a x^7 \left (a+b x^2\right )}-\frac{\left (9 b^3\right ) \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{2 a^4}\\ &=-\frac{9}{14 a^2 x^7}+\frac{9 b}{10 a^3 x^5}-\frac{3 b^2}{2 a^4 x^3}+\frac{9 b^3}{2 a^5 x}+\frac{1}{2 a x^7 \left (a+b x^2\right )}+\frac{\left (9 b^4\right ) \int \frac{1}{a+b x^2} \, dx}{2 a^5}\\ &=-\frac{9}{14 a^2 x^7}+\frac{9 b}{10 a^3 x^5}-\frac{3 b^2}{2 a^4 x^3}+\frac{9 b^3}{2 a^5 x}+\frac{1}{2 a x^7 \left (a+b x^2\right )}+\frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.0532909, size = 91, normalized size = 0.97 \[ \frac{b^4 x}{2 a^5 \left (a+b x^2\right )}-\frac{b^2}{a^4 x^3}+\frac{4 b^3}{a^5 x}+\frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{11/2}}+\frac{2 b}{5 a^3 x^5}-\frac{1}{7 a^2 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 81, normalized size = 0.9 \begin{align*} -{\frac{1}{7\,{a}^{2}{x}^{7}}}+4\,{\frac{{b}^{3}}{{a}^{5}x}}-{\frac{{b}^{2}}{{a}^{4}{x}^{3}}}+{\frac{2\,b}{5\,{a}^{3}{x}^{5}}}+{\frac{{b}^{4}x}{2\,{a}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{9\,{b}^{4}}{2\,{a}^{5}}\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 = 1.1774, size = 470, normalized size = 5. \begin{align*} \left [\frac{630 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} - 84 \, a^{2} b^{2} x^{4} + 36 \, a^{3} b x^{2} - 20 \, a^{4} + 315 \,{\left (b^{4} x^{9} + a b^{3} x^{7}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{140 \,{\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}, \frac{315 \, b^{4} x^{8} + 210 \, a b^{3} x^{6} - 42 \, a^{2} b^{2} x^{4} + 18 \, a^{3} b x^{2} - 10 \, a^{4} + 315 \,{\left (b^{4} x^{9} + a b^{3} x^{7}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{70 \,{\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.00635, size = 138, normalized size = 1.47 \begin{align*} - \frac{9 \sqrt{- \frac{b^{7}}{a^{11}}} \log{\left (- \frac{a^{6} \sqrt{- \frac{b^{7}}{a^{11}}}}{b^{4}} + x \right )}}{4} + \frac{9 \sqrt{- \frac{b^{7}}{a^{11}}} \log{\left (\frac{a^{6} \sqrt{- \frac{b^{7}}{a^{11}}}}{b^{4}} + x \right )}}{4} + \frac{- 10 a^{4} + 18 a^{3} b x^{2} - 42 a^{2} b^{2} x^{4} + 210 a b^{3} x^{6} + 315 b^{4} x^{8}}{70 a^{6} x^{7} + 70 a^{5} b x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.32981, size = 109, normalized size = 1.16 \begin{align*} \frac{9 \, b^{4} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{5}} + \frac{b^{4} x}{2 \,{\left (b x^{2} + a\right )} a^{5}} + \frac{140 \, b^{3} x^{6} - 35 \, a b^{2} x^{4} + 14 \, a^{2} b x^{2} - 5 \, a^{3}}{35 \, a^{5} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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