Optimal. Leaf size=113 \[ -\frac{33 b^2}{8 a^5 x^3}+\frac{99 b^3}{8 a^6 x}+\frac{99 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{13/2}}+\frac{99 b}{40 a^4 x^5}+\frac{11}{8 a^2 x^7 \left (a+b x^2\right )}-\frac{99}{56 a^3 x^7}+\frac{1}{4 a x^7 \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.0558603, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {290, 325, 205} \[ -\frac{33 b^2}{8 a^5 x^3}+\frac{99 b^3}{8 a^6 x}+\frac{99 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{13/2}}+\frac{99 b}{40 a^4 x^5}+\frac{11}{8 a^2 x^7 \left (a+b x^2\right )}-\frac{99}{56 a^3 x^7}+\frac{1}{4 a x^7 \left (a+b x^2\right )^2} \]
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 )^3} \, dx &=\frac{1}{4 a x^7 \left (a+b x^2\right )^2}+\frac{11 \int \frac{1}{x^8 \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=\frac{1}{4 a x^7 \left (a+b x^2\right )^2}+\frac{11}{8 a^2 x^7 \left (a+b x^2\right )}+\frac{99 \int \frac{1}{x^8 \left (a+b x^2\right )} \, dx}{8 a^2}\\ &=-\frac{99}{56 a^3 x^7}+\frac{1}{4 a x^7 \left (a+b x^2\right )^2}+\frac{11}{8 a^2 x^7 \left (a+b x^2\right )}-\frac{(99 b) \int \frac{1}{x^6 \left (a+b x^2\right )} \, dx}{8 a^3}\\ &=-\frac{99}{56 a^3 x^7}+\frac{99 b}{40 a^4 x^5}+\frac{1}{4 a x^7 \left (a+b x^2\right )^2}+\frac{11}{8 a^2 x^7 \left (a+b x^2\right )}+\frac{\left (99 b^2\right ) \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx}{8 a^4}\\ &=-\frac{99}{56 a^3 x^7}+\frac{99 b}{40 a^4 x^5}-\frac{33 b^2}{8 a^5 x^3}+\frac{1}{4 a x^7 \left (a+b x^2\right )^2}+\frac{11}{8 a^2 x^7 \left (a+b x^2\right )}-\frac{\left (99 b^3\right ) \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{8 a^5}\\ &=-\frac{99}{56 a^3 x^7}+\frac{99 b}{40 a^4 x^5}-\frac{33 b^2}{8 a^5 x^3}+\frac{99 b^3}{8 a^6 x}+\frac{1}{4 a x^7 \left (a+b x^2\right )^2}+\frac{11}{8 a^2 x^7 \left (a+b x^2\right )}+\frac{\left (99 b^4\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^6}\\ &=-\frac{99}{56 a^3 x^7}+\frac{99 b}{40 a^4 x^5}-\frac{33 b^2}{8 a^5 x^3}+\frac{99 b^3}{8 a^6 x}+\frac{1}{4 a x^7 \left (a+b x^2\right )^2}+\frac{11}{8 a^2 x^7 \left (a+b x^2\right )}+\frac{99 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.0560113, size = 101, normalized size = 0.89 \[ \frac{1848 a^2 b^3 x^6-264 a^3 b^2 x^4+88 a^4 b x^2-40 a^5+5775 a b^4 x^8+3465 b^5 x^{10}}{280 a^6 x^7 \left (a+b x^2\right )^2}+\frac{99 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{13/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 101, normalized size = 0.9 \begin{align*} -{\frac{1}{7\,{a}^{3}{x}^{7}}}+10\,{\frac{{b}^{3}}{{a}^{6}x}}-2\,{\frac{{b}^{2}}{{a}^{5}{x}^{3}}}+{\frac{3\,b}{5\,{a}^{4}{x}^{5}}}+{\frac{19\,{b}^{5}{x}^{3}}{8\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{21\,{b}^{4}x}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{99\,{b}^{4}}{8\,{a}^{6}}\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 = 1.278, size = 630, normalized size = 5.58 \begin{align*} \left [\frac{6930 \, b^{5} x^{10} + 11550 \, a b^{4} x^{8} + 3696 \, a^{2} b^{3} x^{6} - 528 \, a^{3} b^{2} x^{4} + 176 \, a^{4} b x^{2} - 80 \, a^{5} + 3465 \,{\left (b^{5} x^{11} + 2 \, a b^{4} x^{9} + a^{2} 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 )}{560 \,{\left (a^{6} b^{2} x^{11} + 2 \, a^{7} b x^{9} + a^{8} x^{7}\right )}}, \frac{3465 \, b^{5} x^{10} + 5775 \, a b^{4} x^{8} + 1848 \, a^{2} b^{3} x^{6} - 264 \, a^{3} b^{2} x^{4} + 88 \, a^{4} b x^{2} - 40 \, a^{5} + 3465 \,{\left (b^{5} x^{11} + 2 \, a b^{4} x^{9} + a^{2} b^{3} x^{7}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{280 \,{\left (a^{6} b^{2} x^{11} + 2 \, a^{7} b x^{9} + a^{8} x^{7}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.24989, size = 162, normalized size = 1.43 \begin{align*} - \frac{99 \sqrt{- \frac{b^{7}}{a^{13}}} \log{\left (- \frac{a^{7} \sqrt{- \frac{b^{7}}{a^{13}}}}{b^{4}} + x \right )}}{16} + \frac{99 \sqrt{- \frac{b^{7}}{a^{13}}} \log{\left (\frac{a^{7} \sqrt{- \frac{b^{7}}{a^{13}}}}{b^{4}} + x \right )}}{16} + \frac{- 40 a^{5} + 88 a^{4} b x^{2} - 264 a^{3} b^{2} x^{4} + 1848 a^{2} b^{3} x^{6} + 5775 a b^{4} x^{8} + 3465 b^{5} x^{10}}{280 a^{8} x^{7} + 560 a^{7} b x^{9} + 280 a^{6} b^{2} x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.98601, size = 126, normalized size = 1.12 \begin{align*} \frac{99 \, b^{4} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{6}} + \frac{19 \, b^{5} x^{3} + 21 \, a b^{4} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{6}} + \frac{350 \, b^{3} x^{6} - 70 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} - 5 \, a^{3}}{35 \, a^{6} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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