Optimal. Leaf size=76 \[ \frac{5}{8 b^2 x \left (a x^2+b\right )}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{7/2}}+\frac{1}{4 b x \left (a x^2+b\right )^2}-\frac{15}{8 b^3 x} \]
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Rubi [A] time = 0.0269846, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {263, 290, 325, 205} \[ \frac{5}{8 b^2 x \left (a x^2+b\right )}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{7/2}}+\frac{1}{4 b x \left (a x^2+b\right )^2}-\frac{15}{8 b^3 x} \]
Antiderivative was successfully verified.
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Rule 263
Rule 290
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^8} \, dx &=\int \frac{1}{x^2 \left (b+a x^2\right )^3} \, dx\\ &=\frac{1}{4 b x \left (b+a x^2\right )^2}+\frac{5 \int \frac{1}{x^2 \left (b+a x^2\right )^2} \, dx}{4 b}\\ &=\frac{1}{4 b x \left (b+a x^2\right )^2}+\frac{5}{8 b^2 x \left (b+a x^2\right )}+\frac{15 \int \frac{1}{x^2 \left (b+a x^2\right )} \, dx}{8 b^2}\\ &=-\frac{15}{8 b^3 x}+\frac{1}{4 b x \left (b+a x^2\right )^2}+\frac{5}{8 b^2 x \left (b+a x^2\right )}-\frac{(15 a) \int \frac{1}{b+a x^2} \, dx}{8 b^3}\\ &=-\frac{15}{8 b^3 x}+\frac{1}{4 b x \left (b+a x^2\right )^2}+\frac{5}{8 b^2 x \left (b+a x^2\right )}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0395061, size = 68, normalized size = 0.89 \[ -\frac{15 a^2 x^4+25 a b x^2+8 b^2}{8 b^3 x \left (a x^2+b\right )^2}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 66, normalized size = 0.9 \begin{align*} -{\frac{7\,{x}^{3}{a}^{2}}{8\,{b}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{9\,ax}{8\,{b}^{2} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{15\,a}{8\,{b}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{{b}^{3}x}} \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 = 1.52676, size = 428, normalized size = 5.63 \begin{align*} \left [-\frac{30 \, a^{2} x^{4} + 50 \, a b x^{2} - 15 \,{\left (a^{2} x^{5} + 2 \, a b x^{3} + b^{2} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) + 16 \, b^{2}}{16 \,{\left (a^{2} b^{3} x^{5} + 2 \, a b^{4} x^{3} + b^{5} x\right )}}, -\frac{15 \, a^{2} x^{4} + 25 \, a b x^{2} + 15 \,{\left (a^{2} x^{5} + 2 \, a b x^{3} + b^{2} x\right )} \sqrt{\frac{a}{b}} \arctan \left (x \sqrt{\frac{a}{b}}\right ) + 8 \, b^{2}}{8 \,{\left (a^{2} b^{3} x^{5} + 2 \, a b^{4} x^{3} + b^{5} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.623701, size = 114, normalized size = 1.5 \begin{align*} \frac{15 \sqrt{- \frac{a}{b^{7}}} \log{\left (x - \frac{b^{4} \sqrt{- \frac{a}{b^{7}}}}{a} \right )}}{16} - \frac{15 \sqrt{- \frac{a}{b^{7}}} \log{\left (x + \frac{b^{4} \sqrt{- \frac{a}{b^{7}}}}{a} \right )}}{16} - \frac{15 a^{2} x^{4} + 25 a b x^{2} + 8 b^{2}}{8 a^{2} b^{3} x^{5} + 16 a b^{4} x^{3} + 8 b^{5} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15729, size = 77, normalized size = 1.01 \begin{align*} -\frac{15 \, a \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{3}} - \frac{7 \, a^{2} x^{3} + 9 \, a b x}{8 \,{\left (a x^{2} + b\right )}^{2} b^{3}} - \frac{1}{b^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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