Optimal. Leaf size=65 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2}}+\frac{x}{8 a b \left (a x^2+b\right )}-\frac{x}{4 a \left (a x^2+b\right )^2} \]
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Rubi [A] time = 0.0199228, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {263, 288, 199, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2}}+\frac{x}{8 a b \left (a x^2+b\right )}-\frac{x}{4 a \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 288
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^4} \, dx &=\int \frac{x^2}{\left (b+a x^2\right )^3} \, dx\\ &=-\frac{x}{4 a \left (b+a x^2\right )^2}+\frac{\int \frac{1}{\left (b+a x^2\right )^2} \, dx}{4 a}\\ &=-\frac{x}{4 a \left (b+a x^2\right )^2}+\frac{x}{8 a b \left (b+a x^2\right )}+\frac{\int \frac{1}{b+a x^2} \, dx}{8 a b}\\ &=-\frac{x}{4 a \left (b+a x^2\right )^2}+\frac{x}{8 a b \left (b+a x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0275988, size = 58, normalized size = 0.89 \[ \frac{\frac{\sqrt{a} \sqrt{b} x \left (a x^2-b\right )}{\left (a x^2+b\right )^2}+\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 49, normalized size = 0.8 \begin{align*}{\frac{1}{ \left ( a{x}^{2}+b \right ) ^{2}} \left ({\frac{{x}^{3}}{8\,b}}-{\frac{x}{8\,a}} \right ) }+{\frac{1}{8\,ab}\arctan \left ({ax{\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 = 1.44977, size = 394, normalized size = 6.06 \begin{align*} \left [\frac{2 \, a^{2} b x^{3} - 2 \, a b^{2} x -{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{-a b} \log \left (\frac{a x^{2} - 2 \, \sqrt{-a b} x - b}{a x^{2} + b}\right )}{16 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{2} b^{4}\right )}}, \frac{a^{2} b x^{3} - a b^{2} x +{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{b}\right )}{8 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{2} b^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.599687, size = 110, normalized size = 1.69 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \log{\left (- a b^{2} \sqrt{- \frac{1}{a^{3} b^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \log{\left (a b^{2} \sqrt{- \frac{1}{a^{3} b^{3}}} + x \right )}}{16} + \frac{a x^{3} - b x}{8 a^{3} b x^{4} + 16 a^{2} b^{2} x^{2} + 8 a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18017, size = 68, normalized size = 1.05 \begin{align*} \frac{\arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a b} + \frac{a x^{3} - b x}{8 \,{\left (a x^{2} + b\right )}^{2} a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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