Optimal. Leaf size=46 \[ \frac{x}{2 b \left (a-b x^2\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}} \]
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Rubi [A] time = 0.0135534, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {288, 208} \[ \frac{x}{2 b \left (a-b x^2\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 208
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a-b x^2\right )^2} \, dx &=\frac{x}{2 b \left (a-b x^2\right )}-\frac{\int \frac{1}{a-b x^2} \, dx}{2 b}\\ &=\frac{x}{2 b \left (a-b x^2\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.029005, size = 47, normalized size = 1.02 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left (b x^2-a\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 38, normalized size = 0.8 \begin{align*} -{\frac{x}{2\,b \left ( b{x}^{2}-a \right ) }}-{\frac{1}{2\,b}{\it Artanh} \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 = 1.30328, size = 263, normalized size = 5.72 \begin{align*} \left [-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{a b} x + a}{b x^{2} - a}\right )}{4 \,{\left (a b^{3} x^{2} - a^{2} b^{2}\right )}}, -\frac{a b x -{\left (b x^{2} - a\right )} \sqrt{-a b} \arctan \left (\frac{\sqrt{-a b} x}{a}\right )}{2 \,{\left (a b^{3} x^{2} - a^{2} b^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.352352, size = 71, normalized size = 1.54 \begin{align*} - \frac{x}{- 2 a b + 2 b^{2} x^{2}} + \frac{\sqrt{\frac{1}{a b^{3}}} \log{\left (- a b \sqrt{\frac{1}{a b^{3}}} + x \right )}}{4} - \frac{\sqrt{\frac{1}{a b^{3}}} \log{\left (a b \sqrt{\frac{1}{a b^{3}}} + x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.56562, size = 53, normalized size = 1.15 \begin{align*} \frac{\arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{2 \, \sqrt{-a b} b} - \frac{x}{2 \,{\left (b x^{2} - a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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