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 = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 208
Rule 288
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.83, size = 127, normalized size = 2.76 \[ \left [-\frac {2 \, a b x - {\left (b x^{2} - a\right )} \sqrt {a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {a b} x + a}{b x^{2} - a}\right )}{4 \, {\left (a b^{3} x^{2} - a^{2} b^{2}\right )}}, -\frac {a b x - {\left (b x^{2} - a\right )} \sqrt {-a b} \arctan \left (\frac {\sqrt {-a b} x}{a}\right )}{2 \, {\left (a b^{3} x^{2} - a^{2} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 39, normalized size = 0.85 \[ \frac {\arctan \left (\frac {b x}{\sqrt {-a b}}\right )}{2 \, \sqrt {-a b} b} - \frac {x}{2 \, {\left (b x^{2} - a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.83 \[ -\frac {x}{2 \left (b \,x^{2}-a \right ) b}-\frac {\arctanh \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 52, normalized size = 1.13 \[ -\frac {x}{2 \, {\left (b^{2} x^{2} - a b\right )}} + \frac {\log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{4 \, \sqrt {a b} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.68, size = 34, normalized size = 0.74 \[ \frac {x}{2\,b\,\left (a-b\,x^2\right )}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,b^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 71, normalized size = 1.54 \[ - \frac {x}{- 2 a b + 2 b^{2} x^{2}} + \frac {\sqrt {\frac {1}{a b^{3}}} \log {\left (- a b \sqrt {\frac {1}{a b^{3}}} + x \right )}}{4} - \frac {\sqrt {\frac {1}{a b^{3}}} \log {\left (a b \sqrt {\frac {1}{a b^{3}}} + x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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