Optimal. Leaf size=45 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {x}{2 a \left (a+b x^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1584, 199, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {x}{2 a \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a x+b x^3\right )^2} \, dx &=\int \frac {1}{\left (a+b x^2\right )^2} \, dx\\ &=\frac {x}{2 a \left (a+b x^2\right )}+\frac {\int \frac {1}{a+b x^2} \, dx}{2 a}\\ &=\frac {x}{2 a \left (a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 45, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {x}{2 a \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 120, normalized size = 2.67 \[ \left [\frac {2 \, a b x - {\left (b x^{2} + a\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{4 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}, \frac {a b x + {\left (b x^{2} + a\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{2 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 35, normalized size = 0.78 \[ \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} + \frac {x}{2 \, {\left (b x^{2} + a\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 36, normalized size = 0.80 \[ \frac {x}{2 \left (b \,x^{2}+a \right ) a}+\frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 35, normalized size = 0.78 \[ \frac {x}{2 \, {\left (a b x^{2} + a^{2}\right )}} + \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.96, size = 33, normalized size = 0.73 \[ \frac {x}{2\,a\,\left (b\,x^2+a\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.24, size = 78, normalized size = 1.73 \[ \frac {x}{2 a^{2} + 2 a b x^{2}} - \frac {\sqrt {- \frac {1}{a^{3} b}} \log {\left (- a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b}} \log {\left (a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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