Optimal. Leaf size=57 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3}{2 a^2 x}+\frac{1}{2 a x \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0168707, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {290, 325, 205} \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3}{2 a^2 x}+\frac{1}{2 a x \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b x^2\right )^2} \, dx &=\frac{1}{2 a x \left (a+b x^2\right )}+\frac{3 \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{2 a}\\ &=-\frac{3}{2 a^2 x}+\frac{1}{2 a x \left (a+b x^2\right )}-\frac{(3 b) \int \frac{1}{a+b x^2} \, dx}{2 a^2}\\ &=-\frac{3}{2 a^2 x}+\frac{1}{2 a x \left (a+b x^2\right )}-\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0348756, size = 54, normalized size = 0.95 \[ -\frac{b x}{2 a^2 \left (a+b x^2\right )}-\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{1}{a^2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 46, normalized size = 0.8 \begin{align*} -{\frac{1}{{a}^{2}x}}-{\frac{bx}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,b}{2\,{a}^{2}}\arctan \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 = 1.27124, size = 288, normalized size = 5.05 \begin{align*} \left [-\frac{6 \, b x^{2} - 3 \,{\left (b x^{3} + a x\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 4 \, a}{4 \,{\left (a^{2} b x^{3} + a^{3} x\right )}}, -\frac{3 \, b x^{2} + 3 \,{\left (b x^{3} + a x\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 2 \, a}{2 \,{\left (a^{2} b x^{3} + a^{3} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.444057, size = 90, normalized size = 1.58 \begin{align*} \frac{3 \sqrt{- \frac{b}{a^{5}}} \log{\left (- \frac{a^{3} \sqrt{- \frac{b}{a^{5}}}}{b} + x \right )}}{4} - \frac{3 \sqrt{- \frac{b}{a^{5}}} \log{\left (\frac{a^{3} \sqrt{- \frac{b}{a^{5}}}}{b} + x \right )}}{4} - \frac{2 a + 3 b x^{2}}{2 a^{3} x + 2 a^{2} b x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.56312, size = 63, normalized size = 1.11 \begin{align*} -\frac{3 \, b \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2}} - \frac{3 \, b x^{2} + 2 \, a}{2 \,{\left (b x^{3} + a x\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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