Optimal. Leaf size=43 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b}{a^2 x}-\frac{1}{3 a x^3} \]
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Rubi [A] time = 0.0172627, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {325, 205} \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b}{a^2 x}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx &=-\frac{1}{3 a x^3}-\frac{b \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{a}\\ &=-\frac{1}{3 a x^3}+\frac{b}{a^2 x}+\frac{b^2 \int \frac{1}{a+b x^2} \, dx}{a^2}\\ &=-\frac{1}{3 a x^3}+\frac{b}{a^2 x}+\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0194677, size = 43, normalized size = 1. \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b}{a^2 x}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 39, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,a{x}^{3}}}+{\frac{b}{{a}^{2}x}}+{\frac{{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 = 1.30642, size = 234, normalized size = 5.44 \begin{align*} \left [\frac{3 \, b x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 6 \, b x^{2} - 2 \, a}{6 \, a^{2} x^{3}}, \frac{3 \, b x^{3} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 3 \, b x^{2} - a}{3 \, a^{2} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.3655, size = 87, normalized size = 2.02 \begin{align*} - \frac{\sqrt{- \frac{b^{3}}{a^{5}}} \log{\left (- \frac{a^{3} \sqrt{- \frac{b^{3}}{a^{5}}}}{b^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{3}}{a^{5}}} \log{\left (\frac{a^{3} \sqrt{- \frac{b^{3}}{a^{5}}}}{b^{2}} + x \right )}}{2} + \frac{- a + 3 b x^{2}}{3 a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.95686, size = 54, normalized size = 1.26 \begin{align*} \frac{b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{3 \, b x^{2} - a}{3 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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