Optimal. Leaf size=55 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{5/2}}-\frac{x^3}{2 b \left (a+b x^2\right )}+\frac{3 x}{2 b^2} \]
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Rubi [A] time = 0.016912, antiderivative size = 55, 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 = {288, 321, 205} \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{5/2}}-\frac{x^3}{2 b \left (a+b x^2\right )}+\frac{3 x}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 288
Rule 321
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+b x^2\right )^2} \, dx &=-\frac{x^3}{2 b \left (a+b x^2\right )}+\frac{3 \int \frac{x^2}{a+b x^2} \, dx}{2 b}\\ &=\frac{3 x}{2 b^2}-\frac{x^3}{2 b \left (a+b x^2\right )}-\frac{(3 a) \int \frac{1}{a+b x^2} \, dx}{2 b^2}\\ &=\frac{3 x}{2 b^2}-\frac{x^3}{2 b \left (a+b x^2\right )}-\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0314233, size = 51, normalized size = 0.93 \[ \frac{a x}{2 b^2 \left (a+b x^2\right )}-\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{5/2}}+\frac{x}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 43, normalized size = 0.8 \begin{align*}{\frac{x}{{b}^{2}}}+{\frac{ax}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,a}{2\,{b}^{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 = 1.33109, size = 285, normalized size = 5.18 \begin{align*} \left [\frac{4 \, b x^{3} + 3 \,{\left (b x^{2} + a\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 6 \, a x}{4 \,{\left (b^{3} x^{2} + a b^{2}\right )}}, \frac{2 \, b x^{3} - 3 \,{\left (b x^{2} + a\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 3 \, a x}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.387008, size = 83, normalized size = 1.51 \begin{align*} \frac{a x}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{3 \sqrt{- \frac{a}{b^{5}}} \log{\left (- b^{2} \sqrt{- \frac{a}{b^{5}}} + x \right )}}{4} - \frac{3 \sqrt{- \frac{a}{b^{5}}} \log{\left (b^{2} \sqrt{- \frac{a}{b^{5}}} + x \right )}}{4} + \frac{x}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.84691, size = 57, normalized size = 1.04 \begin{align*} -\frac{3 \, a \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{2}} + \frac{a x}{2 \,{\left (b x^{2} + a\right )} b^{2}} + \frac{x}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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