Optimal. Leaf size=37 \[ -\frac{x^3}{4 \left (x^2+9\right )^2}-\frac{3 x}{8 \left (x^2+9\right )}+\frac{1}{8} \tan ^{-1}\left (\frac{x}{3}\right ) \]
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Rubi [A] time = 0.0082532, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {288, 203} \[ -\frac{x^3}{4 \left (x^2+9\right )^2}-\frac{3 x}{8 \left (x^2+9\right )}+\frac{1}{8} \tan ^{-1}\left (\frac{x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 288
Rule 203
Rubi steps
\begin{align*} \int \frac{x^4}{\left (9+x^2\right )^3} \, dx &=-\frac{x^3}{4 \left (9+x^2\right )^2}+\frac{3}{4} \int \frac{x^2}{\left (9+x^2\right )^2} \, dx\\ &=-\frac{x^3}{4 \left (9+x^2\right )^2}-\frac{3 x}{8 \left (9+x^2\right )}+\frac{3}{8} \int \frac{1}{9+x^2} \, dx\\ &=-\frac{x^3}{4 \left (9+x^2\right )^2}-\frac{3 x}{8 \left (9+x^2\right )}+\frac{1}{8} \tan ^{-1}\left (\frac{x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0127404, size = 28, normalized size = 0.76 \[ \frac{1}{8} \left (\tan ^{-1}\left (\frac{x}{3}\right )-\frac{x \left (5 x^2+27\right )}{\left (x^2+9\right )^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 25, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ({x}^{2}+9 \right ) ^{2}} \left ( -{\frac{5\,{x}^{3}}{8}}-{\frac{27\,x}{8}} \right ) }+{\frac{1}{8}\arctan \left ({\frac{x}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41383, size = 41, normalized size = 1.11 \begin{align*} -\frac{5 \, x^{3} + 27 \, x}{8 \,{\left (x^{4} + 18 \, x^{2} + 81\right )}} + \frac{1}{8} \, \arctan \left (\frac{1}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92872, size = 104, normalized size = 2.81 \begin{align*} -\frac{5 \, x^{3} -{\left (x^{4} + 18 \, x^{2} + 81\right )} \arctan \left (\frac{1}{3} \, x\right ) + 27 \, x}{8 \,{\left (x^{4} + 18 \, x^{2} + 81\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.116243, size = 26, normalized size = 0.7 \begin{align*} - \frac{5 x^{3} + 27 x}{8 x^{4} + 144 x^{2} + 648} + \frac{\operatorname{atan}{\left (\frac{x}{3} \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0556, size = 34, normalized size = 0.92 \begin{align*} -\frac{5 \, x^{3} + 27 \, x}{8 \,{\left (x^{2} + 9\right )}^{2}} + \frac{1}{8} \, \arctan \left (\frac{1}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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