Optimal. Leaf size=27 \[ \frac{7}{8} \tan ^{-1}\left (\frac{x}{2}\right )-\frac{5 x+24}{4 \left (x^2+4\right )} \]
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Rubi [A] time = 0.0139091, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1814, 12, 203} \[ \frac{7}{8} \tan ^{-1}\left (\frac{x}{2}\right )-\frac{5 x+24}{4 \left (x^2+4\right )} \]
Antiderivative was successfully verified.
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Rule 1814
Rule 12
Rule 203
Rubi steps
\begin{align*} \int \frac{2+12 x+3 x^2}{\left (4+x^2\right )^2} \, dx &=-\frac{24+5 x}{4 \left (4+x^2\right )}-\frac{1}{8} \int -\frac{14}{4+x^2} \, dx\\ &=-\frac{24+5 x}{4 \left (4+x^2\right )}+\frac{7}{4} \int \frac{1}{4+x^2} \, dx\\ &=-\frac{24+5 x}{4 \left (4+x^2\right )}+\frac{7}{8} \tan ^{-1}\left (\frac{x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.010329, size = 27, normalized size = 1. \[ \frac{-5 x-24}{4 \left (x^2+4\right )}+\frac{7}{8} \tan ^{-1}\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 21, normalized size = 0.8 \begin{align*}{\frac{1}{{x}^{2}+4} \left ( -{\frac{5\,x}{4}}-6 \right ) }+{\frac{7}{8}\arctan \left ({\frac{x}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49413, size = 28, normalized size = 1.04 \begin{align*} -\frac{5 \, x + 24}{4 \,{\left (x^{2} + 4\right )}} + \frac{7}{8} \, \arctan \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.0288, size = 74, normalized size = 2.74 \begin{align*} \frac{7 \,{\left (x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, x\right ) - 10 \, x - 48}{8 \,{\left (x^{2} + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.11676, size = 19, normalized size = 0.7 \begin{align*} - \frac{5 x + 24}{4 x^{2} + 16} + \frac{7 \operatorname{atan}{\left (\frac{x}{2} \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17825, size = 28, normalized size = 1.04 \begin{align*} -\frac{5 \, x + 24}{4 \,{\left (x^{2} + 4\right )}} + \frac{7}{8} \, \arctan \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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