Optimal. Leaf size=12 \[ c x+\frac{d x^2}{2} \]
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Rubi [A] time = 0.0485936, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 54, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.019, Rules used = {1586} \[ c x+\frac{d x^2}{2} \]
Antiderivative was successfully verified.
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Rule 1586
Rubi steps
\begin{align*} \int \frac{a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^2} \, dx &=\int \frac{a c+a d x+b c x^2+b d x^3}{a+b x^2} \, dx\\ &=\int (c+d x) \, dx\\ &=c x+\frac{d x^2}{2}\\ \end{align*}
Mathematica [A] time = 0.00072, size = 12, normalized size = 1. \[ c x+\frac{d x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 11, normalized size = 0.9 \begin{align*} cx+{\frac{d{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01943, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2} \, d x^{2} + c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24135, size = 23, normalized size = 1.92 \begin{align*} \frac{1}{2} \, d x^{2} + c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.081643, size = 8, normalized size = 0.67 \begin{align*} c x + \frac{d x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24447, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2} \, d x^{2} + c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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