Optimal. Leaf size=18 \[ \frac{\sqrt{a+c x^4}}{2 c} \]
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Rubi [A] time = 0.0044898, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {4, 261} \[ \frac{\sqrt{a+c x^4}}{2 c} \]
Antiderivative was successfully verified.
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Rule 4
Rule 261
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx &=\int \frac{x^3}{\sqrt{a+c x^4}} \, dx\\ &=\frac{\sqrt{a+c x^4}}{2 c}\\ \end{align*}
Mathematica [A] time = 0.003454, size = 18, normalized size = 1. \[ \frac{\sqrt{a+c x^4}}{2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{2\,c}\sqrt{c{x}^{4}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967832, size = 19, normalized size = 1.06 \begin{align*} \frac{\sqrt{c x^{4} + a}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44087, size = 31, normalized size = 1.72 \begin{align*} \frac{\sqrt{c x^{4} + a}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.457962, size = 22, normalized size = 1.22 \begin{align*} \begin{cases} \frac{\sqrt{a + c x^{4}}}{2 c} & \text{for}\: c \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11893, size = 19, normalized size = 1.06 \begin{align*} \frac{\sqrt{c x^{4} + a}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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