Optimal. Leaf size=18 \[ \frac{\left (a+c x^4\right )^{5/2}}{10 c} \]
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Rubi [A] time = 0.0048617, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{\left (a+c x^4\right )^{5/2}}{10 c} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int x^3 \left (a+c x^4\right )^{3/2} \, dx &=\frac{\left (a+c x^4\right )^{5/2}}{10 c}\\ \end{align*}
Mathematica [A] time = 0.0045495, size = 18, normalized size = 1. \[ \frac{\left (a+c x^4\right )^{5/2}}{10 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{10\,c} \left ( c{x}^{4}+a \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969078, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}}}{10 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.44254, size = 70, normalized size = 3.89 \begin{align*} \frac{{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt{c x^{4} + a}}{10 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.28536, size = 60, normalized size = 3.33 \begin{align*} \begin{cases} \frac{a^{2} \sqrt{a + c x^{4}}}{10 c} + \frac{a x^{4} \sqrt{a + c x^{4}}}{5} + \frac{c x^{8} \sqrt{a + c x^{4}}}{10} & \text{for}\: c \neq 0 \\\frac{a^{\frac{3}{2}} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10052, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}}}{10 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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