Optimal. Leaf size=21 \[ \frac{2}{5} a x^{5/2}+\frac{2}{13} c x^{13/2} \]
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Rubi [A] time = 0.0042359, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{2}{5} a x^{5/2}+\frac{2}{13} c x^{13/2} \]
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin{align*} \int x^{3/2} \left (a+c x^4\right ) \, dx &=\int \left (a x^{3/2}+c x^{11/2}\right ) \, dx\\ &=\frac{2}{5} a x^{5/2}+\frac{2}{13} c x^{13/2}\\ \end{align*}
Mathematica [A] time = 0.0045733, size = 21, normalized size = 1. \[ \frac{2}{5} a x^{5/2}+\frac{2}{13} c x^{13/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 16, normalized size = 0.8 \begin{align*}{\frac{10\,c{x}^{4}+26\,a}{65}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99622, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{13} \, c x^{\frac{13}{2}} + \frac{2}{5} \, a x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42448, size = 47, normalized size = 2.24 \begin{align*} \frac{2}{65} \,{\left (5 \, c x^{6} + 13 \, a x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.53658, size = 19, normalized size = 0.9 \begin{align*} \frac{2 a x^{\frac{5}{2}}}{5} + \frac{2 c x^{\frac{13}{2}}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14751, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{13} \, c x^{\frac{13}{2}} + \frac{2}{5} \, a x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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