Optimal. Leaf size=18 \[ \frac {\left (a+c x^4\right )^{5/2}}{10 c} \]
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Rubi [A] time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ \frac {\left (a+c x^4\right )^{5/2}}{10 c} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 32, normalized size = 1.78 \[ \frac {{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {c x^{4} + a}}{10 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 14, normalized size = 0.78 \[ \frac {{\left (c x^{4} + a\right )}^{\frac {5}{2}}}{10 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.83 \[ \frac {\left (c \,x^{4}+a \right )^{\frac {5}{2}}}{10 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 14, normalized size = 0.78 \[ \frac {{\left (c x^{4} + a\right )}^{\frac {5}{2}}}{10 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 14, normalized size = 0.78 \[ \frac {{\left (c\,x^4+a\right )}^{5/2}}{10\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.21, size = 60, normalized size = 3.33 \[ \begin {cases} \frac {a^{2} \sqrt {a + c x^{4}}}{10 c} + \frac {a x^{4} \sqrt {a + c x^{4}}}{5} + \frac {c x^{8} \sqrt {a + c x^{4}}}{10} & \text {for}\: c \neq 0 \\\frac {a^{\frac {3}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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