Optimal. Leaf size=18 \[ \frac {\left (a+c x^4\right )^{3/2}}{6 c} \]
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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} \[ \frac {\left (a+c x^4\right )^{3/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin {align*} \int x^3 \sqrt {a+c x^4} \, dx &=\frac {\left (a+c x^4\right )^{3/2}}{6 c}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \[ \frac {\left (a+c x^4\right )^{3/2}}{6 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 14, normalized size = 0.78 \[ \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 14, normalized size = 0.78 \[ \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, c} \]
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 \[ \frac {\left (c \,x^{4}+a \right )^{\frac {3}{2}}}{6 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 14, normalized size = 0.78 \[ \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 14, normalized size = 0.78 \[ \frac {{\left (c\,x^4+a\right )}^{3/2}}{6\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 39, normalized size = 2.17 \[ \begin {cases} \frac {a \sqrt {a + c x^{4}}}{6 c} + \frac {x^{4} \sqrt {a + c x^{4}}}{6} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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