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 = {267}
\begin {gather*} \frac {\left (a+c x^4\right )^{5/2}}{10 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
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 = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {\left (a+c x^4\right )^{5/2}}{10 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 15, normalized size = 0.83
method | result | size |
gosper | \(\frac {\left (x^{4} c +a \right )^{\frac {5}{2}}}{10 c}\) | \(15\) |
derivativedivides | \(\frac {\left (x^{4} c +a \right )^{\frac {5}{2}}}{10 c}\) | \(15\) |
default | \(\frac {\left (x^{4} c +a \right )^{\frac {5}{2}}}{10 c}\) | \(15\) |
trager | \(\frac {\left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right ) \sqrt {x^{4} c +a}}{10 c}\) | \(33\) |
risch | \(\frac {\left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right ) \sqrt {x^{4} c +a}}{10 c}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (c x^{4} + a\right )}^{\frac {5}{2}}}{10 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs.
\(2 (14) = 28\).
time = 0.36, size = 32, normalized size = 1.78 \begin {gather*} \frac {{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {c x^{4} + a}}{10 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (12) = 24\).
time = 0.17, size = 60, normalized size = 3.33 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (c x^{4} + a\right )}^{\frac {5}{2}}}{10 \, c} \end {gather*}
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 \begin {gather*} \frac {{\left (c\,x^4+a\right )}^{5/2}}{10\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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