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 = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, 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}
\begin {gather*} \frac {2}{5} a x^{5/2}+\frac {2}{13} c x^{13/2} \end {gather*}
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*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 0.90 \begin {gather*} \frac {2}{65} x^{5/2} \left (13 a+5 c x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 14, normalized size = 0.67
method | result | size |
derivativedivides | \(\frac {2 a \,x^{\frac {5}{2}}}{5}+\frac {2 c \,x^{\frac {13}{2}}}{13}\) | \(14\) |
default | \(\frac {2 a \,x^{\frac {5}{2}}}{5}+\frac {2 c \,x^{\frac {13}{2}}}{13}\) | \(14\) |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (5 x^{4} c +13 a \right )}{65}\) | \(16\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (5 x^{4} c +13 a \right )}{65}\) | \(16\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (5 x^{4} c +13 a \right )}{65}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{13} \, c x^{\frac {13}{2}} + \frac {2}{5} \, a x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 18, normalized size = 0.86 \begin {gather*} \frac {2}{65} \, {\left (5 \, c x^{6} + 13 \, a x^{2}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 a x^{\frac {5}{2}}}{5} + \frac {2 c x^{\frac {13}{2}}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.04, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{13} \, c x^{\frac {13}{2}} + \frac {2}{5} \, a x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 15, normalized size = 0.71 \begin {gather*} \frac {2\,x^{5/2}\,\left (5\,c\,x^4+13\,a\right )}{65} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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