Optimal. Leaf size=21 \[ -\frac {2 a}{3 x^{3/2}}+\frac {2}{5} c x^{5/2} \]
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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} c x^{5/2}-\frac {2 a}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int \frac {a+c x^4}{x^{5/2}} \, dx &=\int \left (\frac {a}{x^{5/2}}+c x^{3/2}\right ) \, dx\\ &=-\frac {2 a}{3 x^{3/2}}+\frac {2}{5} c x^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2 \left (5 a-3 c x^4\right )}{15 x^{3/2}} \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 | \(-\frac {2 a}{3 x^{\frac {3}{2}}}+\frac {2 c \,x^{\frac {5}{2}}}{5}\) | \(14\) |
default | \(-\frac {2 a}{3 x^{\frac {3}{2}}}+\frac {2 c \,x^{\frac {5}{2}}}{5}\) | \(14\) |
gosper | \(-\frac {2 \left (-3 x^{4} c +5 a \right )}{15 x^{\frac {3}{2}}}\) | \(16\) |
trager | \(-\frac {2 \left (-3 x^{4} c +5 a \right )}{15 x^{\frac {3}{2}}}\) | \(16\) |
risch | \(-\frac {2 \left (-3 x^{4} c +5 a \right )}{15 x^{\frac {3}{2}}}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{5} \, c x^{\frac {5}{2}} - \frac {2 \, a}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 15, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (3 \, c x^{4} - 5 \, a\right )}}{15 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 19, normalized size = 0.90 \begin {gather*} - \frac {2 a}{3 x^{\frac {3}{2}}} + \frac {2 c x^{\frac {5}{2}}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.84, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{5} \, c x^{\frac {5}{2}} - \frac {2 \, a}{3 \, x^{\frac {3}{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 {10\,a-6\,c\,x^4}{15\,x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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