Optimal. Leaf size=21 \[ \frac {2}{3} b x^{3/2}+\frac {2}{5} a x^{5/2} \]
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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} a x^{5/2}+\frac {2}{3} b x^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right ) x^{3/2} \, dx &=\int \left (b \sqrt {x}+a x^{3/2}\right ) \, dx\\ &=\frac {2}{3} b x^{3/2}+\frac {2}{5} a x^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 17, normalized size = 0.81 \begin {gather*} \frac {2}{15} x^{3/2} (5 b+3 a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 14, normalized size = 0.67
method | result | size |
gosper | \(\frac {2 \left (3 a x +5 b \right ) x^{\frac {3}{2}}}{15}\) | \(14\) |
derivativedivides | \(\frac {2 b \,x^{\frac {3}{2}}}{3}+\frac {2 a \,x^{\frac {5}{2}}}{5}\) | \(14\) |
default | \(\frac {2 b \,x^{\frac {3}{2}}}{3}+\frac {2 a \,x^{\frac {5}{2}}}{5}\) | \(14\) |
trager | \(\frac {2 \left (3 a x +5 b \right ) x^{\frac {3}{2}}}{15}\) | \(14\) |
risch | \(\frac {2 \left (3 a x +5 b \right ) x^{\frac {3}{2}}}{15}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 15, normalized size = 0.71 \begin {gather*} \frac {2}{15} \, {\left (3 \, a + \frac {5 \, b}{x}\right )} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 16, normalized size = 0.76 \begin {gather*} \frac {2}{15} \, {\left (3 \, a x^{2} + 5 \, b x\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 a x^{\frac {5}{2}}}{5} + \frac {2 b x^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{5} \, a x^{\frac {5}{2}} + \frac {2}{3} \, b 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 = 13, normalized size = 0.62 \begin {gather*} \frac {2\,x^{3/2}\,\left (5\,b+3\,a\,x\right )}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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