Optimal. Leaf size=16 \[ \frac {2 (a+b x)^{5/2}}{5 b} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32}
\begin {gather*} \frac {2 (a+b x)^{5/2}}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rubi steps
\begin {align*} \int (a+b x)^{3/2} \, dx &=\frac {2 (a+b x)^{5/2}}{5 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 (a+b x)^{5/2}}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.57, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (a+b x\right )^{\frac {5}{2}}}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 13, normalized size = 0.81
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5 b}\) | \(13\) |
derivativedivides | \(\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5 b}\) | \(13\) |
default | \(\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5 b}\) | \(13\) |
trager | \(\frac {2 \left (x^{2} b^{2}+2 a b x +a^{2}\right ) \sqrt {b x +a}}{5 b}\) | \(29\) |
risch | \(\frac {2 \left (x^{2} b^{2}+2 a b x +a^{2}\right ) \sqrt {b x +a}}{5 b}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}}}{5 \, b} \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. 28 vs.
\(2 (12) = 24\).
time = 0.30, size = 28, normalized size = 1.75 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {b x + a}}{5 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (a + b x\right )^{\frac {5}{2}}}{5 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (12) = 24\).
time = 0.00, size = 113, normalized size = 7.06 \begin {gather*} \frac {\frac {2 b^{2} \left (\frac {1}{5} \sqrt {a+b x} \left (a+b x\right )^{2}-\frac {2}{3} \sqrt {a+b x} \left (a+b x\right ) a+\sqrt {a+b x} a^{2}\right )}{b^{2}}+4 a \left (\frac {1}{3} \sqrt {a+b x} \left (a+b x\right )-a \sqrt {a+b x}\right )+2 a^{2} \sqrt {a+b x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 12, normalized size = 0.75 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{5/2}}{5\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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