Optimal. Leaf size=16 \[ \frac {2 (c+d x)^{5/2}}{5 d} \]
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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 (c+d x)^{5/2}}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rubi steps
\begin {align*} \int (c+d x)^{3/2} \, dx &=\frac {2 (c+d x)^{5/2}}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 (c+d x)^{5/2}}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 13, normalized size = 0.81
method | result | size |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {5}{2}}}{5 d}\) | \(13\) |
derivativedivides | \(\frac {2 \left (d x +c \right )^{\frac {5}{2}}}{5 d}\) | \(13\) |
default | \(\frac {2 \left (d x +c \right )^{\frac {5}{2}}}{5 d}\) | \(13\) |
trager | \(\frac {2 \left (d^{2} x^{2}+2 c d x +c^{2}\right ) \sqrt {d x +c}}{5 d}\) | \(29\) |
risch | \(\frac {2 \left (d^{2} x^{2}+2 c d x +c^{2}\right ) \sqrt {d x +c}}{5 d}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (d x + c\right )}^{\frac {5}{2}}}{5 \, d} \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 = 0.40, size = 28, normalized size = 1.75 \begin {gather*} \frac {2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \sqrt {d x + c}}{5 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (c + d x\right )^{\frac {5}{2}}}{5 d} \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 = 1.10, size = 58, normalized size = 3.62 \begin {gather*} \frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 30 \, \sqrt {d x + c} c^{2} + 10 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} c\right )}}{15 \, d} \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 (c+d\,x\right )}^{5/2}}{5\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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