Optimal. Leaf size=21 \[ \frac {2 x^{3/2}}{3 a (a+b x)^{3/2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37}
\begin {gather*} \frac {2 x^{3/2}}{3 a (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{(a+b x)^{5/2}} \, dx &=\frac {2 x^{3/2}}{3 a (a+b x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 21, normalized size = 1.00 \begin {gather*} \frac {2 x^{3/2}}{3 a (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs.
\(2(15)=30\).
time = 0.12, size = 54, normalized size = 2.57
method | result | size |
gosper | \(\frac {2 x^{\frac {3}{2}}}{3 a \left (b x +a \right )^{\frac {3}{2}}}\) | \(16\) |
default | \(-\frac {\sqrt {x}}{b \left (b x +a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {2 \sqrt {x}}{3 a \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 \sqrt {x}}{3 a^{2} \sqrt {b x +a}}\right )}{2 b}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 15, normalized size = 0.71 \begin {gather*} \frac {2 \, x^{\frac {3}{2}}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a} \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. 33 vs.
\(2 (15) = 30\).
time = 0.81, size = 33, normalized size = 1.57 \begin {gather*} \frac {2 \, \sqrt {b x + a} x^{\frac {3}{2}}}{3 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs.
\(2 (17) = 34\).
time = 0.70, size = 42, normalized size = 2.00 \begin {gather*} \frac {2 x^{\frac {3}{2}}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {3}{2}} b x \sqrt {1 + \frac {b x}{a}}} \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. 86 vs.
\(2 (15) = 30\).
time = 1.75, size = 86, normalized size = 4.10 \begin {gather*} \frac {4 \, {\left (3 \, {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt {b} + a^{2} b^{\frac {5}{2}}\right )} {\left | b \right |}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 36, normalized size = 1.71 \begin {gather*} \frac {2\,x^{3/2}\,\sqrt {a+b\,x}}{3\,\left (a^3+2\,a^2\,b\,x+a\,b^2\,x^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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