Optimal. Leaf size=38 \[ -\frac {\sqrt {2+b x}}{3 x^{3/2}}+\frac {b \sqrt {2+b x}}{3 \sqrt {x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37}
\begin {gather*} \frac {b \sqrt {b x+2}}{3 \sqrt {x}}-\frac {\sqrt {b x+2}}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx &=-\frac {\sqrt {2+b x}}{3 x^{3/2}}-\frac {1}{3} b \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx\\ &=-\frac {\sqrt {2+b x}}{3 x^{3/2}}+\frac {b \sqrt {2+b x}}{3 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 23, normalized size = 0.61 \begin {gather*} \frac {(-1+b x) \sqrt {2+b x}}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 27, normalized size = 0.71
method | result | size |
gosper | \(\frac {\sqrt {b x +2}\, \left (b x -1\right )}{3 x^{\frac {3}{2}}}\) | \(18\) |
meijerg | \(-\frac {\sqrt {2}\, \left (-b x +1\right ) \sqrt {\frac {b x}{2}+1}}{3 x^{\frac {3}{2}}}\) | \(23\) |
risch | \(\frac {x^{2} b^{2}+b x -2}{3 x^{\frac {3}{2}} \sqrt {b x +2}}\) | \(25\) |
default | \(-\frac {\sqrt {b x +2}}{3 x^{\frac {3}{2}}}+\frac {b \sqrt {b x +2}}{3 \sqrt {x}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 26, normalized size = 0.68 \begin {gather*} \frac {\sqrt {b x + 2} b}{2 \, \sqrt {x}} - \frac {{\left (b x + 2\right )}^{\frac {3}{2}}}{6 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 17, normalized size = 0.45 \begin {gather*} \frac {\sqrt {b x + 2} {\left (b x - 1\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.01, size = 34, normalized size = 0.89 \begin {gather*} \frac {b^{\frac {3}{2}} \sqrt {1 + \frac {2}{b x}}}{3} - \frac {\sqrt {b} \sqrt {1 + \frac {2}{b x}}}{3 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.29, size = 42, normalized size = 1.11 \begin {gather*} \frac {{\left ({\left (b x + 2\right )} b^{3} - 3 \, b^{3}\right )} \sqrt {b x + 2} b}{3 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {3}{2}} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.32, size = 17, normalized size = 0.45 \begin {gather*} \frac {\sqrt {b\,x+2}\,\left (\frac {b\,x}{3}-\frac {1}{3}\right )}{x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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