Optimal. Leaf size=37 \[ \frac {\sqrt {x}}{3 \sqrt {b x+2}}+\frac {\sqrt {x}}{3 (b x+2)^{3/2}} \]
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Rubi [A] time = 0.00, antiderivative size = 37, 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 = {45, 37} \[ \frac {\sqrt {x}}{3 \sqrt {b x+2}}+\frac {\sqrt {x}}{3 (b x+2)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} (2+b x)^{5/2}} \, dx &=\frac {\sqrt {x}}{3 (2+b x)^{3/2}}+\frac {1}{3} \int \frac {1}{\sqrt {x} (2+b x)^{3/2}} \, dx\\ &=\frac {\sqrt {x}}{3 (2+b x)^{3/2}}+\frac {\sqrt {x}}{3 \sqrt {2+b x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.62 \[ \frac {\sqrt {x} (b x+3)}{3 (b x+2)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 32, normalized size = 0.86 \[ \frac {{\left (b x + 3\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{2} x^{2} + 4 \, b x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.22, size = 79, normalized size = 2.14 \[ \frac {8 \, {\left (3 \, {\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} b^{\frac {5}{2}}}{3 \, {\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.49 \[ \frac {\left (b x +3\right ) \sqrt {x}}{3 \left (b x +2\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 24, normalized size = 0.65 \[ -\frac {{\left (b - \frac {3 \, {\left (b x + 2\right )}}{x}\right )} x^{\frac {3}{2}}}{6 \, {\left (b x + 2\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 42, normalized size = 1.14 \[ \frac {3\,\sqrt {x}\,\sqrt {b\,x+2}+b\,x^{3/2}\,\sqrt {b\,x+2}}{3\,b^2\,x^2+12\,b\,x+12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.84, size = 75, normalized size = 2.03 \[ \frac {b x}{3 b^{\frac {3}{2}} x \sqrt {1 + \frac {2}{b x}} + 6 \sqrt {b} \sqrt {1 + \frac {2}{b x}}} + \frac {3}{3 b^{\frac {3}{2}} x \sqrt {1 + \frac {2}{b x}} + 6 \sqrt {b} \sqrt {1 + \frac {2}{b x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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