Optimal. Leaf size=32 \[ \frac {1}{\sqrt {x} \sqrt {b x+2}}-\frac {\sqrt {b x+2}}{\sqrt {x}} \]
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Rubi [A] time = 0.00, antiderivative size = 32, 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} \[ \frac {1}{\sqrt {x} \sqrt {b x+2}}-\frac {\sqrt {b x+2}}{\sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} (2+b x)^{3/2}} \, dx &=\frac {1}{\sqrt {x} \sqrt {2+b x}}+\int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx\\ &=\frac {1}{\sqrt {x} \sqrt {2+b x}}-\frac {\sqrt {2+b x}}{\sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.66 \[ \frac {-b x-1}{\sqrt {x} \sqrt {b x+2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 28, normalized size = 0.88 \[ -\frac {\sqrt {b x + 2} {\left (b x + 1\right )} \sqrt {x}}{b x^{2} + 2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.12, size = 74, normalized size = 2.31 \[ -\frac {\sqrt {b x + 2} b^{2}}{2 \, \sqrt {{\left (b x + 2\right )} b - 2 \, b} {\left | b \right |}} - \frac {2 \, b^{\frac {5}{2}}}{{\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.56 \[ -\frac {b x +1}{\sqrt {b x +2}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 26, normalized size = 0.81 \[ -\frac {b \sqrt {x}}{2 \, \sqrt {b x + 2}} - \frac {\sqrt {b x + 2}}{2 \, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 17, normalized size = 0.53 \[ -\frac {b\,x+1}{\sqrt {x}\,\sqrt {b\,x+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.54, size = 34, normalized size = 1.06 \[ - \frac {\sqrt {b}}{\sqrt {1 + \frac {2}{b x}}} - \frac {1}{\sqrt {b} x \sqrt {1 + \frac {2}{b x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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