Optimal. Leaf size=58 \[ -\frac {2 (b x+2)^{3/2}}{\sqrt {x}}+3 b \sqrt {x} \sqrt {b x+2}+6 \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {47, 50, 54, 215} \begin {gather*} -\frac {2 (b x+2)^{3/2}}{\sqrt {x}}+3 b \sqrt {x} \sqrt {b x+2}+6 \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 54
Rule 215
Rubi steps
\begin {align*} \int \frac {(2+b x)^{3/2}}{x^{3/2}} \, dx &=-\frac {2 (2+b x)^{3/2}}{\sqrt {x}}+(3 b) \int \frac {\sqrt {2+b x}}{\sqrt {x}} \, dx\\ &=3 b \sqrt {x} \sqrt {2+b x}-\frac {2 (2+b x)^{3/2}}{\sqrt {x}}+(3 b) \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx\\ &=3 b \sqrt {x} \sqrt {2+b x}-\frac {2 (2+b x)^{3/2}}{\sqrt {x}}+(6 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=3 b \sqrt {x} \sqrt {2+b x}-\frac {2 (2+b x)^{3/2}}{\sqrt {x}}+6 \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.00, size = 28, normalized size = 0.48 \begin {gather*} -\frac {4 \sqrt {2} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {b x}{2}\right )}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 51, normalized size = 0.88 \begin {gather*} \frac {(b x-4) \sqrt {b x+2}}{\sqrt {x}}-6 \sqrt {b} \log \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.23, size = 99, normalized size = 1.71 \begin {gather*} \left [\frac {3 \, \sqrt {b} x \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right ) + \sqrt {b x + 2} {\left (b x - 4\right )} \sqrt {x}}{x}, -\frac {6 \, \sqrt {-b} x \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right ) - \sqrt {b x + 2} {\left (b x - 4\right )} \sqrt {x}}{x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 72, normalized size = 1.24 \begin {gather*} \frac {3 \sqrt {\left (b x +2\right ) x}\, \sqrt {b}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{\sqrt {b x +2}\, \sqrt {x}}+\frac {b^{2} x^{2}-2 b x -8}{\sqrt {b x +2}\, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.91, size = 81, normalized size = 1.40 \begin {gather*} -3 \, \sqrt {b} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right ) - \frac {4 \, \sqrt {b x + 2}}{\sqrt {x}} - \frac {2 \, \sqrt {b x + 2} b}{{\left (b - \frac {b x + 2}{x}\right )} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (b\,x+2\right )}^{3/2}}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.44, size = 73, normalized size = 1.26 \begin {gather*} 6 \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} + \frac {b^{2} x^{\frac {3}{2}}}{\sqrt {b x + 2}} - \frac {2 b \sqrt {x}}{\sqrt {b x + 2}} - \frac {8}{\sqrt {x} \sqrt {b x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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