Optimal. Leaf size=63 \[ -\frac {2 x^{3/2}}{b \sqrt {2+b x}}+\frac {3 \sqrt {x} \sqrt {2+b x}}{b^2}-\frac {6 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 63, 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 = {49, 52, 56, 221}
\begin {gather*} -\frac {6 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}+\frac {3 \sqrt {x} \sqrt {b x+2}}{b^2}-\frac {2 x^{3/2}}{b \sqrt {b x+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 56
Rule 221
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{(2+b x)^{3/2}} \, dx &=-\frac {2 x^{3/2}}{b \sqrt {2+b x}}+\frac {3 \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx}{b}\\ &=-\frac {2 x^{3/2}}{b \sqrt {2+b x}}+\frac {3 \sqrt {x} \sqrt {2+b x}}{b^2}-\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{b^2}\\ &=-\frac {2 x^{3/2}}{b \sqrt {2+b x}}+\frac {3 \sqrt {x} \sqrt {2+b x}}{b^2}-\frac {6 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {2 x^{3/2}}{b \sqrt {2+b x}}+\frac {3 \sqrt {x} \sqrt {2+b x}}{b^2}-\frac {6 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 54, normalized size = 0.86 \begin {gather*} \frac {\sqrt {x} (6+b x)}{b^2 \sqrt {2+b x}}+\frac {6 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 3.43, size = 63, normalized size = 1.00 \begin {gather*} \frac {-6 b^3 \text {ArcSinh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ] \left (2+b x\right )+6 b^{\frac {7}{2}} \sqrt {x} \sqrt {2+b x}+b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {2+b x}}{b^{\frac {11}{2}} \left (2+b x\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 55, normalized size = 0.87
method | result | size |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (\frac {5 b x}{2}+15\right )}{5 \sqrt {\frac {b x}{2}+1}}-6 \sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {5}{2}} \sqrt {\pi }}\) | \(55\) |
risch | \(\frac {\sqrt {x}\, \sqrt {b x +2}}{b^{2}}+\frac {\left (-\frac {3 \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{b^{\frac {5}{2}}}+\frac {4 \sqrt {\left (x +\frac {2}{b}\right )^{2} b -2 x -\frac {4}{b}}}{b^{3} \left (x +\frac {2}{b}\right )}\right ) \sqrt {x \left (b x +2\right )}}{\sqrt {x}\, \sqrt {b x +2}}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 90, normalized size = 1.43 \begin {gather*} \frac {2 \, {\left (2 \, b - \frac {3 \, {\left (b x + 2\right )}}{x}\right )}}{\frac {\sqrt {b x + 2} b^{3}}{\sqrt {x}} - \frac {{\left (b x + 2\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}}} + \frac {3 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 134, normalized size = 2.13 \begin {gather*} \left [\frac {3 \, {\left (b x + 2\right )} \sqrt {b} \log \left (b x - \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right ) + {\left (b^{2} x + 6 \, b\right )} \sqrt {b x + 2} \sqrt {x}}{b^{4} x + 2 \, b^{3}}, \frac {6 \, {\left (b x + 2\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (b^{2} x + 6 \, b\right )} \sqrt {b x + 2} \sqrt {x}}{b^{4} x + 2 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.82, size = 58, normalized size = 0.92 \begin {gather*} \frac {x^{\frac {3}{2}}}{b \sqrt {b x + 2}} + \frac {6 \sqrt {x}}{b^{2} \sqrt {b x + 2}} - \frac {6 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 92, normalized size = 1.46 \begin {gather*} 2 \left (\frac {2 \left (\frac {\frac {1}{4} b^{2} \sqrt {x} \sqrt {x}}{b^{3}}+\frac {\frac {1}{4}\cdot 6 b}{b^{3}}\right ) \sqrt {x} \sqrt {b x+2}}{b x+2}+\frac {3 \ln \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )}{b^{2} \sqrt {b}}\right ) \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 {x^{3/2}}{{\left (b\,x+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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