Optimal. Leaf size=58 \[ 3 b \sqrt {x} \sqrt {2+b x}-\frac {2 (2+b x)^{3/2}}{\sqrt {x}}+6 \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]
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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 = {49, 52, 56, 221}
\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 49
Rule 52
Rule 56
Rule 221
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) \text {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 [A]
time = 0.07, size = 51, normalized size = 0.88 \begin {gather*} \frac {(-4+b x) \sqrt {2+b x}}{\sqrt {x}}-6 \sqrt {b} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 3.18, size = 61, normalized size = 1.05 \begin {gather*} \frac {6 \sqrt {b} \sqrt {x} \text {ArcSinh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ] \left (2+b x\right )+b x \left (-2+b x\right ) \sqrt {2+b x}-8 \sqrt {2+b x}}{\sqrt {x} \left (2+b x\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 55, normalized size = 0.95
method | result | size |
meijerg | \(\frac {3 \sqrt {b}\, \left (-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {b x}{4}+1\right ) \sqrt {\frac {b x}{2}+1}}{3 \sqrt {x}\, \sqrt {b}}+4 \sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )\right )}{2 \sqrt {\pi }}\) | \(55\) |
risch | \(\frac {x^{2} b^{2}-2 b x -8}{\sqrt {x}\, \sqrt {b x +2}}+\frac {3 \sqrt {b}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right ) \sqrt {x \left (b x +2\right )}}{\sqrt {x}\, \sqrt {b x +2}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, 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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Fricas [A]
time = 0.31, 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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Sympy [A]
time = 1.39, 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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Giac [A]
time = 1.12, size = 118, normalized size = 2.03 \begin {gather*} \frac {b b^{2} \left (\frac {2 \left (\frac {1}{2} \sqrt {b x+2} \sqrt {b x+2}-3\right ) \sqrt {b x+2} \sqrt {b \left (b x+2\right )-2 b}}{b \left (b x+2\right )-2 b}-\frac {6 \ln \left |\sqrt {b \left (b x+2\right )-2 b}-\sqrt {b} \sqrt {b x+2}\right |}{\sqrt {b}}\right )}{\left |b\right | b} \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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