Optimal. Leaf size=61 \[ \frac {3}{2} \sqrt {x} \sqrt {2+b x}+\frac {1}{2} \sqrt {x} (2+b x)^{3/2}+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 56, 221}
\begin {gather*} \frac {1}{2} \sqrt {x} (b x+2)^{3/2}+\frac {3}{2} \sqrt {x} \sqrt {b x+2}+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 221
Rubi steps
\begin {align*} \int \frac {(2+b x)^{3/2}}{\sqrt {x}} \, dx &=\frac {1}{2} \sqrt {x} (2+b x)^{3/2}+\frac {3}{2} \int \frac {\sqrt {2+b x}}{\sqrt {x}} \, dx\\ &=\frac {3}{2} \sqrt {x} \sqrt {2+b x}+\frac {1}{2} \sqrt {x} (2+b x)^{3/2}+\frac {3}{2} \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx\\ &=\frac {3}{2} \sqrt {x} \sqrt {2+b x}+\frac {1}{2} \sqrt {x} (2+b x)^{3/2}+3 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {3}{2} \sqrt {x} \sqrt {2+b x}+\frac {1}{2} \sqrt {x} (2+b x)^{3/2}+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 54, normalized size = 0.89 \begin {gather*} \frac {1}{2} \sqrt {x} \sqrt {2+b x} (5+b x)-\frac {3 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 3.32, size = 54, normalized size = 0.89 \begin {gather*} \frac {\sqrt {b} \sqrt {x} \left (10+7 b x+b^2 x^2\right )+6 \text {ArcSinh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ] \sqrt {2+b x}}{2 \sqrt {b} \sqrt {2+b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 72, normalized size = 1.18
method | result | size |
meijerg | \(\frac {4 \sqrt {\pi }\, \sqrt {b}\, \sqrt {x}\, \sqrt {2}\, \left (\frac {b x}{8}+\frac {5}{8}\right ) \sqrt {\frac {b x}{2}+1}+3 \sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {b}\, \sqrt {\pi }}\) | \(54\) |
risch | \(\frac {\left (b x +5\right ) \sqrt {x}\, \sqrt {b x +2}}{2}+\frac {3 \sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{2 \sqrt {b x +2}\, \sqrt {x}\, \sqrt {b}}\) | \(65\) |
default | \(\frac {\left (b x +2\right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 \sqrt {x}\, \sqrt {b x +2}}{2}+\frac {3 \sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{2 \sqrt {b x +2}\, \sqrt {x}\, \sqrt {b}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs.
\(2 (42) = 84\).
time = 0.35, size = 98, normalized size = 1.61 \begin {gather*} -\frac {3 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{2 \, \sqrt {b}} - \frac {\frac {3 \, \sqrt {b x + 2} b}{\sqrt {x}} - \frac {5 \, {\left (b x + 2\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}}{b^{2} - \frac {2 \, {\left (b x + 2\right )} b}{x} + \frac {{\left (b x + 2\right )}^{2}}{x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 105, normalized size = 1.72 \begin {gather*} \left [\frac {{\left (b^{2} x + 5 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 3 \, \sqrt {b} \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{2 \, b}, \frac {{\left (b^{2} x + 5 \, b\right )} \sqrt {b x + 2} \sqrt {x} - 6 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{2 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.59, size = 76, normalized size = 1.25 \begin {gather*} \frac {b^{2} x^{\frac {5}{2}}}{2 \sqrt {b x + 2}} + \frac {7 b x^{\frac {3}{2}}}{2 \sqrt {b x + 2}} + \frac {5 \sqrt {x}}{\sqrt {b x + 2}} + \frac {3 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 113, normalized size = 1.85 \begin {gather*} \frac {b^{2} \left (2 \left (\frac {\frac {1}{4} \sqrt {b x+2} \sqrt {b x+2}}{b}+\frac {\frac {1}{4}\cdot 3}{b}\right ) \sqrt {b x+2} \sqrt {b \left (b x+2\right )-2 b}-\frac {3 \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}}{\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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