Optimal. Leaf size=82 \[ \frac {\sqrt {x} \sqrt {2+b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2+b x}+\frac {1}{3} x^{3/2} (2+b x)^{3/2}-\frac {\sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {\sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}+\frac {1}{3} x^{3/2} (b x+2)^{3/2}+\frac {1}{2} x^{3/2} \sqrt {b x+2}+\frac {\sqrt {x} \sqrt {b x+2}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 221
Rubi steps
\begin {align*} \int \sqrt {x} (2+b x)^{3/2} \, dx &=\frac {1}{3} x^{3/2} (2+b x)^{3/2}+\int \sqrt {x} \sqrt {2+b x} \, dx\\ &=\frac {1}{2} x^{3/2} \sqrt {2+b x}+\frac {1}{3} x^{3/2} (2+b x)^{3/2}+\frac {1}{2} \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx\\ &=\frac {\sqrt {x} \sqrt {2+b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2+b x}+\frac {1}{3} x^{3/2} (2+b x)^{3/2}-\frac {\int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{2 b}\\ &=\frac {\sqrt {x} \sqrt {2+b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2+b x}+\frac {1}{3} x^{3/2} (2+b x)^{3/2}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=\frac {\sqrt {x} \sqrt {2+b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2+b x}+\frac {1}{3} x^{3/2} (2+b x)^{3/2}-\frac {\sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 65, normalized size = 0.79 \begin {gather*} \frac {\sqrt {x} \sqrt {2+b x} \left (3+7 b x+2 b^2 x^2\right )}{6 b}+\frac {\log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 87, normalized size = 1.06
method | result | size |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (2 x^{2} b^{2}+7 b x +3\right ) \sqrt {\frac {b x}{2}+1}}{6}-\sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {3}{2}} \sqrt {\pi }}\) | \(63\) |
risch | \(\frac {\left (2 x^{2} b^{2}+7 b x +3\right ) \sqrt {x}\, \sqrt {b x +2}}{6 b}-\frac {\sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{2 b^{\frac {3}{2}} \sqrt {b x +2}\, \sqrt {x}}\) | \(77\) |
default | \(\frac {x^{\frac {3}{2}} \left (b x +2\right )^{\frac {3}{2}}}{3}+\frac {x^{\frac {3}{2}} \sqrt {b x +2}}{2}+\frac {\sqrt {x}\, \sqrt {b x +2}}{2 b}-\frac {\sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{2 b^{\frac {3}{2}} \sqrt {b x +2}\, \sqrt {x}}\) | \(87\) |
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. 132 vs.
\(2 (57) = 114\).
time = 0.52, size = 132, normalized size = 1.61 \begin {gather*} \frac {\frac {3 \, \sqrt {b x + 2} b^{2}}{\sqrt {x}} - \frac {8 \, {\left (b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} - \frac {3 \, {\left (b x + 2\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}}{3 \, {\left (b^{4} - \frac {3 \, {\left (b x + 2\right )} b^{3}}{x} + \frac {3 \, {\left (b x + 2\right )}^{2} b^{2}}{x^{2}} - \frac {{\left (b x + 2\right )}^{3} b}{x^{3}}\right )}} + \frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{2 \, b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.60, size = 124, normalized size = 1.51 \begin {gather*} \left [\frac {{\left (2 \, b^{3} x^{2} + 7 \, b^{2} x + 3 \, 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 )}{6 \, b^{2}}, \frac {{\left (2 \, b^{3} x^{2} + 7 \, b^{2} x + 3 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 6 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{6 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.74, size = 92, normalized size = 1.12 \begin {gather*} \frac {b^{2} x^{\frac {7}{2}}}{3 \sqrt {b x + 2}} + \frac {11 b x^{\frac {5}{2}}}{6 \sqrt {b x + 2}} + \frac {17 x^{\frac {3}{2}}}{6 \sqrt {b x + 2}} + \frac {\sqrt {x}}{b \sqrt {b x + 2}} - \frac {\operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {3}{2}}} \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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {x}\,{\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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