Optimal. Leaf size=84 \[ -\frac {\sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{6 b}+\frac {1}{3} x^{5/2} \sqrt {2+b x}+\frac {\sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 84, 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^{5/2}}-\frac {\sqrt {x} \sqrt {b x+2}}{2 b^2}+\frac {1}{3} x^{5/2} \sqrt {b x+2}+\frac {x^{3/2} \sqrt {b x+2}}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 221
Rubi steps
\begin {align*} \int x^{3/2} \sqrt {2+b x} \, dx &=\frac {1}{3} x^{5/2} \sqrt {2+b x}+\frac {1}{3} \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx\\ &=\frac {x^{3/2} \sqrt {2+b x}}{6 b}+\frac {1}{3} x^{5/2} \sqrt {2+b x}-\frac {\int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx}{2 b}\\ &=-\frac {\sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{6 b}+\frac {1}{3} x^{5/2} \sqrt {2+b x}+\frac {\int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{2 b^2}\\ &=-\frac {\sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{6 b}+\frac {1}{3} x^{5/2} \sqrt {2+b x}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {\sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{6 b}+\frac {1}{3} x^{5/2} \sqrt {2+b x}+\frac {\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 = 65, normalized size = 0.77 \begin {gather*} \frac {\sqrt {x} \sqrt {2+b x} \left (-3+b x+2 b^2 x^2\right )}{6 b^2}-\frac {\log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 100, normalized size = 1.19
method | result | size |
meijerg | \(-\frac {4 \left (\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (-10 x^{2} b^{2}-5 b x +15\right ) \sqrt {\frac {b x}{2}+1}}{120}-\frac {\sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{4}\right )}{b^{\frac {5}{2}} \sqrt {\pi }}\) | \(63\) |
risch | \(\frac {\left (2 x^{2} b^{2}+b x -3\right ) \sqrt {x}\, \sqrt {b x +2}}{6 b^{2}}+\frac {\ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right ) \sqrt {x \left (b x +2\right )}}{2 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {b x +2}}\) | \(76\) |
default | \(\frac {x^{\frac {3}{2}} \left (b x +2\right )^{\frac {3}{2}}}{3 b}-\frac {\frac {\sqrt {x}\, \left (b x +2\right )^{\frac {3}{2}}}{2 b}-\frac {\sqrt {x}\, \sqrt {b x +2}+\frac {\sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{\sqrt {b x +2}\, \sqrt {x}\, \sqrt {b}}}{2 b}}{b}\) | \(100\) |
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. 134 vs.
\(2 (59) = 118\).
time = 0.52, size = 134, normalized size = 1.60 \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^{5} - \frac {3 \, {\left (b x + 2\right )} b^{4}}{x} + \frac {3 \, {\left (b x + 2\right )}^{2} b^{3}}{x^{2}} - \frac {{\left (b x + 2\right )}^{3} b^{2}}{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 {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.53, size = 121, normalized size = 1.44 \begin {gather*} \left [\frac {{\left (2 \, b^{3} x^{2} + 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^{3}}, \frac {{\left (2 \, b^{3} x^{2} + 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^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.14, size = 90, normalized size = 1.07 \begin {gather*} \frac {b x^{\frac {7}{2}}}{3 \sqrt {b x + 2}} + \frac {5 x^{\frac {5}{2}}}{6 \sqrt {b x + 2}} - \frac {x^{\frac {3}{2}}}{6 b \sqrt {b x + 2}} - \frac {\sqrt {x}}{b^{2} \sqrt {b x + 2}} + \frac {\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 [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 x^{3/2}\,\sqrt {b\,x+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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