Optimal. Leaf size=105 \[ -\frac {3 \sqrt {x} \sqrt {2+b x}}{8 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2+b x}+\frac {1}{4} x^{5/2} (2+b x)^{3/2}+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{5/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{5/2}}-\frac {3 \sqrt {x} \sqrt {b x+2}}{8 b^2}+\frac {1}{4} x^{5/2} (b x+2)^{3/2}+\frac {1}{4} x^{5/2} \sqrt {b x+2}+\frac {x^{3/2} \sqrt {b x+2}}{8 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} (2+b x)^{3/2} \, dx &=\frac {1}{4} x^{5/2} (2+b x)^{3/2}+\frac {3}{4} \int x^{3/2} \sqrt {2+b x} \, dx\\ &=\frac {1}{4} x^{5/2} \sqrt {2+b x}+\frac {1}{4} x^{5/2} (2+b x)^{3/2}+\frac {1}{4} \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx\\ &=\frac {x^{3/2} \sqrt {2+b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2+b x}+\frac {1}{4} x^{5/2} (2+b x)^{3/2}-\frac {3 \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx}{8 b}\\ &=-\frac {3 \sqrt {x} \sqrt {2+b x}}{8 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2+b x}+\frac {1}{4} x^{5/2} (2+b x)^{3/2}+\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{8 b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2+b x}}{8 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2+b x}+\frac {1}{4} x^{5/2} (2+b x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2+b x}}{8 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2+b x}+\frac {1}{4} x^{5/2} (2+b x)^{3/2}+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 75, normalized size = 0.71 \begin {gather*} \frac {\sqrt {x} \sqrt {2+b x} \left (-3+b x+6 b^2 x^2+2 b^3 x^3\right )}{8 b^2}-\frac {3 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{4 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 114, normalized size = 1.09
method | result | size |
meijerg | \(\frac {-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (-10 b^{3} x^{3}-30 x^{2} b^{2}-5 b x +15\right ) \sqrt {\frac {b x}{2}+1}}{40}+\frac {3 \sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{4}}{b^{\frac {5}{2}} \sqrt {\pi }}\) | \(71\) |
risch | \(\frac {\left (2 b^{3} x^{3}+6 x^{2} b^{2}+b x -3\right ) \sqrt {x}\, \sqrt {b x +2}}{8 b^{2}}+\frac {3 \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right ) \sqrt {x \left (b x +2\right )}}{8 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {b x +2}}\) | \(84\) |
default | \(\frac {x^{\frac {3}{2}} \left (b x +2\right )^{\frac {5}{2}}}{4 b}-\frac {3 \left (\frac {\sqrt {x}\, \left (b x +2\right )^{\frac {5}{2}}}{3 b}-\frac {\frac {\left (b x +2\right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 \sqrt {x}\, \sqrt {b x +2}}{2}+\frac {3 \sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{2 \sqrt {b x +2}\, \sqrt {x}\, \sqrt {b}}}{3 b}\right )}{4 b}\) | \(114\) |
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. 163 vs.
\(2 (72) = 144\).
time = 0.53, size = 163, normalized size = 1.55 \begin {gather*} -\frac {\frac {3 \, \sqrt {b x + 2} b^{3}}{\sqrt {x}} - \frac {11 \, {\left (b x + 2\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}} - \frac {11 \, {\left (b x + 2\right )}^{\frac {5}{2}} b}{x^{\frac {5}{2}}} + \frac {3 \, {\left (b x + 2\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}}}{4 \, {\left (b^{6} - \frac {4 \, {\left (b x + 2\right )} b^{5}}{x} + \frac {6 \, {\left (b x + 2\right )}^{2} b^{4}}{x^{2}} - \frac {4 \, {\left (b x + 2\right )}^{3} b^{3}}{x^{3}} + \frac {{\left (b x + 2\right )}^{4} b^{2}}{x^{4}}\right )}} - \frac {3 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{8 \, b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 137, normalized size = 1.30 \begin {gather*} \left [\frac {{\left (2 \, b^{4} x^{3} + 6 \, 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 )}{8 \, b^{3}}, \frac {{\left (2 \, b^{4} x^{3} + 6 \, 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 )}{8 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.62, size = 117, normalized size = 1.11 \begin {gather*} \frac {b^{2} x^{\frac {9}{2}}}{4 \sqrt {b x + 2}} + \frac {5 b x^{\frac {7}{2}}}{4 \sqrt {b x + 2}} + \frac {13 x^{\frac {5}{2}}}{8 \sqrt {b x + 2}} - \frac {x^{\frac {3}{2}}}{8 b \sqrt {b x + 2}} - \frac {3 \sqrt {x}}{4 b^{2} \sqrt {b x + 2}} + \frac {3 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 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}\,{\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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