Optimal. Leaf size=102 \[ \frac {5 \sqrt {x} \sqrt {2+b x}}{8 b}+\frac {5}{8} x^{3/2} \sqrt {2+b x}+\frac {5}{12} x^{3/2} (2+b x)^{3/2}+\frac {1}{4} x^{3/2} (2+b x)^{5/2}-\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 102, 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 {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{3/2}}+\frac {1}{4} x^{3/2} (b x+2)^{5/2}+\frac {5}{12} x^{3/2} (b x+2)^{3/2}+\frac {5}{8} x^{3/2} \sqrt {b x+2}+\frac {5 \sqrt {x} \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 \sqrt {x} (2+b x)^{5/2} \, dx &=\frac {1}{4} x^{3/2} (2+b x)^{5/2}+\frac {5}{4} \int \sqrt {x} (2+b x)^{3/2} \, dx\\ &=\frac {5}{12} x^{3/2} (2+b x)^{3/2}+\frac {1}{4} x^{3/2} (2+b x)^{5/2}+\frac {5}{4} \int \sqrt {x} \sqrt {2+b x} \, dx\\ &=\frac {5}{8} x^{3/2} \sqrt {2+b x}+\frac {5}{12} x^{3/2} (2+b x)^{3/2}+\frac {1}{4} x^{3/2} (2+b x)^{5/2}+\frac {5}{8} \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx\\ &=\frac {5 \sqrt {x} \sqrt {2+b x}}{8 b}+\frac {5}{8} x^{3/2} \sqrt {2+b x}+\frac {5}{12} x^{3/2} (2+b x)^{3/2}+\frac {1}{4} x^{3/2} (2+b x)^{5/2}-\frac {5 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{8 b}\\ &=\frac {5 \sqrt {x} \sqrt {2+b x}}{8 b}+\frac {5}{8} x^{3/2} \sqrt {2+b x}+\frac {5}{12} x^{3/2} (2+b x)^{3/2}+\frac {1}{4} x^{3/2} (2+b x)^{5/2}-\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{4 b}\\ &=\frac {5 \sqrt {x} \sqrt {2+b x}}{8 b}+\frac {5}{8} x^{3/2} \sqrt {2+b x}+\frac {5}{12} x^{3/2} (2+b x)^{3/2}+\frac {1}{4} x^{3/2} (2+b x)^{5/2}-\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 76, normalized size = 0.75 \begin {gather*} \frac {\sqrt {x} \sqrt {2+b x} \left (15+59 b x+34 b^2 x^2+6 b^3 x^3\right )}{24 b}+\frac {5 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{4 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 99, normalized size = 0.97
method | result | size |
meijerg | \(-\frac {30 \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (6 b^{3} x^{3}+34 x^{2} b^{2}+59 b x +15\right ) \sqrt {\frac {b x}{2}+1}}{720}+\frac {\sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{24}\right )}{b^{\frac {3}{2}} \sqrt {\pi }}\) | \(71\) |
risch | \(\frac {\left (6 b^{3} x^{3}+34 x^{2} b^{2}+59 b x +15\right ) \sqrt {x}\, \sqrt {b x +2}}{24 b}-\frac {5 \sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{8 b^{\frac {3}{2}} \sqrt {b x +2}\, \sqrt {x}}\) | \(85\) |
default | \(\frac {x^{\frac {3}{2}} \left (b x +2\right )^{\frac {5}{2}}}{4}+\frac {5 x^{\frac {3}{2}} \left (b x +2\right )^{\frac {3}{2}}}{12}+\frac {5 x^{\frac {3}{2}} \sqrt {b x +2}}{8}+\frac {5 \sqrt {x}\, \sqrt {b x +2}}{8 b}-\frac {5 \sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{8 b^{\frac {3}{2}} \sqrt {b x +2}\, \sqrt {x}}\) | \(99\) |
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. 161 vs.
\(2 (69) = 138\).
time = 0.60, size = 161, normalized size = 1.58 \begin {gather*} \frac {\frac {15 \, \sqrt {b x + 2} b^{3}}{\sqrt {x}} - \frac {55 \, {\left (b x + 2\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}} + \frac {73 \, {\left (b x + 2\right )}^{\frac {5}{2}} b}{x^{\frac {5}{2}}} + \frac {15 \, {\left (b x + 2\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}}}{12 \, {\left (b^{5} - \frac {4 \, {\left (b x + 2\right )} b^{4}}{x} + \frac {6 \, {\left (b x + 2\right )}^{2} b^{3}}{x^{2}} - \frac {4 \, {\left (b x + 2\right )}^{3} b^{2}}{x^{3}} + \frac {{\left (b x + 2\right )}^{4} b}{x^{4}}\right )}} + \frac {5 \, \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 {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 140, normalized size = 1.37 \begin {gather*} \left [\frac {{\left (6 \, b^{4} x^{3} + 34 \, b^{3} x^{2} + 59 \, b^{2} x + 15 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 15 \, \sqrt {b} \log \left (b x - \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{24 \, b^{2}}, \frac {{\left (6 \, b^{4} x^{3} + 34 \, b^{3} x^{2} + 59 \, b^{2} x + 15 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 30 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{24 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 8.33, size = 119, normalized size = 1.17 \begin {gather*} \frac {b^{3} x^{\frac {9}{2}}}{4 \sqrt {b x + 2}} + \frac {23 b^{2} x^{\frac {7}{2}}}{12 \sqrt {b x + 2}} + \frac {127 b x^{\frac {5}{2}}}{24 \sqrt {b x + 2}} + \frac {133 x^{\frac {3}{2}}}{24 \sqrt {b x + 2}} + \frac {5 \sqrt {x}}{4 b \sqrt {b x + 2}} - \frac {5 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 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 )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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