Optimal. Leaf size=53 \[ \frac {1}{x^{3/2} \sqrt {2+b x}}-\frac {2 \sqrt {2+b x}}{3 x^{3/2}}+\frac {2 b \sqrt {2+b x}}{3 \sqrt {x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37}
\begin {gather*} -\frac {2 \sqrt {b x+2}}{3 x^{3/2}}+\frac {1}{x^{3/2} \sqrt {b x+2}}+\frac {2 b \sqrt {b x+2}}{3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} (2+b x)^{3/2}} \, dx &=\frac {1}{x^{3/2} \sqrt {2+b x}}+2 \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx\\ &=\frac {1}{x^{3/2} \sqrt {2+b x}}-\frac {2 \sqrt {2+b x}}{3 x^{3/2}}-\frac {1}{3} (2 b) \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx\\ &=\frac {1}{x^{3/2} \sqrt {2+b x}}-\frac {2 \sqrt {2+b x}}{3 x^{3/2}}+\frac {2 b \sqrt {2+b x}}{3 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 32, normalized size = 0.60 \begin {gather*} \frac {-1+2 b x+2 b^2 x^2}{3 x^{3/2} \sqrt {2+b x}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 4.85, size = 57, normalized size = 1.08 \begin {gather*} \frac {\sqrt {b} \left (-2+3 b x \left (1+2 b x\right )+2 b^3 x^3\right ) \sqrt {\frac {2+b x}{b x}}}{3 x \left (4+4 b x+b^2 x^2\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 43, normalized size = 0.81
method | result | size |
gosper | \(\frac {2 x^{2} b^{2}+2 b x -1}{3 x^{\frac {3}{2}} \sqrt {b x +2}}\) | \(27\) |
meijerg | \(-\frac {\sqrt {2}\, \left (-2 x^{2} b^{2}-2 b x +1\right )}{6 x^{\frac {3}{2}} \sqrt {\frac {b x}{2}+1}}\) | \(31\) |
default | \(-\frac {1}{3 x^{\frac {3}{2}} \sqrt {b x +2}}-\frac {2 b \left (-\frac {1}{\sqrt {x}\, \sqrt {b x +2}}-\frac {b \sqrt {x}}{\sqrt {b x +2}}\right )}{3}\) | \(43\) |
risch | \(\frac {5 x^{2} b^{2}+8 b x -4}{12 x^{\frac {3}{2}} \sqrt {b x +2}}+\frac {b^{2} \sqrt {x}}{4 \sqrt {b x +2}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 41, normalized size = 0.77 \begin {gather*} \frac {b^{2} \sqrt {x}}{4 \, \sqrt {b x + 2}} + \frac {\sqrt {b x + 2} b}{2 \, \sqrt {x}} - \frac {{\left (b x + 2\right )}^{\frac {3}{2}}}{12 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 39, normalized size = 0.74 \begin {gather*} \frac {{\left (2 \, b^{2} x^{2} + 2 \, b x - 1\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b x^{3} + 2 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 170 vs.
\(2 (49) = 98\).
time = 2.44, size = 170, normalized size = 3.21 \begin {gather*} \frac {2 b^{\frac {15}{2}} x^{3} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} + \frac {6 b^{\frac {13}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} + \frac {3 b^{\frac {11}{2}} x \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} - \frac {2 b^{\frac {9}{2}} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs.
\(2 (37) = 74\).
time = 0.01, size = 138, normalized size = 2.60 \begin {gather*} 2 \left (\frac {2 b^{2} \sqrt {x} \sqrt {b x+2}}{16 \left (b x+2\right )}+\frac {2 \left (-3 b \sqrt {b} \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )^{4}+24 b \sqrt {b} \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )^{2}-20 b \sqrt {b}\right )}{12 \left (\left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )^{2}-2\right )^{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 37, normalized size = 0.70 \begin {gather*} \frac {\sqrt {b\,x+2}\,\left (\frac {2\,x}{3}+\frac {2\,b\,x^2}{3}-\frac {1}{3\,b}\right )}{x^{5/2}+\frac {2\,x^{3/2}}{b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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