Optimal. Leaf size=38 \[ -\frac {(2+b x)^{3/2}}{5 x^{5/2}}+\frac {b (2+b x)^{3/2}}{15 x^{3/2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, 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 {b (b x+2)^{3/2}}{15 x^{3/2}}-\frac {(b x+2)^{3/2}}{5 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {\sqrt {2+b x}}{x^{7/2}} \, dx &=-\frac {(2+b x)^{3/2}}{5 x^{5/2}}-\frac {1}{5} b \int \frac {\sqrt {2+b x}}{x^{5/2}} \, dx\\ &=-\frac {(2+b x)^{3/2}}{5 x^{5/2}}+\frac {b (2+b x)^{3/2}}{15 x^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 31, normalized size = 0.82 \begin {gather*} \frac {\sqrt {2+b x} \left (-6-b x+b^2 x^2\right )}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 4.35, size = 35, normalized size = 0.92 \begin {gather*} \frac {\sqrt {b} \left (-6-b x+b^2 x^2\right ) \sqrt {\frac {2+b x}{b x}}}{15 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.12, size = 43, normalized size = 1.13
method | result | size |
gosper | \(\frac {\left (b x +2\right )^{\frac {3}{2}} \left (b x -3\right )}{15 x^{\frac {5}{2}}}\) | \(18\) |
meijerg | \(-\frac {2 \sqrt {2}\, \left (-\frac {1}{6} x^{2} b^{2}+\frac {1}{6} b x +1\right ) \sqrt {\frac {b x}{2}+1}}{5 x^{\frac {5}{2}}}\) | \(31\) |
risch | \(\frac {b^{3} x^{3}+x^{2} b^{2}-8 b x -12}{15 x^{\frac {5}{2}} \sqrt {b x +2}}\) | \(33\) |
default | \(-\frac {2 \sqrt {b x +2}}{5 x^{\frac {5}{2}}}+\frac {b \left (-\frac {\sqrt {b x +2}}{3 x^{\frac {3}{2}}}+\frac {b \sqrt {b x +2}}{3 \sqrt {x}}\right )}{5}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 26, normalized size = 0.68 \begin {gather*} \frac {{\left (b x + 2\right )}^{\frac {3}{2}} b}{6 \, x^{\frac {3}{2}}} - \frac {{\left (b x + 2\right )}^{\frac {5}{2}}}{10 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 25, normalized size = 0.66 \begin {gather*} \frac {{\left (b^{2} x^{2} - b x - 6\right )} \sqrt {b x + 2}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.88, size = 56, normalized size = 1.47 \begin {gather*} \frac {b^{\frac {5}{2}} \sqrt {1 + \frac {2}{b x}}}{15} - \frac {b^{\frac {3}{2}} \sqrt {1 + \frac {2}{b x}}}{15 x} - \frac {2 \sqrt {b} \sqrt {1 + \frac {2}{b x}}}{5 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 102, normalized size = 2.68 \begin {gather*} \frac {2 b^{2} \left (\frac {15}{450} b^{5} \sqrt {b x+2} \sqrt {b x+2}-\frac {75}{450} b^{5}\right ) \sqrt {b x+2} \sqrt {b x+2} \sqrt {b x+2} \sqrt {b \left (b x+2\right )-2 b}}{\left |b\right | b \left (b \left (b x+2\right )-2 b\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 26, normalized size = 0.68 \begin {gather*} -\frac {\sqrt {b\,x+2}\,\left (-\frac {b^2\,x^2}{15}+\frac {b\,x}{15}+\frac {2}{5}\right )}{x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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