Integrand size = 15, antiderivative size = 38 \[ \int \frac {\sqrt {2+b x}}{x^{7/2}} \, dx=-\frac {(2+b x)^{3/2}}{5 x^{5/2}}+\frac {b (2+b x)^{3/2}}{15 x^{3/2}} \]
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Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {\sqrt {2+b x}}{x^{7/2}} \, dx=\frac {b (b x+2)^{3/2}}{15 x^{3/2}}-\frac {(b x+2)^{3/2}}{5 x^{5/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {2+b x}}{x^{7/2}} \, dx=\frac {\sqrt {2+b x} \left (-6-b x+b^2 x^2\right )}{15 x^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.47
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 | \(-\frac {2 \sqrt {2}\, \left (-\frac {1}{6} b^{2} x^{2}+\frac {1}{6} b x +1\right ) \sqrt {\frac {b x}{2}+1}}{5 x^{\frac {5}{2}}}\) | \(31\) |
risch | \(\frac {b^{3} x^{3}+b^{2} x^{2}-8 b x -12}{15 x^{\frac {5}{2}} \sqrt {b x +2}}\) | \(33\) |
default | \(-\frac {2 \sqrt {b x +2}}{5 x^{\frac {5}{2}}}+\frac {b \left (-\frac {\sqrt {b x +2}}{3 x^{\frac {3}{2}}}+\frac {b \sqrt {b x +2}}{3 \sqrt {x}}\right )}{5}\) | \(43\) |
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Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {2+b x}}{x^{7/2}} \, dx=\frac {{\left (b^{2} x^{2} - b x - 6\right )} \sqrt {b x + 2}}{15 \, x^{\frac {5}{2}}} \]
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Time = 2.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt {2+b x}}{x^{7/2}} \, dx=\frac {b^{\frac {5}{2}} \sqrt {1 + \frac {2}{b x}}}{15} - \frac {b^{\frac {3}{2}} \sqrt {1 + \frac {2}{b x}}}{15 x} - \frac {2 \sqrt {b} \sqrt {1 + \frac {2}{b x}}}{5 x^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {2+b x}}{x^{7/2}} \, dx=\frac {{\left (b x + 2\right )}^{\frac {3}{2}} b}{6 \, x^{\frac {3}{2}}} - \frac {{\left (b x + 2\right )}^{\frac {5}{2}}}{10 \, x^{\frac {5}{2}}} \]
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Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {2+b x}}{x^{7/2}} \, dx=\frac {{\left ({\left (b x + 2\right )} b^{5} - 5 \, b^{5}\right )} {\left (b x + 2\right )}^{\frac {3}{2}} b}{15 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {5}{2}} {\left | b \right |}} \]
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {2+b x}}{x^{7/2}} \, dx=-\frac {\sqrt {b\,x+2}\,\left (-\frac {b^2\,x^2}{15}+\frac {b\,x}{15}+\frac {2}{5}\right )}{x^{5/2}} \]
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