Integrand size = 16, antiderivative size = 46 \[ \int \frac {\sqrt {a-b x}}{x^{7/2}} \, dx=-\frac {2 (a-b x)^{3/2}}{5 a x^{5/2}}-\frac {4 b (a-b x)^{3/2}}{15 a^2 x^{3/2}} \]
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Time = 0.00 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37} \[ \int \frac {\sqrt {a-b x}}{x^{7/2}} \, dx=-\frac {4 b (a-b x)^{3/2}}{15 a^2 x^{3/2}}-\frac {2 (a-b x)^{3/2}}{5 a x^{5/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a-b x)^{3/2}}{5 a x^{5/2}}+\frac {(2 b) \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx}{5 a} \\ & = -\frac {2 (a-b x)^{3/2}}{5 a x^{5/2}}-\frac {4 b (a-b x)^{3/2}}{15 a^2 x^{3/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {a-b x}}{x^{7/2}} \, dx=-\frac {2 \sqrt {a-b x} \left (3 a^2-a b x-2 b^2 x^2\right )}{15 a^2 x^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(-\frac {2 \left (-b x +a \right )^{\frac {3}{2}} \left (2 b x +3 a \right )}{15 x^{\frac {5}{2}} a^{2}}\) | \(25\) |
risch | \(-\frac {2 \sqrt {-b x +a}\, \left (-2 b^{2} x^{2}-a b x +3 a^{2}\right )}{15 x^{\frac {5}{2}} a^{2}}\) | \(36\) |
default | \(-\frac {\sqrt {-b x +a}}{2 x^{\frac {5}{2}}}-\frac {a \left (-\frac {2 \sqrt {-b x +a}}{5 a \,x^{\frac {5}{2}}}+\frac {4 b \left (-\frac {2 \sqrt {-b x +a}}{3 a \,x^{\frac {3}{2}}}-\frac {4 b \sqrt {-b x +a}}{3 a^{2} \sqrt {x}}\right )}{5 a}\right )}{4}\) | \(75\) |
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none
Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {a-b x}}{x^{7/2}} \, dx=\frac {2 \, {\left (2 \, b^{2} x^{2} + a b x - 3 \, a^{2}\right )} \sqrt {-b x + a}}{15 \, a^{2} x^{\frac {5}{2}}} \]
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Result contains complex when optimal does not.
Time = 3.53 (sec) , antiderivative size = 241, normalized size of antiderivative = 5.24 \[ \int \frac {\sqrt {a-b x}}{x^{7/2}} \, dx=\begin {cases} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{5 x^{2}} + \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}}{15 a x} + \frac {4 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} - 1}}{15 a^{2}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {6 i a^{3} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{x \left (- 15 a^{3} b x + 15 a^{2} b^{2} x^{2}\right )} - \frac {8 i a^{2} b^{\frac {5}{2}} \sqrt {- \frac {a}{b x} + 1}}{- 15 a^{3} b x + 15 a^{2} b^{2} x^{2}} - \frac {2 i a b^{\frac {7}{2}} x \sqrt {- \frac {a}{b x} + 1}}{- 15 a^{3} b x + 15 a^{2} b^{2} x^{2}} + \frac {4 i b^{\frac {9}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{- 15 a^{3} b x + 15 a^{2} b^{2} x^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {a-b x}}{x^{7/2}} \, dx=-\frac {2 \, {\left (\frac {5 \, {\left (-b x + a\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {3 \, {\left (-b x + a\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}\right )}}{15 \, a^{2}} \]
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none
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {a-b x}}{x^{7/2}} \, dx=\frac {2 \, {\left (\frac {2 \, {\left (b x - a\right )} b^{5}}{a^{2}} + \frac {5 \, b^{5}}{a}\right )} {\left (b x - a\right )} \sqrt {-b x + a} b}{15 \, {\left ({\left (b x - a\right )} b + a b\right )}^{\frac {5}{2}} {\left | b \right |}} \]
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Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a-b x}}{x^{7/2}} \, dx=\frac {\sqrt {a-b\,x}\,\left (\frac {4\,b^2\,x^2}{15\,a^2}+\frac {2\,b\,x}{15\,a}-\frac {2}{5}\right )}{x^{5/2}} \]
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