Integrand size = 16, antiderivative size = 46 \[ \int \frac {1}{x^{5/2} \sqrt {a-b x}} \, dx=-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}}-\frac {4 b \sqrt {a-b x}}{3 a^2 \sqrt {x}} \]
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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} \[ \int \frac {1}{x^{5/2} \sqrt {a-b x}} \, dx=-\frac {4 b \sqrt {a-b x}}{3 a^2 \sqrt {x}}-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a-b x}}{3 a x^{3/2}}+\frac {(2 b) \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx}{3 a} \\ & = -\frac {2 \sqrt {a-b x}}{3 a x^{3/2}}-\frac {4 b \sqrt {a-b x}}{3 a^2 \sqrt {x}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^{5/2} \sqrt {a-b x}} \, dx=-\frac {2 \sqrt {a-b x} (a+2 b x)}{3 a^2 x^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(-\frac {2 \sqrt {-b x +a}\, \left (2 b x +a \right )}{3 x^{\frac {3}{2}} a^{2}}\) | \(23\) |
risch | \(-\frac {2 \sqrt {-b x +a}\, \left (2 b x +a \right )}{3 x^{\frac {3}{2}} a^{2}}\) | \(23\) |
default | \(-\frac {2 \sqrt {-b x +a}}{3 a \,x^{\frac {3}{2}}}-\frac {4 b \sqrt {-b x +a}}{3 a^{2} \sqrt {x}}\) | \(35\) |
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none
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x^{5/2} \sqrt {a-b x}} \, dx=-\frac {2 \, {\left (2 \, b x + a\right )} \sqrt {-b x + a}}{3 \, a^{2} x^{\frac {3}{2}}} \]
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Result contains complex when optimal does not.
Time = 1.19 (sec) , antiderivative size = 177, normalized size of antiderivative = 3.85 \[ \int \frac {1}{x^{5/2} \sqrt {a-b x}} \, dx=\begin {cases} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{3 a x} - \frac {4 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}}{3 a^{2}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {2 i a^{2} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{3} b x + 3 a^{2} b^{2} x^{2}} + \frac {2 i a b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{3} b x + 3 a^{2} b^{2} x^{2}} - \frac {4 i b^{\frac {7}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{3} b x + 3 a^{2} b^{2} x^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^{5/2} \sqrt {a-b x}} \, dx=-\frac {2 \, {\left (\frac {3 \, \sqrt {-b x + a} b}{\sqrt {x}} + \frac {{\left (-b x + a\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}\right )}}{3 \, a^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^{5/2} \sqrt {a-b x}} \, dx=-\frac {2 \, {\left (\frac {2 \, {\left (b x - a\right )} b^{3}}{a^{2}} + \frac {3 \, b^{3}}{a}\right )} \sqrt {-b x + a} b}{3 \, {\left ({\left (b x - a\right )} b + a b\right )}^{\frac {3}{2}} {\left | b \right |}} \]
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Time = 0.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^{5/2} \sqrt {a-b x}} \, dx=-\frac {\left (\frac {2}{3\,a}+\frac {4\,b\,x}{3\,a^2}\right )\,\sqrt {a-b\,x}}{x^{3/2}} \]
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