Integrand size = 15, antiderivative size = 47 \[ \int \frac {\sqrt {x}}{(-a+b x)^2} \, dx=\frac {\sqrt {x}}{b (a-b x)}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {43, 65, 214} \[ \int \frac {\sqrt {x}}{(-a+b x)^2} \, dx=\frac {\sqrt {x}}{b (a-b x)}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
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Rule 43
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x}}{b (a-b x)}+\frac {\int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{2 b} \\ & = \frac {\sqrt {x}}{b (a-b x)}+\frac {\text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{b} \\ & = \frac {\sqrt {x}}{b (a-b x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {x}}{(-a+b x)^2} \, dx=-\frac {\sqrt {x}}{b (-a+b x)}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
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Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {\sqrt {x}}{b \left (-b x +a \right )}-\frac {\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(38\) |
default | \(\frac {\sqrt {x}}{b \left (-b x +a \right )}-\frac {\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(38\) |
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Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.62 \[ \int \frac {\sqrt {x}}{(-a+b x)^2} \, dx=\left [-\frac {2 \, a b \sqrt {x} - \sqrt {a b} {\left (b x - a\right )} \log \left (\frac {b x + a - 2 \, \sqrt {a b} \sqrt {x}}{b x - a}\right )}{2 \, {\left (a b^{3} x - a^{2} b^{2}\right )}}, -\frac {a b \sqrt {x} - \sqrt {-a b} {\left (b x - a\right )} \arctan \left (\frac {\sqrt {-a b}}{b \sqrt {x}}\right )}{a b^{3} x - a^{2} b^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (37) = 74\).
Time = 2.61 (sec) , antiderivative size = 243, normalized size of antiderivative = 5.17 \[ \int \frac {\sqrt {x}}{(-a+b x)^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a^{2}} & \text {for}\: b = 0 \\- \frac {2}{b^{2} \sqrt {x}} & \text {for}\: a = 0 \\- \frac {a \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{2} \sqrt {\frac {a}{b}} + 2 b^{3} x \sqrt {\frac {a}{b}}} + \frac {a \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{2} \sqrt {\frac {a}{b}} + 2 b^{3} x \sqrt {\frac {a}{b}}} - \frac {2 b \sqrt {x} \sqrt {\frac {a}{b}}}{- 2 a b^{2} \sqrt {\frac {a}{b}} + 2 b^{3} x \sqrt {\frac {a}{b}}} + \frac {b x \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{2} \sqrt {\frac {a}{b}} + 2 b^{3} x \sqrt {\frac {a}{b}}} - \frac {b x \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{2} \sqrt {\frac {a}{b}} + 2 b^{3} x \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {x}}{(-a+b x)^2} \, dx=-\frac {\sqrt {x}}{b^{2} x - a b} + \frac {\log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{2 \, \sqrt {a b} b} \]
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Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {x}}{(-a+b x)^2} \, dx=\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} b} - \frac {\sqrt {x}}{{\left (b x - a\right )} b} \]
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Time = 0.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {x}}{(-a+b x)^2} \, dx=\frac {\sqrt {x}}{b\,\left (a-b\,x\right )}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a}\,b^{3/2}} \]
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