Integrand size = 15, antiderivative size = 75 \[ \int \frac {\sqrt {x}}{(-a+b x)^3} \, dx=-\frac {\sqrt {x}}{2 b (a-b x)^2}+\frac {\sqrt {x}}{4 a b (a-b x)}+\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {43, 44, 65, 214} \[ \int \frac {\sqrt {x}}{(-a+b x)^3} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {\sqrt {x}}{4 a b (a-b x)}-\frac {\sqrt {x}}{2 b (a-b x)^2} \]
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Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {x}}{2 b (a-b x)^2}+\frac {\int \frac {1}{\sqrt {x} (-a+b x)^2} \, dx}{4 b} \\ & = -\frac {\sqrt {x}}{2 b (a-b x)^2}+\frac {\sqrt {x}}{4 a b (a-b x)}-\frac {\int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{8 a b} \\ & = -\frac {\sqrt {x}}{2 b (a-b x)^2}+\frac {\sqrt {x}}{4 a b (a-b x)}-\frac {\text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a b} \\ & = -\frac {\sqrt {x}}{2 b (a-b x)^2}+\frac {\sqrt {x}}{4 a b (a-b x)}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {x}}{(-a+b x)^3} \, dx=-\frac {\sqrt {x} (a+b x)}{4 a b (a-b x)^2}+\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}} \]
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Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {x^{\frac {3}{2}}}{8 a}+\frac {\sqrt {x}}{8 b}\right )}{\left (-b x +a \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b a \sqrt {a b}}\) | \(53\) |
default | \(-\frac {2 \left (\frac {x^{\frac {3}{2}}}{8 a}+\frac {\sqrt {x}}{8 b}\right )}{\left (-b x +a \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b a \sqrt {a b}}\) | \(53\) |
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Time = 0.23 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.44 \[ \int \frac {\sqrt {x}}{(-a+b x)^3} \, dx=\left [\frac {{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {a b} \log \left (\frac {b x + a + 2 \, \sqrt {a b} \sqrt {x}}{b x - a}\right ) - 2 \, {\left (a b^{2} x + a^{2} b\right )} \sqrt {x}}{8 \, {\left (a^{2} b^{4} x^{2} - 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}, -\frac {{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {-a b} \arctan \left (\frac {\sqrt {-a b}}{b \sqrt {x}}\right ) + {\left (a b^{2} x + a^{2} b\right )} \sqrt {x}}{4 \, {\left (a^{2} b^{4} x^{2} - 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (58) = 116\).
Time = 10.40 (sec) , antiderivative size = 575, normalized size of antiderivative = 7.67 \[ \int \frac {\sqrt {x}}{(-a+b x)^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2 x^{\frac {3}{2}}}{3 a^{3}} & \text {for}\: b = 0 \\- \frac {2}{3 b^{3} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\- \frac {a^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} + \frac {a^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} - \frac {2 a b \sqrt {x} \sqrt {\frac {a}{b}}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} + \frac {2 a b x \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} - \frac {2 a b x \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} - \frac {2 b^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} - \frac {b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} + \frac {b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {x}}{(-a+b x)^3} \, dx=-\frac {b x^{\frac {3}{2}} + a \sqrt {x}}{4 \, {\left (a b^{3} x^{2} - 2 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {\log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} a b} \]
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Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {x}}{(-a+b x)^3} \, dx=-\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 \, \sqrt {-a b} a b} - \frac {b x^{\frac {3}{2}} + a \sqrt {x}}{4 \, {\left (b x - a\right )}^{2} a b} \]
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Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {x}}{(-a+b x)^3} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{3/2}\,b^{3/2}}-\frac {\frac {x^{3/2}}{4\,a}+\frac {\sqrt {x}}{4\,b}}{a^2-2\,a\,b\,x+b^2\,x^2} \]
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