Integrand size = 15, antiderivative size = 57 \[ \int \frac {x^{3/2}}{(-a+b x)^2} \, dx=\frac {3 \sqrt {x}}{b^2}+\frac {x^{3/2}}{b (a-b x)}-\frac {3 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \]
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Time = 0.01 (sec) , antiderivative size = 57, 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, 52, 65, 214} \[ \int \frac {x^{3/2}}{(-a+b x)^2} \, dx=-\frac {3 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}+\frac {x^{3/2}}{b (a-b x)}+\frac {3 \sqrt {x}}{b^2} \]
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Rule 43
Rule 52
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {x^{3/2}}{b (a-b x)}+\frac {3 \int \frac {\sqrt {x}}{-a+b x} \, dx}{2 b} \\ & = \frac {3 \sqrt {x}}{b^2}+\frac {x^{3/2}}{b (a-b x)}+\frac {(3 a) \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{2 b^2} \\ & = \frac {3 \sqrt {x}}{b^2}+\frac {x^{3/2}}{b (a-b x)}+\frac {(3 a) \text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{b^2} \\ & = \frac {3 \sqrt {x}}{b^2}+\frac {x^{3/2}}{b (a-b x)}-\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int \frac {x^{3/2}}{(-a+b x)^2} \, dx=\frac {\sqrt {x} (-3 a+2 b x)}{b^2 (-a+b x)}-\frac {3 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {2 \sqrt {x}}{b^{2}}-\frac {2 a \left (-\frac {\sqrt {x}}{2 \left (-b x +a \right )}+\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{2}}\) | \(48\) |
default | \(\frac {2 \sqrt {x}}{b^{2}}-\frac {2 a \left (-\frac {\sqrt {x}}{2 \left (-b x +a \right )}+\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{2}}\) | \(48\) |
risch | \(\frac {2 \sqrt {x}}{b^{2}}+\frac {a \left (-\frac {\sqrt {x}}{b x -a}-\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{b^{2}}\) | \(48\) |
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Time = 0.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.42 \[ \int \frac {x^{3/2}}{(-a+b x)^2} \, dx=\left [\frac {3 \, {\left (b x - a\right )} \sqrt {\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {\frac {a}{b}} + a}{b x - a}\right ) + 2 \, {\left (2 \, b x - 3 \, a\right )} \sqrt {x}}{2 \, {\left (b^{3} x - a b^{2}\right )}}, \frac {3 \, {\left (b x - a\right )} \sqrt {-\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {-\frac {a}{b}}}{a}\right ) + {\left (2 \, b x - 3 \, a\right )} \sqrt {x}}{b^{3} x - a b^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (49) = 98\).
Time = 5.09 (sec) , antiderivative size = 301, normalized size of antiderivative = 5.28 \[ \int \frac {x^{3/2}}{(-a+b x)^2} \, dx=\begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 a^{2}} & \text {for}\: b = 0 \\\frac {2 \sqrt {x}}{b^{2}} & \text {for}\: a = 0 \\- \frac {3 a^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{3} \sqrt {\frac {a}{b}} + 2 b^{4} x \sqrt {\frac {a}{b}}} + \frac {3 a^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{3} \sqrt {\frac {a}{b}} + 2 b^{4} x \sqrt {\frac {a}{b}}} - \frac {6 a b \sqrt {x} \sqrt {\frac {a}{b}}}{- 2 a b^{3} \sqrt {\frac {a}{b}} + 2 b^{4} x \sqrt {\frac {a}{b}}} + \frac {3 a b x \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{3} \sqrt {\frac {a}{b}} + 2 b^{4} x \sqrt {\frac {a}{b}}} - \frac {3 a b x \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{3} \sqrt {\frac {a}{b}} + 2 b^{4} x \sqrt {\frac {a}{b}}} + \frac {4 b^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{- 2 a b^{3} \sqrt {\frac {a}{b}} + 2 b^{4} x \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.19 \[ \int \frac {x^{3/2}}{(-a+b x)^2} \, dx=-\frac {a \sqrt {x}}{b^{3} x - a b^{2}} + \frac {3 \, a \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} + \frac {2 \, \sqrt {x}}{b^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {x^{3/2}}{(-a+b x)^2} \, dx=\frac {3 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} b^{2}} - \frac {a \sqrt {x}}{{\left (b x - a\right )} b^{2}} + \frac {2 \, \sqrt {x}}{b^{2}} \]
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Time = 0.14 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82 \[ \int \frac {x^{3/2}}{(-a+b x)^2} \, dx=\frac {2\,\sqrt {x}}{b^2}+\frac {a\,\sqrt {x}}{a\,b^2-b^3\,x}-\frac {3\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \]
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