Integrand size = 15, antiderivative size = 72 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=-\frac {x^{3/2}}{2 b (a-b x)^2}+\frac {3 \sqrt {x}}{4 b^2 (a-b x)}-\frac {3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}} \]
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Time = 0.01 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {43, 65, 214} \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=-\frac {3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}}+\frac {3 \sqrt {x}}{4 b^2 (a-b x)}-\frac {x^{3/2}}{2 b (a-b x)^2} \]
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Rule 43
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{3/2}}{2 b (a-b x)^2}+\frac {3 \int \frac {\sqrt {x}}{(-a+b x)^2} \, dx}{4 b} \\ & = -\frac {x^{3/2}}{2 b (a-b x)^2}+\frac {3 \sqrt {x}}{4 b^2 (a-b x)}+\frac {3 \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{8 b^2} \\ & = -\frac {x^{3/2}}{2 b (a-b x)^2}+\frac {3 \sqrt {x}}{4 b^2 (a-b x)}+\frac {3 \text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{4 b^2} \\ & = -\frac {x^{3/2}}{2 b (a-b x)^2}+\frac {3 \sqrt {x}}{4 b^2 (a-b x)}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.83 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=\frac {\sqrt {x} (3 a-5 b x)}{4 b^2 (a-b x)^2}-\frac {3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {5 x^{\frac {3}{2}}}{8 b}-\frac {3 a \sqrt {x}}{8 b^{2}}\right )}{\left (-b x +a \right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b^{2} \sqrt {a b}}\) | \(51\) |
default | \(-\frac {2 \left (\frac {5 x^{\frac {3}{2}}}{8 b}-\frac {3 a \sqrt {x}}{8 b^{2}}\right )}{\left (-b x +a \right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b^{2} \sqrt {a b}}\) | \(51\) |
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Time = 0.24 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.58 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=\left [\frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {a b} \log \left (\frac {b x + a - 2 \, \sqrt {a b} \sqrt {x}}{b x - a}\right ) - 2 \, {\left (5 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt {x}}{8 \, {\left (a b^{5} x^{2} - 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}, \frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {-a b} \arctan \left (\frac {\sqrt {-a b}}{b \sqrt {x}}\right ) - {\left (5 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt {x}}{4 \, {\left (a b^{5} x^{2} - 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (61) = 122\).
Time = 15.79 (sec) , antiderivative size = 552, normalized size of antiderivative = 7.67 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2 x^{\frac {5}{2}}}{5 a^{3}} & \text {for}\: b = 0 \\- \frac {2}{b^{3} \sqrt {x}} & \text {for}\: a = 0 \\\frac {3 a^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} - \frac {3 a^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} + \frac {6 a b \sqrt {x} \sqrt {\frac {a}{b}}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} - \frac {6 a b x \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} + \frac {6 a b x \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} - \frac {10 b^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} + \frac {3 b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} - \frac {3 b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.08 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=-\frac {5 \, b x^{\frac {3}{2}} - 3 \, a \sqrt {x}}{4 \, {\left (b^{4} x^{2} - 2 \, a b^{3} x + a^{2} b^{2}\right )}} + \frac {3 \, \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.71 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=\frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 \, \sqrt {-a b} b^{2}} - \frac {5 \, b x^{\frac {3}{2}} - 3 \, a \sqrt {x}}{4 \, {\left (b x - a\right )}^{2} b^{2}} \]
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Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.81 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=-\frac {\frac {5\,x^{3/2}}{4\,b}-\frac {3\,a\,\sqrt {x}}{4\,b^2}}{a^2-2\,a\,b\,x+b^2\,x^2}-\frac {3\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,\sqrt {a}\,b^{5/2}} \]
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