Integrand size = 16, antiderivative size = 72 \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\frac {2 x^{3/2}}{3 b (a-b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a-b x}}+\frac {2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{5/2}} \]
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Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {49, 65, 223, 209} \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{5/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a-b x}}+\frac {2 x^{3/2}}{3 b (a-b x)^{3/2}} \]
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Rule 49
Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{3/2}}{3 b (a-b x)^{3/2}}-\frac {\int \frac {\sqrt {x}}{(a-b x)^{3/2}} \, dx}{b} \\ & = \frac {2 x^{3/2}}{3 b (a-b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a-b x}}+\frac {\int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{b^2} \\ & = \frac {2 x^{3/2}}{3 b (a-b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a-b x}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{b^2} \\ & = \frac {2 x^{3/2}}{3 b (a-b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a-b x}}+\frac {2 \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{b^2} \\ & = \frac {2 x^{3/2}}{3 b (a-b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a-b x}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\frac {2 \sqrt {x} (-3 a+4 b x)}{3 b^2 (a-b x)^{3/2}}+\frac {4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{b^{5/2}} \]
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\[\int \frac {x^{\frac {3}{2}}}{\left (-b x +a \right )^{\frac {5}{2}}}d x\]
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none
Time = 0.24 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.61 \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\left [-\frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (4 \, b^{2} x - 3 \, a b\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac {2 \, {\left (3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (4 \, b^{2} x - 3 \, a b\right )} \sqrt {-b x + a} \sqrt {x}\right )}}{3 \, {\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]
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Result contains complex when optimal does not.
Time = 2.49 (sec) , antiderivative size = 833, normalized size of antiderivative = 11.57 \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\begin {cases} - \frac {6 i a^{\frac {39}{2}} b^{11} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}} + \frac {3 \pi a^{\frac {39}{2}} b^{11} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}} + \frac {6 i a^{\frac {37}{2}} b^{12} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}} - \frac {3 \pi a^{\frac {37}{2}} b^{12} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}} + \frac {6 i a^{19} b^{\frac {23}{2}} x^{14}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}} - \frac {8 i a^{18} b^{\frac {25}{2}} x^{15}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {6 a^{\frac {39}{2}} b^{11} x^{\frac {27}{2}} \sqrt {1 - \frac {b x}{a}} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 - \frac {b x}{a}}} - \frac {6 a^{\frac {37}{2}} b^{12} x^{\frac {29}{2}} \sqrt {1 - \frac {b x}{a}} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 - \frac {b x}{a}}} - \frac {6 a^{19} b^{\frac {23}{2}} x^{14}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 - \frac {b x}{a}}} + \frac {8 a^{18} b^{\frac {25}{2}} x^{15}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.72 \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\frac {2 \, {\left (b + \frac {3 \, {\left (b x - a\right )}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (-b x + a\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {5}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (54) = 108\).
Time = 16.02 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.69 \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\frac {{\left (\frac {3 \, \log \left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{\sqrt {-b}} + \frac {8 \, {\left (3 \, a {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{4} \sqrt {-b} - 3 \, a^{2} {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} \sqrt {-b} b + 2 \, a^{3} \sqrt {-b} b^{2}\right )}}{{\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3}}\right )} {\left | b \right |}}{3 \, b^{3}} \]
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Timed out. \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\int \frac {x^{3/2}}{{\left (a-b\,x\right )}^{5/2}} \,d x \]
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