Integrand size = 16, antiderivative size = 67 \[ \int \frac {(a-b x)^{3/2}}{x^{5/2}} \, dx=\frac {2 b \sqrt {a-b x}}{\sqrt {x}}-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}+2 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 67, 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} \[ \int \frac {(a-b x)^{3/2}}{x^{5/2}} \, dx=2 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}+\frac {2 b \sqrt {a-b x}}{\sqrt {x}} \]
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Rule 49
Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}-b \int \frac {\sqrt {a-b x}}{x^{3/2}} \, dx \\ & = \frac {2 b \sqrt {a-b x}}{\sqrt {x}}-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}+b^2 \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx \\ & = \frac {2 b \sqrt {a-b x}}{\sqrt {x}}-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 b \sqrt {a-b x}}{\sqrt {x}}-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right ) \\ & = \frac {2 b \sqrt {a-b x}}{\sqrt {x}}-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}+2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {(a-b x)^{3/2}}{x^{5/2}} \, dx=\frac {2 \sqrt {a-b x} (-a+4 b x)}{3 x^{3/2}}+4 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06
method | result | size |
risch | \(-\frac {2 \sqrt {-b x +a}\, \left (-4 b x +a \right )}{3 x^{\frac {3}{2}}}+\frac {b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{\sqrt {x}\, \sqrt {-b x +a}}\) | \(71\) |
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Time = 0.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.72 \[ \int \frac {(a-b x)^{3/2}}{x^{5/2}} \, dx=\left [\frac {3 \, \sqrt {-b} b x^{2} \log \left (-2 \, b x - 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (4 \, b x - a\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, x^{2}}, -\frac {2 \, {\left (3 \, b^{\frac {3}{2}} x^{2} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (4 \, b x - a\right )} \sqrt {-b x + a} \sqrt {x}\right )}}{3 \, x^{2}}\right ] \]
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Result contains complex when optimal does not.
Time = 2.44 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.79 \[ \int \frac {(a-b x)^{3/2}}{x^{5/2}} \, dx=\begin {cases} - \frac {2 a \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{3 x} + \frac {8 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}}{3} - 2 i b^{\frac {3}{2}} \log {\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} + i b^{\frac {3}{2}} \log {\left (\frac {a}{b x} \right )} + 2 b^{\frac {3}{2}} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i a \sqrt {b} \sqrt {- \frac {a}{b x} + 1}}{3 x} + \frac {8 i b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{3} + i b^{\frac {3}{2}} \log {\left (\frac {a}{b x} \right )} - 2 i b^{\frac {3}{2}} \log {\left (\sqrt {- \frac {a}{b x} + 1} + 1 \right )} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.73 \[ \int \frac {(a-b x)^{3/2}}{x^{5/2}} \, dx=-2 \, b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + \frac {2 \, \sqrt {-b x + a} b}{\sqrt {x}} - \frac {2 \, {\left (-b x + a\right )}^{\frac {3}{2}}}{3 \, x^{\frac {3}{2}}} \]
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Time = 76.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.40 \[ \int \frac {(a-b x)^{3/2}}{x^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, b^{2} \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b}} + \frac {{\left (4 \, {\left (b x - a\right )} b^{3} + 3 \, a b^{3}\right )} \sqrt {-b x + a}}{{\left ({\left (b x - a\right )} b + a b\right )}^{\frac {3}{2}}}\right )} b}{3 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a-b x)^{3/2}}{x^{5/2}} \, dx=\int \frac {{\left (a-b\,x\right )}^{3/2}}{x^{5/2}} \,d x \]
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