Integrand size = 16, antiderivative size = 66 \[ \int \frac {(a-b x)^{3/2}}{x^{3/2}} \, dx=-3 b \sqrt {x} \sqrt {a-b x}-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-3 a \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {49, 52, 65, 223, 209} \[ \int \frac {(a-b x)^{3/2}}{x^{3/2}} \, dx=-3 a \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-3 b \sqrt {x} \sqrt {a-b x} \]
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Rule 49
Rule 52
Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-(3 b) \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx \\ & = -3 b \sqrt {x} \sqrt {a-b x}-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-\frac {1}{2} (3 a b) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx \\ & = -3 b \sqrt {x} \sqrt {a-b x}-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-(3 a b) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = -3 b \sqrt {x} \sqrt {a-b x}-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-(3 a b) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right ) \\ & = -3 b \sqrt {x} \sqrt {a-b x}-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-3 a \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int \frac {(a-b x)^{3/2}}{x^{3/2}} \, dx=\frac {(-2 a-b x) \sqrt {a-b x}}{\sqrt {x}}-6 a \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.12
method | result | size |
risch | \(-\frac {\sqrt {-b x +a}\, \left (b x +2 a \right )}{\sqrt {x}}-\frac {3 a \sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{2 \sqrt {x}\, \sqrt {-b x +a}}\) | \(74\) |
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none
Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.65 \[ \int \frac {(a-b x)^{3/2}}{x^{3/2}} \, dx=\left [\frac {3 \, a \sqrt {-b} x \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (b x + 2 \, a\right )} \sqrt {-b x + a} \sqrt {x}}{2 \, x}, \frac {3 \, a \sqrt {b} x \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (b x + 2 \, a\right )} \sqrt {-b x + a} \sqrt {x}}{x}\right ] \]
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Result contains complex when optimal does not.
Time = 2.90 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.98 \[ \int \frac {(a-b x)^{3/2}}{x^{3/2}} \, dx=\begin {cases} \frac {2 i a^{\frac {3}{2}}}{\sqrt {x} \sqrt {-1 + \frac {b x}{a}}} - \frac {i \sqrt {a} b \sqrt {x}}{\sqrt {-1 + \frac {b x}{a}}} + 3 i a \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - \frac {i b^{2} x^{\frac {3}{2}}}{\sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {2 a^{\frac {3}{2}}}{\sqrt {x} \sqrt {1 - \frac {b x}{a}}} + \frac {\sqrt {a} b \sqrt {x}}{\sqrt {1 - \frac {b x}{a}}} - 3 a \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} + \frac {b^{2} x^{\frac {3}{2}}}{\sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int \frac {(a-b x)^{3/2}}{x^{3/2}} \, dx=3 \, a \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - \frac {2 \, \sqrt {-b x + a} a}{\sqrt {x}} - \frac {\sqrt {-b x + a} a b}{{\left (b - \frac {b x - a}{x}\right )} \sqrt {x}} \]
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Time = 78.77 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.24 \[ \int \frac {(a-b x)^{3/2}}{x^{3/2}} \, dx=-\frac {{\left (\frac {3 \, a \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b}} + \frac {{\left (b x + 2 \, a\right )} \sqrt {-b x + a}}{\sqrt {{\left (b x - a\right )} b + a b}}\right )} b^{2}}{{\left | b \right |}} \]
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Timed out. \[ \int \frac {(a-b x)^{3/2}}{x^{3/2}} \, dx=\int \frac {{\left (a-b\,x\right )}^{3/2}}{x^{3/2}} \,d x \]
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