Integrand size = 15, antiderivative size = 74 \[ \int \frac {1}{x^3 \sqrt {-a+b x}} \, dx=\frac {\sqrt {-a+b x}}{2 a x^2}+\frac {3 b \sqrt {-a+b x}}{4 a^2 x}+\frac {3 b^2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{5/2}} \]
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Time = 0.01 (sec) , antiderivative size = 74, 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 = {44, 65, 211} \[ \int \frac {1}{x^3 \sqrt {-a+b x}} \, dx=\frac {3 b^2 \arctan \left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{5/2}}+\frac {3 b \sqrt {b x-a}}{4 a^2 x}+\frac {\sqrt {b x-a}}{2 a x^2} \]
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Rule 44
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-a+b x}}{2 a x^2}+\frac {(3 b) \int \frac {1}{x^2 \sqrt {-a+b x}} \, dx}{4 a} \\ & = \frac {\sqrt {-a+b x}}{2 a x^2}+\frac {3 b \sqrt {-a+b x}}{4 a^2 x}+\frac {\left (3 b^2\right ) \int \frac {1}{x \sqrt {-a+b x}} \, dx}{8 a^2} \\ & = \frac {\sqrt {-a+b x}}{2 a x^2}+\frac {3 b \sqrt {-a+b x}}{4 a^2 x}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{4 a^2} \\ & = \frac {\sqrt {-a+b x}}{2 a x^2}+\frac {3 b \sqrt {-a+b x}}{4 a^2 x}+\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{5/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^3 \sqrt {-a+b x}} \, dx=\frac {\sqrt {-a+b x} (2 a+3 b x)}{4 a^2 x^2}+\frac {3 b^2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74
method | result | size |
risch | \(-\frac {\left (-b x +a \right ) \left (3 b x +2 a \right )}{4 a^{2} x^{2} \sqrt {b x -a}}+\frac {3 b^{2} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{4 a^{\frac {5}{2}}}\) | \(55\) |
pseudoelliptic | \(\frac {3 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right ) b^{2} x^{2}+3 b x \sqrt {b x -a}\, \sqrt {a}+2 \sqrt {b x -a}\, a^{\frac {3}{2}}}{4 a^{\frac {5}{2}} x^{2}}\) | \(62\) |
derivativedivides | \(2 b^{2} \left (\frac {\sqrt {b x -a}}{4 a \,b^{2} x^{2}}+\frac {\frac {3 \sqrt {b x -a}}{8 a b x}+\frac {3 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}}{a}\right )\) | \(72\) |
default | \(2 b^{2} \left (\frac {\sqrt {b x -a}}{4 a \,b^{2} x^{2}}+\frac {\frac {3 \sqrt {b x -a}}{8 a b x}+\frac {3 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}}{a}\right )\) | \(72\) |
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Time = 0.23 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.73 \[ \int \frac {1}{x^3 \sqrt {-a+b x}} \, dx=\left [-\frac {3 \, \sqrt {-a} b^{2} x^{2} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) - 2 \, {\left (3 \, a b x + 2 \, a^{2}\right )} \sqrt {b x - a}}{8 \, a^{3} x^{2}}, \frac {3 \, \sqrt {a} b^{2} x^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (3 \, a b x + 2 \, a^{2}\right )} \sqrt {b x - a}}{4 \, a^{3} x^{2}}\right ] \]
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Result contains complex when optimal does not.
Time = 3.53 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.92 \[ \int \frac {1}{x^3 \sqrt {-a+b x}} \, dx=\begin {cases} \frac {i}{2 \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} - 1}} + \frac {i \sqrt {b}}{4 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}} - \frac {3 i b^{\frac {3}{2}}}{4 a^{2} \sqrt {x} \sqrt {\frac {a}{b x} - 1}} + \frac {3 i b^{2} \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {5}{2}}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {1}{2 \sqrt {b} x^{\frac {5}{2}} \sqrt {- \frac {a}{b x} + 1}} - \frac {\sqrt {b}}{4 a x^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}} + \frac {3 b^{\frac {3}{2}}}{4 a^{2} \sqrt {x} \sqrt {- \frac {a}{b x} + 1}} - \frac {3 b^{2} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^3 \sqrt {-a+b x}} \, dx=\frac {3 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {5}{2}}} + \frac {3 \, {\left (b x - a\right )}^{\frac {3}{2}} b^{2} + 5 \, \sqrt {b x - a} a b^{2}}{4 \, {\left ({\left (b x - a\right )}^{2} a^{2} + 2 \, {\left (b x - a\right )} a^{3} + a^{4}\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^3 \sqrt {-a+b x}} \, dx=\frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {3 \, {\left (b x - a\right )}^{\frac {3}{2}} b^{3} + 5 \, \sqrt {b x - a} a b^{3}}{a^{2} b^{2} x^{2}}}{4 \, b} \]
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Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^3 \sqrt {-a+b x}} \, dx=\frac {3\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{4\,a^{5/2}}+\frac {5\,\sqrt {b\,x-a}}{4\,a\,x^2}+\frac {3\,{\left (b\,x-a\right )}^{3/2}}{4\,a^2\,x^2} \]
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