Integrand size = 15, antiderivative size = 95 \[ \int \frac {1}{x^3 (-a+b x)^{3/2}} \, dx=-\frac {15 b^2}{4 a^3 \sqrt {-a+b x}}+\frac {1}{2 a x^2 \sqrt {-a+b x}}+\frac {5 b}{4 a^2 x \sqrt {-a+b x}}-\frac {15 b^2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{7/2}} \]
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Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {44, 53, 65, 211} \[ \int \frac {1}{x^3 (-a+b x)^{3/2}} \, dx=-\frac {15 b^2 \arctan \left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{7/2}}-\frac {15 b^2}{4 a^3 \sqrt {b x-a}}+\frac {5 b}{4 a^2 x \sqrt {b x-a}}+\frac {1}{2 a x^2 \sqrt {b x-a}} \]
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Rule 44
Rule 53
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2 a x^2 \sqrt {-a+b x}}+\frac {(5 b) \int \frac {1}{x^2 (-a+b x)^{3/2}} \, dx}{4 a} \\ & = \frac {1}{2 a x^2 \sqrt {-a+b x}}+\frac {5 b}{4 a^2 x \sqrt {-a+b x}}+\frac {\left (15 b^2\right ) \int \frac {1}{x (-a+b x)^{3/2}} \, dx}{8 a^2} \\ & = -\frac {15 b^2}{4 a^3 \sqrt {-a+b x}}+\frac {1}{2 a x^2 \sqrt {-a+b x}}+\frac {5 b}{4 a^2 x \sqrt {-a+b x}}-\frac {\left (15 b^2\right ) \int \frac {1}{x \sqrt {-a+b x}} \, dx}{8 a^3} \\ & = -\frac {15 b^2}{4 a^3 \sqrt {-a+b x}}+\frac {1}{2 a x^2 \sqrt {-a+b x}}+\frac {5 b}{4 a^2 x \sqrt {-a+b x}}-\frac {(15 b) \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{4 a^3} \\ & = -\frac {15 b^2}{4 a^3 \sqrt {-a+b x}}+\frac {1}{2 a x^2 \sqrt {-a+b x}}+\frac {5 b}{4 a^2 x \sqrt {-a+b x}}-\frac {15 b^2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{7/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^3 (-a+b x)^{3/2}} \, dx=\frac {2 a^2+5 a b x-15 b^2 x^2}{4 a^3 x^2 \sqrt {-a+b x}}-\frac {15 b^2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{7/2}} \]
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Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.75
method | result | size |
pseudoelliptic | \(-\frac {15 \left (\arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right ) b^{2} x^{2} \sqrt {b x -a}+\sqrt {a}\, b^{2} x^{2}-\frac {a^{\frac {3}{2}} b x}{3}-\frac {2 a^{\frac {5}{2}}}{15}\right )}{4 \sqrt {b x -a}\, a^{\frac {7}{2}} x^{2}}\) | \(71\) |
risch | \(\frac {\left (-b x +a \right ) \left (7 b x +2 a \right )}{4 a^{3} x^{2} \sqrt {b x -a}}-\frac {2 b^{2}}{a^{3} \sqrt {b x -a}}-\frac {15 b^{2} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{4 a^{\frac {7}{2}}}\) | \(72\) |
derivativedivides | \(2 b^{2} \left (-\frac {1}{a^{3} \sqrt {b x -a}}-\frac {\frac {\frac {7 \left (b x -a \right )^{\frac {3}{2}}}{8}+\frac {9 a \sqrt {b x -a}}{8}}{b^{2} x^{2}}+\frac {15 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{3}}\right )\) | \(77\) |
default | \(2 b^{2} \left (-\frac {1}{a^{3} \sqrt {b x -a}}-\frac {\frac {\frac {7 \left (b x -a \right )^{\frac {3}{2}}}{8}+\frac {9 a \sqrt {b x -a}}{8}}{b^{2} x^{2}}+\frac {15 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{3}}\right )\) | \(77\) |
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Time = 0.23 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.08 \[ \int \frac {1}{x^3 (-a+b x)^{3/2}} \, dx=\left [-\frac {15 \, {\left (b^{3} x^{3} - a b^{2} x^{2}\right )} \sqrt {-a} \log \left (\frac {b x + 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, {\left (15 \, a b^{2} x^{2} - 5 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt {b x - a}}{8 \, {\left (a^{4} b x^{3} - a^{5} x^{2}\right )}}, -\frac {15 \, {\left (b^{3} x^{3} - a b^{2} x^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (15 \, a b^{2} x^{2} - 5 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt {b x - a}}{4 \, {\left (a^{4} b x^{3} - a^{5} x^{2}\right )}}\right ] \]
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Result contains complex when optimal does not.
Time = 5.50 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.38 \[ \int \frac {1}{x^3 (-a+b x)^{3/2}} \, dx=\begin {cases} - \frac {i}{2 a \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} - 1}} - \frac {5 i \sqrt {b}}{4 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}} + \frac {15 i b^{\frac {3}{2}}}{4 a^{3} \sqrt {x} \sqrt {\frac {a}{b x} - 1}} - \frac {15 i b^{2} \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {7}{2}}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {1}{2 a \sqrt {b} x^{\frac {5}{2}} \sqrt {- \frac {a}{b x} + 1}} + \frac {5 \sqrt {b}}{4 a^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}} - \frac {15 b^{\frac {3}{2}}}{4 a^{3} \sqrt {x} \sqrt {- \frac {a}{b x} + 1}} + \frac {15 b^{2} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 (-a+b x)^{3/2}} \, dx=-\frac {15 \, {\left (b x - a\right )}^{2} b^{2} + 25 \, {\left (b x - a\right )} a b^{2} + 8 \, a^{2} b^{2}}{4 \, {\left ({\left (b x - a\right )}^{\frac {5}{2}} a^{3} + 2 \, {\left (b x - a\right )}^{\frac {3}{2}} a^{4} + \sqrt {b x - a} a^{5}\right )}} - \frac {15 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {7}{2}}} \]
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Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^3 (-a+b x)^{3/2}} \, dx=-\frac {15 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {7}{2}}} - \frac {2 \, b^{2}}{\sqrt {b x - a} a^{3}} - \frac {7 \, {\left (b x - a\right )}^{\frac {3}{2}} b^{2} + 9 \, \sqrt {b x - a} a b^{2}}{4 \, a^{3} b^{2} x^{2}} \]
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Time = 0.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^3 (-a+b x)^{3/2}} \, dx=-\frac {\frac {2\,b^2}{a}+\frac {15\,b^2\,{\left (a-b\,x\right )}^2}{4\,a^3}-\frac {25\,b^2\,\left (a-b\,x\right )}{4\,a^2}}{2\,a\,{\left (b\,x-a\right )}^{3/2}+{\left (b\,x-a\right )}^{5/2}+a^2\,\sqrt {b\,x-a}}-\frac {15\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{4\,a^{7/2}} \]
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