Integrand size = 15, antiderivative size = 116 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=-\frac {35 b^2}{12 a^3 (-a+b x)^{3/2}}+\frac {1}{2 a x^2 (-a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (-a+b x)^{3/2}}+\frac {35 b^2}{4 a^4 \sqrt {-a+b x}}+\frac {35 b^2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
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Time = 0.02 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {44, 53, 65, 211} \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\frac {35 b^2 \arctan \left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{9/2}}+\frac {35 b^2}{4 a^4 \sqrt {b x-a}}-\frac {35 b^2}{12 a^3 (b x-a)^{3/2}}+\frac {7 b}{4 a^2 x (b x-a)^{3/2}}+\frac {1}{2 a x^2 (b x-a)^{3/2}} \]
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Rule 44
Rule 53
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2 a x^2 (-a+b x)^{3/2}}+\frac {(7 b) \int \frac {1}{x^2 (-a+b x)^{5/2}} \, dx}{4 a} \\ & = \frac {1}{2 a x^2 (-a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (-a+b x)^{3/2}}+\frac {\left (35 b^2\right ) \int \frac {1}{x (-a+b x)^{5/2}} \, dx}{8 a^2} \\ & = -\frac {35 b^2}{12 a^3 (-a+b x)^{3/2}}+\frac {1}{2 a x^2 (-a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (-a+b x)^{3/2}}-\frac {\left (35 b^2\right ) \int \frac {1}{x (-a+b x)^{3/2}} \, dx}{8 a^3} \\ & = -\frac {35 b^2}{12 a^3 (-a+b x)^{3/2}}+\frac {1}{2 a x^2 (-a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (-a+b x)^{3/2}}+\frac {35 b^2}{4 a^4 \sqrt {-a+b x}}+\frac {\left (35 b^2\right ) \int \frac {1}{x \sqrt {-a+b x}} \, dx}{8 a^4} \\ & = -\frac {35 b^2}{12 a^3 (-a+b x)^{3/2}}+\frac {1}{2 a x^2 (-a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (-a+b x)^{3/2}}+\frac {35 b^2}{4 a^4 \sqrt {-a+b x}}+\frac {(35 b) \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{4 a^4} \\ & = -\frac {35 b^2}{12 a^3 (-a+b x)^{3/2}}+\frac {1}{2 a x^2 (-a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (-a+b x)^{3/2}}+\frac {35 b^2}{4 a^4 \sqrt {-a+b x}}+\frac {35 b^2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\frac {6 a^3+21 a^2 b x-140 a b^2 x^2+105 b^3 x^3}{12 a^4 x^2 (-a+b x)^{3/2}}+\frac {35 b^2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
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Time = 0.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77
method | result | size |
risch | \(-\frac {\left (-b x +a \right ) \left (11 b x +2 a \right )}{4 a^{4} x^{2} \sqrt {b x -a}}+\frac {35 b^{2} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{4 a^{\frac {9}{2}}}+\frac {6 b^{2}}{a^{4} \sqrt {b x -a}}-\frac {2 b^{2}}{3 a^{3} \left (b x -a \right )^{\frac {3}{2}}}\) | \(89\) |
pseudoelliptic | \(-\frac {35 \left (\sqrt {b x -a}\, b^{2} x^{2} \left (-b x +a \right ) \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )-\sqrt {a}\, b^{3} x^{3}+\frac {4 a^{\frac {3}{2}} b^{2} x^{2}}{3}-\frac {a^{\frac {5}{2}} b x}{5}-\frac {2 a^{\frac {7}{2}}}{35}\right )}{4 \left (b x -a \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{2}}\) | \(89\) |
derivativedivides | \(2 b^{2} \left (-\frac {1}{3 a^{3} \left (b x -a \right )^{\frac {3}{2}}}+\frac {3}{a^{4} \sqrt {b x -a}}+\frac {\frac {\frac {11 \left (b x -a \right )^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {b x -a}}{8}}{b^{2} x^{2}}+\frac {35 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}\right )\) | \(90\) |
default | \(2 b^{2} \left (-\frac {1}{3 a^{3} \left (b x -a \right )^{\frac {3}{2}}}+\frac {3}{a^{4} \sqrt {b x -a}}+\frac {\frac {\frac {11 \left (b x -a \right )^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {b x -a}}{8}}{b^{2} x^{2}}+\frac {35 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}\right )\) | \(90\) |
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Time = 0.24 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.24 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\left [-\frac {105 \, {\left (b^{4} x^{4} - 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {-a} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) - 2 \, {\left (105 \, a b^{3} x^{3} - 140 \, a^{2} b^{2} x^{2} + 21 \, a^{3} b x + 6 \, a^{4}\right )} \sqrt {b x - a}}{24 \, {\left (a^{5} b^{2} x^{4} - 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}, \frac {105 \, {\left (b^{4} x^{4} - 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (105 \, a b^{3} x^{3} - 140 \, a^{2} b^{2} x^{2} + 21 \, a^{3} b x + 6 \, a^{4}\right )} \sqrt {b x - a}}{12 \, {\left (a^{5} b^{2} x^{4} - 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}\right ] \]
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Result contains complex when optimal does not.
Time = 17.58 (sec) , antiderivative size = 1108, normalized size of antiderivative = 9.55 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\text {Too large to display} \]
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Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\frac {105 \, {\left (b x - a\right )}^{3} b^{2} + 175 \, {\left (b x - a\right )}^{2} a b^{2} + 56 \, {\left (b x - a\right )} a^{2} b^{2} - 8 \, a^{3} b^{2}}{12 \, {\left ({\left (b x - a\right )}^{\frac {7}{2}} a^{4} + 2 \, {\left (b x - a\right )}^{\frac {5}{2}} a^{5} + {\left (b x - a\right )}^{\frac {3}{2}} a^{6}\right )}} + \frac {35 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {9}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\frac {35 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {9}{2}}} + \frac {2 \, {\left (9 \, {\left (b x - a\right )} b^{2} - a b^{2}\right )}}{3 \, {\left (b x - a\right )}^{\frac {3}{2}} a^{4}} + \frac {11 \, {\left (b x - a\right )}^{\frac {3}{2}} b^{2} + 13 \, \sqrt {b x - a} a b^{2}}{4 \, a^{4} b^{2} x^{2}} \]
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Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\frac {35\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{4\,a^{9/2}}-\frac {\frac {2\,b^2}{3\,a}-\frac {175\,b^2\,{\left (a-b\,x\right )}^2}{12\,a^3}+\frac {35\,b^2\,{\left (a-b\,x\right )}^3}{4\,a^4}+\frac {14\,b^2\,\left (a-b\,x\right )}{3\,a^2}}{2\,a\,{\left (b\,x-a\right )}^{5/2}+{\left (b\,x-a\right )}^{7/2}+a^2\,{\left (b\,x-a\right )}^{3/2}} \]
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