Integrand size = 15, antiderivative size = 69 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=-\frac {5}{3 b^2 x^{3/2}}+\frac {5 a}{b^3 \sqrt {x}}+\frac {1}{b x^{3/2} (b+a x)}+\frac {5 a^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \]
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Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {269, 44, 53, 65, 211} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\frac {5 a^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}}+\frac {5 a}{b^3 \sqrt {x}}+\frac {1}{b x^{3/2} (a x+b)}-\frac {5}{3 b^2 x^{3/2}} \]
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Rule 44
Rule 53
Rule 65
Rule 211
Rule 269
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^{5/2} (b+a x)^2} \, dx \\ & = \frac {1}{b x^{3/2} (b+a x)}+\frac {5 \int \frac {1}{x^{5/2} (b+a x)} \, dx}{2 b} \\ & = -\frac {5}{3 b^2 x^{3/2}}+\frac {1}{b x^{3/2} (b+a x)}-\frac {(5 a) \int \frac {1}{x^{3/2} (b+a x)} \, dx}{2 b^2} \\ & = -\frac {5}{3 b^2 x^{3/2}}+\frac {5 a}{b^3 \sqrt {x}}+\frac {1}{b x^{3/2} (b+a x)}+\frac {\left (5 a^2\right ) \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{2 b^3} \\ & = -\frac {5}{3 b^2 x^{3/2}}+\frac {5 a}{b^3 \sqrt {x}}+\frac {1}{b x^{3/2} (b+a x)}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{b^3} \\ & = -\frac {5}{3 b^2 x^{3/2}}+\frac {5 a}{b^3 \sqrt {x}}+\frac {1}{b x^{3/2} (b+a x)}+\frac {5 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\frac {-2 b^2+10 a b x+15 a^2 x^2}{3 b^3 x^{3/2} (b+a x)}+\frac {5 a^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \]
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Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {4 a x -\frac {2 b}{3}}{b^{3} x^{\frac {3}{2}}}+\frac {a^{2} \left (\frac {\sqrt {x}}{a x +b}+\frac {5 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{b^{3}}\) | \(55\) |
derivativedivides | \(-\frac {2}{3 b^{2} x^{\frac {3}{2}}}+\frac {4 a}{b^{3} \sqrt {x}}+\frac {2 a^{2} \left (\frac {\sqrt {x}}{2 a x +2 b}+\frac {5 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{3}}\) | \(58\) |
default | \(-\frac {2}{3 b^{2} x^{\frac {3}{2}}}+\frac {4 a}{b^{3} \sqrt {x}}+\frac {2 a^{2} \left (\frac {\sqrt {x}}{2 a x +2 b}+\frac {5 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{3}}\) | \(58\) |
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Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.67 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\left [\frac {15 \, {\left (a^{2} x^{3} + a b x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {a x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - b}{a x + b}\right ) + 2 \, {\left (15 \, a^{2} x^{2} + 10 \, a b x - 2 \, b^{2}\right )} \sqrt {x}}{6 \, {\left (a b^{3} x^{3} + b^{4} x^{2}\right )}}, -\frac {15 \, {\left (a^{2} x^{3} + a b x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {\frac {a}{b}}}{a \sqrt {x}}\right ) - {\left (15 \, a^{2} x^{2} + 10 \, a b x - 2 \, b^{2}\right )} \sqrt {x}}{3 \, {\left (a b^{3} x^{3} + b^{4} x^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (65) = 130\).
Time = 116.53 (sec) , antiderivative size = 452, normalized size of antiderivative = 6.55 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 b^{2} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\- \frac {2}{7 a^{2} x^{\frac {7}{2}}} & \text {for}\: b = 0 \\\frac {15 a^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} - \frac {15 a^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} + \frac {30 a^{2} x^{2} \sqrt {- \frac {b}{a}}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} + \frac {15 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} - \frac {15 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} + \frac {20 a b x \sqrt {- \frac {b}{a}}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} - \frac {4 b^{2} \sqrt {- \frac {b}{a}}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\frac {a^{2}}{{\left (a b^{3} + \frac {b^{4}}{x}\right )} \sqrt {x}} - \frac {5 \, a^{2} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} b^{3}} + \frac {2 \, {\left (\frac {6 \, a}{\sqrt {x}} - \frac {b}{x^{\frac {3}{2}}}\right )}}{3 \, b^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\frac {5 \, a^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {a^{2} \sqrt {x}}{{\left (a x + b\right )} b^{3}} + \frac {2 \, {\left (6 \, a x - b\right )}}{3 \, b^{3} x^{\frac {3}{2}}} \]
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Time = 5.67 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\frac {\frac {5\,a^2\,x^2}{b^3}-\frac {2}{3\,b}+\frac {10\,a\,x}{3\,b^2}}{a\,x^{5/2}+b\,x^{3/2}}+\frac {5\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \]
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