Integrand size = 15, antiderivative size = 98 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=-\frac {9 b^3 \sqrt {x}}{a^5}+\frac {3 b^2 x^{3/2}}{a^4}-\frac {9 b x^{5/2}}{5 a^3}+\frac {9 x^{7/2}}{7 a^2}-\frac {x^{9/2}}{a (b+a x)}+\frac {9 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{11/2}} \]
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Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {269, 43, 52, 65, 211} \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {9 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{11/2}}-\frac {9 b^3 \sqrt {x}}{a^5}+\frac {3 b^2 x^{3/2}}{a^4}-\frac {9 b x^{5/2}}{5 a^3}+\frac {9 x^{7/2}}{7 a^2}-\frac {x^{9/2}}{a (a x+b)} \]
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Rule 43
Rule 52
Rule 65
Rule 211
Rule 269
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{9/2}}{(b+a x)^2} \, dx \\ & = -\frac {x^{9/2}}{a (b+a x)}+\frac {9 \int \frac {x^{7/2}}{b+a x} \, dx}{2 a} \\ & = \frac {9 x^{7/2}}{7 a^2}-\frac {x^{9/2}}{a (b+a x)}-\frac {(9 b) \int \frac {x^{5/2}}{b+a x} \, dx}{2 a^2} \\ & = -\frac {9 b x^{5/2}}{5 a^3}+\frac {9 x^{7/2}}{7 a^2}-\frac {x^{9/2}}{a (b+a x)}+\frac {\left (9 b^2\right ) \int \frac {x^{3/2}}{b+a x} \, dx}{2 a^3} \\ & = \frac {3 b^2 x^{3/2}}{a^4}-\frac {9 b x^{5/2}}{5 a^3}+\frac {9 x^{7/2}}{7 a^2}-\frac {x^{9/2}}{a (b+a x)}-\frac {\left (9 b^3\right ) \int \frac {\sqrt {x}}{b+a x} \, dx}{2 a^4} \\ & = -\frac {9 b^3 \sqrt {x}}{a^5}+\frac {3 b^2 x^{3/2}}{a^4}-\frac {9 b x^{5/2}}{5 a^3}+\frac {9 x^{7/2}}{7 a^2}-\frac {x^{9/2}}{a (b+a x)}+\frac {\left (9 b^4\right ) \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{2 a^5} \\ & = -\frac {9 b^3 \sqrt {x}}{a^5}+\frac {3 b^2 x^{3/2}}{a^4}-\frac {9 b x^{5/2}}{5 a^3}+\frac {9 x^{7/2}}{7 a^2}-\frac {x^{9/2}}{a (b+a x)}+\frac {\left (9 b^4\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{a^5} \\ & = -\frac {9 b^3 \sqrt {x}}{a^5}+\frac {3 b^2 x^{3/2}}{a^4}-\frac {9 b x^{5/2}}{5 a^3}+\frac {9 x^{7/2}}{7 a^2}-\frac {x^{9/2}}{a (b+a x)}+\frac {9 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{11/2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.92 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {\sqrt {x} \left (-315 b^4-210 a b^3 x+42 a^2 b^2 x^2-18 a^3 b x^3+10 a^4 x^4\right )}{35 a^5 (b+a x)}+\frac {9 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{11/2}} \]
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Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {2 \left (5 a^{3} x^{3}-14 a^{2} b \,x^{2}+35 a \,b^{2} x -140 b^{3}\right ) \sqrt {x}}{35 a^{5}}+\frac {b^{4} \left (-\frac {\sqrt {x}}{a x +b}+\frac {9 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{a^{5}}\) | \(78\) |
derivativedivides | \(\frac {\frac {2 a^{3} x^{\frac {7}{2}}}{7}-\frac {4 a^{2} b \,x^{\frac {5}{2}}}{5}+2 a \,b^{2} x^{\frac {3}{2}}-8 b^{3} \sqrt {x}}{a^{5}}+\frac {2 b^{4} \left (-\frac {\sqrt {x}}{2 \left (a x +b \right )}+\frac {9 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{5}}\) | \(80\) |
default | \(\frac {\frac {2 a^{3} x^{\frac {7}{2}}}{7}-\frac {4 a^{2} b \,x^{\frac {5}{2}}}{5}+2 a \,b^{2} x^{\frac {3}{2}}-8 b^{3} \sqrt {x}}{a^{5}}+\frac {2 b^{4} \left (-\frac {\sqrt {x}}{2 \left (a x +b \right )}+\frac {9 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{5}}\) | \(80\) |
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Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.13 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\left [\frac {315 \, {\left (a b^{3} x + b^{4}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, {\left (10 \, a^{4} x^{4} - 18 \, a^{3} b x^{3} + 42 \, a^{2} b^{2} x^{2} - 210 \, a b^{3} x - 315 \, b^{4}\right )} \sqrt {x}}{70 \, {\left (a^{6} x + a^{5} b\right )}}, \frac {315 \, {\left (a b^{3} x + b^{4}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) + {\left (10 \, a^{4} x^{4} - 18 \, a^{3} b x^{3} + 42 \, a^{2} b^{2} x^{2} - 210 \, a b^{3} x - 315 \, b^{4}\right )} \sqrt {x}}{35 \, {\left (a^{6} x + a^{5} b\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (92) = 184\).
Time = 25.58 (sec) , antiderivative size = 495, normalized size of antiderivative = 5.05 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\begin {cases} \tilde {\infty } x^{\frac {11}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {11}{2}}}{11 b^{2}} & \text {for}\: a = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a^{2}} & \text {for}\: b = 0 \\\frac {20 a^{5} x^{\frac {9}{2}} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {36 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} + \frac {84 a^{3} b^{2} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {420 a^{2} b^{3} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {630 a b^{4} \sqrt {x} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} + \frac {315 a b^{4} x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {315 a b^{4} x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} + \frac {315 b^{5} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {315 b^{5} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.90 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {10 \, a^{4} - \frac {18 \, a^{3} b}{x} + \frac {42 \, a^{2} b^{2}}{x^{2}} - \frac {210 \, a b^{3}}{x^{3}} - \frac {315 \, b^{4}}{x^{4}}}{35 \, {\left (\frac {a^{6}}{x^{\frac {7}{2}}} + \frac {a^{5} b}{x^{\frac {9}{2}}}\right )}} - \frac {9 \, b^{4} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} a^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.90 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {9 \, b^{4} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} - \frac {b^{4} \sqrt {x}}{{\left (a x + b\right )} a^{5}} + \frac {2 \, {\left (5 \, a^{12} x^{\frac {7}{2}} - 14 \, a^{11} b x^{\frac {5}{2}} + 35 \, a^{10} b^{2} x^{\frac {3}{2}} - 140 \, a^{9} b^{3} \sqrt {x}\right )}}{35 \, a^{14}} \]
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Time = 5.60 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {2\,x^{7/2}}{7\,a^2}-\frac {4\,b\,x^{5/2}}{5\,a^3}+\frac {2\,b^2\,x^{3/2}}{a^4}-\frac {8\,b^3\,\sqrt {x}}{a^5}-\frac {b^4\,\sqrt {x}}{x\,a^6+b\,a^5}+\frac {9\,b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{a^{11/2}} \]
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