Integrand size = 13, antiderivative size = 74 \[ \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx=-\frac {5 b}{3 a^2 (a+b x)^{3/2}}-\frac {1}{a x (a+b x)^{3/2}}-\frac {5 b}{a^3 \sqrt {a+b x}}+\frac {5 b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}} \]
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Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {44, 53, 65, 214} \[ \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx=\frac {5 b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {5 b}{a^3 \sqrt {a+b x}}-\frac {5 b}{3 a^2 (a+b x)^{3/2}}-\frac {1}{a x (a+b x)^{3/2}} \]
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Rule 44
Rule 53
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a x (a+b x)^{3/2}}-\frac {(5 b) \int \frac {1}{x (a+b x)^{5/2}} \, dx}{2 a} \\ & = -\frac {5 b}{3 a^2 (a+b x)^{3/2}}-\frac {1}{a x (a+b x)^{3/2}}-\frac {(5 b) \int \frac {1}{x (a+b x)^{3/2}} \, dx}{2 a^2} \\ & = -\frac {5 b}{3 a^2 (a+b x)^{3/2}}-\frac {1}{a x (a+b x)^{3/2}}-\frac {5 b}{a^3 \sqrt {a+b x}}-\frac {(5 b) \int \frac {1}{x \sqrt {a+b x}} \, dx}{2 a^3} \\ & = -\frac {5 b}{3 a^2 (a+b x)^{3/2}}-\frac {1}{a x (a+b x)^{3/2}}-\frac {5 b}{a^3 \sqrt {a+b x}}-\frac {5 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a^3} \\ & = -\frac {5 b}{3 a^2 (a+b x)^{3/2}}-\frac {1}{a x (a+b x)^{3/2}}-\frac {5 b}{a^3 \sqrt {a+b x}}+\frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx=\frac {-3 a^2-20 a b x-15 b^2 x^2}{3 a^3 x (a+b x)^{3/2}}+\frac {5 b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}} \]
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Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81
method | result | size |
risch | \(-\frac {\sqrt {b x +a}}{a^{3} x}-\frac {b \left (-\frac {10 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {8}{\sqrt {b x +a}}+\frac {4 a}{3 \left (b x +a \right )^{\frac {3}{2}}}\right )}{2 a^{3}}\) | \(60\) |
pseudoelliptic | \(\frac {5 \left (b x +a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b x -5 \sqrt {a}\, b^{2} x^{2}-\frac {20 a^{\frac {3}{2}} b x}{3}-a^{\frac {5}{2}}}{x \,a^{\frac {7}{2}} \left (b x +a \right )^{\frac {3}{2}}}\) | \(62\) |
derivativedivides | \(2 b \left (-\frac {1}{3 a^{2} \left (b x +a \right )^{\frac {3}{2}}}-\frac {2}{a^{3} \sqrt {b x +a}}+\frac {-\frac {\sqrt {b x +a}}{2 b x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}}{a^{3}}\right )\) | \(66\) |
default | \(2 b \left (-\frac {1}{3 a^{2} \left (b x +a \right )^{\frac {3}{2}}}-\frac {2}{a^{3} \sqrt {b x +a}}+\frac {-\frac {\sqrt {b x +a}}{2 b x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}}{a^{3}}\right )\) | \(66\) |
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Time = 0.23 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.99 \[ \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx=\left [\frac {15 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (15 \, a b^{2} x^{2} + 20 \, a^{2} b x + 3 \, a^{3}\right )} \sqrt {b x + a}}{6 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}}, -\frac {15 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (15 \, a b^{2} x^{2} + 20 \, a^{2} b x + 3 \, a^{3}\right )} \sqrt {b x + a}}{3 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (68) = 136\).
Time = 4.23 (sec) , antiderivative size = 818, normalized size of antiderivative = 11.05 \[ \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx=- \frac {6 a^{17} \sqrt {1 + \frac {b x}{a}}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {46 a^{16} b x \sqrt {1 + \frac {b x}{a}}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {15 a^{16} b x \log {\left (\frac {b x}{a} \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} + \frac {30 a^{16} b x \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {70 a^{15} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {45 a^{15} b^{2} x^{2} \log {\left (\frac {b x}{a} \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} + \frac {90 a^{15} b^{2} x^{2} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {30 a^{14} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {45 a^{14} b^{3} x^{3} \log {\left (\frac {b x}{a} \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} + \frac {90 a^{14} b^{3} x^{3} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {15 a^{13} b^{4} x^{4} \log {\left (\frac {b x}{a} \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} + \frac {30 a^{13} b^{4} x^{4} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx=-\frac {15 \, {\left (b x + a\right )}^{2} b - 10 \, {\left (b x + a\right )} a b - 2 \, a^{2} b}{3 \, {\left ({\left (b x + a\right )}^{\frac {5}{2}} a^{3} - {\left (b x + a\right )}^{\frac {3}{2}} a^{4}\right )}} - \frac {5 \, b \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{2 \, a^{\frac {7}{2}}} \]
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Time = 0.32 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx=-\frac {5 \, b \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} - \frac {2 \, {\left (6 \, {\left (b x + a\right )} b + a b\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3}} - \frac {\sqrt {b x + a}}{a^{3} x} \]
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Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx=\frac {5\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {\frac {2\,b}{3\,a}+\frac {10\,b\,\left (a+b\,x\right )}{3\,a^2}-\frac {5\,b\,{\left (a+b\,x\right )}^2}{a^3}}{a\,{\left (a+b\,x\right )}^{3/2}-{\left (a+b\,x\right )}^{5/2}} \]
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