Integrand size = 13, antiderivative size = 54 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {2}{3 a (a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {a+b x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {53, 65, 214} \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2}{a^2 \sqrt {a+b x}}+\frac {2}{3 a (a+b x)^{3/2}} \]
[In]
[Out]
Rule 53
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3 a (a+b x)^{3/2}}+\frac {\int \frac {1}{x (a+b x)^{3/2}} \, dx}{a} \\ & = \frac {2}{3 a (a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {a+b x}}+\frac {\int \frac {1}{x \sqrt {a+b x}} \, dx}{a^2} \\ & = \frac {2}{3 a (a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {a+b x}}+\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a^2 b} \\ & = \frac {2}{3 a (a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {a+b x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {2 (a+3 (a+b x))}{3 a^2 (a+b x)^{3/2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {2}{3 a \left (b x +a \right )^{\frac {3}{2}}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2}{a^{2} \sqrt {b x +a}}\) | \(43\) |
default | \(\frac {2}{3 a \left (b x +a \right )^{\frac {3}{2}}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2}{a^{2} \sqrt {b x +a}}\) | \(43\) |
pseudoelliptic | \(-\frac {2 \left (\left (b x +a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )-\sqrt {a}\, b x -\frac {4 a^{\frac {3}{2}}}{3}\right )}{a^{\frac {5}{2}} \left (b x +a \right )^{\frac {3}{2}}}\) | \(46\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (42) = 84\).
Time = 0.24 (sec) , antiderivative size = 177, normalized size of antiderivative = 3.28 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\left [\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (3 \, a b x + 4 \, a^{2}\right )} \sqrt {b x + a}}{3 \, {\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )}}, \frac {2 \, {\left (3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b x + 4 \, a^{2}\right )} \sqrt {b x + a}\right )}}{3 \, {\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (48) = 96\).
Time = 1.92 (sec) , antiderivative size = 697, normalized size of antiderivative = 12.91 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {8 a^{7} \sqrt {1 + \frac {b x}{a}}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {3 a^{7} \log {\left (\frac {b x}{a} \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} - \frac {6 a^{7} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {14 a^{6} b x \sqrt {1 + \frac {b x}{a}}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {9 a^{6} b x \log {\left (\frac {b x}{a} \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} - \frac {18 a^{6} b x \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {6 a^{5} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {9 a^{5} b^{2} x^{2} \log {\left (\frac {b x}{a} \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} - \frac {18 a^{5} b^{2} x^{2} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {3 a^{4} b^{3} x^{3} \log {\left (\frac {b x}{a} \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} - \frac {6 a^{4} b^{3} x^{3} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {\log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x + 4 \, a\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {2 \, {\left (3 \, b x + 4 \, a\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {\frac {2\,\left (a+b\,x\right )}{a^2}+\frac {2}{3\,a}}{{\left (a+b\,x\right )}^{3/2}}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{a^{5/2}} \]
[In]
[Out]