Integrand size = 13, antiderivative size = 65 \[ \int \frac {(a+b x)^{5/2}}{x} \, dx=2 a^2 \sqrt {a+b x}+\frac {2}{3} a (a+b x)^{3/2}+\frac {2}{5} (a+b x)^{5/2}-2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {52, 65, 214} \[ \int \frac {(a+b x)^{5/2}}{x} \, dx=-2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+2 a^2 \sqrt {a+b x}+\frac {2}{3} a (a+b x)^{3/2}+\frac {2}{5} (a+b x)^{5/2} \]
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Rule 52
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {2}{5} (a+b x)^{5/2}+a \int \frac {(a+b x)^{3/2}}{x} \, dx \\ & = \frac {2}{3} a (a+b x)^{3/2}+\frac {2}{5} (a+b x)^{5/2}+a^2 \int \frac {\sqrt {a+b x}}{x} \, dx \\ & = 2 a^2 \sqrt {a+b x}+\frac {2}{3} a (a+b x)^{3/2}+\frac {2}{5} (a+b x)^{5/2}+a^3 \int \frac {1}{x \sqrt {a+b x}} \, dx \\ & = 2 a^2 \sqrt {a+b x}+\frac {2}{3} a (a+b x)^{3/2}+\frac {2}{5} (a+b x)^{5/2}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b} \\ & = 2 a^2 \sqrt {a+b x}+\frac {2}{3} a (a+b x)^{3/2}+\frac {2}{5} (a+b x)^{5/2}-2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^{5/2}}{x} \, dx=\frac {2}{15} \sqrt {a+b x} \left (23 a^2+11 a b x+3 b^2 x^2\right )-2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.72
method | result | size |
pseudoelliptic | \(-2 a^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+\frac {2 \sqrt {b x +a}\, \left (3 b^{2} x^{2}+11 a b x +23 a^{2}\right )}{15}\) | \(47\) |
derivativedivides | \(\frac {2 a \left (b x +a \right )^{\frac {3}{2}}}{3}+\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-2 a^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+2 a^{2} \sqrt {b x +a}\) | \(50\) |
default | \(\frac {2 a \left (b x +a \right )^{\frac {3}{2}}}{3}+\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-2 a^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+2 a^{2} \sqrt {b x +a}\) | \(50\) |
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Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.75 \[ \int \frac {(a+b x)^{5/2}}{x} \, dx=\left [a^{\frac {5}{2}} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + \frac {2}{15} \, {\left (3 \, b^{2} x^{2} + 11 \, a b x + 23 \, a^{2}\right )} \sqrt {b x + a}, 2 \, \sqrt {-a} a^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + \frac {2}{15} \, {\left (3 \, b^{2} x^{2} + 11 \, a b x + 23 \, a^{2}\right )} \sqrt {b x + a}\right ] \]
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Time = 3.85 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.49 \[ \int \frac {(a+b x)^{5/2}}{x} \, dx=\frac {46 a^{\frac {5}{2}} \sqrt {1 + \frac {b x}{a}}}{15} + a^{\frac {5}{2}} \log {\left (\frac {b x}{a} \right )} - 2 a^{\frac {5}{2}} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )} + \frac {22 a^{\frac {3}{2}} b x \sqrt {1 + \frac {b x}{a}}}{15} + \frac {2 \sqrt {a} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{5} \]
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Time = 0.33 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^{5/2}}{x} \, dx=a^{\frac {5}{2}} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2}{5} \, {\left (b x + a\right )}^{\frac {5}{2}} + \frac {2}{3} \, {\left (b x + a\right )}^{\frac {3}{2}} a + 2 \, \sqrt {b x + a} a^{2} \]
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Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^{5/2}}{x} \, dx=\frac {2 \, a^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {2}{5} \, {\left (b x + a\right )}^{\frac {5}{2}} + \frac {2}{3} \, {\left (b x + a\right )}^{\frac {3}{2}} a + 2 \, \sqrt {b x + a} a^{2} \]
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Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^{5/2}}{x} \, dx=\frac {2\,a\,{\left (a+b\,x\right )}^{3/2}}{3}+\frac {2\,{\left (a+b\,x\right )}^{5/2}}{5}+2\,a^2\,\sqrt {a+b\,x}+a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,2{}\mathrm {i} \]
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