Integrand size = 13, antiderivative size = 66 \[ \int \frac {(a+b x)^{5/2}}{x^2} \, dx=5 a b \sqrt {a+b x}+\frac {5}{3} b (a+b x)^{3/2}-\frac {(a+b x)^{5/2}}{x}-5 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 66, 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 = {43, 52, 65, 214} \[ \int \frac {(a+b x)^{5/2}}{x^2} \, dx=-5 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )-\frac {(a+b x)^{5/2}}{x}+\frac {5}{3} b (a+b x)^{3/2}+5 a b \sqrt {a+b x} \]
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Rule 43
Rule 52
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{5/2}}{x}+\frac {1}{2} (5 b) \int \frac {(a+b x)^{3/2}}{x} \, dx \\ & = \frac {5}{3} b (a+b x)^{3/2}-\frac {(a+b x)^{5/2}}{x}+\frac {1}{2} (5 a b) \int \frac {\sqrt {a+b x}}{x} \, dx \\ & = 5 a b \sqrt {a+b x}+\frac {5}{3} b (a+b x)^{3/2}-\frac {(a+b x)^{5/2}}{x}+\frac {1}{2} \left (5 a^2 b\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx \\ & = 5 a b \sqrt {a+b x}+\frac {5}{3} b (a+b x)^{3/2}-\frac {(a+b x)^{5/2}}{x}+\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right ) \\ & = 5 a b \sqrt {a+b x}+\frac {5}{3} b (a+b x)^{3/2}-\frac {(a+b x)^{5/2}}{x}-5 a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^{5/2}}{x^2} \, dx=\frac {\sqrt {a+b x} \left (-3 a^2+14 a b x+2 b^2 x^2\right )}{3 x}-5 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86
method | result | size |
risch | \(-\frac {a^{2} \sqrt {b x +a}}{x}+\frac {b \left (\frac {4 \left (b x +a \right )^{\frac {3}{2}}}{3}+8 a \sqrt {b x +a}-10 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right )}{2}\) | \(57\) |
pseudoelliptic | \(-\frac {5 \left (\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a^{2} b x -\frac {2 \left (\sqrt {a}\, b^{2} x^{2}+7 a^{\frac {3}{2}} b x -\frac {3 a^{\frac {5}{2}}}{2}\right ) \sqrt {b x +a}}{15}\right )}{\sqrt {a}\, x}\) | \(60\) |
derivativedivides | \(2 b \left (\frac {\left (b x +a \right )^{\frac {3}{2}}}{3}+2 a \sqrt {b x +a}-a^{2} \left (\frac {\sqrt {b x +a}}{2 b x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )\right )\) | \(62\) |
default | \(2 b \left (\frac {\left (b x +a \right )^{\frac {3}{2}}}{3}+2 a \sqrt {b x +a}-a^{2} \left (\frac {\sqrt {b x +a}}{2 b x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )\right )\) | \(62\) |
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Time = 0.23 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.91 \[ \int \frac {(a+b x)^{5/2}}{x^2} \, dx=\left [\frac {15 \, a^{\frac {3}{2}} b x \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (2 \, b^{2} x^{2} + 14 \, a b x - 3 \, a^{2}\right )} \sqrt {b x + a}}{6 \, x}, \frac {15 \, \sqrt {-a} a b x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (2 \, b^{2} x^{2} + 14 \, a b x - 3 \, a^{2}\right )} \sqrt {b x + a}}{3 \, x}\right ] \]
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Time = 3.33 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.50 \[ \int \frac {(a+b x)^{5/2}}{x^2} \, dx=- \frac {a^{\frac {5}{2}} \sqrt {1 + \frac {b x}{a}}}{x} + \frac {14 a^{\frac {3}{2}} b \sqrt {1 + \frac {b x}{a}}}{3} + \frac {5 a^{\frac {3}{2}} b \log {\left (\frac {b x}{a} \right )}}{2} - 5 a^{\frac {3}{2}} b \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )} + \frac {2 \sqrt {a} b^{2} x \sqrt {1 + \frac {b x}{a}}}{3} \]
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Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^{5/2}}{x^2} \, dx=\frac {5}{2} \, a^{\frac {3}{2}} b \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2}{3} \, {\left (b x + a\right )}^{\frac {3}{2}} b + 4 \, \sqrt {b x + a} a b - \frac {\sqrt {b x + a} a^{2}}{x} \]
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Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^{5/2}}{x^2} \, dx=\frac {\frac {15 \, a^{2} b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2} + 12 \, \sqrt {b x + a} a b^{2} - \frac {3 \, \sqrt {b x + a} a^{2} b}{x}}{3 \, b} \]
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Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^{5/2}}{x^2} \, dx=\frac {2\,b\,{\left (a+b\,x\right )}^{3/2}}{3}-\frac {a^2\,\sqrt {a+b\,x}}{x}+4\,a\,b\,\sqrt {a+b\,x}+a^{3/2}\,b\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i} \]
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