Integrand size = 15, antiderivative size = 60 \[ \int \frac {(2+b x)^{3/2}}{x^{5/2}} \, dx=-\frac {2 b \sqrt {2+b x}}{\sqrt {x}}-\frac {2 (2+b x)^{3/2}}{3 x^{3/2}}+2 b^{3/2} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {49, 56, 221} \[ \int \frac {(2+b x)^{3/2}}{x^{5/2}} \, dx=2 b^{3/2} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )-\frac {2 (b x+2)^{3/2}}{3 x^{3/2}}-\frac {2 b \sqrt {b x+2}}{\sqrt {x}} \]
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Rule 49
Rule 56
Rule 221
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (2+b x)^{3/2}}{3 x^{3/2}}+b \int \frac {\sqrt {2+b x}}{x^{3/2}} \, dx \\ & = -\frac {2 b \sqrt {2+b x}}{\sqrt {x}}-\frac {2 (2+b x)^{3/2}}{3 x^{3/2}}+b^2 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx \\ & = -\frac {2 b \sqrt {2+b x}}{\sqrt {x}}-\frac {2 (2+b x)^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 b \sqrt {2+b x}}{\sqrt {x}}-\frac {2 (2+b x)^{3/2}}{3 x^{3/2}}+2 b^{3/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.92 \[ \int \frac {(2+b x)^{3/2}}{x^{5/2}} \, dx=-\frac {4 \sqrt {2+b x} (1+2 b x)}{3 x^{3/2}}-2 b^{3/2} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.92
method | result | size |
meijerg | \(\frac {3 b^{\frac {3}{2}} \left (-\frac {16 \sqrt {\pi }\, \sqrt {2}\, \left (2 b x +1\right ) \sqrt {\frac {b x}{2}+1}}{9 x^{\frac {3}{2}} b^{\frac {3}{2}}}+\frac {8 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{3}\right )}{4 \sqrt {\pi }}\) | \(55\) |
risch | \(-\frac {4 \left (2 b^{2} x^{2}+5 b x +2\right )}{3 x^{\frac {3}{2}} \sqrt {b x +2}}+\frac {b^{\frac {3}{2}} \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right ) \sqrt {x \left (b x +2\right )}}{\sqrt {x}\, \sqrt {b x +2}}\) | \(73\) |
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Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.80 \[ \int \frac {(2+b x)^{3/2}}{x^{5/2}} \, dx=\left [\frac {3 \, b^{\frac {3}{2}} x^{2} \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right ) - 4 \, {\left (2 \, b x + 1\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, x^{2}}, -\frac {2 \, {\left (3 \, \sqrt {-b} b x^{2} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right ) + 2 \, {\left (2 \, b x + 1\right )} \sqrt {b x + 2} \sqrt {x}\right )}}{3 \, x^{2}}\right ] \]
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Time = 1.75 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.17 \[ \int \frac {(2+b x)^{3/2}}{x^{5/2}} \, dx=- \frac {8 b^{\frac {3}{2}} \sqrt {1 + \frac {2}{b x}}}{3} - b^{\frac {3}{2}} \log {\left (\frac {1}{b x} \right )} + 2 b^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {2}{b x}} + 1 \right )} - \frac {4 \sqrt {b} \sqrt {1 + \frac {2}{b x}}}{3 x} \]
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Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.12 \[ \int \frac {(2+b x)^{3/2}}{x^{5/2}} \, dx=-b^{\frac {3}{2}} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right ) - \frac {2 \, \sqrt {b x + 2} b}{\sqrt {x}} - \frac {2 \, {\left (b x + 2\right )}^{\frac {3}{2}}}{3 \, x^{\frac {3}{2}}} \]
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Time = 5.75 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.32 \[ \int \frac {(2+b x)^{3/2}}{x^{5/2}} \, dx=-\frac {2 \, {\left (3 \, b^{\frac {3}{2}} \log \left ({\left | -\sqrt {b x + 2} \sqrt {b} + \sqrt {{\left (b x + 2\right )} b - 2 \, b} \right |}\right ) + \frac {2 \, {\left (2 \, {\left (b x + 2\right )} b^{3} - 3 \, b^{3}\right )} \sqrt {b x + 2}}{{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {3}{2}}}\right )} b}{3 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(2+b x)^{3/2}}{x^{5/2}} \, dx=\int \frac {{\left (b\,x+2\right )}^{3/2}}{x^{5/2}} \,d x \]
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