Integrand size = 15, antiderivative size = 63 \[ \int \frac {(a+b x)^{3/2}}{x^{3/2}} \, dx=3 b \sqrt {x} \sqrt {a+b x}-\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+3 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {49, 52, 65, 223, 212} \[ \int \frac {(a+b x)^{3/2}}{x^{3/2}} \, dx=3 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+3 b \sqrt {x} \sqrt {a+b x} \]
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Rule 49
Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+(3 b) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx \\ & = 3 b \sqrt {x} \sqrt {a+b x}-\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+\frac {1}{2} (3 a b) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx \\ & = 3 b \sqrt {x} \sqrt {a+b x}-\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+(3 a b) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = 3 b \sqrt {x} \sqrt {a+b x}-\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+(3 a b) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right ) \\ & = 3 b \sqrt {x} \sqrt {a+b x}-\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+3 a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^{3/2}}{x^{3/2}} \, dx=\frac {(-2 a+b x) \sqrt {a+b x}}{\sqrt {x}}+6 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.13
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-b x +2 a \right )}{\sqrt {x}}+\frac {3 a \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{2 \sqrt {x}\, \sqrt {b x +a}}\) | \(71\) |
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Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.73 \[ \int \frac {(a+b x)^{3/2}}{x^{3/2}} \, dx=\left [\frac {3 \, a \sqrt {b} x \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, \sqrt {b x + a} {\left (b x - 2 \, a\right )} \sqrt {x}}{2 \, x}, -\frac {3 \, a \sqrt {-b} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - \sqrt {b x + a} {\left (b x - 2 \, a\right )} \sqrt {x}}{x}\right ] \]
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Time = 2.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b x)^{3/2}}{x^{3/2}} \, dx=- \frac {2 a^{\frac {3}{2}}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} - \frac {\sqrt {a} b \sqrt {x}}{\sqrt {1 + \frac {b x}{a}}} + 3 a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} + \frac {b^{2} x^{\frac {3}{2}}}{\sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]
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Time = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b x)^{3/2}}{x^{3/2}} \, dx=-\frac {3}{2} \, a \sqrt {b} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right ) - \frac {2 \, \sqrt {b x + a} a}{\sqrt {x}} - \frac {\sqrt {b x + a} a b}{{\left (b - \frac {b x + a}{x}\right )} \sqrt {x}} \]
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Time = 77.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^{3/2}}{x^{3/2}} \, dx=-\frac {{\left (\frac {3 \, a \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{\sqrt {b}} - \frac {\sqrt {b x + a} {\left (b x - 2 \, a\right )}}{\sqrt {{\left (b x + a\right )} b - a b}}\right )} b^{2}}{{\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{3/2}}{x^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}}{x^{3/2}} \,d x \]
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