Integrand size = 15, antiderivative size = 77 \[ \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx=-\frac {3 a \sqrt {x} \sqrt {a+b x}}{4 b^2}+\frac {x^{3/2} \sqrt {a+b x}}{2 b}+\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{5/2}} \]
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Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {52, 65, 223, 212} \[ \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx=\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{5/2}}-\frac {3 a \sqrt {x} \sqrt {a+b x}}{4 b^2}+\frac {x^{3/2} \sqrt {a+b x}}{2 b} \]
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {(3 a) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{4 b} \\ & = -\frac {3 a \sqrt {x} \sqrt {a+b x}}{4 b^2}+\frac {x^{3/2} \sqrt {a+b x}}{2 b}+\frac {\left (3 a^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{8 b^2} \\ & = -\frac {3 a \sqrt {x} \sqrt {a+b x}}{4 b^2}+\frac {x^{3/2} \sqrt {a+b x}}{2 b}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^2} \\ & = -\frac {3 a \sqrt {x} \sqrt {a+b x}}{4 b^2}+\frac {x^{3/2} \sqrt {a+b x}}{2 b}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^2} \\ & = -\frac {3 a \sqrt {x} \sqrt {a+b x}}{4 b^2}+\frac {x^{3/2} \sqrt {a+b x}}{2 b}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{5/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.96 \[ \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} (-3 a+2 b x)+6 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{4 b^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99
method | result | size |
risch | \(-\frac {\left (-2 b x +3 a \right ) \sqrt {x}\, \sqrt {b x +a}}{4 b^{2}}+\frac {3 a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{8 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(76\) |
default | \(\frac {x^{\frac {3}{2}} \sqrt {b x +a}}{2 b}-\frac {3 a \left (\frac {\sqrt {x}\, \sqrt {b x +a}}{b}-\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {b x +a}}\right )}{4 b}\) | \(87\) |
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Time = 0.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.55 \[ \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx=\left [\frac {3 \, a^{2} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{8 \, b^{3}}, -\frac {3 \, a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{4 \, b^{3}}\right ] \]
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Time = 3.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.30 \[ \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx=- \frac {3 a^{\frac {3}{2}} \sqrt {x}}{4 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {\sqrt {a} x^{\frac {3}{2}}}{4 b \sqrt {1 + \frac {b x}{a}}} + \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {5}{2}}} + \frac {x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (55) = 110\).
Time = 0.41 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.45 \[ \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx=-\frac {3 \, a^{2} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{8 \, b^{\frac {5}{2}}} + \frac {\frac {5 \, \sqrt {b x + a} a^{2} b}{\sqrt {x}} - \frac {3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {3}{2}}}}{4 \, {\left (b^{4} - \frac {2 \, {\left (b x + a\right )} b^{3}}{x} + \frac {{\left (b x + a\right )}^{2} b^{2}}{x^{2}}\right )}} \]
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Time = 76.58 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx=-\frac {{\left (3 \, a^{2} \sqrt {b} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right ) - \sqrt {{\left (b x + a\right )} b - a b} {\left (2 \, b x - 3 \, a\right )} \sqrt {b x + a}\right )} {\left | b \right |}}{4 \, b^{4}} \]
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Timed out. \[ \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx=\int \frac {x^{3/2}}{\sqrt {a+b\,x}} \,d x \]
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