Integrand size = 15, antiderivative size = 71 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx=\frac {3}{4} a \sqrt {x} \sqrt {a+b x}+\frac {1}{2} \sqrt {x} (a+b x)^{3/2}+\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 \sqrt {b}} \]
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Time = 0.02 (sec) , antiderivative size = 71, 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} \[ \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx=\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 \sqrt {b}}+\frac {3}{4} a \sqrt {x} \sqrt {a+b x}+\frac {1}{2} \sqrt {x} (a+b x)^{3/2} \]
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \sqrt {x} (a+b x)^{3/2}+\frac {1}{4} (3 a) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx \\ & = \frac {3}{4} a \sqrt {x} \sqrt {a+b x}+\frac {1}{2} \sqrt {x} (a+b x)^{3/2}+\frac {1}{8} \left (3 a^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx \\ & = \frac {3}{4} a \sqrt {x} \sqrt {a+b x}+\frac {1}{2} \sqrt {x} (a+b x)^{3/2}+\frac {1}{4} \left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {3}{4} a \sqrt {x} \sqrt {a+b x}+\frac {1}{2} \sqrt {x} (a+b x)^{3/2}+\frac {1}{4} \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 ) \\ & = \frac {3}{4} a \sqrt {x} \sqrt {a+b x}+\frac {1}{2} \sqrt {x} (a+b x)^{3/2}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 \sqrt {b}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx=\frac {1}{4} \sqrt {x} \sqrt {a+b x} (5 a+2 b x)-\frac {3 a^2 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{4 \sqrt {b}} \]
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Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {\left (2 b x +5 a \right ) \sqrt {x}\, \sqrt {b x +a}}{4}+\frac {3 a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{8 \sqrt {b}\, \sqrt {x}\, \sqrt {b x +a}}\) | \(73\) |
default | \(\frac {\left (b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {b x +a}+\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\) | \(78\) |
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Time = 0.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.68 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx=\left [\frac {3 \, a^{2} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{2} x + 5 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{8 \, b}, -\frac {3 \, a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (2 \, b^{2} x + 5 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{4 \, b}\right ] \]
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Time = 2.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx=\frac {5 a^{\frac {3}{2}} \sqrt {x} \sqrt {1 + \frac {b x}{a}}}{4} + \frac {\sqrt {a} b x^{\frac {3}{2}} \sqrt {1 + \frac {b x}{a}}}{2} + \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 \sqrt {b}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (49) = 98\).
Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.51 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx=-\frac {3 \, a^{2} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{8 \, \sqrt {b}} - \frac {\frac {3 \, \sqrt {b x + a} a^{2} b}{\sqrt {x}} - \frac {5 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {3}{2}}}}{4 \, {\left (b^{2} - \frac {2 \, {\left (b x + a\right )} b}{x} + \frac {{\left (b x + a\right )}^{2}}{x^{2}}\right )}} \]
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Time = 79.38 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx=-\frac {{\left (\frac {3 \, a^{2} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{\sqrt {b}} - \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )}}{b} + \frac {3 \, a}{b}\right )}\right )} b}{4 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}}{\sqrt {x}} \,d x \]
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