Integrand size = 22, antiderivative size = 66 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {a+b x}}{c \sqrt {c+d x}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 95, 214} \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {a+b x}}{c \sqrt {c+d x}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}} \]
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Rule 95
Rule 96
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {a+b x}}{c \sqrt {c+d x}}+\frac {a \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c} \\ & = \frac {2 \sqrt {a+b x}}{c \sqrt {c+d x}}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c} \\ & = \frac {2 \sqrt {a+b x}}{c \sqrt {c+d x}}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {a+b x}}{c \sqrt {c+d x}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(142\) vs. \(2(50)=100\).
Time = 0.54 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.17
method | result | size |
default | \(\frac {\left (-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a d x -\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right ) \sqrt {b x +a}}{\sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}\, c}\) | \(143\) |
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Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (50) = 100\).
Time = 0.30 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.76 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx=\left [\frac {{\left (d x + c\right )} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c d x + c^{2}\right )}}, \frac {{\left (d x + c\right )} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) + 2 \, \sqrt {b x + a} \sqrt {d x + c}}{c d x + c^{2}}\right ] \]
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\[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx=\int \frac {\sqrt {a + b x}}{x \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (50) = 100\).
Time = 0.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.97 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx=-\frac {2 \, \sqrt {b d} a b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c {\left | b \right |}} + \frac {2 \, \sqrt {b x + a} b^{2}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} c {\left | b \right |}} \]
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Timed out. \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx=\int \frac {\sqrt {a+b\,x}}{x\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
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