Integrand size = 23, antiderivative size = 122 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=-\frac {(b c-3 a d) \sqrt {a+\frac {b}{x}}}{a c^2 \sqrt {c+\frac {d}{x}}}+\frac {\left (a+\frac {b}{x}\right )^{3/2} x}{a c \sqrt {c+\frac {d}{x}}}+\frac {(b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} c^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {382, 98, 96, 95, 214} \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\frac {(b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} c^{5/2}}-\frac {\sqrt {a+\frac {b}{x}} (b c-3 a d)}{a c^2 \sqrt {c+\frac {d}{x}}}+\frac {x \left (a+\frac {b}{x}\right )^{3/2}}{a c \sqrt {c+\frac {d}{x}}} \]
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Rule 95
Rule 96
Rule 98
Rule 214
Rule 382
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{3/2}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\left (a+\frac {b}{x}\right )^{3/2} x}{a c \sqrt {c+\frac {d}{x}}}+\frac {\left (-\frac {b c}{2}+\frac {3 a d}{2}\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = -\frac {(b c-3 a d) \sqrt {a+\frac {b}{x}}}{a c^2 \sqrt {c+\frac {d}{x}}}+\frac {\left (a+\frac {b}{x}\right )^{3/2} x}{a c \sqrt {c+\frac {d}{x}}}-\frac {(b c-3 a d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\frac {1}{x}\right )}{2 c^2} \\ & = -\frac {(b c-3 a d) \sqrt {a+\frac {b}{x}}}{a c^2 \sqrt {c+\frac {d}{x}}}+\frac {\left (a+\frac {b}{x}\right )^{3/2} x}{a c \sqrt {c+\frac {d}{x}}}-\frac {(b c-3 a d) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}}\right )}{c^2} \\ & = -\frac {(b c-3 a d) \sqrt {a+\frac {b}{x}}}{a c^2 \sqrt {c+\frac {d}{x}}}+\frac {\left (a+\frac {b}{x}\right )^{3/2} x}{a c \sqrt {c+\frac {d}{x}}}+\frac {(b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} c^{5/2}} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x \left (\sqrt {a} \sqrt {c} \sqrt {b+a x} (3 d+c x)+(b c-3 a d) \sqrt {d+c x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {b+a x}}{\sqrt {a} \sqrt {d+c x}}\right )\right )}{\sqrt {a} c^{5/2} \sqrt {b+a x} (d+c x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(279\) vs. \(2(102)=204\).
Time = 0.16 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.30
method | result | size |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \sqrt {\frac {c x +d}{x}}\, \left (-3 \ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a c d x +\ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) b \,c^{2} x +2 c x \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}-3 \ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a \,d^{2}+\ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) b c d +6 d \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}\right )}{2 \sqrt {a c}\, \left (c x +d \right ) \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, c^{2}}\) | \(280\) |
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Time = 0.39 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.61 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\left [-\frac {{\left (b c d - 3 \, a d^{2} + {\left (b c^{2} - 3 \, a c d\right )} x\right )} \sqrt {a c} \log \left (-8 \, a^{2} c^{2} x^{2} - b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2} + 4 \, {\left (2 \, a c x^{2} + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - 8 \, {\left (a b c^{2} + a^{2} c d\right )} x\right ) - 4 \, {\left (a c^{2} x^{2} + 3 \, a c d x\right )} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{4 \, {\left (a c^{4} x + a c^{3} d\right )}}, -\frac {{\left (b c d - 3 \, a d^{2} + {\left (b c^{2} - 3 \, a c d\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {2 \, \sqrt {-a c} x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, a c x + b c + a d}\right ) - 2 \, {\left (a c^{2} x^{2} + 3 \, a c d x\right )} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, {\left (a c^{4} x + a c^{3} d\right )}}\right ] \]
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\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\int \frac {\sqrt {a + \frac {b}{x}}}{\left (c + \frac {d}{x}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\int { \frac {\sqrt {a + \frac {b}{x}}}{{\left (c + \frac {d}{x}\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\int { \frac {\sqrt {a + \frac {b}{x}}}{{\left (c + \frac {d}{x}\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\int \frac {\sqrt {a+\frac {b}{x}}}{{\left (c+\frac {d}{x}\right )}^{3/2}} \,d x \]
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