Integrand size = 23, antiderivative size = 81 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x}{c}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} c^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {382, 96, 95, 214} \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} c^{3/2}}+\frac {x \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}}}{c} \]
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Rule 95
Rule 96
Rule 214
Rule 382
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2 \sqrt {c+d x}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x}{c}-\frac {(b c-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} \\ & = \frac {\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x}{c}-\frac {(b c-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} \\ & = \frac {\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x}{c}+\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} c^{3/2}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {d+c x} \left (\frac {\sqrt {b+a x} \sqrt {d+c x}}{c}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d+c x}}{\sqrt {c} \sqrt {b+a x}}\right )}{\sqrt {a} c^{3/2}}\right )}{\sqrt {c+\frac {d}{x}} \sqrt {b+a x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(154\) vs. \(2(65)=130\).
Time = 0.13 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.91
method | result | size |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \sqrt {\frac {c x +d}{x}}\, \left (-\ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a d +\ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) b c +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}\right )}{2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, c \sqrt {a c}}\) | \(155\) |
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none
Time = 0.41 (sec) , antiderivative size = 247, normalized size of antiderivative = 3.05 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\left [\frac {4 \, a c x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - \sqrt {a c} {\left (b c - a d\right )} \log \left (-8 \, a^{2} c^{2} x^{2} - b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2} + 4 \, {\left (2 \, a c x^{2} + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - 8 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )}{4 \, a c^{2}}, \frac {2 \, a c x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - \sqrt {-a c} {\left (b c - a d\right )} \arctan \left (\frac {2 \, \sqrt {-a c} x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, a c x + b c + a d}\right )}{2 \, a c^{2}}\right ] \]
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\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\int \frac {\sqrt {a + \frac {b}{x}}}{\sqrt {c + \frac {d}{x}}}\, dx \]
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\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\int { \frac {\sqrt {a + \frac {b}{x}}}{\sqrt {c + \frac {d}{x}}} \,d x } \]
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\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\int { \frac {\sqrt {a + \frac {b}{x}}}{\sqrt {c + \frac {d}{x}}} \,d x } \]
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Time = 10.77 (sec) , antiderivative size = 478, normalized size of antiderivative = 5.90 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}} \, dx=\frac {d\,\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )}{4\,c\,\left (\sqrt {c+\frac {d}{x}}-\sqrt {c}\right )}-\frac {\frac {\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )\,\left (\frac {c\,b^2}{4}+\frac {a\,d\,b}{4}\right )}{\sqrt {a}\,c^{3/2}\,d\,\left (\sqrt {c+\frac {d}{x}}-\sqrt {c}\right )}-\frac {b^2}{4\,c\,d}+\frac {{\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )}^2\,\left (\frac {a^2\,d^2}{4}-\frac {3\,a\,b\,c\,d}{4}+\frac {b^2\,c^2}{4}\right )}{a\,c^2\,d\,{\left (\sqrt {c+\frac {d}{x}}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )}^3}{{\left (\sqrt {c+\frac {d}{x}}-\sqrt {c}\right )}^3}+\frac {b\,\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )}{d\,\left (\sqrt {c+\frac {d}{x}}-\sqrt {c}\right )}-\frac {{\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )}^2\,\left (a\,d+b\,c\right )}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {c+\frac {d}{x}}-\sqrt {c}\right )}^2}}+\frac {\ln \left (\frac {\sqrt {a+\frac {b}{x}}-\sqrt {a}}{\sqrt {c+\frac {d}{x}}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}-a^{3/2}\,\sqrt {c}\,d\right )}{2\,a\,c^2}-\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {a+\frac {b}{x}}-\sqrt {a}\,\sqrt {c+\frac {d}{x}}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )}{\sqrt {c+\frac {d}{x}}-\sqrt {c}}\right )}{\sqrt {c+\frac {d}{x}}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}-a^{3/2}\,\sqrt {c}\,d\right )}{2\,a\,c^2} \]
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