Integrand size = 20, antiderivative size = 106 \[ \int \frac {\sqrt {a+b \sqrt {x}+c x}}{x} \, dx=2 \sqrt {a+b \sqrt {x}+c x}-2 \sqrt {a} \text {arctanh}\left (\frac {2 a+b \sqrt {x}}{2 \sqrt {a} \sqrt {a+b \sqrt {x}+c x}}\right )+\frac {b \text {arctanh}\left (\frac {b+2 c \sqrt {x}}{2 \sqrt {c} \sqrt {a+b \sqrt {x}+c x}}\right )}{\sqrt {c}} \]
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Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1371, 748, 857, 635, 212, 738} \[ \int \frac {\sqrt {a+b \sqrt {x}+c x}}{x} \, dx=-2 \sqrt {a} \text {arctanh}\left (\frac {2 a+b \sqrt {x}}{2 \sqrt {a} \sqrt {a+b \sqrt {x}+c x}}\right )+\frac {b \text {arctanh}\left (\frac {b+2 c \sqrt {x}}{2 \sqrt {c} \sqrt {a+b \sqrt {x}+c x}}\right )}{\sqrt {c}}+2 \sqrt {a+b \sqrt {x}+c x} \]
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Rule 212
Rule 635
Rule 738
Rule 748
Rule 857
Rule 1371
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \sqrt {a+b \sqrt {x}+c x}-\text {Subst}\left (\int \frac {-2 a-b x}{x \sqrt {a+b x+c x^2}} \, dx,x,\sqrt {x}\right ) \\ & = 2 \sqrt {a+b \sqrt {x}+c x}+(2 a) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\sqrt {x}\right )+b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\sqrt {x}\right ) \\ & = 2 \sqrt {a+b \sqrt {x}+c x}-(4 a) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \sqrt {x}}{\sqrt {a+b \sqrt {x}+c x}}\right )+(2 b) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \sqrt {x}}{\sqrt {a+b \sqrt {x}+c x}}\right ) \\ & = 2 \sqrt {a+b \sqrt {x}+c x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b \sqrt {x}}{2 \sqrt {a} \sqrt {a+b \sqrt {x}+c x}}\right )+\frac {b \tanh ^{-1}\left (\frac {b+2 c \sqrt {x}}{2 \sqrt {c} \sqrt {a+b \sqrt {x}+c x}}\right )}{\sqrt {c}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {a+b \sqrt {x}+c x}}{x} \, dx=2 \sqrt {a+b \sqrt {x}+c x}+4 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}-\sqrt {a+b \sqrt {x}+c x}}{\sqrt {a}}\right )-\frac {b \log \left (b+2 c \sqrt {x}-2 \sqrt {c} \sqrt {a+b \sqrt {x}+c x}\right )}{\sqrt {c}} \]
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Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(2 \sqrt {a +c x +b \sqrt {x}}+\frac {b \ln \left (\frac {\frac {b}{2}+c \sqrt {x}}{\sqrt {c}}+\sqrt {a +c x +b \sqrt {x}}\right )}{\sqrt {c}}-2 \sqrt {a}\, \ln \left (\frac {2 a +b \sqrt {x}+2 \sqrt {a}\, \sqrt {a +c x +b \sqrt {x}}}{\sqrt {x}}\right )\) | \(84\) |
default | \(2 \sqrt {a +c x +b \sqrt {x}}+\frac {b \ln \left (\frac {\frac {b}{2}+c \sqrt {x}}{\sqrt {c}}+\sqrt {a +c x +b \sqrt {x}}\right )}{\sqrt {c}}-2 \sqrt {a}\, \ln \left (\frac {2 a +b \sqrt {x}+2 \sqrt {a}\, \sqrt {a +c x +b \sqrt {x}}}{\sqrt {x}}\right )\) | \(84\) |
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Timed out. \[ \int \frac {\sqrt {a+b \sqrt {x}+c x}}{x} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {a+b \sqrt {x}+c x}}{x} \, dx=\int \frac {\sqrt {a + b \sqrt {x} + c x}}{x}\, dx \]
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\[ \int \frac {\sqrt {a+b \sqrt {x}+c x}}{x} \, dx=\int { \frac {\sqrt {c x + b \sqrt {x} + a}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {a+b \sqrt {x}+c x}}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {a+b \sqrt {x}+c x}}{x} \, dx=\int \frac {\sqrt {a+c\,x+b\,\sqrt {x}}}{x} \,d x \]
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