Integrand size = 22, antiderivative size = 73 \[ \int \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} \, dx=\frac {a \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} x}{a+\frac {b}{x}}-\frac {b \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} \log \left (\frac {1}{x}\right )}{a+\frac {b}{x}} \]
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Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1356, 660, 45} \[ \int \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} \, dx=\frac {a x \sqrt {a^2+\frac {2 a b}{x}+\frac {b^2}{x^2}}}{a+\frac {b}{x}}-\frac {b \log \left (\frac {1}{x}\right ) \sqrt {a^2+\frac {2 a b}{x}+\frac {b^2}{x^2}}}{a+\frac {b}{x}} \]
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Rule 45
Rule 660
Rule 1356
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} \text {Subst}\left (\int \frac {a b+b^2 x}{x^2} \, dx,x,\frac {1}{x}\right )}{a b+\frac {b^2}{x}} \\ & = -\frac {\sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} \text {Subst}\left (\int \left (\frac {a b}{x^2}+\frac {b^2}{x}\right ) \, dx,x,\frac {1}{x}\right )}{a b+\frac {b^2}{x}} \\ & = \frac {a \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} x}{a+\frac {b}{x}}+\frac {b \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} \log (x)}{a+\frac {b}{x}} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.44 \[ \int \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} \, dx=\frac {x \sqrt {\frac {(b+a x)^2}{x^2}} (a x+b \log (x))}{b+a x} \]
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Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.55
method | result | size |
default | \(\frac {\sqrt {\frac {a^{2} x^{2}+2 a b x +b^{2}}{x^{2}}}\, x \left (a x +b \ln \left (x \right )\right )}{a x +b}\) | \(40\) |
risch | \(\frac {\sqrt {\frac {\left (a x +b \right )^{2}}{x^{2}}}\, x^{2} a}{a x +b}+\frac {\sqrt {\frac {\left (a x +b \right )^{2}}{x^{2}}}\, x b \ln \left (x \right )}{a x +b}\) | \(52\) |
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Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.11 \[ \int \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} \, dx=a x + b \log \left (x\right ) \]
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\[ \int \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} \, dx=\int \sqrt {a^{2} + \frac {2 a b}{x} + \frac {b^{2}}{x^{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.11 \[ \int \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} \, dx=a x + b \log \left (x\right ) \]
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Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.40 \[ \int \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} \, dx=a x \mathrm {sgn}\left (a x^{2} + b x\right ) + b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x^{2} + b x\right ) \]
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Time = 8.39 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.84 \[ \int \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} \, dx=x\,\sqrt {\frac {1}{x^2}}\,\sqrt {a^2\,x^2+2\,a\,b\,x+b^2}-x\,\ln \left (\frac {2\,\sqrt {b^2}\,\sqrt {a^2\,x^2+2\,a\,b\,x+b^2}+2\,b^2+2\,a\,b\,x}{x}\right )\,\sqrt {b^2}\,\sqrt {\frac {1}{x^2}}+\frac {a\,b\,x\,\ln \left (\frac {a\,b+\sqrt {a^2}\,\sqrt {a^2\,x^2+2\,a\,b\,x+b^2}+a^2\,x}{\sqrt {a^2}}\right )\,\sqrt {\frac {1}{x^2}}}{\sqrt {a^2}} \]
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