Integrand size = 23, antiderivative size = 61 \[ \int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a} \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2142, 14} \[ \int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a}-\frac {b^2}{3 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}} \]
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Rule 14
Rule 2142
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b^2+x^2}{x^{5/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a} \\ & = \frac {\text {Subst}\left (\int \left (\frac {b^2}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a} \\ & = -\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2 \left (b^2+3 a x \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}} \]
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\[\int \frac {1}{\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {2 \, {\left (a^{2} x^{2} - \sqrt {a^{2} x^{2} + b^{2}} a x - b^{2}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{3 \, a b^{2}} \]
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\[ \int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]
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\[ \int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]
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\[ \int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {1}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \]
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