Integrand size = 23, antiderivative size = 62 \[ \int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a} \]
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Time = 0.02 (sec) , antiderivative size = 62, 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} \[ \int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx=\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{3 a}-\frac {b^2}{a \sqrt {\sqrt {a^2 x^2+b^2}+a x}} \]
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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^{3/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a} \\ & = \frac {\text {Subst}\left (\int \left (\frac {b^2}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a} \\ & = -\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.95 \[ \int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx=\frac {-2 b^2+2 a x \left (a x+\sqrt {b^2+a^2 x^2}\right )}{3 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \]
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\[\int \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.71 \[ \int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx=\frac {2 \, {\left (2 \, a x - \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{3 \, a} \]
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\[ \int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx=\int \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\, dx \]
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\[ \int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx=\int { \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \,d x } \]
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\[ \int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx=\int { \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \,d x } \]
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Timed out. \[ \int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx=\int \sqrt {a\,x+\sqrt {a^2\,x^2+b^2}} \,d x \]
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