Integrand size = 22, antiderivative size = 22 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arcsinh}(c x)} \, dx=\text {Int}\left (\frac {\sqrt {d+e x^2}}{a+b \text {arcsinh}(c x)},x\right ) \]
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Not integrable
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {\sqrt {d+e x^2}}{a+b \text {arcsinh}(c x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {d+e x^2}}{a+b \text {arcsinh}(c x)} \, dx \\ \end{align*}
Not integrable
Time = 1.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {\sqrt {d+e x^2}}{a+b \text {arcsinh}(c x)} \, dx \]
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Not integrable
Time = 0.60 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
\[\int \frac {\sqrt {e \,x^{2}+d}}{a +b \,\operatorname {arcsinh}\left (c x \right )}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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Not integrable
Time = 0.37 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {\sqrt {d + e x^{2}}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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Not integrable
Time = 2.61 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {\sqrt {e\,x^2+d}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]
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