Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}} \, dx=\text {Int}\left (\frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}},x\right ) \]
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Not integrable
Time = 0.04 (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 = {} \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}} \, dx \\ \end{align*}
Not integrable
Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}} \, dx \]
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Not integrable
Time = 0.90 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
\[\int \frac {1}{\left (e \,x^{2}+d \right )^{2} \sqrt {a +b \,\operatorname {arcsinh}\left (c x \right )}}d x\]
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Exception generated. \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 41.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {1}{\sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \left (d + e x^{2}\right )^{2}}\, dx \]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{2} \sqrt {b \operatorname {arsinh}\left (c x\right ) + a}} \,d x } \]
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Not integrable
Time = 1.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{2} \sqrt {b \operatorname {arsinh}\left (c x\right ) + a}} \,d x } \]
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Not integrable
Time = 2.63 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {1}{\sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )}\,{\left (e\,x^2+d\right )}^2} \,d x \]
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