Integrand size = 27, antiderivative size = 27 \[ \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x^2}{b c \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}+\frac {2 \text {Int}\left (\frac {x}{\left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))},x\right )}{b c} \]
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Not integrable
Time = 0.15 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {x^2}{b c \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}+\frac {2 \int \frac {x}{\left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))} \, dx}{b c} \\ \end{align*}
Not integrable
Time = 3.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx \]
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Not integrable
Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
\[\int \frac {x^{2}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.89 \[ \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{2}}{{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Not integrable
Time = 1.80 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}\, dx \]
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Not integrable
Time = 0.46 (sec) , antiderivative size = 447, normalized size of antiderivative = 16.56 \[ \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{2}}{{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{2}}{{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Not integrable
Time = 2.60 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (c^2\,x^2+1\right )}^{3/2}} \,d x \]
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