Integrand size = 21, antiderivative size = 102 \[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {x^{1+m} \text {arcsinh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-a^2 x^2\right )}{1+m}-\frac {a x^{2+m} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};-a^2 x^2\right )}{2+3 m+m^2} \]
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Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5817} \[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {x^{m+1} \text {arcsinh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},-a^2 x^2\right )}{m+1}-\frac {a x^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;-a^2 x^2\right )}{m^2+3 m+2} \]
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Rule 5817
Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} \text {arcsinh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-a^2 x^2\right )}{1+m}-\frac {a x^{2+m} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};-a^2 x^2\right )}{2+3 m+m^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.95 \[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {x^{1+m} \left ((2+m) \text {arcsinh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-a^2 x^2\right )-a x \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};-a^2 x^2\right )\right )}{(1+m) (2+m)} \]
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\[\int \frac {x^{m} \operatorname {arcsinh}\left (a x \right )}{\sqrt {a^{2} x^{2}+1}}d x\]
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\[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{m} \operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1}} \,d x } \]
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\[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^{m} \operatorname {asinh}{\left (a x \right )}}{\sqrt {a^{2} x^{2} + 1}}\, dx \]
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\[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{m} \operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1}} \,d x } \]
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\[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{m} \operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^m\,\mathrm {asinh}\left (a\,x\right )}{\sqrt {a^2\,x^2+1}} \,d x \]
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