Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\frac {3 c d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{1+n}}{2 b (1+n) \sqrt {d+c^2 d x^2}}+\frac {2^{-3-n} c d^2 e^{-\frac {2 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {d+c^2 d x^2}}-\frac {2^{-3-n} c d^2 e^{\frac {2 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {d+c^2 d x^2}}+d^2 \text {Int}\left (\frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}},x\right ) \]
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Not integrable
Time = 0.53 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 c^2 d^2 (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}}+\frac {d^2 (a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}}+\frac {c^4 d^2 x^2 (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}}\right ) \, dx \\ & = d^2 \int \frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}} \, dx+\left (2 c^2 d^2\right ) \int \frac {(a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}} \, dx+\left (c^4 d^2\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}} \, dx \\ & = \frac {2 c d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{1+n}}{b (1+n) \sqrt {d+c^2 d x^2}}+d^2 \int \frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}} \, dx+\frac {\left (c d^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b \sqrt {d+c^2 d x^2}} \\ & = \frac {2 c d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{1+n}}{b (1+n) \sqrt {d+c^2 d x^2}}+d^2 \int \frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}} \, dx-\frac {\left (c d^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {x^n}{2}-\frac {1}{2} x^n \cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b \sqrt {d+c^2 d x^2}} \\ & = \frac {3 c d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{1+n}}{2 b (1+n) \sqrt {d+c^2 d x^2}}+d^2 \int \frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}} \, dx+\frac {\left (c d^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b \sqrt {d+c^2 d x^2}} \\ & = \frac {3 c d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{1+n}}{2 b (1+n) \sqrt {d+c^2 d x^2}}+d^2 \int \frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}} \, dx+\frac {\left (c d^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b \sqrt {d+c^2 d x^2}}+\frac {\left (c d^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b \sqrt {d+c^2 d x^2}} \\ & = \frac {3 c d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{1+n}}{2 b (1+n) \sqrt {d+c^2 d x^2}}+\frac {2^{-3-n} c d^2 e^{-\frac {2 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {d+c^2 d x^2}}-\frac {2^{-3-n} c d^2 e^{\frac {2 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {d+c^2 d x^2}}+d^2 \int \frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}} \, dx \\ \end{align*}
Not integrable
Time = 0.95 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
\[\int \frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{n}}{x^{2}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}}{x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 2.56 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{x^2} \,d x \]
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