Integrand size = 28, antiderivative size = 235 \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-2 (3+n)} e^{-\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-2 (3+n)} e^{\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}} \]
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Time = 0.25 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5819, 5556, 3388, 2212} \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^{n+1}}{8 b c^3 (n+1) \sqrt {c^2 x^2+1}}+\frac {2^{-2 (n+3)} e^{-\frac {4 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {2^{-2 (n+3)} e^{\frac {4 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}} \]
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Rule 2212
Rule 3388
Rule 5556
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int x^n \cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^3 \sqrt {1+c^2 x^2}} \\ & = \frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int \left (-\frac {x^n}{8}+\frac {1}{8} x^n \cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int x^n \cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int e^{-i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b c^3 \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int e^{i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {4^{-3-n} e^{-\frac {4 a}{b}} \sqrt {d+c^2 d 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 {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {4^{-3-n} e^{\frac {4 a}{b}} \sqrt {d+c^2 d 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 {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}
Time = 0.75 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.73 \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {8 (a+b \text {arcsinh}(c x))}{b (1+n)}+4^{-n} e^{-\frac {4 a}{b}} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^{-n} \left (\left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{64 c^3 \sqrt {d \left (1+c^2 x^2\right )}} \]
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\[\int x^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{n} \sqrt {c^{2} d \,x^{2}+d}d x\]
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\[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\int { \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \]
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\[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\int x^{2} \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{n}\, dx \]
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\[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\int { \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \]
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\[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\int { \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,\sqrt {d\,c^2\,x^2+d} \,d x \]
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