Integrand size = 16, antiderivative size = 545 \[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {3^{-1-n} e^{-\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 d^3}-\frac {2^{-2-n} c e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^3}-\frac {e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 d^3}+\frac {c^2 e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^3}+\frac {e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 d^3}-\frac {c^2 e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^3}-\frac {2^{-2-n} c e^{\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^3}-\frac {3^{-1-n} e^{\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 d^3} \]
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Time = 0.85 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5859, 5830, 6873, 12, 6874, 3388, 2212, 5556, 3389} \[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {c^2 e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^3}-\frac {c^2 e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^3}+\frac {3^{-n-1} e^{-\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 d^3}-\frac {c 2^{-n-2} e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^3}-\frac {e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 d^3}+\frac {e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 d^3}-\frac {c 2^{-n-2} e^{\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^3}-\frac {3^{-n-1} e^{\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 d^3} \]
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Rule 12
Rule 2212
Rule 3388
Rule 3389
Rule 5556
Rule 5830
Rule 5859
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {c}{d}+\frac {x}{d}\right )^2 (a+b \text {arcsinh}(x))^n \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int (a+b x)^n \cosh (x) \left (-\frac {c}{d}+\frac {\sinh (x)}{d}\right )^2 \, dx,x,\text {arcsinh}(c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b x)^n \cosh (x) (c-\sinh (x))^2}{d^2} \, dx,x,\text {arcsinh}(c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int (a+b x)^n \cosh (x) (c-\sinh (x))^2 \, dx,x,\text {arcsinh}(c+d x)\right )}{d^3} \\ & = \frac {\text {Subst}\left (\int \left (c^2 (a+b x)^n \cosh (x)-2 c (a+b x)^n \cosh (x) \sinh (x)+(a+b x)^n \cosh (x) \sinh ^2(x)\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d^3} \\ & = \frac {\text {Subst}\left (\int (a+b x)^n \cosh (x) \sinh ^2(x) \, dx,x,\text {arcsinh}(c+d x)\right )}{d^3}-\frac {(2 c) \text {Subst}\left (\int (a+b x)^n \cosh (x) \sinh (x) \, dx,x,\text {arcsinh}(c+d x)\right )}{d^3}+\frac {c^2 \text {Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\text {arcsinh}(c+d x)\right )}{d^3} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{4} (a+b x)^n \cosh (x)+\frac {1}{4} (a+b x)^n \cosh (3 x)\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d^3}-\frac {(2 c) \text {Subst}\left (\int \frac {1}{2} (a+b x)^n \sinh (2 x) \, dx,x,\text {arcsinh}(c+d x)\right )}{d^3}+\frac {c^2 \text {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\text {arcsinh}(c+d x)\right )}{2 d^3}+\frac {c^2 \text {Subst}\left (\int e^x (a+b x)^n \, dx,x,\text {arcsinh}(c+d x)\right )}{2 d^3} \\ & = \frac {c^2 e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^3}-\frac {c^2 e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^3}-\frac {\text {Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\text {arcsinh}(c+d x)\right )}{4 d^3}+\frac {\text {Subst}\left (\int (a+b x)^n \cosh (3 x) \, dx,x,\text {arcsinh}(c+d x)\right )}{4 d^3}-\frac {c \text {Subst}\left (\int (a+b x)^n \sinh (2 x) \, dx,x,\text {arcsinh}(c+d x)\right )}{d^3} \\ & = \frac {c^2 e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^3}-\frac {c^2 e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^3}+\frac {\text {Subst}\left (\int e^{-3 x} (a+b x)^n \, dx,x,\text {arcsinh}(c+d x)\right )}{8 d^3}-\frac {\text {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\text {arcsinh}(c+d x)\right )}{8 d^3}-\frac {\text {Subst}\left (\int e^x (a+b x)^n \, dx,x,\text {arcsinh}(c+d x)\right )}{8 d^3}+\frac {\text {Subst}\left (\int e^{3 x} (a+b x)^n \, dx,x,\text {arcsinh}(c+d x)\right )}{8 d^3}+\frac {c \text {Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\text {arcsinh}(c+d x)\right )}{2 d^3}-\frac {c \text {Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\text {arcsinh}(c+d x)\right )}{2 d^3} \\ & = \frac {3^{-1-n} e^{-\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 d^3}-\frac {2^{-2-n} c e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^3}-\frac {e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 d^3}+\frac {c^2 e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^3}+\frac {e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 d^3}-\frac {c^2 e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^3}-\frac {2^{-2-n} c e^{\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^3}-\frac {3^{-1-n} e^{\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 d^3} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.63 \[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {2^{-3-n} 3^{-1-n} e^{-\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{b^2}\right )^{-n} \left (-2^n 3^{1+n} \left (-1+4 c^2\right ) e^{\frac {4 a}{b}} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^n \Gamma \left (1+n,\frac {a}{b}+\text {arcsinh}(c+d x)\right )+2^n \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )^n \Gamma \left (1+n,-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-2\ 3^{1+n} c e^{a/b} \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )^n \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+2^n 3^{1+n} \left (-1+4 c^2\right ) e^{\frac {2 a}{b}} \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )^n \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-e^{\frac {5 a}{b}} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^n \left (2\ 3^{1+n} c \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+2^n e^{a/b} \Gamma \left (1+n,\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )}{d^3} \]
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\[\int x^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{n}d x\]
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\[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x^{2} \,d x } \]
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\[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\int x^{2} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{n}\, dx \]
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\[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x^{2} \,d x } \]
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\[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x^{2} \,d x } \]
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Timed out. \[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^n \,d x \]
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