Integrand size = 14, antiderivative size = 267 \[ \int x (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {2^{-3-n} e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^2}-\frac {c e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^2}+\frac {c e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^2}+\frac {2^{-3-n} e^{\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^2} \]
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Time = 0.41 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5859, 5830, 6873, 12, 6874, 3388, 2212, 5556, 3389} \[ \int x (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {2^{-n-3} e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^2}-\frac {c e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^2}+\frac {c e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^2}+\frac {2^{-n-3} e^{\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^2} \]
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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 ) (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 ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b x)^n \cosh (x) (-c+\sinh (x))}{d} \, dx,x,\text {arcsinh}(c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int (a+b x)^n \cosh (x) (-c+\sinh (x)) \, dx,x,\text {arcsinh}(c+d x)\right )}{d^2} \\ & = \frac {\text {Subst}\left (\int \left (-c (a+b x)^n \cosh (x)+(a+b x)^n \cosh (x) \sinh (x)\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d^2} \\ & = \frac {\text {Subst}\left (\int (a+b x)^n \cosh (x) \sinh (x) \, dx,x,\text {arcsinh}(c+d x)\right )}{d^2}-\frac {c \text {Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\text {arcsinh}(c+d x)\right )}{d^2} \\ & = \frac {\text {Subst}\left (\int \frac {1}{2} (a+b x)^n \sinh (2 x) \, dx,x,\text {arcsinh}(c+d x)\right )}{d^2}-\frac {c \text {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\text {arcsinh}(c+d x)\right )}{2 d^2}-\frac {c \text {Subst}\left (\int e^x (a+b x)^n \, dx,x,\text {arcsinh}(c+d x)\right )}{2 d^2} \\ & = -\frac {c 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^2}+\frac {c 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^2}+\frac {\text {Subst}\left (\int (a+b x)^n \sinh (2 x) \, dx,x,\text {arcsinh}(c+d x)\right )}{2 d^2} \\ & = -\frac {c 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^2}+\frac {c 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^2}-\frac {\text {Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\text {arcsinh}(c+d x)\right )}{4 d^2}+\frac {\text {Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\text {arcsinh}(c+d x)\right )}{4 d^2} \\ & = \frac {2^{-3-n} 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^2}-\frac {c 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^2}+\frac {c 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^2}+\frac {2^{-3-n} 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^2} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.85 \[ \int x (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {2^{-3-n} e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{b^2}\right )^{-n} \left (2^{2+n} c e^{\frac {3 a}{b}} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^n \Gamma \left (1+n,\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )^n \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-2^{2+n} c e^{a/b} \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )^n \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+e^{\frac {4 a}{b}} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^n \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{d^2} \]
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\[\int x \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{n}d x\]
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\[ \int x (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x \,d x } \]
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\[ \int x (a+b \text {arcsinh}(c+d x))^n \, dx=\int x \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{n}\, dx \]
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\[ \int x (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x \,d x } \]
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\[ \int x (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x \,d x } \]
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Timed out. \[ \int x (a+b \text {arcsinh}(c+d x))^n \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^n \,d x \]
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