Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\frac {5^{-1-n} d^3 e^{-\frac {5 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 {5 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {d+c^2 d x^2}}-\frac {5\ 3^{-1-n} d^3 e^{-\frac {3 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 {3 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {d+c^2 d x^2}}+\frac {3^{-n} d^3 e^{-\frac {3 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 {3 (a+b \text {arcsinh}(c x))}{b}\right )}{8 \sqrt {d+c^2 d x^2}}+\frac {11 d^3 e^{-\frac {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 {a+b \text {arcsinh}(c x)}{b}\right )}{16 \sqrt {d+c^2 d x^2}}+\frac {11 d^3 e^{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 {a+b \text {arcsinh}(c x)}{b}\right )}{16 \sqrt {d+c^2 d x^2}}-\frac {5\ 3^{-1-n} d^3 e^{\frac {3 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 {3 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {d+c^2 d x^2}}+\frac {3^{-n} d^3 e^{\frac {3 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 {3 (a+b \text {arcsinh}(c x))}{b}\right )}{8 \sqrt {d+c^2 d x^2}}+\frac {5^{-1-n} d^3 e^{\frac {5 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 {5 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {d+c^2 d x^2}}+d^3 \text {Int}\left (\frac {(a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}},x\right ) \]
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Not integrable
Time = 1.17 (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 = {} \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^3 (a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}}+\frac {3 c^2 d^3 x (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}}+\frac {3 c^4 d^3 x^3 (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}}+\frac {c^6 d^3 x^5 (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}}\right ) \, dx \\ & = d^3 \int \frac {(a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}} \, dx+\left (3 c^2 d^3\right ) \int \frac {x (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}} \, dx+\left (3 c^4 d^3\right ) \int \frac {x^3 (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}} \, dx+\left (c^6 d^3\right ) \int \frac {x^5 (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}} \, dx \\ & = d^3 \int \frac {(a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}} \, dx-\frac {\left (d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \sinh ^5\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 {\left (3 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \sinh \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 {\left (3 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \sinh ^3\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}} \\ & = d^3 \int \frac {(a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}} \, dx+\frac {\left (i d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{16} i x^n \sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )-\frac {5}{16} i x^n \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )+\frac {5}{8} i x^n \sinh \left (\frac {a}{b}-\frac {x}{b}\right )\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b \sqrt {d+c^2 d x^2}}-\frac {\left (3 i d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{4} i x^n \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )+\frac {3}{4} i x^n \sinh \left (\frac {a}{b}-\frac {x}{b}\right )\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b \sqrt {d+c^2 d x^2}}-\frac {\left (3 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b \sqrt {d+c^2 d x^2}}+\frac {\left (3 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b \sqrt {d+c^2 d x^2}} \\ & = \frac {3 d^3 e^{-\frac {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 {a+b \text {arcsinh}(c x)}{b}\right )}{2 \sqrt {d+c^2 d x^2}}+\frac {3 d^3 e^{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 {a+b \text {arcsinh}(c x)}{b}\right )}{2 \sqrt {d+c^2 d x^2}}+d^3 \int \frac {(a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}} \, dx-\frac {\left (d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b \sqrt {d+c^2 d x^2}}+\frac {\left (5 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b \sqrt {d+c^2 d x^2}}-\frac {\left (5 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \sinh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b \sqrt {d+c^2 d x^2}}-\frac {\left (3 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b \sqrt {d+c^2 d x^2}}+\frac {\left (9 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \sinh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b \sqrt {d+c^2 d x^2}} \\ & = \frac {3 d^3 e^{-\frac {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 {a+b \text {arcsinh}(c x)}{b}\right )}{2 \sqrt {d+c^2 d x^2}}+\frac {3 d^3 e^{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 {a+b \text {arcsinh}(c x)}{b}\right )}{2 \sqrt {d+c^2 d x^2}}+d^3 \int \frac {(a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}} \, dx-\frac {\left (d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b \sqrt {d+c^2 d x^2}}+\frac {\left (d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b \sqrt {d+c^2 d x^2}}+\frac {\left (5 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b \sqrt {d+c^2 d x^2}}-\frac {\left (5 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b \sqrt {d+c^2 d x^2}}-\frac {\left (5 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b \sqrt {d+c^2 d x^2}}+\frac {\left (5 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b \sqrt {d+c^2 d x^2}}-\frac {\left (3 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b \sqrt {d+c^2 d x^2}}+\frac {\left (3 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b \sqrt {d+c^2 d x^2}}+\frac {\left (9 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b \sqrt {d+c^2 d x^2}}-\frac {\left (9 d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b \sqrt {d+c^2 d x^2}} \\ & = \frac {5^{-1-n} d^3 e^{-\frac {5 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 {5 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {d+c^2 d x^2}}-\frac {5\ 3^{-1-n} d^3 e^{-\frac {3 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 {3 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {d+c^2 d x^2}}+\frac {3^{-n} d^3 e^{-\frac {3 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 {3 (a+b \text {arcsinh}(c x))}{b}\right )}{8 \sqrt {d+c^2 d x^2}}+\frac {11 d^3 e^{-\frac {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 {a+b \text {arcsinh}(c x)}{b}\right )}{16 \sqrt {d+c^2 d x^2}}+\frac {11 d^3 e^{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 {a+b \text {arcsinh}(c x)}{b}\right )}{16 \sqrt {d+c^2 d x^2}}-\frac {5\ 3^{-1-n} d^3 e^{\frac {3 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 {3 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {d+c^2 d x^2}}+\frac {3^{-n} d^3 e^{\frac {3 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 {3 (a+b \text {arcsinh}(c x))}{b}\right )}{8 \sqrt {d+c^2 d x^2}}+\frac {5^{-1-n} d^3 e^{\frac {5 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 {5 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {d+c^2 d x^2}}+d^3 \int \frac {(a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}} \, dx \\ \end{align*}
Not integrable
Time = 0.40 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x} \, dx \]
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Not integrable
Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
\[\int \frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{n}}{x}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 2.44 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x} \,d x \]
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