Integrand size = 28, antiderivative size = 616 \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{16 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} 3^{-1-n} d e^{-\frac {6 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-2 n} d e^{-\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} d e^{-\frac {2 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} d e^{\frac {2 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-2 n} d e^{\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} 3^{-1-n} d e^{\frac {6 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}} \]
[Out]
Time = 0.45 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5819, 5556, 3388, 2212} \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {d \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^{n+1}}{16 b c^3 (n+1) \sqrt {c^2 x^2+1}}+\frac {d 2^{-n-7} 3^{-n-1} e^{-\frac {6 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}+\frac {d 2^{-2 n-7} e^{-\frac {4 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {d 2^{-n-7} e^{-\frac {2 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}+\frac {d 2^{-n-7} e^{\frac {2 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {d 2^{-2 n-7} e^{\frac {4 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {d 2^{-n-7} 3^{-n-1} e^{\frac {6 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}} \]
[In]
[Out]
Rule 2212
Rule 3388
Rule 5556
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x^n \cosh ^4\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 {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (-\frac {x^n}{16}+\frac {1}{32} x^n \cosh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )+\frac {1}{16} x^n \cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )-\frac {1}{32} x^n \cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{16 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {6 a}{b}-\frac {6 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3 \sqrt {1+c^2 x^2}}-\frac {\left (d \sqrt {d+c^2 d 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 )}{32 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \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 )}{16 b c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{16 b c^3 (1+n) \sqrt {1+c^2 x^2}}-\frac {\left (d \sqrt {d+c^2 d 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 )}{64 b c^3 \sqrt {1+c^2 x^2}}-\frac {\left (d \sqrt {d+c^2 d 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 )}{64 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {6 i a}{b}-\frac {6 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {6 i a}{b}-\frac {6 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \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 )}{32 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \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 )}{32 b c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{16 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} 3^{-1-n} d e^{-\frac {6 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 {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-2 n} d 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^{-7-n} d e^{-\frac {2 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 {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} d e^{\frac {2 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 {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-2 n} d 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^{-7-n} 3^{-1-n} d e^{\frac {6 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 {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}
Time = 2.34 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.70 \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {2^{-7-2 n} 3^{-1-n} d^2 e^{-\frac {6 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^{-n} \left (-2^n b (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )-3^{1+n} b e^{\frac {2 a}{b}} (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+2^n 3^{1+n} b e^{\frac {4 a}{b}} (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-2^n 3^{1+n} b e^{\frac {8 a}{b}} (1+n) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+3^{1+n} b e^{\frac {10 a}{b}} (1+n) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+2^n e^{\frac {6 a}{b}} \left (2^{3+n} 3^{1+n} (a+b \text {arcsinh}(c x)) \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^n+b e^{\frac {6 a}{b}} (1+n) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{b c^3 (1+n) \sqrt {d+c^2 d x^2}} \]
[In]
[Out]
\[\int x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{n}d x\]
[In]
[Out]
\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \]
[In]
[Out]
Exception generated. \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\text {Exception raised: HeuristicGCDFailed} \]
[In]
[Out]
\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \]
[In]
[Out]
\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]
[In]
[Out]