Integrand size = 25, antiderivative size = 420 \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b c (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-2 (3+n)} d e^{-\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c \sqrt {1+c^2 x^2}}+\frac {2^{-3-n} d e^{-\frac {2 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c \sqrt {1+c^2 x^2}}-\frac {2^{-3-n} d e^{\frac {2 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c \sqrt {1+c^2 x^2}}-\frac {2^{-2 (3+n)} d e^{\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c \sqrt {1+c^2 x^2}} \]
3/8*d*(a+b*arcsinh(c*x))^(1+n)*(c^2*d*x^2+d)^(1/2)/b/c/(1+n)/(c^2*x^2+1)^( 1/2)+d*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-4*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+ d)^(1/2)/(2^(6+2*n))/c/exp(4*a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^ (1/2)+2^(-3-n)*d*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-2*(a+b*arcsinh(c*x))/b)*( c^2*d*x^2+d)^(1/2)/c/exp(2*a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1 /2)-2^(-3-n)*d*exp(2*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,2*(a+b*arcsinh(c* x))/b)*(c^2*d*x^2+d)^(1/2)/c/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)- d*exp(4*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,4*(a+b*arcsinh(c*x))/b)*(c^2*d *x^2+d)^(1/2)/(2^(6+2*n))/c/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)
Time = 1.09 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.69 \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {8 (a+b \text {arcsinh}(c x))}{b (1+n)}+8 \left (\frac {4 a+4 b \text {arcsinh}(c x)}{b+b n}+2^{-n} e^{-\frac {2 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-2^{-n} e^{\frac {2 a}{b}} \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )\right )+4^{-n} e^{-\frac {4 a}{b}} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^{-n} \left (\left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{64 c \sqrt {d+c^2 d x^2}} \]
(d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*((-8*(a + b*ArcSinh[c*x]))/( b*(1 + n)) + 8*((4*a + 4*b*ArcSinh[c*x])/(b + b*n) + Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b]/(2^n*E^((2*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n) - (E^ ((2*a)/b)*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/(2^n*(a/b + ArcSinh[c* x])^n)) + ((a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b ] - E^((8*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcSi nh[c*x]))/b])/(4^n*E^((4*a)/b)*(-((a + b*ArcSinh[c*x])^2/b^2))^n)))/(64*c* Sqrt[d + c^2*d*x^2])
Time = 0.61 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.72, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6206, 3042, 3793, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx\) |
| \(\Big \downarrow \) 6206 |
| \(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int (a+b \text {arcsinh}(c x))^n \cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b c \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int (a+b \text {arcsinh}(c x))^n \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )^4d(a+b \text {arcsinh}(c x))}{b c \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int \left (\frac {1}{8} \cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right ) (a+b \text {arcsinh}(c x))^n+\frac {1}{2} \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right ) (a+b \text {arcsinh}(c x))^n+\frac {3}{8} (a+b \text {arcsinh}(c x))^n\right )d(a+b \text {arcsinh}(c x))}{b c \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \left (\frac {3 (a+b \text {arcsinh}(c x))^{n+1}}{8 (n+1)}+b 2^{-2 (n+3)} e^{-\frac {4 a}{b}} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+b 2^{-n-3} e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-b 2^{-n-3} e^{\frac {2 a}{b}} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-b 2^{-2 (n+3)} e^{\frac {4 a}{b}} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b c \sqrt {c^2 x^2+1}}\) |
(d*Sqrt[d + c^2*d*x^2]*((3*(a + b*ArcSinh[c*x])^(1 + n))/(8*(1 + n)) + (b* (a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(3 + n))*E^((4*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-3 - n)*b*(a + b*A rcSinh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b])/(E^((2*a)/b)*(-( (a + b*ArcSinh[c*x])/b))^n) - (2^(-3 - n)*b*E^((2*a)/b)*(a + b*ArcSinh[c*x ])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/((a + b*ArcSinh[c*x])/b)^n - (b*E^((4*a)/b)*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x ]))/b])/(2^(2*(3 + n))*((a + b*ArcSinh[c*x])/b)^n)))/(b*c*Sqrt[1 + c^2*x^2 ])
3.6.19.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Subst[Int [x^n*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0]
\[\int \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{n}d x\]
\[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} \,d x } \]
Timed out. \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\text {Timed out} \]
\[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} \,d x } \]
Exception generated. \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]