Integrand size = 23, antiderivative size = 271 \[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2} \, dx=-\frac {3 \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}}{16 a \sqrt {1+a^2 x^2}}-\frac {3 a x^2 \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}}{8 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}+\frac {\sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{5/2}}{5 a \sqrt {1+a^2 x^2}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}} \]
1/2*x*arcsinh(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)+1/5*arcsinh(a*x)^(5/2)*(a^2*c *x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+3/128*erf(2^(1/2)*arcsinh(a*x)^(1/2))*2^ (1/2)*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+3/128*erfi(2^(1/2)* arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/ 2)-3/16*(a^2*c*x^2+c)^(1/2)*arcsinh(a*x)^(1/2)/a/(a^2*x^2+1)^(1/2)-3/8*a*x ^2*(a^2*c*x^2+c)^(1/2)*arcsinh(a*x)^(1/2)/(a^2*x^2+1)^(1/2)
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.46 \[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2} \, dx=\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (15 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+15 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+8 \sqrt {\text {arcsinh}(a x)} (-15 \cosh (2 \text {arcsinh}(a x))+4 \text {arcsinh}(a x) (4 \text {arcsinh}(a x)+5 \sinh (2 \text {arcsinh}(a x))))\right )}{640 a \sqrt {1+a^2 x^2}} \]
(Sqrt[c*(1 + a^2*x^2)]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + 15 *Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + 8*Sqrt[ArcSinh[a*x]]*(-15*C osh[2*ArcSinh[a*x]] + 4*ArcSinh[a*x]*(4*ArcSinh[a*x] + 5*Sinh[2*ArcSinh[a* x]]))))/(640*a*Sqrt[1 + a^2*x^2])
Time = 0.92 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.71, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6200, 6192, 6198, 6234, 3042, 25, 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.
\(\displaystyle \int \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c} \, dx\) |
\(\Big \downarrow \) 6200 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \int x \sqrt {\text {arcsinh}(a x)}dx}{4 \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\) |
\(\Big \downarrow \) 6192 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {\int \frac {a^2 x^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^2}\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {\int -\frac {\sin (i \text {arcsinh}(a x))^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^2}\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}+\frac {\int \frac {\sin (i \text {arcsinh}(a x))^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^2}\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {\int \left (\frac {1}{2 \sqrt {\text {arcsinh}(a x)}}-\frac {\cosh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{4 a^2}+\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\sqrt {\text {arcsinh}(a x)}}{4 a^2}\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\) |
(x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/2 + (Sqrt[c + a^2*c*x^2]*ArcSin h[a*x]^(5/2))/(5*a*Sqrt[1 + a^2*x^2]) - (3*a*Sqrt[c + a^2*c*x^2]*((x^2*Sqr t[ArcSinh[a*x]])/2 - (-Sqrt[ArcSinh[a*x]] + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[A rcSinh[a*x]]])/4 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/4)/(4*a^2 )))/(4*Sqrt[1 + a^2*x^2])
3.5.78.3.1 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_)*(x_)^(m_.), x_Symbol] :> Simp[ x^(m + 1)*((a + b*ArcSinh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int [x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; Free Q[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*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 + 2, 0] && IGtQ[m, 0]
\[\int \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {a^{2} c \,x^{2}+c}d x\]
Exception generated. \[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2} \, dx=\int \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
\[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2} \, dx=\int { \sqrt {a^{2} c x^{2} + c} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \]
Exception generated. \[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2} \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^{3/2}\,\sqrt {c\,a^2\,x^2+c} \,d x \]