Integrand size = 10, antiderivative size = 152 \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\frac {15 \sqrt {\text {arcsinh}(a x)}}{64 a^2}+\frac {15}{32} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{8 a}+\frac {\text {arcsinh}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{256 a^2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{256 a^2} \]
1/4*arcsinh(a*x)^(5/2)/a^2+1/2*x^2*arcsinh(a*x)^(5/2)-15/512*erf(2^(1/2)*a rcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^2-15/512*erfi(2^(1/2)*arcsinh(a*x)^( 1/2))*2^(1/2)*Pi^(1/2)/a^2-5/8*x*arcsinh(a*x)^(3/2)*(a^2*x^2+1)^(1/2)/a+15 /64*arcsinh(a*x)^(1/2)/a^2+15/32*x^2*arcsinh(a*x)^(1/2)
Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.34 \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\frac {\frac {\sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-2 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\Gamma \left (\frac {7}{2},2 \text {arcsinh}(a x)\right )}{32 \sqrt {2} a^2} \]
((Sqrt[ArcSinh[a*x]]*Gamma[7/2, -2*ArcSinh[a*x]])/Sqrt[-ArcSinh[a*x]] + Ga mma[7/2, 2*ArcSinh[a*x]])/(32*Sqrt[2]*a^2)
Time = 0.93 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6192, 6227, 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.
\(\displaystyle \int x \text {arcsinh}(a x)^{5/2} \, dx\) |
\(\Big \downarrow \) 6192 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \int \frac {x^2 \text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {\int \frac {\text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {3 \int x \sqrt {\text {arcsinh}(a x)}dx}{4 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\) |
\(\Big \downarrow \) 6192 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {3 \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 a}-\frac {\int \frac {\text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {3 \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 a}-\frac {\text {arcsinh}(a x)^{5/2}}{5 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {3 \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {\int \frac {a^2 x^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^2}\right )}{4 a}-\frac {\text {arcsinh}(a x)^{5/2}}{5 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {3 \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {\int -\frac {\sin (i \text {arcsinh}(a x))^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^2}\right )}{4 a}-\frac {\text {arcsinh}(a x)^{5/2}}{5 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {3 \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}+\frac {\int \frac {\sin (i \text {arcsinh}(a x))^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^2}\right )}{4 a}-\frac {\text {arcsinh}(a x)^{5/2}}{5 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {3 \left (\frac {\int \left (\frac {1}{2 \sqrt {\text {arcsinh}(a x)}}-\frac {\cosh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{4 a^2}+\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}\right )}{4 a}-\frac {\text {arcsinh}(a x)^{5/2}}{5 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {\text {arcsinh}(a x)^{5/2}}{5 a^3}-\frac {3 \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\sqrt {\text {arcsinh}(a x)}}{4 a^2}\right )}{4 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\) |
(x^2*ArcSinh[a*x]^(5/2))/2 - (5*a*((x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2) )/(2*a^2) - ArcSinh[a*x]^(5/2)/(5*a^3) - (3*((x^2*Sqrt[ArcSinh[a*x]])/2 - (-Sqrt[ArcSinh[a*x]] + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/4 + (S qrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/4)/(4*a^2)))/(4*a)))/4
3.1.89.3.1 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_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 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]
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (-128 \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+160 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, a x -120 \sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}-64 \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }-60 \sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }+15 \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )+15 \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )\right )}{512 \sqrt {\pi }\, a^{2}}\) | \(136\) |
-1/512*2^(1/2)*(-128*arcsinh(a*x)^(5/2)*2^(1/2)*Pi^(1/2)*a^2*x^2+160*arcsi nh(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*a*x-120*2^(1/2)*arcsinh(a *x)^(1/2)*Pi^(1/2)*a^2*x^2-64*arcsinh(a*x)^(5/2)*2^(1/2)*Pi^(1/2)-60*2^(1/ 2)*arcsinh(a*x)^(1/2)*Pi^(1/2)+15*Pi*erf(2^(1/2)*arcsinh(a*x)^(1/2))+15*Pi *erfi(2^(1/2)*arcsinh(a*x)^(1/2)))/Pi^(1/2)/a^2
Exception generated. \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\int x \operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}\, dx \]
\[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}} \,d x } \]
Exception generated. \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\int x\,{\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \]