Integrand size = 12, antiderivative size = 179 \[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=\frac {\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{6 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{96 a^3}-\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{96 a^3} \] Output:
1/3*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^(1/2)/a^3-1/6*x^2*(a^2*x^2+1)^(1/2)*arc sinh(a*x)^(1/2)/a+1/3*x^3*arcsinh(a*x)^(3/2)-3/32*Pi^(1/2)*erf(arcsinh(a*x )^(1/2))/a^3+1/288*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*arcsinh(a*x)^(1/2))/a^3-3/ 32*Pi^(1/2)*erfi(arcsinh(a*x)^(1/2))/a^3+1/288*3^(1/2)*Pi^(1/2)*erfi(3^(1/ 2)*arcsinh(a*x)^(1/2))/a^3
Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.55 \[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=\frac {\frac {\sqrt {3} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-3 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\frac {27 \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-\text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+27 \Gamma \left (\frac {5}{2},\text {arcsinh}(a x)\right )-\sqrt {3} \Gamma \left (\frac {5}{2},3 \text {arcsinh}(a x)\right )}{216 a^3} \] Input:
Integrate[x^2*ArcSinh[a*x]^(3/2),x]
Output:
((Sqrt[3]*Sqrt[-ArcSinh[a*x]]*Gamma[5/2, -3*ArcSinh[a*x]])/Sqrt[ArcSinh[a* x]] + (27*Sqrt[ArcSinh[a*x]]*Gamma[5/2, -ArcSinh[a*x]])/Sqrt[-ArcSinh[a*x] ] + 27*Gamma[5/2, ArcSinh[a*x]] - Sqrt[3]*Gamma[5/2, 3*ArcSinh[a*x]])/(216 *a^3)
Time = 1.78 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.28, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6192, 6227, 6195, 5971, 2009, 6213, 6189, 3042, 3788, 26, 2611, 2633, 2634}
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.
\(\displaystyle \int x^2 \text {arcsinh}(a x)^{3/2} \, dx\) |
\(\Big \downarrow \) 6192 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \int \frac {x^3 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {\int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}}dx}{6 a}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\) |
\(\Big \downarrow \) 6195 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {\int \frac {a^2 x^2 \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {\int \left (\frac {\cosh (3 \text {arcsinh}(a x))}{4 \sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {a^2 x^2+1}}{4 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int \frac {1}{\sqrt {\text {arcsinh}(a x)}}dx}{2 a}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\) |
\(\Big \downarrow \) 6189 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int \frac {\sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int \frac {\sin \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\frac {1}{2} i \int -\frac {i e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {i e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\frac {1}{2} \int \frac {e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)+\frac {1}{2} \int \frac {e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\int e^{\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a^2}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\) |
Input:
Int[x^2*ArcSinh[a*x]^(3/2),x]
Output:
(x^3*ArcSinh[a*x]^(3/2))/3 - (a*((x^2*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]] )/(3*a^2) - (2*((Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/a^2 - ((Sqrt[Pi]*Er f[Sqrt[ArcSinh[a*x]]])/2 + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/2)/(2*a^2)) )/(3*a^2) - (-1/8*(Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]]) + (Sqrt[Pi/3]*Erf[Sqr t[3]*Sqrt[ArcSinh[a*x]]])/8 - (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/8 + (Sqr t[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/8)/(6*a^4)))/2
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [I/2 Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e , f, m}, x] && IntegerQ[2*k]
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
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_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -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]
\[\int x^{2} \operatorname {arcsinh}\left (x a \right )^{\frac {3}{2}}d x\]
Input:
int(x^2*arcsinh(x*a)^(3/2),x)
Output:
int(x^2*arcsinh(x*a)^(3/2),x)
Exception generated. \[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2*arcsinh(a*x)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=\int x^{2} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}\, dx \] Input:
integrate(x**2*asinh(a*x)**(3/2),x)
Output:
Integral(x**2*asinh(a*x)**(3/2), x)
\[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=\int { x^{2} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(x^2*arcsinh(a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(x^2*arcsinh(a*x)^(3/2), x)
\[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=\int { x^{2} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(x^2*arcsinh(a*x)^(3/2),x, algorithm="giac")
Output:
integrate(x^2*arcsinh(a*x)^(3/2), x)
Timed out. \[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=\int x^2\,{\mathrm {asinh}\left (a\,x\right )}^{3/2} \,d x \] Input:
int(x^2*asinh(a*x)^(3/2),x)
Output:
int(x^2*asinh(a*x)^(3/2), x)
\[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=\int \sqrt {\mathit {asinh} \left (a x \right )}\, \mathit {asinh} \left (a x \right ) x^{2}d x \] Input:
int(x^2*asinh(a*x)^(3/2),x)
Output:
int(sqrt(asinh(a*x))*asinh(a*x)*x**2,x)