\(\int \frac {x^2 (d+c^2 d x^2)^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\) [468]

Optimal result

Integrand size = 28, antiderivative size = 457 \[ \int \frac {x^2 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d^2 x^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {5 d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {d^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {3 d^2 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {d^2 e^{\frac {7 a}{b}} \sqrt {7 \pi } \text {erf}\left (\frac {\sqrt {7} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {3 d^2 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{-\frac {7 a}{b}} \sqrt {7 \pi } \text {erfi}\left (\frac {\sqrt {7} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3} \]

[Out]

5/64*d^2*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c^3-5/64*d^2*erfi((a+b*arcsinh(c*x))^
(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c^3/exp(a/b)-1/64*d^2*exp(3*a/b)*erf(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))
*3^(1/2)*Pi^(1/2)/b^(3/2)/c^3+1/64*d^2*erfi(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/b^(3/2)
/c^3/exp(3*a/b)-3/64*d^2*exp(5*a/b)*erf(5^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(3/2)/c^3
+3/64*d^2*erfi(5^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(3/2)/c^3/exp(5*a/b)-1/64*d^2*exp(
7*a/b)*erf(7^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*7^(1/2)*Pi^(1/2)/b^(3/2)/c^3+1/64*d^2*erfi(7^(1/2)*(a+b*a
rcsinh(c*x))^(1/2)/b^(1/2))*7^(1/2)*Pi^(1/2)/b^(3/2)/c^3/exp(7*a/b)-2*d^2*x^2*(c^2*x^2+1)^(5/2)/b/c/(a+b*arcsi
nh(c*x))^(1/2)

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5814, 5819, 5556, 3389, 2211, 2236, 2235} \[ \int \frac {x^2 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {5 \sqrt {\pi } d^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {\sqrt {3 \pi } d^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {3 \sqrt {5 \pi } d^2 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {\sqrt {7 \pi } d^2 e^{\frac {7 a}{b}} \text {erf}\left (\frac {\sqrt {7} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {5 \sqrt {\pi } d^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {\sqrt {3 \pi } d^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {3 \sqrt {5 \pi } d^2 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {\sqrt {7 \pi } d^2 e^{-\frac {7 a}{b}} \text {erfi}\left (\frac {\sqrt {7} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {2 d^2 x^2 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}} \]

[In]

Int[(x^2*(d + c^2*d*x^2)^2)/(a + b*ArcSinh[c*x])^(3/2),x]

[Out]

(-2*d^2*x^2*(1 + c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (5*d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSi
nh[c*x]]/Sqrt[b]])/(64*b^(3/2)*c^3) - (d^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[
b]])/(64*b^(3/2)*c^3) - (3*d^2*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*b^(
3/2)*c^3) - (d^2*E^((7*a)/b)*Sqrt[7*Pi]*Erf[(Sqrt[7]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3) - (5
*d^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(64*b^(3/2)*c^3*E^(a/b)) + (d^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]
*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3*E^((3*a)/b)) + (3*d^2*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b
*ArcSinh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3*E^((5*a)/b)) + (d^2*Sqrt[7*Pi]*Erfi[(Sqrt[7]*Sqrt[a + b*ArcSinh[c*x]
])/Sqrt[b]])/(64*b^(3/2)*c^3*E^((7*a)/b))

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 2235

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]

Rule 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 5556

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]

Rule 5814

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[(f*x)^m*Sqrt[1 + c^2*x^2]*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[f*(m/(b*c*(
n + 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] - Dist[c*((m + 2*p + 1)/(b*f*(n + 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}, x] && EqQ[e, c^2*d]
&& LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]

Rule 5819

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 d^2 x^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\left (4 d^2\right ) \int \frac {x \left (1+c^2 x^2\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b c}+\frac {\left (14 c d^2\right ) \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b} \\ & = -\frac {2 d^2 x^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {\cosh ^4\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^3}-\frac {\left (14 d^2\right ) \text {Subst}\left (\int \frac {\cosh ^4\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^3} \\ & = -\frac {2 d^2 x^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\left (4 d^2\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 \sqrt {x}}+\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 \sqrt {x}}+\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^3}-\frac {\left (14 d^2\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{64 \sqrt {x}}+\frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{64 \sqrt {x}}-\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{64 \sqrt {x}}-\frac {3 \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{64 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^3} \\ & = -\frac {2 d^2 x^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\left (7 d^2\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b^2 c^3}-\frac {\left (7 d^2\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b^2 c^3}-\frac {d^2 \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2 c^3}-\frac {d^2 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b^2 c^3}+\frac {\left (21 d^2\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b^2 c^3}+\frac {\left (21 d^2\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b^2 c^3}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2 c^3} \\ & = -\frac {2 d^2 x^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\left (7 d^2\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^3}+\frac {\left (7 d^2\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^3}-\frac {\left (7 d^2\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {7 i a}{b}-\frac {7 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^3}+\frac {\left (7 d^2\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {7 i a}{b}-\frac {7 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^3}-\frac {d^2 \text {Subst}\left (\int \frac {e^{-i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3}+\frac {d^2 \text {Subst}\left (\int \frac {e^{i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3}-\frac {d^2 \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2 c^3}+\frac {d^2 \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2 c^3}+\frac {\left (21 d^2\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^3}-\frac {\left (21 d^2\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^3}+\frac {\left (21 d^2\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^3}-\frac {\left (21 d^2\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^3}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3} \\ & = -\frac {2 d^2 x^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\left (7 d^2\right ) \text {Subst}\left (\int e^{\frac {7 a}{b}-\frac {7 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{32 b^2 c^3}-\frac {\left (7 d^2\right ) \text {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{32 b^2 c^3}+\frac {\left (7 d^2\right ) \text {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{32 b^2 c^3}+\frac {\left (7 d^2\right ) \text {Subst}\left (\int e^{-\frac {7 a}{b}+\frac {7 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{32 b^2 c^3}-\frac {d^2 \text {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 b^2 c^3}+\frac {d^2 \text {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 b^2 c^3}-\frac {d^2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b^2 c^3}+\frac {d^2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b^2 c^3}+\frac {\left (21 d^2\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{32 b^2 c^3}+\frac {\left (21 d^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{32 b^2 c^3}-\frac {\left (21 d^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{32 b^2 c^3}-\frac {\left (21 d^2\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{32 b^2 c^3}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 b^2 c^3}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 b^2 c^3} \\ & = -\frac {2 d^2 x^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {5 d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {d^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {3 d^2 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {d^2 e^{\frac {7 a}{b}} \sqrt {7 \pi } \text {erf}\left (\frac {\sqrt {7} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {3 d^2 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{-\frac {7 a}{b}} \sqrt {7 \pi } \text {erfi}\left (\frac {\sqrt {7} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.26 \[ \int \frac {x^2 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {d^2 e^{-7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \left (e^{\frac {7 a}{b}}+3 e^{\frac {7 a}{b}+2 \text {arcsinh}(c x)}+e^{\frac {7 a}{b}+4 \text {arcsinh}(c x)}-5 e^{\frac {7 a}{b}+6 \text {arcsinh}(c x)}-5 e^{\frac {7 a}{b}+8 \text {arcsinh}(c x)}+e^{\frac {7 a}{b}+10 \text {arcsinh}(c x)}+3 e^{\frac {7 a}{b}+12 \text {arcsinh}(c x)}+e^{\frac {7 a}{b}+14 \text {arcsinh}(c x)}+5 e^{\frac {8 a}{b}+7 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )-\sqrt {7} e^{7 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )-3 \sqrt {5} e^{\frac {2 a}{b}+7 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\sqrt {3} e^{\frac {4 a}{b}+7 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+5 e^{\frac {6 a}{b}+7 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-\sqrt {3} e^{\frac {10 a}{b}+7 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-3 \sqrt {5} e^{\frac {12 a}{b}+7 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\sqrt {7} e^{7 \left (\frac {2 a}{b}+\text {arcsinh}(c x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{64 b c^3 \sqrt {a+b \text {arcsinh}(c x)}} \]

[In]

Integrate[(x^2*(d + c^2*d*x^2)^2)/(a + b*ArcSinh[c*x])^(3/2),x]

[Out]

-1/64*(d^2*(E^((7*a)/b) + 3*E^((7*a)/b + 2*ArcSinh[c*x]) + E^((7*a)/b + 4*ArcSinh[c*x]) - 5*E^((7*a)/b + 6*Arc
Sinh[c*x]) - 5*E^((7*a)/b + 8*ArcSinh[c*x]) + E^((7*a)/b + 10*ArcSinh[c*x]) + 3*E^((7*a)/b + 12*ArcSinh[c*x])
+ E^((7*a)/b + 14*ArcSinh[c*x]) + 5*E^((8*a)/b + 7*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + Arc
Sinh[c*x]] - Sqrt[7]*E^(7*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-7*(a + b*ArcSinh[c*x]))/b
] - 3*Sqrt[5]*E^((2*a)/b + 7*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-5*(a + b*ArcSinh[c*x])
)/b] - Sqrt[3]*E^((4*a)/b + 7*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]
))/b] + 5*E^((6*a)/b + 7*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] -
Sqrt[3]*E^((10*a)/b + 7*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b] - 3*Sqr
t[5]*E^((12*a)/b + 7*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (5*(a + b*ArcSinh[c*x]))/b] - Sqrt[7]*E
^(7*((2*a)/b + ArcSinh[c*x]))*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (7*(a + b*ArcSinh[c*x]))/b]))/(b*c^3*E^(7*(a
/b + ArcSinh[c*x]))*Sqrt[a + b*ArcSinh[c*x]])

Maple [F]

\[\int \frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{2}}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]

[In]

int(x^2*(c^2*d*x^2+d)^2/(a+b*arcsinh(c*x))^(3/2),x)

[Out]

int(x^2*(c^2*d*x^2+d)^2/(a+b*arcsinh(c*x))^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^2 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^2*(c^2*d*x^2+d)^2/(a+b*arcsinh(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \frac {x^2 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=d^{2} \left (\int \frac {x^{2}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {2 c^{2} x^{4}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {c^{4} x^{6}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \]

[In]

integrate(x**2*(c**2*d*x**2+d)**2/(a+b*asinh(c*x))**(3/2),x)

[Out]

d**2*(Integral(x**2/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a + b*asinh(c*x))*asinh(c*x)), x) + Integral(2*c**2*x**
4/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a + b*asinh(c*x))*asinh(c*x)), x) + Integral(c**4*x**6/(a*sqrt(a + b*asin
h(c*x)) + b*sqrt(a + b*asinh(c*x))*asinh(c*x)), x))

Maxima [F]

\[ \int \frac {x^2 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{2} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^2*(c^2*d*x^2+d)^2/(a+b*arcsinh(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate((c^2*d*x^2 + d)^2*x^2/(b*arcsinh(c*x) + a)^(3/2), x)

Giac [F]

\[ \int \frac {x^2 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{2} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^2*(c^2*d*x^2+d)^2/(a+b*arcsinh(c*x))^(3/2),x, algorithm="giac")

[Out]

integrate((c^2*d*x^2 + d)^2*x^2/(b*arcsinh(c*x) + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {x^2\,{\left (d\,c^2\,x^2+d\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \]

[In]

int((x^2*(d + c^2*d*x^2)^2)/(a + b*asinh(c*x))^(3/2),x)

[Out]

int((x^2*(d + c^2*d*x^2)^2)/(a + b*asinh(c*x))^(3/2), x)