\(\int (d+e x^2) (a+b \text {arcsinh}(c x))^{3/2} \, dx\) [634]

Optimal result

Integrand size = 20, antiderivative size = 427 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2} \, dx=-\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+d x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b^{3/2} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b^{3/2} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3} \]

[Out]

d*x*(a+b*arcsinh(c*x))^(3/2)+1/3*e*x^3*(a+b*arcsinh(c*x))^(3/2)+1/288*b^(3/2)*e*exp(3*a/b)*erf(3^(1/2)*(a+b*ar
csinh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/c^3+1/288*b^(3/2)*e*erfi(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))
*3^(1/2)*Pi^(1/2)/c^3/exp(3*a/b)+3/8*b^(3/2)*d*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c-3/32*
b^(3/2)*e*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c^3+3/8*b^(3/2)*d*erfi((a+b*arcsinh(c*x))^(1
/2)/b^(1/2))*Pi^(1/2)/c/exp(a/b)-3/32*b^(3/2)*e*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c^3/exp(a/b)-3
/2*b*d*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^(1/2)/c+1/3*b*e*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^(1/2)/c^3-1/6
*b*e*x^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^(1/2)/c

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5793, 5772, 5798, 5774, 3388, 2211, 2236, 2235, 5777, 5812, 5780, 5556} \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2} \, dx=-\frac {3 \sqrt {\pi } b^{3/2} e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3}-\frac {3 \sqrt {\pi } b^{3/2} e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 \sqrt {\pi } b^{3/2} d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 \sqrt {\pi } b^{3/2} d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b d \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}-\frac {b e x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+\frac {b e \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}+d x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^{3/2} \]

[In]

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

[Out]

(-3*b*d*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(2*c) + (b*e*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(
3*c^3) - (b*e*x^2*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(6*c) + d*x*(a + b*ArcSinh[c*x])^(3/2) + (e*x^3*
(a + b*ArcSinh[c*x])^(3/2))/3 + (3*b^(3/2)*d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c) - (
3*b^(3/2)*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^3) + (b^(3/2)*e*E^((3*a)/b)*Sqrt[Pi/
3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(96*c^3) + (3*b^(3/2)*d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c
*x]]/Sqrt[b]])/(8*c*E^(a/b)) - (3*b^(3/2)*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^3*E^(a/b))
+ (b^(3/2)*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(96*c^3*E^((3*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 3388

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

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 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5774

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cosh[-a/b + x/b], x], x
, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 5777

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcSinh[c*x])^n/(
m + 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /;
FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 5780

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

Rule 5793

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] &&
(p > 0 || IGtQ[n, 0])

Rule 5798

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] - Dist[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]

Rule 5812

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] + (-Dist[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] - Dist[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
]

Rubi steps \begin{align*} \text {integral}& = \int \left (d (a+b \text {arcsinh}(c x))^{3/2}+e x^2 (a+b \text {arcsinh}(c x))^{3/2}\right ) \, dx \\ & = d \int (a+b \text {arcsinh}(c x))^{3/2} \, dx+e \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx \\ & = d x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} (3 b c d) \int \frac {x \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{2} (b c e) \int \frac {x^3 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+d x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{4} \left (3 b^2 d\right ) \int \frac {1}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx+\frac {1}{12} \left (b^2 e\right ) \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx+\frac {(b e) \int \frac {x \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {1+c^2 x^2}} \, dx}{3 c} \\ & = -\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+d x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {(3 b d) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 c}+\frac {(b e) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{12 c^3}-\frac {\left (b^2 e\right ) \int \frac {1}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{6 c^2} \\ & = -\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+d x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {(3 b d) \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 )}{8 c}+\frac {(3 b d) \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 )}{8 c}+\frac {(b e) \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}-\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{12 c^3}-\frac {(b e) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{6 c^3} \\ & = -\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+d x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {(3 b d) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 c}+\frac {(3 b d) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 c}+\frac {(b e) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{48 c^3}-\frac {(b e) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{48 c^3}-\frac {(b e) \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 )}{12 c^3}-\frac {(b e) \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 )}{12 c^3} \\ & = -\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+d x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {(b e) \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 )}{96 c^3}-\frac {(b e) \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 )}{96 c^3}+\frac {(b e) \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 )}{96 c^3}+\frac {(b e) \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 )}{96 c^3}-\frac {(b e) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{6 c^3}-\frac {(b e) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{6 c^3} \\ & = -\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+d x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {b^{3/2} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {b^{3/2} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {(b e) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{48 c^3}-\frac {(b e) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{48 c^3}-\frac {(b e) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{48 c^3}+\frac {(b e) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{48 c^3} \\ & = -\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+d x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b^{3/2} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b^{3/2} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.29 (sec) , antiderivative size = 770, normalized size of antiderivative = 1.80 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2} \, dx=\frac {a d e^{-\frac {a}{b}} \sqrt {a+b \text {arcsinh}(c x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )}{\sqrt {\frac {a}{b}+\text {arcsinh}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}}\right )}{2 c}+\frac {a e e^{-\frac {3 a}{b}} \sqrt {a+b \text {arcsinh}(c x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{72 c^3 \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}}}+\frac {\sqrt {b} d \left (4 \sqrt {b} \sqrt {a+b \text {arcsinh}(c x)} \left (-3 \sqrt {1+c^2 x^2}+2 c x \text {arcsinh}(c x)\right )+(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(-2 a+3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )}{8 c}+\frac {\sqrt {b} e \left (-9 \left (4 \sqrt {b} \sqrt {a+b \text {arcsinh}(c x)} \left (-3 \sqrt {1+c^2 x^2}+2 c x \text {arcsinh}(c x)\right )+(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(-2 a+3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+(2 a+b) \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right )+(-2 a+b) \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )+\sinh \left (\frac {3 a}{b}\right )\right )+12 \sqrt {b} \sqrt {a+b \text {arcsinh}(c x)} (-\cosh (3 \text {arcsinh}(c x))+2 \text {arcsinh}(c x) \sinh (3 \text {arcsinh}(c x)))\right )}{288 c^3} \]

[In]

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

[Out]

(a*d*Sqrt[a + b*ArcSinh[c*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c*x]])/Sqrt[a/b + ArcSinh[c*x]]) + Gamm
a[3/2, -((a + b*ArcSinh[c*x])/b)]/Sqrt[-((a + b*ArcSinh[c*x])/b)]))/(2*c*E^(a/b)) + (a*e*Sqrt[a + b*ArcSinh[c*
x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[3/2, a/b + ArcSinh[c*x]] + Sqrt[3]*Sqrt[a/b + ArcSinh
[c*x]]*Gamma[3/2, (-3*(a + b*ArcSinh[c*x]))/b] - 9*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[3/2, -((a + b*Ar
cSinh[c*x])/b)] - Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[3/2, (3*(a + b*ArcSinh[c*x]))/b]))
/(72*c^3*E^((3*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)]) + (Sqrt[b]*d*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c*x]]*(-3
*Sqrt[1 + c^2*x^2] + 2*c*x*ArcSinh[c*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a
/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])))/(8*c)
+ (Sqrt[b]*e*(-9*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c*x]]*(-3*Sqrt[1 + c^2*x^2] + 2*c*x*ArcSinh[c*x]) + (2*a + 3*b
)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a +
b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + (2*a + b)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x
]])/Sqrt[b]]*(Cosh[(3*a)/b] - Sinh[(3*a)/b]) + (-2*a + b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sq
rt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) + 12*Sqrt[b]*Sqrt[a + b*ArcSinh[c*x]]*(-Cosh[3*ArcSinh[c*x]] + 2*ArcSin
h[c*x]*Sinh[3*ArcSinh[c*x]])))/(288*c^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((e*x^2+d)*(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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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