3.7.29 \(\int (d+e x^2)^2 \sqrt {a+b \sinh ^{-1}(c x)} \, dx\) [629]

Optimal. Leaf size=672 \[ d^2 x \sqrt {a+b \sinh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {\sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {b} e^2 e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^5}+\frac {\sqrt {b} d e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {b} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^5}+\frac {\sqrt {b} e^2 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}+\frac {\sqrt {b} d e e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {b} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {b} d e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {\sqrt {b} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {b} e^2 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{320 c^5} \]

[Out]

1/1600*e^2*exp(5*a/b)*erf(5^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*b^(1/2)*5^(1/2)*Pi^(1/2)/c^5-1/1600*e^2*er
fi(5^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*b^(1/2)*5^(1/2)*Pi^(1/2)/c^5/exp(5*a/b)+1/72*d*e*exp(3*a/b)*erf(3
^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*b^(1/2)*3^(1/2)*Pi^(1/2)/c^3-1/192*e^2*exp(3*a/b)*erf(3^(1/2)*(a+b*ar
csinh(c*x))^(1/2)/b^(1/2))*b^(1/2)*3^(1/2)*Pi^(1/2)/c^5-1/72*d*e*erfi(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2)
)*b^(1/2)*3^(1/2)*Pi^(1/2)/c^3/exp(3*a/b)+1/192*e^2*erfi(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*b^(1/2)*3^(
1/2)*Pi^(1/2)/c^5/exp(3*a/b)+1/4*d^2*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c-1/8*d*e
*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c^3+1/32*e^2*exp(a/b)*erf((a+b*arcsinh(c*x))^
(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c^5-1/4*d^2*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c/exp(a/b)
+1/8*d*e*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c^3/exp(a/b)-1/32*e^2*erfi((a+b*arcsinh(c*x))
^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c^5/exp(a/b)+d^2*x*(a+b*arcsinh(c*x))^(1/2)+2/3*d*e*x^3*(a+b*arcsinh(c*x))^(1
/2)+1/5*e^2*x^5*(a+b*arcsinh(c*x))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.20, antiderivative size = 672, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5793, 5772, 5819, 3389, 2211, 2236, 2235, 5777, 3393} \begin {gather*} \frac {\sqrt {\pi } \sqrt {b} e^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^5}+\frac {\sqrt {\frac {\pi }{5}} \sqrt {b} e^2 e^{\frac {5 a}{b}} \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {\pi } \sqrt {b} e^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^5}+\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {\frac {\pi }{5}} \sqrt {b} e^2 e^{-\frac {5 a}{b}} \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {\pi } \sqrt {b} d e e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} d e e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {\sqrt {\pi } \sqrt {b} d e e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} d e e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {\sqrt {\pi } \sqrt {b} d^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {\pi } \sqrt {b} d^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}+d^2 x \sqrt {a+b \sinh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sinh ^{-1}(c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

d^2*x*Sqrt[a + b*ArcSinh[c*x]] + (2*d*e*x^3*Sqrt[a + b*ArcSinh[c*x]])/3 + (e^2*x^5*Sqrt[a + b*ArcSinh[c*x]])/5
+ (Sqrt[b]*d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*d*e*E^(a/b)*Sqrt[Pi]*
Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c^3) + (Sqrt[b]*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sq
rt[b]])/(32*c^5) + (Sqrt[b]*d*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(24*c^
3) - (Sqrt[b]*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*c^5) + (Sqrt[b]*
e^2*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(320*c^5) - (Sqrt[b]*d^2*Sqrt[Pi]*
Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c*E^(a/b)) + (Sqrt[b]*d*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sq
rt[b]])/(8*c^3*E^(a/b)) - (Sqrt[b]*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^5*E^(a/b)) - (Sq
rt[b]*d*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(24*c^3*E^((3*a)/b)) + (Sqrt[b]*e^2*Sqr
t[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*c^5*E^((3*a)/b)) - (Sqrt[b]*e^2*Sqrt[Pi/5]*Erfi[
(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(320*c^5*E^((5*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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[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])
)

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 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 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 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*} \int \left (d+e x^2\right )^2 \sqrt {a+b \sinh ^{-1}(c x)} \, dx &=\int \left (d^2 \sqrt {a+b \sinh ^{-1}(c x)}+2 d e x^2 \sqrt {a+b \sinh ^{-1}(c x)}+e^2 x^4 \sqrt {a+b \sinh ^{-1}(c x)}\right ) \, dx\\ &=d^2 \int \sqrt {a+b \sinh ^{-1}(c x)} \, dx+(2 d e) \int x^2 \sqrt {a+b \sinh ^{-1}(c x)} \, dx+e^2 \int x^4 \sqrt {a+b \sinh ^{-1}(c x)} \, dx\\ &=d^2 x \sqrt {a+b \sinh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {1}{2} \left (b c d^2\right ) \int \frac {x}{\sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx-\frac {1}{3} (b c d e) \int \frac {x^3}{\sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx-\frac {1}{10} \left (b c e^2\right ) \int \frac {x^5}{\sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx\\ &=d^2 x \sqrt {a+b \sinh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 c}-\frac {(b d e) \text {Subst}\left (\int \frac {\sinh ^3(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 c^3}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\sinh ^5(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{10 c^5}\\ &=d^2 x \sqrt {a+b \sinh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c}-\frac {(i b d e) \text {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 \sqrt {a+b x}}-\frac {i \sinh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c^3}+\frac {\left (i b e^2\right ) \text {Subst}\left (\int \left (\frac {5 i \sinh (x)}{8 \sqrt {a+b x}}-\frac {5 i \sinh (3 x)}{16 \sqrt {a+b x}}+\frac {i \sinh (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{10 c^5}\\ &=d^2 x \sqrt {a+b \sinh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {d^2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{2 c}-\frac {d^2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{2 c}-\frac {(b d e) \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{12 c^3}+\frac {(b d e) \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^3}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\sinh (5 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{160 c^5}+\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^5}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^5}\\ &=d^2 x \sqrt {a+b \sinh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {\sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}+\frac {(b d e) \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{24 c^3}-\frac {(b d e) \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{24 c^3}-\frac {(b d e) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^3}+\frac {(b d e) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^3}+\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{320 c^5}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{320 c^5}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^5}+\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^5}+\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^5}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^5}\\ &=d^2 x \sqrt {a+b \sinh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {\sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}+\frac {(d e) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{12 c^3}-\frac {(d e) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{12 c^3}-\frac {(d e) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{4 c^3}+\frac {(d e) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{4 c^3}+\frac {e^2 \text {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{160 c^5}-\frac {e^2 \text {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{160 c^5}-\frac {e^2 \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{32 c^5}+\frac {e^2 \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{32 c^5}+\frac {e^2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{16 c^5}-\frac {e^2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{16 c^5}\\ &=d^2 x \sqrt {a+b \sinh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {\sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {b} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^5}+\frac {\sqrt {b} d e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {b} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^5}+\frac {\sqrt {b} e^2 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}+\frac {\sqrt {b} d e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {b} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {b} d e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {\sqrt {b} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {b} e^2 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{320 c^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 4.17, size = 535, normalized size = 0.80 \begin {gather*} -\frac {b e^{-\frac {5 a}{b}} \left (450 e^{\frac {6 a}{b}} \left (8 a c^4 d^2 \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)}+8 b c^4 d^2 \sinh ^{-1}(c x) \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)}+b \left (4 c^2 d-e\right ) e \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \sqrt {-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}}\right ) \Gamma \left (\frac {3}{2},\frac {a}{b}+\sinh ^{-1}(c x)\right )+9 \sqrt {5} b e^2 \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \sqrt {-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}} \Gamma \left (\frac {3}{2},-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+25 \sqrt {3} b \left (8 c^2 d-3 e\right ) e e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \sqrt {-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}} \Gamma \left (\frac {3}{2},-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+450 e^{\frac {4 a}{b}} \left (8 a c^4 d^2 \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}}+8 b c^4 d^2 \sinh ^{-1}(c x) \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}}+b e \left (-4 c^2 d+e\right ) \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \sqrt {-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}}\right ) \Gamma \left (\frac {3}{2},-\frac {a+b \sinh ^{-1}(c x)}{b}\right )-b e e^{\frac {8 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \sqrt {-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}} \left (25 \sqrt {3} \left (8 c^2 d-3 e\right ) \Gamma \left (\frac {3}{2},\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+9 \sqrt {5} e e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )\right )}{7200 c^5 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7200*(b*(450*E^((6*a)/b)*(8*a*c^4*d^2*Sqrt[a/b + ArcSinh[c*x]] + 8*b*c^4*d^2*ArcSinh[c*x]*Sqrt[a/b + ArcSin
h[c*x]] + b*(4*c^2*d - e)*e*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, a/
b + ArcSinh[c*x]] + 9*Sqrt[5]*b*e^2*Sqrt[a/b + ArcSinh[c*x]]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)]*Gamma[3/2, (-
5*(a + b*ArcSinh[c*x]))/b] + 25*Sqrt[3]*b*(8*c^2*d - 3*e)*e*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Sqrt[-((a + b
*ArcSinh[c*x])^2/b^2)]*Gamma[3/2, (-3*(a + b*ArcSinh[c*x]))/b] + 450*E^((4*a)/b)*(8*a*c^4*d^2*Sqrt[-((a + b*Ar
cSinh[c*x])/b)] + 8*b*c^4*d^2*ArcSinh[c*x]*Sqrt[-((a + b*ArcSinh[c*x])/b)] + b*e*(-4*c^2*d + e)*Sqrt[a/b + Arc
Sinh[c*x]]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, -((a + b*ArcSinh[c*x])/b)] - b*e*E^((8*a)/b)*Sqrt[-
((a + b*ArcSinh[c*x])/b)]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)]*(25*Sqrt[3]*(8*c^2*d - 3*e)*Gamma[3/2, (3*(a + b
*ArcSinh[c*x]))/b] + 9*Sqrt[5]*e*E^((2*a)/b)*Gamma[3/2, (5*(a + b*ArcSinh[c*x]))/b])))/(c^5*E^((5*a)/b)*(a + b
*ArcSinh[c*x])^(3/2))

________________________________________________________________________________________

Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (e \,x^{2}+d \right )^{2} \sqrt {a +b \arcsinh \left (c x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x)

[Out]

int((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \left (d + e x^{2}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**2*(a+b*asinh(c*x))**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )}\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c*x))^(1/2)*(d + e*x^2)^2,x)

[Out]

int((a + b*asinh(c*x))^(1/2)*(d + e*x^2)^2, x)

________________________________________________________________________________________