\(\int (f+g x)^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx\) [34]

Optimal result

Integrand size = 30, antiderivative size = 640 \[ \int (f+g x)^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b f^2 g x \sqrt {d+c^2 d x^2}}{c \sqrt {1+c^2 x^2}}+\frac {2 b g^3 x \sqrt {d+c^2 d x^2}}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {b c f^3 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}-\frac {3 b f g^2 x^2 \sqrt {d+c^2 d x^2}}{16 c \sqrt {1+c^2 x^2}}-\frac {b c f^2 g x^3 \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b g^3 x^3 \sqrt {d+c^2 d x^2}}{45 c \sqrt {1+c^2 x^2}}-\frac {3 b c f g^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b c g^3 x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}+\frac {1}{2} f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {f^2 g \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{c^2}-\frac {g^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {g^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {f^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}}-\frac {3 f g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c^3 \sqrt {1+c^2 x^2}} \]

[Out]

1/2*f^3*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+3/8*f*g^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+3/4*f*
g^2*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+f^2*g*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2-1/
3*g^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^4+1/5*g^3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*d*x
^2+d)^(1/2)/c^4-b*f^2*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)+2/15*b*g^3*x*(c^2*d*x^2+d)^(1/2)/c^3/(c^2*x^
2+1)^(1/2)-1/4*b*c*f^3*x^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-3/16*b*f*g^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x
^2+1)^(1/2)-1/3*b*c*f^2*g*x^3*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/45*b*g^3*x^3*(c^2*d*x^2+d)^(1/2)/c/(c^2*
x^2+1)^(1/2)-3/16*b*c*f*g^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/25*b*c*g^3*x^5*(c^2*d*x^2+d)^(1/2)/(c^
2*x^2+1)^(1/2)+1/4*f^3*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c/(c^2*x^2+1)^(1/2)-3/16*f*g^2*(a+b*arcsinh(
c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c^3/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5845, 5838, 5785, 5783, 30, 5798, 5806, 5812, 272, 45, 5804, 12} \[ \int (f+g x)^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{2} f^3 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {f^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {c^2 x^2+1}}+\frac {f^2 g \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{c^2}+\frac {3 f g^2 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {g^3 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {g^3 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 c^4}-\frac {3 f g^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{16 b c^3 \sqrt {c^2 x^2+1}}-\frac {b c f^3 x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}}-\frac {b f^2 g x \sqrt {c^2 d x^2+d}}{c \sqrt {c^2 x^2+1}}-\frac {b c f^2 g x^3 \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}}-\frac {3 b f g^2 x^2 \sqrt {c^2 d x^2+d}}{16 c \sqrt {c^2 x^2+1}}-\frac {3 b c f g^2 x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}-\frac {b c g^3 x^5 \sqrt {c^2 d x^2+d}}{25 \sqrt {c^2 x^2+1}}-\frac {b g^3 x^3 \sqrt {c^2 d x^2+d}}{45 c \sqrt {c^2 x^2+1}}+\frac {2 b g^3 x \sqrt {c^2 d x^2+d}}{15 c^3 \sqrt {c^2 x^2+1}} \]

[In]

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

[Out]

-((b*f^2*g*x*Sqrt[d + c^2*d*x^2])/(c*Sqrt[1 + c^2*x^2])) + (2*b*g^3*x*Sqrt[d + c^2*d*x^2])/(15*c^3*Sqrt[1 + c^
2*x^2]) - (b*c*f^3*x^2*Sqrt[d + c^2*d*x^2])/(4*Sqrt[1 + c^2*x^2]) - (3*b*f*g^2*x^2*Sqrt[d + c^2*d*x^2])/(16*c*
Sqrt[1 + c^2*x^2]) - (b*c*f^2*g*x^3*Sqrt[d + c^2*d*x^2])/(3*Sqrt[1 + c^2*x^2]) - (b*g^3*x^3*Sqrt[d + c^2*d*x^2
])/(45*c*Sqrt[1 + c^2*x^2]) - (3*b*c*f*g^2*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (b*c*g^3*x^5*Sqrt
[d + c^2*d*x^2])/(25*Sqrt[1 + c^2*x^2]) + (f^3*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/2 + (3*f*g^2*x*Sqrt
[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*c^2) + (3*f*g^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/4 + (f^
2*g*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/c^2 - (g^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a +
b*ArcSinh[c*x]))/(3*c^4) + (g^3*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(5*c^4) + (f^3*Sqrt[
d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2*x^2]) - (3*f*g^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh
[c*x])^2)/(16*b*c^3*Sqrt[1 + c^2*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[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]

Rule 5785

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcSinh[c*x])^n/2), x] + (Dist[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[(a + b*ArcSinh[c*x])^
n/Sqrt[1 + c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[x*(a + b*ArcSinh[c*x
])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[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 5804

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[
SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegerQ[p -
1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rule 5806

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

Rule 5838

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g
}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p,
-1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))

Rule 5845

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Dist[Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f + g*x)^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] /;
FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IntegerQ[p - 1/2] &&  !GtQ[d, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d+c^2 d x^2} \int (f+g x)^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\sqrt {d+c^2 d x^2} \int \left (f^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+3 f^2 g x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+3 f g^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+g^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (f^3 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 f^2 g \sqrt {d+c^2 d x^2}\right ) \int x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (g^3 \sqrt {d+c^2 d x^2}\right ) \int x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {1}{2} f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{4} f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {f^2 g \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{c^2}-\frac {g^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {g^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {\left (f^3 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b c f^3 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b f^2 g \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{c \sqrt {1+c^2 x^2}}+\frac {\left (3 f g^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (b c g^3 \sqrt {d+c^2 d x^2}\right ) \int \frac {-2+c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\frac {b f^2 g x \sqrt {d+c^2 d x^2}}{c \sqrt {1+c^2 x^2}}-\frac {b c f^3 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}-\frac {b c f^2 g x^3 \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {3 b c f g^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}+\frac {1}{2} f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {f^2 g \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{c^2}-\frac {g^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {g^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {f^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}}-\frac {\left (3 f g^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{8 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (3 b f g^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt {1+c^2 x^2}}-\frac {\left (b g^3 \sqrt {d+c^2 d x^2}\right ) \int \left (-2+c^2 x^2+3 c^4 x^4\right ) \, dx}{15 c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {b f^2 g x \sqrt {d+c^2 d x^2}}{c \sqrt {1+c^2 x^2}}+\frac {2 b g^3 x \sqrt {d+c^2 d x^2}}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {b c f^3 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}-\frac {3 b f g^2 x^2 \sqrt {d+c^2 d x^2}}{16 c \sqrt {1+c^2 x^2}}-\frac {b c f^2 g x^3 \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b g^3 x^3 \sqrt {d+c^2 d x^2}}{45 c \sqrt {1+c^2 x^2}}-\frac {3 b c f g^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b c g^3 x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}+\frac {1}{2} f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {f^2 g \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{c^2}-\frac {g^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {g^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {f^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}}-\frac {3 f g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.64 \[ \int (f+g x)^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {240 a d \left (1+c^2 x^2\right )^{3/2} \left (-16 g^3+c^2 g \left (120 f^2+45 f g x+8 g^2 x^2\right )+6 c^4 x \left (10 f^3+20 f^2 g x+15 f g^2 x^2+4 g^3 x^3\right )\right )-9600 b c^2 d f^2 g \left (3 c x+4 c^3 x^3+c^5 x^5-3 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)\right )+128 b d g^3 \left (1+c^2 x^2\right ) \left (30 c x-5 c^3 x^3-9 c^5 x^5+15 \sqrt {1+c^2 x^2} \left (-2+c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)\right )+3600 a c \sqrt {d} f \left (4 c^2 f^2-3 g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-3600 b c^3 d f^3 \left (1+c^2 x^2\right ) (\cosh (2 \text {arcsinh}(c x))-2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))-675 b c d f g^2 \left (1+c^2 x^2\right ) \left (8 \text {arcsinh}(c x)^2+\cosh (4 \text {arcsinh}(c x))-4 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )}{28800 c^4 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \]

[In]

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

[Out]

(240*a*d*(1 + c^2*x^2)^(3/2)*(-16*g^3 + c^2*g*(120*f^2 + 45*f*g*x + 8*g^2*x^2) + 6*c^4*x*(10*f^3 + 20*f^2*g*x
+ 15*f*g^2*x^2 + 4*g^3*x^3)) - 9600*b*c^2*d*f^2*g*(3*c*x + 4*c^3*x^3 + c^5*x^5 - 3*(1 + c^2*x^2)^(5/2)*ArcSinh
[c*x]) + 128*b*d*g^3*(1 + c^2*x^2)*(30*c*x - 5*c^3*x^3 - 9*c^5*x^5 + 15*Sqrt[1 + c^2*x^2]*(-2 + c^2*x^2 + 3*c^
4*x^4)*ArcSinh[c*x]) + 3600*a*c*Sqrt[d]*f*(4*c^2*f^2 - 3*g^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*Log[c*d*x
+ Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 3600*b*c^3*d*f^3*(1 + c^2*x^2)*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSin
h[c*x] + Sinh[2*ArcSinh[c*x]])) - 675*b*c*d*f*g^2*(1 + c^2*x^2)*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*A
rcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/(28800*c^4*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1308\) vs. \(2(560)=1120\).

Time = 0.86 (sec) , antiderivative size = 1309, normalized size of antiderivative = 2.05

method	result	size
default	\(\text {Expression too large to display}\)	\(1309\)
parts	\(\text {Expression too large to display}\)	\(1309\)

[In]

int((g*x+f)^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

a*(f^3*(1/2*x*(c^2*d*x^2+d)^(1/2)+1/2*d*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2))+g^3*(1/5*
x^2*(c^2*d*x^2+d)^(3/2)/c^2/d-2/15/d/c^4*(c^2*d*x^2+d)^(3/2))+3*f*g^2*(1/4*x*(c^2*d*x^2+d)^(3/2)/c^2/d-1/4/c^2
*(1/2*x*(c^2*d*x^2+d)^(1/2)+1/2*d*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)))+f^2*g/c^2/d*(c
^2*d*x^2+d)^(3/2))+b*(1/16*(d*(c^2*x^2+1))^(1/2)*f*arcsinh(c*x)^2*(4*c^2*f^2-3*g^2)/(c^2*x^2+1)^(1/2)/c^3+1/80
0*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6+16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4+20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^
2*x^2+5*c*x*(c^2*x^2+1)^(1/2)+1)*g^3*(-1+5*arcsinh(c*x))/c^4/(c^2*x^2+1)+3/256*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^
5+8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3+8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x+(c^2*x^2+1)^(1/2))*f*g^2*(-1+4*arcs
inh(c*x))/c^3/(c^2*x^2+1)+1/288*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2+3*c*x*(
c^2*x^2+1)^(1/2)+1)*g*(36*arcsinh(c*x)*c^2*f^2-12*c^2*f^2-3*arcsinh(c*x)*g^2+g^2)/c^4/(c^2*x^2+1)+1/16*(d*(c^2
*x^2+1))^(1/2)*(2*c^3*x^3+2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x+(c^2*x^2+1)^(1/2))*f^3*(-1+2*arcsinh(c*x))/c/(c^2*
x^2+1)+1/16*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*g*(6*arcsinh(c*x)*c^2*f^2-6*c^2*f^2-arcsin
h(c*x)*g^2+g^2)/c^4/(c^2*x^2+1)+1/16*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*g*(6*arcsinh(c*x)
*c^2*f^2+6*c^2*f^2-arcsinh(c*x)*g^2-g^2)/c^4/(c^2*x^2+1)+1/16*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2*x^2*(c^2*
x^2+1)^(1/2)+2*c*x-(c^2*x^2+1)^(1/2))*f^3*(1+2*arcsinh(c*x))/c/(c^2*x^2+1)+1/288*(d*(c^2*x^2+1))^(1/2)*(4*c^4*
x^4-4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*g*(36*arcsinh(c*x)*c^2*f^2+12*c^2*f^2-3*a
rcsinh(c*x)*g^2-g^2)/c^4/(c^2*x^2+1)+3/256*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5-8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3
*x^3-8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x-(c^2*x^2+1)^(1/2))*f*g^2*(1+4*arcsinh(c*x))/c^3/(c^2*x^2+1)+1/800*(d*(c
^2*x^2+1))^(1/2)*(16*c^6*x^6-16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4-20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2-5
*c*x*(c^2*x^2+1)^(1/2)+1)*g^3*(1+5*arcsinh(c*x))/c^4/(c^2*x^2+1))

Fricas [F]

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

[In]

integrate((g*x+f)^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral((a*g^3*x^3 + 3*a*f*g^2*x^2 + 3*a*f^2*g*x + a*f^3 + (b*g^3*x^3 + 3*b*f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*
arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)

Sympy [F]

\[ \int (f+g x)^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}\, dx \]

[In]

integrate((g*x+f)**3*(a+b*asinh(c*x))*(c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))*(f + g*x)**3, x)

Maxima [F(-2)]

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

[In]

integrate((g*x+f)^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F(-2)]

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

[In]

integrate((g*x+f)^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2),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. \[ \int (f+g x)^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d} \,d x \]

[In]

int((f + g*x)^3*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(1/2),x)

[Out]

int((f + g*x)^3*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(1/2), x)