\(\int (f+g x)^3 (d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) [39]

Optimal result

Integrand size = 30, antiderivative size = 918 \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {3 b d f^2 g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}+\frac {2 b d g^3 x \sqrt {d+c^2 d x^2}}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {5 b c d f^3 x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {3 b d f g^2 x^2 \sqrt {d+c^2 d x^2}}{32 c \sqrt {1+c^2 x^2}}-\frac {2 b c d f^2 g x^3 \sqrt {d+c^2 d x^2}}{5 \sqrt {1+c^2 x^2}}-\frac {b d g^3 x^3 \sqrt {d+c^2 d x^2}}{105 c \sqrt {1+c^2 x^2}}-\frac {b c^3 d f^3 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {7 b c d f g^2 x^4 \sqrt {d+c^2 d x^2}}{32 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d f^2 g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {8 b c d g^3 x^5 \sqrt {d+c^2 d x^2}}{175 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f g^2 x^6 \sqrt {d+c^2 d x^2}}{12 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g^3 x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}+\frac {3}{8} d f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} d f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{2} d f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d f^2 g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {d g^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {d g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {3 d f^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}}-\frac {3 d f g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c^3 \sqrt {1+c^2 x^2}} \]

[Out]

3/8*d*f^3*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+3/16*d*f*g^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+3
/8*d*f*g^2*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/4*d*f^3*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)
^(1/2)+1/2*d*f*g^2*x^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+3/5*d*f^2*g*(c^2*x^2+1)^2*(a+b*arcsi
nh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2-1/5*d*g^3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^4+1/7*d*g^3*
(c^2*x^2+1)^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^4-3/5*b*d*f^2*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/
2)+2/35*b*d*g^3*x*(c^2*d*x^2+d)^(1/2)/c^3/(c^2*x^2+1)^(1/2)-5/16*b*c*d*f^3*x^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)
^(1/2)-3/32*b*d*f*g^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-2/5*b*c*d*f^2*g*x^3*(c^2*d*x^2+d)^(1/2)/(c^2
*x^2+1)^(1/2)-1/105*b*d*g^3*x^3*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-1/16*b*c^3*d*f^3*x^4*(c^2*d*x^2+d)^(1/
2)/(c^2*x^2+1)^(1/2)-7/32*b*c*d*f*g^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-3/25*b*c^3*d*f^2*g*x^5*(c^2*d*
x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-8/175*b*c*d*g^3*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/12*b*c^3*d*f*g^2*x^
6*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/49*b*c^3*d*g^3*x^7*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+3/16*d*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)-3/32*d*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 = 0.66 (sec) , antiderivative size = 918, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.567, Rules used = {5845, 5838, 5786, 5785, 5783, 30, 14, 5798, 200, 5808, 5806, 5812, 272, 45, 5804, 12, 380} \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b c^3 d g^3 \sqrt {c^2 d x^2+d} x^7}{49 \sqrt {c^2 x^2+1}}-\frac {b c^3 d f g^2 \sqrt {c^2 d x^2+d} x^6}{12 \sqrt {c^2 x^2+1}}-\frac {8 b c d g^3 \sqrt {c^2 d x^2+d} x^5}{175 \sqrt {c^2 x^2+1}}-\frac {3 b c^3 d f^2 g \sqrt {c^2 d x^2+d} x^5}{25 \sqrt {c^2 x^2+1}}-\frac {b c^3 d f^3 \sqrt {c^2 d x^2+d} x^4}{16 \sqrt {c^2 x^2+1}}-\frac {7 b c d f g^2 \sqrt {c^2 d x^2+d} x^4}{32 \sqrt {c^2 x^2+1}}+\frac {3}{8} d f g^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x^3+\frac {1}{2} d f g^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x^3-\frac {b d g^3 \sqrt {c^2 d x^2+d} x^3}{105 c \sqrt {c^2 x^2+1}}-\frac {2 b c d f^2 g \sqrt {c^2 d x^2+d} x^3}{5 \sqrt {c^2 x^2+1}}-\frac {5 b c d f^3 \sqrt {c^2 d x^2+d} x^2}{16 \sqrt {c^2 x^2+1}}-\frac {3 b d f g^2 \sqrt {c^2 d x^2+d} x^2}{32 c \sqrt {c^2 x^2+1}}+\frac {3}{8} d f^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x+\frac {3 d f g^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x}{16 c^2}+\frac {1}{4} d f^3 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x+\frac {2 b d g^3 \sqrt {c^2 d x^2+d} x}{35 c^3 \sqrt {c^2 x^2+1}}-\frac {3 b d f^2 g \sqrt {c^2 d x^2+d} x}{5 c \sqrt {c^2 x^2+1}}+\frac {3 d f^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {c^2 x^2+1}}-\frac {3 d f g^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{32 b c^3 \sqrt {c^2 x^2+1}}+\frac {d g^3 \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {d g^3 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {3 d f^2 g \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{5 c^2} \]

[In]

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

[Out]

(-3*b*d*f^2*g*x*Sqrt[d + c^2*d*x^2])/(5*c*Sqrt[1 + c^2*x^2]) + (2*b*d*g^3*x*Sqrt[d + c^2*d*x^2])/(35*c^3*Sqrt[
1 + c^2*x^2]) - (5*b*c*d*f^3*x^2*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (3*b*d*f*g^2*x^2*Sqrt[d + c^2*d
*x^2])/(32*c*Sqrt[1 + c^2*x^2]) - (2*b*c*d*f^2*g*x^3*Sqrt[d + c^2*d*x^2])/(5*Sqrt[1 + c^2*x^2]) - (b*d*g^3*x^3
*Sqrt[d + c^2*d*x^2])/(105*c*Sqrt[1 + c^2*x^2]) - (b*c^3*d*f^3*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2])
- (7*b*c*d*f*g^2*x^4*Sqrt[d + c^2*d*x^2])/(32*Sqrt[1 + c^2*x^2]) - (3*b*c^3*d*f^2*g*x^5*Sqrt[d + c^2*d*x^2])/
(25*Sqrt[1 + c^2*x^2]) - (8*b*c*d*g^3*x^5*Sqrt[d + c^2*d*x^2])/(175*Sqrt[1 + c^2*x^2]) - (b*c^3*d*f*g^2*x^6*Sq
rt[d + c^2*d*x^2])/(12*Sqrt[1 + c^2*x^2]) - (b*c^3*d*g^3*x^7*Sqrt[d + c^2*d*x^2])/(49*Sqrt[1 + c^2*x^2]) + (3*
d*f^3*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/8 + (3*d*f*g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(
16*c^2) + (3*d*f*g^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/8 + (d*f^3*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x
^2]*(a + b*ArcSinh[c*x]))/4 + (d*f*g^2*x^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/2 + (3*d*f^
2*g*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(5*c^2) - (d*g^3*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*
x^2]*(a + b*ArcSinh[c*x]))/(5*c^4) + (d*g^3*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^4)
+ (3*d*f^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*c*Sqrt[1 + c^2*x^2]) - (3*d*f*g^2*Sqrt[d + c^2*d*
x^2]*(a + b*ArcSinh[c*x])^2)/(32*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 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 380

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

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 5786

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*(
(a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Dist[2*d*(p/(2*p + 1)), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^
n, x], x] - Dist[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*A
rcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 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 5808

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(f*(m + 2*p + 1))), x] + (Dist[2*d*(p/(m + 2*p + 1)), Int
[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*c*(n/(f*(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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -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 {\left (d \sqrt {d+c^2 d x^2}\right ) \int (f+g x)^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \left (f^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+3 f^2 g x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+3 f g^2 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+g^3 x^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d f^3 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 d f^2 g \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 d f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (d g^3 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {1}{4} d f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{2} d f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d f^2 g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {d 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 {d g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {\left (3 d f^3 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (b c d f^3 \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (3 b d f^2 g \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^2 \, dx}{5 c \sqrt {1+c^2 x^2}}+\frac {\left (3 d 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}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b c d f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b c d g^3 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^2 \left (-2+5 c^2 x^2\right )}{35 c^4} \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {3}{8} d f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{8} d f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} d f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{2} d f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d f^2 g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {d 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 {d g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {\left (3 d 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}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d f^3 \sqrt {d+c^2 d x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c d f^3 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (3 b d f^2 g \sqrt {d+c^2 d x^2}\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt {1+c^2 x^2}}+\frac {\left (3 d 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}{8 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c d f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d f g^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b d g^3 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^2 \left (-2+5 c^2 x^2\right ) \, dx}{35 c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {3 b d f^2 g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {5 b c d f^3 x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {2 b c d f^2 g x^3 \sqrt {d+c^2 d x^2}}{5 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f^3 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {7 b c d f g^2 x^4 \sqrt {d+c^2 d x^2}}{32 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d f^2 g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f g^2 x^6 \sqrt {d+c^2 d x^2}}{12 \sqrt {1+c^2 x^2}}+\frac {3}{8} d f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} d f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{2} d f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d f^2 g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {d 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 {d g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {3 d f^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}}-\frac {\left (3 d 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}{16 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (3 b d f g^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt {1+c^2 x^2}}-\frac {\left (b d g^3 \sqrt {d+c^2 d x^2}\right ) \int \left (-2+c^2 x^2+8 c^4 x^4+5 c^6 x^6\right ) \, dx}{35 c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {3 b d f^2 g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}+\frac {2 b d g^3 x \sqrt {d+c^2 d x^2}}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {5 b c d f^3 x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {3 b d f g^2 x^2 \sqrt {d+c^2 d x^2}}{32 c \sqrt {1+c^2 x^2}}-\frac {2 b c d f^2 g x^3 \sqrt {d+c^2 d x^2}}{5 \sqrt {1+c^2 x^2}}-\frac {b d g^3 x^3 \sqrt {d+c^2 d x^2}}{105 c \sqrt {1+c^2 x^2}}-\frac {b c^3 d f^3 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {7 b c d f g^2 x^4 \sqrt {d+c^2 d x^2}}{32 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d f^2 g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {8 b c d g^3 x^5 \sqrt {d+c^2 d x^2}}{175 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f g^2 x^6 \sqrt {d+c^2 d x^2}}{12 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g^3 x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}+\frac {3}{8} d f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} d f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{2} d f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d f^2 g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {d 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 {d g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {3 d f^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}}-\frac {3 d f g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.60 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.58 \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^2 \left (1+c^2 x^2\right ) \left (-1680 a \sqrt {1+c^2 x^2} \left (-32 g^3+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )+4 c^6 x^3 \left (35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right )+2 c^4 x \left (175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right )\right )+b \left (-35 c g^2 (245 f+1536 g x)+70 c^3 \left (1785 f^3+8064 f^2 g x+1260 f g^2 x^2+128 g^3 x^3\right )+168 c^5 x^2 \left (1750 f^3+2240 f^2 g x+1225 f g^2 x^2+256 g^3 x^3\right )+16 c^7 x^4 \left (3675 f^3+7056 f^2 g x+4900 f g^2 x^2+1200 g^3 x^3\right )\right )\right )+88200 b c d^2 f \left (2 c^2 f^2-g^2\right ) \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+176400 a c d^{3/2} f \left (2 c^2 f^2-g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+420 b d^2 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (35 c f \left (16 c^2 f^2-3 g^2\right ) \sinh (2 \text {arcsinh}(c x))+35 c f \left (2 c^2 f^2+3 g^2\right ) \sinh (4 \text {arcsinh}(c x))+g \left (64 \left (1+c^2 x^2\right )^{5/2} \left (-2 g^2+c^2 \left (21 f^2+5 g^2 x^2\right )\right )+35 c f g \sinh (6 \text {arcsinh}(c x))\right )\right )}{940800 c^4 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \]

[In]

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

[Out]

(-(d^2*(1 + c^2*x^2)*(-1680*a*Sqrt[1 + c^2*x^2]*(-32*g^3 + c^2*g*(336*f^2 + 105*f*g*x + 16*g^2*x^2) + 4*c^6*x^
3*(35*f^3 + 84*f^2*g*x + 70*f*g^2*x^2 + 20*g^3*x^3) + 2*c^4*x*(175*f^3 + 336*f^2*g*x + 245*f*g^2*x^2 + 64*g^3*
x^3)) + b*(-35*c*g^2*(245*f + 1536*g*x) + 70*c^3*(1785*f^3 + 8064*f^2*g*x + 1260*f*g^2*x^2 + 128*g^3*x^3) + 16
8*c^5*x^2*(1750*f^3 + 2240*f^2*g*x + 1225*f*g^2*x^2 + 256*g^3*x^3) + 16*c^7*x^4*(3675*f^3 + 7056*f^2*g*x + 490
0*f*g^2*x^2 + 1200*g^3*x^3)))) + 88200*b*c*d^2*f*(2*c^2*f^2 - g^2)*(1 + c^2*x^2)*ArcSinh[c*x]^2 + 176400*a*c*d
^(3/2)*f*(2*c^2*f^2 - 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]] + 42
0*b*d^2*(1 + c^2*x^2)*ArcSinh[c*x]*(35*c*f*(16*c^2*f^2 - 3*g^2)*Sinh[2*ArcSinh[c*x]] + 35*c*f*(2*c^2*f^2 + 3*g
^2)*Sinh[4*ArcSinh[c*x]] + g*(64*(1 + c^2*x^2)^(5/2)*(-2*g^2 + c^2*(21*f^2 + 5*g^2*x^2)) + 35*c*f*g*Sinh[6*Arc
Sinh[c*x]])))/(940800*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. \(2078\) vs. \(2(802)=1604\).

Time = 0.88 (sec) , antiderivative size = 2079, normalized size of antiderivative = 2.26

method	result	size
default	\(\text {Expression too large to display}\)	\(2079\)
parts	\(\text {Expression too large to display}\)	\(2079\)

[In]

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

[Out]

a*(f^3*(1/4*x*(c^2*d*x^2+d)^(3/2)+3/4*d*(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*(1/7*x^2*(c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(c^2*d*x^2+d)^(5/2))+3*f*g^2*(1/6*x
*(c^2*d*x^2+d)^(5/2)/c^2/d-1/6/c^2*(1/4*x*(c^2*d*x^2+d)^(3/2)+3/4*d*(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))))+3/5*f^2*g/c^2/d*(c^2*d*x^2+d)^(5/2))+b*(3/32*(d*(c^2*x^2
+1))^(1/2)*f*arcsinh(c*x)^2*(2*c^2*f^2-g^2)*d/(c^2*x^2+1)^(1/2)/c^3+1/6272*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8+6
4*c^7*x^7*(c^2*x^2+1)^(1/2)+144*c^6*x^6+112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4+56*c^3*x^3*(c^2*x^2+1)^(1/2)
+25*c^2*x^2+7*c*x*(c^2*x^2+1)^(1/2)+1)*g^3*(-1+7*arcsinh(c*x))*d/c^4/(c^2*x^2+1)+1/768*(d*(c^2*x^2+1))^(1/2)*(
32*c^7*x^7+32*c^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5+48*c^4*x^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3+18*c^2*x^2*(c^2*x^2
+1)^(1/2)+6*c*x+(c^2*x^2+1)^(1/2))*f*g^2*(-1+6*arcsinh(c*x))*d/c^3/(c^2*x^2+1)+1/3200*(d*(c^2*x^2+1))^(1/2)*(1
6*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*(60*arcsinh(c*x)*c^2*f^2-12*c^2*f^2+5*arcsinh(c*x)*g^2-g^2)*d/c^4/(c^2*x^2+1)+1/512*(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*
(8*arcsinh(c*x)*c^2*f^2-2*c^2*f^2+12*arcsinh(c*x)*g^2-3*g^2)*d/c^3/(c^2*x^2+1)+1/384*(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)*d/c^4/(c^2*x^2+1)+1/256*(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*(32*arcsinh(c*x)*c^2*f^2-16*c^2*f^2-6*arcsinh(c*x)*g^2+3*g^2)*d/c^3/(c^2*x^2+1)+3/1
28*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*g*(8*arcsinh(c*x)*c^2*f^2-8*c^2*f^2-arcsinh(c*x)*g^
2+g^2)*d/c^4/(c^2*x^2+1)+3/128*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*g*(8*arcsinh(c*x)*c^2*f
^2+8*c^2*f^2-arcsinh(c*x)*g^2-g^2)*d/c^4/(c^2*x^2+1)+1/256*(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*(32*arcsinh(c*x)*c^2*f^2+16*c^2*f^2-6*arcsinh(c*x)*g^2-3*g^2)*d/c^3/(c^2*
x^2+1)+1/384*(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)*d/c^4/(c^2*x^2+1)+1/512*(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*(8*arc
sinh(c*x)*c^2*f^2+2*c^2*f^2+12*arcsinh(c*x)*g^2+3*g^2)*d/c^3/(c^2*x^2+1)+1/3200*(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*(60*arcsinh(c*x)*c^2*f^2+12*c^2*f^2+5*arcsinh(c*x)*g^2+g^2)*d/c^4/(c^2*x^2+1)+1/768*(d*(c^2*x^2+1))^(1/2)*(
32*c^7*x^7-32*c^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5-48*c^4*x^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3-18*c^2*x^2*(c^2*x^2
+1)^(1/2)+6*c*x-(c^2*x^2+1)^(1/2))*f*g^2*(1+6*arcsinh(c*x))*d/c^3/(c^2*x^2+1)+1/6272*(d*(c^2*x^2+1))^(1/2)*(64
*c^8*x^8-64*c^7*x^7*(c^2*x^2+1)^(1/2)+144*c^6*x^6-112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4-56*c^3*x^3*(c^2*x^
2+1)^(1/2)+25*c^2*x^2-7*c*x*(c^2*x^2+1)^(1/2)+1)*g^3*(1+7*arcsinh(c*x))*d/c^4/(c^2*x^2+1))

Fricas [F]

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

[In]

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

[Out]

integral((a*c^2*d*g^3*x^5 + 3*a*c^2*d*f*g^2*x^4 + 3*a*d*f^2*g*x + a*d*f^3 + (3*a*c^2*d*f^2*g + a*d*g^3)*x^3 +
(a*c^2*d*f^3 + 3*a*d*f*g^2)*x^2 + (b*c^2*d*g^3*x^5 + 3*b*c^2*d*f*g^2*x^4 + 3*b*d*f^2*g*x + b*d*f^3 + (3*b*c^2*
d*f^2*g + b*d*g^3)*x^3 + (b*c^2*d*f^3 + 3*b*d*f*g^2)*x^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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