\(\int (f+g x)^2 (d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) [44]

Optimal result

Integrand size = 30, antiderivative size = 901 \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {2 b d^2 f g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^2 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 g^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 f g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d+c^2 d x^2}}{768 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d+c^2 d x^2}}{288 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 f g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5 d^2 g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d^2 g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^2 f^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{8} d^2 g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d^2 f g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {5 d^2 f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {5 d^2 g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {1+c^2 x^2}} \]

[Out]

-1/36*b*d^2*f^2*(c^2*x^2+1)^(5/2)*(c^2*d*x^2+d)^(1/2)/c+5/16*d^2*f^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+
5/128*d^2*g^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+5/64*d^2*g^2*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(
1/2)+5/24*d^2*f^2*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+5/48*d^2*g^2*x^3*(c^2*x^2+1)*(a+b*arcsi
nh(c*x))*(c^2*d*x^2+d)^(1/2)+1/6*d^2*f^2*x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/8*d^2*g^2*x^
3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+2/7*d^2*f*g*(c^2*x^2+1)^3*(a+b*arcsinh(c*x))*(c^2*d*x^2
+d)^(1/2)/c^2-2/7*b*d^2*f*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-25/96*b*c*d^2*f^2*x^2*(c^2*d*x^2+d)^(1/2
)/(c^2*x^2+1)^(1/2)-5/256*b*d^2*g^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-2/7*b*c*d^2*f*g*x^3*(c^2*d*x^2
+d)^(1/2)/(c^2*x^2+1)^(1/2)-5/96*b*c^3*d^2*f^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-59/768*b*c*d^2*g^2*x^
4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-6/35*b*c^3*d^2*f*g*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-17/288*b*
c^3*d^2*g^2*x^6*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-2/49*b*c^5*d^2*f*g*x^7*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(
1/2)-1/64*b*c^5*d^2*g^2*x^8*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+5/32*d^2*f^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2
+d)^(1/2)/b/c/(c^2*x^2+1)^(1/2)-5/256*d^2*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 = 0.64 (sec) , antiderivative size = 901, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5845, 5838, 5786, 5785, 5783, 30, 14, 267, 5798, 200, 5808, 5806, 5812, 272, 45} \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b c^5 d^2 g^2 \sqrt {c^2 d x^2+d} x^8}{64 \sqrt {c^2 x^2+1}}-\frac {2 b c^5 d^2 f g \sqrt {c^2 d x^2+d} x^7}{49 \sqrt {c^2 x^2+1}}-\frac {17 b c^3 d^2 g^2 \sqrt {c^2 d x^2+d} x^6}{288 \sqrt {c^2 x^2+1}}-\frac {6 b c^3 d^2 f g \sqrt {c^2 d x^2+d} x^5}{35 \sqrt {c^2 x^2+1}}-\frac {5 b c^3 d^2 f^2 \sqrt {c^2 d x^2+d} x^4}{96 \sqrt {c^2 x^2+1}}-\frac {59 b c d^2 g^2 \sqrt {c^2 d x^2+d} x^4}{768 \sqrt {c^2 x^2+1}}+\frac {5}{64} d^2 g^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x^3+\frac {1}{8} d^2 g^2 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x^3+\frac {5}{48} d^2 g^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x^3-\frac {2 b c d^2 f g \sqrt {c^2 d x^2+d} x^3}{7 \sqrt {c^2 x^2+1}}-\frac {25 b c d^2 f^2 \sqrt {c^2 d x^2+d} x^2}{96 \sqrt {c^2 x^2+1}}-\frac {5 b d^2 g^2 \sqrt {c^2 d x^2+d} x^2}{256 c \sqrt {c^2 x^2+1}}+\frac {5}{16} d^2 f^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x+\frac {5 d^2 g^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x}{128 c^2}+\frac {1}{6} d^2 f^2 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x+\frac {5}{24} d^2 f^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x-\frac {2 b d^2 f g \sqrt {c^2 d x^2+d} x}{7 c \sqrt {c^2 x^2+1}}+\frac {5 d^2 f^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {c^2 x^2+1}}-\frac {5 d^2 g^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {c^2 x^2+1}}+\frac {2 d^2 f g \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {b d^2 f^2 \left (c^2 x^2+1\right )^{5/2} \sqrt {c^2 d x^2+d}}{36 c} \]

[In]

Int[(f + g*x)^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(-2*b*d^2*f*g*x*Sqrt[d + c^2*d*x^2])/(7*c*Sqrt[1 + c^2*x^2]) - (25*b*c*d^2*f^2*x^2*Sqrt[d + c^2*d*x^2])/(96*Sq
rt[1 + c^2*x^2]) - (5*b*d^2*g^2*x^2*Sqrt[d + c^2*d*x^2])/(256*c*Sqrt[1 + c^2*x^2]) - (2*b*c*d^2*f*g*x^3*Sqrt[d
+ c^2*d*x^2])/(7*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*f^2*x^4*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (59*
b*c*d^2*g^2*x^4*Sqrt[d + c^2*d*x^2])/(768*Sqrt[1 + c^2*x^2]) - (6*b*c^3*d^2*f*g*x^5*Sqrt[d + c^2*d*x^2])/(35*S
qrt[1 + c^2*x^2]) - (17*b*c^3*d^2*g^2*x^6*Sqrt[d + c^2*d*x^2])/(288*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*f*g*x^7*
Sqrt[d + c^2*d*x^2])/(49*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*g^2*x^8*Sqrt[d + c^2*d*x^2])/(64*Sqrt[1 + c^2*x^2]) -
(b*d^2*f^2*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2])/(36*c) + (5*d^2*f^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[
c*x]))/16 + (5*d^2*g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(128*c^2) + (5*d^2*g^2*x^3*Sqrt[d + c^2*d*x
^2]*(a + b*ArcSinh[c*x]))/64 + (5*d^2*f^2*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/24 + (5*d^
2*g^2*x^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/48 + (d^2*f^2*x*(1 + c^2*x^2)^2*Sqrt[d + c^2
*d*x^2]*(a + b*ArcSinh[c*x]))/6 + (d^2*g^2*x^3*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/8 + (
2*d^2*f*g*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^2) + (5*d^2*f^2*Sqrt[d + c^2*d*x^2]*(
a + b*ArcSinh[c*x])^2)/(32*b*c*Sqrt[1 + c^2*x^2]) - (5*d^2*g^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(25
6*b*c^3*Sqrt[1 + c^2*x^2])

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 267

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

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 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 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^2 \sqrt {d+c^2 d x^2}\right ) \int (f+g x)^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (f^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+2 f g x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+g^2 x^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d^2 f^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (2 d^2 f g \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (d^2 g^2 \sqrt {d+c^2 d x^2}\right ) \int x^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {1}{6} d^2 f^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{8} d^2 g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d^2 f g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {\left (5 d^2 f^2 \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}{6 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 f^2 \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^2 \, dx}{6 \sqrt {1+c^2 x^2}}-\frac {\left (2 b d^2 f g \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \, dx}{7 c \sqrt {1+c^2 x^2}}+\frac {\left (5 d^2 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}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right )^2 \, dx}{8 \sqrt {1+c^2 x^2}} \\ & = -\frac {b d^2 f^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{24} d^2 f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d^2 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 {1}{6} d^2 f^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{8} d^2 g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d^2 f g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {\left (5 d^2 f^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f^2 \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{24 \sqrt {1+c^2 x^2}}-\frac {\left (2 b d^2 f g \sqrt {d+c^2 d x^2}\right ) \int \left (1+3 c^2 x^2+3 c^4 x^4+c^6 x^6\right ) \, dx}{7 c \sqrt {1+c^2 x^2}}+\frac {\left (5 d^2 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}{16 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 g^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x \left (1+c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx}{48 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b d^2 f g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 f g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 f g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{64} d^2 g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d^2 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 {1}{6} d^2 f^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{8} d^2 g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d^2 f g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {\left (5 d^2 f^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 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{24 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {1+c^2 x^2}}+\frac {\left (5 d^2 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}{64 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 g^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (x+2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \, dx}{64 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 g^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx}{48 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b d^2 f g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^2 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 f g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d+c^2 d x^2}}{768 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d+c^2 d x^2}}{288 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 f g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5 d^2 g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d^2 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 {1}{6} d^2 f^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{8} d^2 g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d^2 f g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {5 d^2 f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {\left (5 d^2 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}{128 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (5 b d^2 g^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b d^2 f g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^2 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 g^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 f g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d+c^2 d x^2}}{768 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d+c^2 d x^2}}{288 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 f g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5 d^2 g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d^2 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 {1}{6} d^2 f^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{8} d^2 g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d^2 f g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {5 d^2 f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {5 d^2 g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.73 (sec) , antiderivative size = 555, normalized size of antiderivative = 0.62 \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^3 \left (1+c^2 x^2\right ) \left (b \left (-87955 g^2+1120 c^2 \left (2093 f^2+4608 f g x+315 g^2 x^2\right )+3360 c^4 x^2 \left (1848 f^2+1536 f g x+413 g^2 x^2\right )+640 c^8 x^6 \left (784 f^2+1152 f g x+441 g^2 x^2\right )+1792 c^6 x^4 \left (1365 f^2+1728 f g x+595 g^2 x^2\right )\right )-6720 a c \sqrt {1+c^2 x^2} \left (768 f g \left (1+c^2 x^2\right )^3+56 c^2 f^2 x \left (33+26 c^2 x^2+8 c^4 x^4\right )+7 g^2 x \left (15+118 c^2 x^2+136 c^4 x^4+48 c^6 x^6\right )\right )\right )+352800 b d^3 \left (8 c^2 f^2-g^2\right ) \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+705600 a d^{5/2} \left (8 c^2 f^2-g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+840 b d^3 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (6144 c f g \sqrt {1+c^2 x^2}+18432 c^3 f g x^2 \sqrt {1+c^2 x^2}+18432 c^5 f g x^4 \sqrt {1+c^2 x^2}+6144 c^7 f g x^6 \sqrt {1+c^2 x^2}+336 \left (15 c^2 f^2-g^2\right ) \sinh (2 \text {arcsinh}(c x))+168 \left (6 c^2 f^2+g^2\right ) \sinh (4 \text {arcsinh}(c x))+112 c^2 f^2 \sinh (6 \text {arcsinh}(c x))+112 g^2 \sinh (6 \text {arcsinh}(c x))+21 g^2 \sinh (8 \text {arcsinh}(c x))\right )}{18063360 c^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \]

[In]

Integrate[(f + g*x)^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(-(d^3*(1 + c^2*x^2)*(b*(-87955*g^2 + 1120*c^2*(2093*f^2 + 4608*f*g*x + 315*g^2*x^2) + 3360*c^4*x^2*(1848*f^2
+ 1536*f*g*x + 413*g^2*x^2) + 640*c^8*x^6*(784*f^2 + 1152*f*g*x + 441*g^2*x^2) + 1792*c^6*x^4*(1365*f^2 + 1728
*f*g*x + 595*g^2*x^2)) - 6720*a*c*Sqrt[1 + c^2*x^2]*(768*f*g*(1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 + 26*c^2*x^2 +
8*c^4*x^4) + 7*g^2*x*(15 + 118*c^2*x^2 + 136*c^4*x^4 + 48*c^6*x^6)))) + 352800*b*d^3*(8*c^2*f^2 - g^2)*(1 + c
^2*x^2)*ArcSinh[c*x]^2 + 705600*a*d^(5/2)*(8*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]] + 840*b*d^3*(1 + c^2*x^2)*ArcSinh[c*x]*(6144*c*f*g*Sqrt[1 + c^2*x^2] + 18432*c^3*
f*g*x^2*Sqrt[1 + c^2*x^2] + 18432*c^5*f*g*x^4*Sqrt[1 + c^2*x^2] + 6144*c^7*f*g*x^6*Sqrt[1 + c^2*x^2] + 336*(15
*c^2*f^2 - g^2)*Sinh[2*ArcSinh[c*x]] + 168*(6*c^2*f^2 + g^2)*Sinh[4*ArcSinh[c*x]] + 112*c^2*f^2*Sinh[6*ArcSinh
[c*x]] + 112*g^2*Sinh[6*ArcSinh[c*x]] + 21*g^2*Sinh[8*ArcSinh[c*x]]))/(18063360*c^3*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. \(2308\) vs. \(2(791)=1582\).

Time = 1.03 (sec) , antiderivative size = 2309, normalized size of antiderivative = 2.56

method	result	size
default	\(\text {Expression too large to display}\)	\(2309\)
parts	\(\text {Expression too large to display}\)	\(2309\)

[In]

int((g*x+f)^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x,method=_RETURNVERBOSE)

[Out]

a*(f^2*(1/6*x*(c^2*d*x^2+d)^(5/2)+5/6*d*(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^2*(1/8*x*(c^2*d*x^2+d)^(7/2)/c^2/d-1/8/c^2*(1/6*x
*(c^2*d*x^2+d)^(5/2)+5/6*d*(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)))))+2/7*f*g*(c^2*d*x^2+d)^(7/2)/c^2/d)+b*(5/256*(d*(c^2*x^2+1))^(1/
2)*arcsinh(c*x)^2*(8*c^2*f^2-g^2)*d^2/(c^2*x^2+1)^(1/2)/c^3+1/16384*(d*(c^2*x^2+1))^(1/2)*(128*c^9*x^9+128*c^8
*x^8*(c^2*x^2+1)^(1/2)+320*c^7*x^7+256*c^6*x^6*(c^2*x^2+1)^(1/2)+272*c^5*x^5+160*c^4*x^4*(c^2*x^2+1)^(1/2)+88*
c^3*x^3+32*c^2*x^2*(c^2*x^2+1)^(1/2)+8*c*x+(c^2*x^2+1)^(1/2))*g^2*(-1+8*arcsinh(c*x))*d^2/c^3/(c^2*x^2+1)+1/31
36*(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)+10
4*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)*f*g*(-1+7*arcsinh(c*x))*d^2/c^2/(
c^2*x^2+1)+1/2304*(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))*(6*arcsinh(c*x)*c^2*f^2-c^2*f^2+6*
arcsinh(c*x)*g^2-g^2)*d^2/c^3/(c^2*x^2+1)+1/320*(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)*f*g*(-1+5*arcsinh(c*x))*d^2/c^2
/(c^2*x^2+1)+1/1024*(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))*(24*arcsinh(c*x)*c^2*f^2-6*c^2*f^2+4*arcsinh(c*x)*g^2-g^2)*d^2/c^3/(c^2*x^2
+1)+1/64*(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)*f*g
*(-1+3*arcsinh(c*x))*d^2/c^2/(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))*(30*arcsinh(c*x)*c^2*f^2-15*c^2*f^2-2*arcsinh(c*x)*g^2+g^2)*d^2/c^3/(c^2*x^2+1)+5/64*(d
*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*f*g*(-1+arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+5/64*(d*(c^2*x
^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*f*g*(arcsinh(c*x)+1)*d^2/c^2/(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))*(30*arcsinh(c*x)*c^2*f^2+15*c^2*f^2-2*ar
csinh(c*x)*g^2-g^2)*d^2/c^3/(c^2*x^2+1)+1/64*(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)*f*g*(3*arcsinh(c*x)+1)*d^2/c^2/(c^2*x^2+1)+1/1024*(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))*(24*arcsinh(
c*x)*c^2*f^2+6*c^2*f^2+4*arcsinh(c*x)*g^2+g^2)*d^2/c^3/(c^2*x^2+1)+1/320*(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)*f*g*(1
+5*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+1/2304*(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))*(6*arcsi
nh(c*x)*c^2*f^2+c^2*f^2+6*arcsinh(c*x)*g^2+g^2)*d^2/c^3/(c^2*x^2+1)+1/3136*(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)*f*g*(1+7*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+1/16384*(d*(c^2*x^2+1))^(1/2
)*(128*c^9*x^9-128*c^8*x^8*(c^2*x^2+1)^(1/2)+320*c^7*x^7-256*c^6*x^6*(c^2*x^2+1)^(1/2)+272*c^5*x^5-160*c^4*x^4
*(c^2*x^2+1)^(1/2)+88*c^3*x^3-32*c^2*x^2*(c^2*x^2+1)^(1/2)+8*c*x-(c^2*x^2+1)^(1/2))*g^2*(1+8*arcsinh(c*x))*d^2
/c^3/(c^2*x^2+1))

Fricas [F]

\[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]

[In]

integrate((g*x+f)^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*g^2*x^6 + 2*a*c^4*d^2*f*g*x^5 + 4*a*c^2*d^2*f*g*x^3 + 2*a*d^2*f*g*x + a*d^2*f^2 + (a*c^4*d
^2*f^2 + 2*a*c^2*d^2*g^2)*x^4 + (2*a*c^2*d^2*f^2 + a*d^2*g^2)*x^2 + (b*c^4*d^2*g^2*x^6 + 2*b*c^4*d^2*f*g*x^5 +
4*b*c^2*d^2*f*g*x^3 + 2*b*d^2*f*g*x + b*d^2*f^2 + (b*c^4*d^2*f^2 + 2*b*c^2*d^2*g^2)*x^4 + (2*b*c^2*d^2*f^2 +
b*d^2*g^2)*x^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)

Sympy [F]

\[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )^{2}\, dx \]

[In]

integrate((g*x+f)**2*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x)),x)

[Out]

Integral((d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))*(f + g*x)**2, x)

Maxima [F(-2)]

Exception generated. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((g*x+f)^2*(c^2*d*x^2+d)^(5/2)*(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. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((g*x+f)^2*(c^2*d*x^2+d)^(5/2)*(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. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \]

[In]

int((f + g*x)^2*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2),x)

[Out]

int((f + g*x)^2*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2), x)