3.39 \(\int (f+g x)^3 (d+c^2 d x^2)^{3/2} (a+b \sinh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=918 \[ -\frac{b c^3 d g^3 \sqrt{c^2 d x^2+d} x^7}{49 \sqrt{c^2 x^2+1}}-\frac{b c^3 d f g^2 \sqrt{c^2 d x^2+d} x^6}{12 \sqrt{c^2 x^2+1}}-\frac{8 b c d g^3 \sqrt{c^2 d x^2+d} x^5}{175 \sqrt{c^2 x^2+1}}-\frac{3 b c^3 d f^2 g \sqrt{c^2 d x^2+d} x^5}{25 \sqrt{c^2 x^2+1}}-\frac{b c^3 d f^3 \sqrt{c^2 d x^2+d} x^4}{16 \sqrt{c^2 x^2+1}}-\frac{7 b c d f g^2 \sqrt{c^2 d x^2+d} x^4}{32 \sqrt{c^2 x^2+1}}+\frac{3}{8} d f g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3+\frac{1}{2} d f g^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3-\frac{b d g^3 \sqrt{c^2 d x^2+d} x^3}{105 c \sqrt{c^2 x^2+1}}-\frac{2 b c d f^2 g \sqrt{c^2 d x^2+d} x^3}{5 \sqrt{c^2 x^2+1}}-\frac{5 b c d f^3 \sqrt{c^2 d x^2+d} x^2}{16 \sqrt{c^2 x^2+1}}-\frac{3 b d f g^2 \sqrt{c^2 d x^2+d} x^2}{32 c \sqrt{c^2 x^2+1}}+\frac{3}{8} d f^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac{3 d f g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x}{16 c^2}+\frac{1}{4} d f^3 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac{2 b d g^3 \sqrt{c^2 d x^2+d} x}{35 c^3 \sqrt{c^2 x^2+1}}-\frac{3 b d f^2 g \sqrt{c^2 d x^2+d} x}{5 c \sqrt{c^2 x^2+1}}+\frac{3 d f^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt{c^2 x^2+1}}-\frac{3 d f g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{c^2 x^2+1}}+\frac{d g^3 \left (c^2 x^2+1\right )^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}-\frac{d g^3 \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^4}+\frac{3 d f^2 g \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.932396, antiderivative size = 918, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 17, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.567, Rules used = {5835, 5821, 5684, 5682, 5675, 30, 14, 5717, 194, 5744, 5742, 5758, 266, 43, 5732, 12, 373} \[ -\frac{b c^3 d g^3 \sqrt{c^2 d x^2+d} x^7}{49 \sqrt{c^2 x^2+1}}-\frac{b c^3 d f g^2 \sqrt{c^2 d x^2+d} x^6}{12 \sqrt{c^2 x^2+1}}-\frac{8 b c d g^3 \sqrt{c^2 d x^2+d} x^5}{175 \sqrt{c^2 x^2+1}}-\frac{3 b c^3 d f^2 g \sqrt{c^2 d x^2+d} x^5}{25 \sqrt{c^2 x^2+1}}-\frac{b c^3 d f^3 \sqrt{c^2 d x^2+d} x^4}{16 \sqrt{c^2 x^2+1}}-\frac{7 b c d f g^2 \sqrt{c^2 d x^2+d} x^4}{32 \sqrt{c^2 x^2+1}}+\frac{3}{8} d f g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3+\frac{1}{2} d f g^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3-\frac{b d g^3 \sqrt{c^2 d x^2+d} x^3}{105 c \sqrt{c^2 x^2+1}}-\frac{2 b c d f^2 g \sqrt{c^2 d x^2+d} x^3}{5 \sqrt{c^2 x^2+1}}-\frac{5 b c d f^3 \sqrt{c^2 d x^2+d} x^2}{16 \sqrt{c^2 x^2+1}}-\frac{3 b d f g^2 \sqrt{c^2 d x^2+d} x^2}{32 c \sqrt{c^2 x^2+1}}+\frac{3}{8} d f^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac{3 d f g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x}{16 c^2}+\frac{1}{4} d f^3 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac{2 b d g^3 \sqrt{c^2 d x^2+d} x}{35 c^3 \sqrt{c^2 x^2+1}}-\frac{3 b d f^2 g \sqrt{c^2 d x^2+d} x}{5 c \sqrt{c^2 x^2+1}}+\frac{3 d f^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt{c^2 x^2+1}}-\frac{3 d f g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{c^2 x^2+1}}+\frac{d g^3 \left (c^2 x^2+1\right )^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}-\frac{d g^3 \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^4}+\frac{3 d f^2 g \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2} \]

Antiderivative was successfully verified.

[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 5835

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 + c^2*x^2)^FracPart[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]

Rule 5821

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 5684

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[x*(1
+ c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && Gt
Q[n, 0] && GtQ[p, 0]

Rule 5682

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[Sqrt[d + e*x^2]/(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*Sqrt[d + e*x^2])/(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 5675

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

Rule 30

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

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 5717

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[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 194

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 5744

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*d^IntPart[p]*(d + e*x^2)^FracPart[p]
)/(f*(m + 2*p + 1)*(1 + c^2*x^2)^FracPart[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]
&& (RationalQ[m] || EqQ[n, 1])

Rule 5742

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[Sqrt[d + e*x^2]/((m + 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*Sqrt[d + e*x^2])/(f*
(m + 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] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 5758

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

Rule 266

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 43

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 5732

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

Rule 12

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

Rule 373

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]

Rubi steps

\begin{align*} \int (f+g x)^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right )+3 f^2 g x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+3 f g^2 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+g^3 x^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )\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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{2} d f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d f^2 g \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}-\frac{d g^3 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^4}+\frac{d g^3 \left (1+c^2 x^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac{\left (3 d f^3 \sqrt{d+c^2 d x^2}\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{8} d f g^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d f^3 x \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{2} d f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d f^2 g \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}-\frac{d g^3 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^4}+\frac{d g^3 \left (1+c^2 x^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac{\left (3 d f^3 \sqrt{d+c^2 d x^2}\right ) \int \frac{a+b \sinh ^{-1}(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 \left (a+b \sinh ^{-1}(c x)\right )}{\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} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d f g^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac{3}{8} d f g^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d f^3 x \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{2} d f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d f^2 g \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}-\frac{d g^3 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^4}+\frac{d g^3 \left (1+c^2 x^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac{3 d f^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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 \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d f g^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac{3}{8} d f g^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d f^3 x \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{2} d f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d f^2 g \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}-\frac{d g^3 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^4}+\frac{d g^3 \left (1+c^2 x^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac{3 d f^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt{1+c^2 x^2}}-\frac{3 d f g^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 4.24489, size = 779, normalized size = 0.85 \[ \frac{529200 a c d^{3/2} f \sqrt{c^2 x^2+1} \left (2 c^2 f^2-g^2\right ) \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )+5040 a d \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (4 c^6 x^3 \left (84 f^2 g x+35 f^3+70 f g^2 x^2+20 g^3 x^3\right )+2 c^4 x \left (336 f^2 g x+175 f^3+245 f g^2 x^2+64 g^3 x^3\right )+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )-32 g^3\right )-940800 b c^2 d f^2 g \sqrt{c^2 d x^2+d} \left (c^3 x^3-3 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)+3 c x\right )-37632 b c^2 d f^2 g \sqrt{c^2 d x^2+d} \left (c x \left (9 c^4 x^4+5 c^2 x^2-30\right )-15 \sqrt{c^2 x^2+1} \left (3 c^4 x^4+c^2 x^2-2\right ) \sinh ^{-1}(c x)\right )-352800 b c^3 d f^3 \sqrt{c^2 d x^2+d} \left (\cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )-22050 b c^3 d f^3 \sqrt{c^2 d x^2+d} \left (8 \sinh ^{-1}(c x)^2-4 \sinh \left (4 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\cosh \left (4 \sinh ^{-1}(c x)\right )\right )-66150 b c d f g^2 \sqrt{c^2 d x^2+d} \left (8 \sinh ^{-1}(c x)^2-4 \sinh \left (4 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\cosh \left (4 \sinh ^{-1}(c x)\right )\right )+3675 b c d f g^2 \sqrt{c^2 d x^2+d} \left (72 \sinh ^{-1}(c x)^2+12 \left (-3 \sinh \left (2 \sinh ^{-1}(c x)\right )-3 \sinh \left (4 \sinh ^{-1}(c x)\right )+\sinh \left (6 \sinh ^{-1}(c x)\right )\right ) \sinh ^{-1}(c x)+18 \cosh \left (2 \sinh ^{-1}(c x)\right )+9 \cosh \left (4 \sinh ^{-1}(c x)\right )-2 \cosh \left (6 \sinh ^{-1}(c x)\right )\right )-12544 b d g^3 \sqrt{c^2 d x^2+d} \left (c x \left (9 c^4 x^4+5 c^2 x^2-30\right )-15 \sqrt{c^2 x^2+1} \left (3 c^4 x^4+c^2 x^2-2\right ) \sinh ^{-1}(c x)\right )-256 b d g^3 \sqrt{c^2 d x^2+d} \left (c x \left (225 c^6 x^6+63 c^4 x^4-140 c^2 x^2+840\right )-105 \sqrt{c^2 x^2+1} \left (15 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8\right ) \sinh ^{-1}(c x)\right )}{2822400 c^4 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5040*a*d*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*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)) - 940800*b*c^2*d*f^2*g*Sqrt[d + c^2*d*x^2]*(3*c*x + c^3*x^3 - 3*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]) - 3763
2*b*c^2*d*f^2*g*Sqrt[d + c^2*d*x^2]*(c*x*(-30 + 5*c^2*x^2 + 9*c^4*x^4) - 15*Sqrt[1 + c^2*x^2]*(-2 + c^2*x^2 +
3*c^4*x^4)*ArcSinh[c*x]) - 12544*b*d*g^3*Sqrt[d + c^2*d*x^2]*(c*x*(-30 + 5*c^2*x^2 + 9*c^4*x^4) - 15*Sqrt[1 +
c^2*x^2]*(-2 + c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x]) - 256*b*d*g^3*Sqrt[d + c^2*d*x^2]*(c*x*(840 - 140*c^2*x^2 +
63*c^4*x^4 + 225*c^6*x^6) - 105*Sqrt[1 + c^2*x^2]*(8 - 4*c^2*x^2 + 3*c^4*x^4 + 15*c^6*x^6)*ArcSinh[c*x]) + 529
200*a*c*d^(3/2)*f*(2*c^2*f^2 - g^2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 352800*b*c^3*
d*f^3*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])) - 2205
0*b*c^3*d*f^3*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x
]]) - 66150*b*c*d*f*g^2*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*A
rcSinh[c*x]]) + 3675*b*c*d*f*g^2*Sqrt[d + c^2*d*x^2]*(72*ArcSinh[c*x]^2 + 18*Cosh[2*ArcSinh[c*x]] + 9*Cosh[4*A
rcSinh[c*x]] - 2*Cosh[6*ArcSinh[c*x]] + 12*ArcSinh[c*x]*(-3*Sinh[2*ArcSinh[c*x]] - 3*Sinh[4*ArcSinh[c*x]] + Si
nh[6*ArcSinh[c*x]])))/(2822400*c^4*Sqrt[1 + c^2*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.493, size = 1510, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5*b*(d*(c^2*x^2+1))^(1/2)*g*d/c/(c^2*x^2+1)^(1/2)*x*f^2-1/12*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d*c^3/(c^2*x^2+1
)^(1/2)*x^6-1/35*b*(d*(c^2*x^2+1))^(1/2)*g^3*d/c^2/(c^2*x^2+1)*arcsinh(c*x)*x^2+3/5*b*(d*(c^2*x^2+1))^(1/2)*g*
d/c^2/(c^2*x^2+1)*arcsinh(c*x)*f^2+1/4*b*(d*(c^2*x^2+1))^(1/2)*f^3*d*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^5+7/8*b*(d
*(c^2*x^2+1))^(1/2)*f^3*d*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^3-3/32*b*(d*(c^2*x^2+1))^(1/2)*f*arcsinh(c*x)^2*d/(c^
2*x^2+1)^(1/2)/c^3*g^2+1/7*b*(d*(c^2*x^2+1))^(1/2)*g^3*d*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^8+13/35*b*(d*(c^2*x^2+
1))^(1/2)*g^3*d*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^6+17/16*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d/(c^2*x^2+1)*arcsinh(c*x
)*x^3+9/5*b*(d*(c^2*x^2+1))^(1/2)*g*d/(c^2*x^2+1)*arcsinh(c*x)*x^2*f^2-7/32*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d*c/
(c^2*x^2+1)^(1/2)*x^4-3/32*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d/c/(c^2*x^2+1)^(1/2)*x^2-3/25*b*(d*(c^2*x^2+1))^(1/2
)*g*d*c^3/(c^2*x^2+1)^(1/2)*x^5*f^2-2/5*b*(d*(c^2*x^2+1))^(1/2)*g*d*c/(c^2*x^2+1)^(1/2)*x^3*f^2+3/8*a*f^3*d^2*
ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/4*a*f^3*x*(c^2*d*x^2+d)^(3/2)+3/8*a*f^3*d*x*(c^2
*d*x^2+d)^(1/2)-2/35*a*g^3/d/c^4*(c^2*d*x^2+d)^(5/2)+3/16*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d/c^2/(c^2*x^2+1)*arcs
inh(c*x)*x+3/5*b*(d*(c^2*x^2+1))^(1/2)*g*d*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^6*f^2+9/5*b*(d*(c^2*x^2+1))^(1/2)*g*
d*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^4*f^2+1/2*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^7+11
/8*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^5+2/35*b*(d*(c^2*x^2+1))^(1/2)*g^3*d/c^3/(c^
2*x^2+1)^(1/2)*x+7/768*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d/c^3/(c^2*x^2+1)^(1/2)-1/16*b*(d*(c^2*x^2+1))^(1/2)*f^3*
d*c^3/(c^2*x^2+1)^(1/2)*x^4-5/16*b*(d*(c^2*x^2+1))^(1/2)*f^3*d*c/(c^2*x^2+1)^(1/2)*x^2-1/49*b*(d*(c^2*x^2+1))^
(1/2)*g^3*d*c^3/(c^2*x^2+1)^(1/2)*x^7-3/16*a*f*g^2/c^2*d*x*(c^2*d*x^2+d)^(1/2)-3/16*a*f*g^2/c^2*d^2*ln(x*c^2*d
/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/2*a*f*g^2*x*(c^2*d*x^2+d)^(5/2)/c^2/d-8/175*b*(d*(c^2*x^2+
1))^(1/2)*g^3*d*c/(c^2*x^2+1)^(1/2)*x^5-1/105*b*(d*(c^2*x^2+1))^(1/2)*g^3*d/c/(c^2*x^2+1)^(1/2)*x^3+3/16*b*(d*
(c^2*x^2+1))^(1/2)*f^3*arcsinh(c*x)^2*d/(c^2*x^2+1)^(1/2)/c-2/35*b*(d*(c^2*x^2+1))^(1/2)*g^3*d/c^4/(c^2*x^2+1)
*arcsinh(c*x)+9/35*b*(d*(c^2*x^2+1))^(1/2)*g^3*d/(c^2*x^2+1)*arcsinh(c*x)*x^4+5/8*b*(d*(c^2*x^2+1))^(1/2)*f^3*
d/(c^2*x^2+1)*arcsinh(c*x)*x-1/8*a*f*g^2/c^2*x*(c^2*d*x^2+d)^(3/2)+3/5*a*f^2*g/c^2/d*(c^2*d*x^2+d)^(5/2)+1/7*a
*g^3*x^2*(c^2*d*x^2+d)^(5/2)/c^2/d-17/128*b*(d*(c^2*x^2+1))^(1/2)*f^3*d/c/(c^2*x^2+1)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (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} +{\left (3 \, a c^{2} d f^{2} g + a d g^{3}\right )} x^{3} +{\left (a c^{2} d f^{3} + 3 \, a d f g^{2}\right )} x^{2} +{\left (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} +{\left (3 \, b c^{2} d f^{2} g + b d g^{3}\right )} x^{3} +{\left (b c^{2} d f^{3} + 3 \, b d f g^{2}\right )} x^{2}\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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