3.43 \(\int (f+g x)^3 (d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=1228 \[ \text{result too large to display} \]

[Out]

(-3*b*d^2*f^2*g*x*Sqrt[d + c^2*d*x^2])/(7*c*Sqrt[1 + c^2*x^2]) + (2*b*d^2*g^3*x*Sqrt[d + c^2*d*x^2])/(63*c^3*S
qrt[1 + c^2*x^2]) - (25*b*c*d^2*f^3*x^2*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (15*b*d^2*f*g^2*x^2*Sqrt
[d + c^2*d*x^2])/(256*c*Sqrt[1 + c^2*x^2]) - (3*b*c*d^2*f^2*g*x^3*Sqrt[d + c^2*d*x^2])/(7*Sqrt[1 + c^2*x^2]) -
(b*d^2*g^3*x^3*Sqrt[d + c^2*d*x^2])/(189*c*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*f^3*x^4*Sqrt[d + c^2*d*x^2])/(96
*Sqrt[1 + c^2*x^2]) - (59*b*c*d^2*f*g^2*x^4*Sqrt[d + c^2*d*x^2])/(256*Sqrt[1 + c^2*x^2]) - (9*b*c^3*d^2*f^2*g*
x^5*Sqrt[d + c^2*d*x^2])/(35*Sqrt[1 + c^2*x^2]) - (b*c*d^2*g^3*x^5*Sqrt[d + c^2*d*x^2])/(21*Sqrt[1 + c^2*x^2])
- (17*b*c^3*d^2*f*g^2*x^6*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (3*b*c^5*d^2*f^2*g*x^7*Sqrt[d + c^2*d
*x^2])/(49*Sqrt[1 + c^2*x^2]) - (19*b*c^3*d^2*g^3*x^7*Sqrt[d + c^2*d*x^2])/(441*Sqrt[1 + c^2*x^2]) - (3*b*c^5*
d^2*f*g^2*x^8*Sqrt[d + c^2*d*x^2])/(64*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*g^3*x^9*Sqrt[d + c^2*d*x^2])/(81*Sqrt[1
+ c^2*x^2]) - (b*d^2*f^3*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2])/(36*c) + (5*d^2*f^3*x*Sqrt[d + c^2*d*x^2]*(
a + b*ArcSinh[c*x]))/16 + (15*d^2*f*g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(128*c^2) + (15*d^2*f*g^2*
x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/64 + (5*d^2*f^3*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSi
nh[c*x]))/24 + (5*d^2*f*g^2*x^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/16 + (d^2*f^3*x*(1 + c
^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/6 + (3*d^2*f*g^2*x^3*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(
a + b*ArcSinh[c*x]))/8 + (3*d^2*f^2*g*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^2) - (d^2
*g^3*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^4) + (d^2*g^3*(1 + c^2*x^2)^4*Sqrt[d + c^2
*d*x^2]*(a + b*ArcSinh[c*x]))/(9*c^4) + (5*d^2*f^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c*Sqrt[1
+ c^2*x^2]) - (15*d^2*f*g^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(256*b*c^3*Sqrt[1 + c^2*x^2])

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Rubi [A]  time = 1.13905, antiderivative size = 1228, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 18, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5835, 5821, 5684, 5682, 5675, 30, 14, 261, 5717, 194, 5744, 5742, 5758, 266, 43, 5732, 12, 373} \[ -\frac{b c^5 d^2 g^3 \sqrt{c^2 d x^2+d} x^9}{81 \sqrt{c^2 x^2+1}}-\frac{3 b c^5 d^2 f g^2 \sqrt{c^2 d x^2+d} x^8}{64 \sqrt{c^2 x^2+1}}-\frac{19 b c^3 d^2 g^3 \sqrt{c^2 d x^2+d} x^7}{441 \sqrt{c^2 x^2+1}}-\frac{3 b c^5 d^2 f^2 g \sqrt{c^2 d x^2+d} x^7}{49 \sqrt{c^2 x^2+1}}-\frac{17 b c^3 d^2 f g^2 \sqrt{c^2 d x^2+d} x^6}{96 \sqrt{c^2 x^2+1}}-\frac{b c d^2 g^3 \sqrt{c^2 d x^2+d} x^5}{21 \sqrt{c^2 x^2+1}}-\frac{9 b c^3 d^2 f^2 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^3 \sqrt{c^2 d x^2+d} x^4}{96 \sqrt{c^2 x^2+1}}-\frac{59 b c d^2 f g^2 \sqrt{c^2 d x^2+d} x^4}{256 \sqrt{c^2 x^2+1}}+\frac{15}{64} d^2 f g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3+\frac{3}{8} d^2 f g^2 \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3+\frac{5}{16} d^2 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^2 g^3 \sqrt{c^2 d x^2+d} x^3}{189 c \sqrt{c^2 x^2+1}}-\frac{3 b c d^2 f^2 g \sqrt{c^2 d x^2+d} x^3}{7 \sqrt{c^2 x^2+1}}-\frac{25 b c d^2 f^3 \sqrt{c^2 d x^2+d} x^2}{96 \sqrt{c^2 x^2+1}}-\frac{15 b d^2 f 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^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac{15 d^2 f g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x}{128 c^2}+\frac{1}{6} d^2 f^3 \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac{5}{24} d^2 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^2 g^3 \sqrt{c^2 d x^2+d} x}{63 c^3 \sqrt{c^2 x^2+1}}-\frac{3 b d^2 f^2 g \sqrt{c^2 d x^2+d} x}{7 c \sqrt{c^2 x^2+1}}+\frac{5 d^2 f^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt{c^2 x^2+1}}-\frac{15 d^2 f g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{c^2 x^2+1}}+\frac{d^2 g^3 \left (c^2 x^2+1\right )^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}-\frac{d^2 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{3 d^2 f^2 g \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^2}-\frac{b d^2 f^3 \left (c^2 x^2+1\right )^{5/2} \sqrt{c^2 d x^2+d}}{36 c} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(-3*b*d^2*f^2*g*x*Sqrt[d + c^2*d*x^2])/(7*c*Sqrt[1 + c^2*x^2]) + (2*b*d^2*g^3*x*Sqrt[d + c^2*d*x^2])/(63*c^3*S
qrt[1 + c^2*x^2]) - (25*b*c*d^2*f^3*x^2*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (15*b*d^2*f*g^2*x^2*Sqrt
[d + c^2*d*x^2])/(256*c*Sqrt[1 + c^2*x^2]) - (3*b*c*d^2*f^2*g*x^3*Sqrt[d + c^2*d*x^2])/(7*Sqrt[1 + c^2*x^2]) -
(b*d^2*g^3*x^3*Sqrt[d + c^2*d*x^2])/(189*c*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*f^3*x^4*Sqrt[d + c^2*d*x^2])/(96
*Sqrt[1 + c^2*x^2]) - (59*b*c*d^2*f*g^2*x^4*Sqrt[d + c^2*d*x^2])/(256*Sqrt[1 + c^2*x^2]) - (9*b*c^3*d^2*f^2*g*
x^5*Sqrt[d + c^2*d*x^2])/(35*Sqrt[1 + c^2*x^2]) - (b*c*d^2*g^3*x^5*Sqrt[d + c^2*d*x^2])/(21*Sqrt[1 + c^2*x^2])
- (17*b*c^3*d^2*f*g^2*x^6*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (3*b*c^5*d^2*f^2*g*x^7*Sqrt[d + c^2*d
*x^2])/(49*Sqrt[1 + c^2*x^2]) - (19*b*c^3*d^2*g^3*x^7*Sqrt[d + c^2*d*x^2])/(441*Sqrt[1 + c^2*x^2]) - (3*b*c^5*
d^2*f*g^2*x^8*Sqrt[d + c^2*d*x^2])/(64*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*g^3*x^9*Sqrt[d + c^2*d*x^2])/(81*Sqrt[1
+ c^2*x^2]) - (b*d^2*f^3*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2])/(36*c) + (5*d^2*f^3*x*Sqrt[d + c^2*d*x^2]*(
a + b*ArcSinh[c*x]))/16 + (15*d^2*f*g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(128*c^2) + (15*d^2*f*g^2*
x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/64 + (5*d^2*f^3*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSi
nh[c*x]))/24 + (5*d^2*f*g^2*x^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/16 + (d^2*f^3*x*(1 + c
^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/6 + (3*d^2*f*g^2*x^3*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(
a + b*ArcSinh[c*x]))/8 + (3*d^2*f^2*g*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^2) - (d^2
*g^3*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^4) + (d^2*g^3*(1 + c^2*x^2)^4*Sqrt[d + c^2
*d*x^2]*(a + b*ArcSinh[c*x]))/(9*c^4) + (5*d^2*f^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c*Sqrt[1
+ c^2*x^2]) - (15*d^2*f*g^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(256*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 261

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 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 )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \int (f+g x)^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \int \left (f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+3 f^2 g x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+3 f g^2 x^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+g^3 x^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\left (d^2 f^3 \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (3 d^2 f^2 g \sqrt{d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (3 d^2 f g^2 \sqrt{d+c^2 d x^2}\right ) \int x^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (d^2 g^3 \sqrt{d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d^2 f^2 g \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^2}-\frac{d^2 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{d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac{\left (5 d^2 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}{6 \sqrt{1+c^2 x^2}}-\frac{\left (b c d^2 f^3 \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 (3 b d^2 f^2 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 (15 d^2 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}{8 \sqrt{1+c^2 x^2}}-\frac{\left (3 b c d^2 f 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{\left (b c d^2 g^3 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right )}{63 c^4} \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt{d+c^2 d x^2}}{36 c}+\frac{5}{24} d^2 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{5}{16} d^2 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{1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d^2 f^2 g \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^2}-\frac{d^2 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{d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac{\left (5 d^2 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}{8 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c d^2 f^3 \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 (3 b d^2 f^2 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 (15 d^2 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}{16 \sqrt{1+c^2 x^2}}-\frac{\left (3 b c d^2 f g^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{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 f g^2 \sqrt{d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx}{16 \sqrt{1+c^2 x^2}}-\frac{\left (b d^2 g^3 \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right ) \, dx}{63 c^3 \sqrt{1+c^2 x^2}}\\ &=-\frac{3 b d^2 f^2 g x \sqrt{d+c^2 d x^2}}{7 c \sqrt{1+c^2 x^2}}-\frac{3 b c d^2 f^2 g x^3 \sqrt{d+c^2 d x^2}}{7 \sqrt{1+c^2 x^2}}-\frac{9 b c^3 d^2 f^2 g x^5 \sqrt{d+c^2 d x^2}}{35 \sqrt{1+c^2 x^2}}-\frac{3 b c^5 d^2 f^2 g x^7 \sqrt{d+c^2 d x^2}}{49 \sqrt{1+c^2 x^2}}-\frac{b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt{d+c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f^3 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{15}{64} d^2 f g^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{24} d^2 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{5}{16} d^2 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{1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d^2 f^2 g \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^2}-\frac{d^2 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{d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac{\left (5 d^2 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}{16 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c d^2 f^3 \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^3 \sqrt{d+c^2 d x^2}\right ) \int x \, dx}{16 \sqrt{1+c^2 x^2}}+\frac{\left (15 d^2 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}{64 \sqrt{1+c^2 x^2}}-\frac{\left (3 b c d^2 f g^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{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 (15 b c d^2 f 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 f g^2 \sqrt{d+c^2 d x^2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx}{16 \sqrt{1+c^2 x^2}}-\frac{\left (b d^2 g^3 \sqrt{d+c^2 d x^2}\right ) \int \left (-2+c^2 x^2+15 c^4 x^4+19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \sqrt{1+c^2 x^2}}\\ &=-\frac{3 b d^2 f^2 g x \sqrt{d+c^2 d x^2}}{7 c \sqrt{1+c^2 x^2}}+\frac{2 b d^2 g^3 x \sqrt{d+c^2 d x^2}}{63 c^3 \sqrt{1+c^2 x^2}}-\frac{25 b c d^2 f^3 x^2 \sqrt{d+c^2 d x^2}}{96 \sqrt{1+c^2 x^2}}-\frac{3 b c d^2 f^2 g x^3 \sqrt{d+c^2 d x^2}}{7 \sqrt{1+c^2 x^2}}-\frac{b d^2 g^3 x^3 \sqrt{d+c^2 d x^2}}{189 c \sqrt{1+c^2 x^2}}-\frac{5 b c^3 d^2 f^3 x^4 \sqrt{d+c^2 d x^2}}{96 \sqrt{1+c^2 x^2}}-\frac{59 b c d^2 f g^2 x^4 \sqrt{d+c^2 d x^2}}{256 \sqrt{1+c^2 x^2}}-\frac{9 b c^3 d^2 f^2 g x^5 \sqrt{d+c^2 d x^2}}{35 \sqrt{1+c^2 x^2}}-\frac{b c d^2 g^3 x^5 \sqrt{d+c^2 d x^2}}{21 \sqrt{1+c^2 x^2}}-\frac{17 b c^3 d^2 f g^2 x^6 \sqrt{d+c^2 d x^2}}{96 \sqrt{1+c^2 x^2}}-\frac{3 b c^5 d^2 f^2 g x^7 \sqrt{d+c^2 d x^2}}{49 \sqrt{1+c^2 x^2}}-\frac{19 b c^3 d^2 g^3 x^7 \sqrt{d+c^2 d x^2}}{441 \sqrt{1+c^2 x^2}}-\frac{3 b c^5 d^2 f g^2 x^8 \sqrt{d+c^2 d x^2}}{64 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 g^3 x^9 \sqrt{d+c^2 d x^2}}{81 \sqrt{1+c^2 x^2}}-\frac{b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt{d+c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f^3 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{15 d^2 f g^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac{15}{64} d^2 f g^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{24} d^2 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{5}{16} d^2 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{1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d^2 f^2 g \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^2}-\frac{d^2 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{d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac{5 d^2 f^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt{1+c^2 x^2}}-\frac{\left (15 d^2 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}{128 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (15 b d^2 f g^2 \sqrt{d+c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt{1+c^2 x^2}}\\ &=-\frac{3 b d^2 f^2 g x \sqrt{d+c^2 d x^2}}{7 c \sqrt{1+c^2 x^2}}+\frac{2 b d^2 g^3 x \sqrt{d+c^2 d x^2}}{63 c^3 \sqrt{1+c^2 x^2}}-\frac{25 b c d^2 f^3 x^2 \sqrt{d+c^2 d x^2}}{96 \sqrt{1+c^2 x^2}}-\frac{15 b d^2 f g^2 x^2 \sqrt{d+c^2 d x^2}}{256 c \sqrt{1+c^2 x^2}}-\frac{3 b c d^2 f^2 g x^3 \sqrt{d+c^2 d x^2}}{7 \sqrt{1+c^2 x^2}}-\frac{b d^2 g^3 x^3 \sqrt{d+c^2 d x^2}}{189 c \sqrt{1+c^2 x^2}}-\frac{5 b c^3 d^2 f^3 x^4 \sqrt{d+c^2 d x^2}}{96 \sqrt{1+c^2 x^2}}-\frac{59 b c d^2 f g^2 x^4 \sqrt{d+c^2 d x^2}}{256 \sqrt{1+c^2 x^2}}-\frac{9 b c^3 d^2 f^2 g x^5 \sqrt{d+c^2 d x^2}}{35 \sqrt{1+c^2 x^2}}-\frac{b c d^2 g^3 x^5 \sqrt{d+c^2 d x^2}}{21 \sqrt{1+c^2 x^2}}-\frac{17 b c^3 d^2 f g^2 x^6 \sqrt{d+c^2 d x^2}}{96 \sqrt{1+c^2 x^2}}-\frac{3 b c^5 d^2 f^2 g x^7 \sqrt{d+c^2 d x^2}}{49 \sqrt{1+c^2 x^2}}-\frac{19 b c^3 d^2 g^3 x^7 \sqrt{d+c^2 d x^2}}{441 \sqrt{1+c^2 x^2}}-\frac{3 b c^5 d^2 f g^2 x^8 \sqrt{d+c^2 d x^2}}{64 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 g^3 x^9 \sqrt{d+c^2 d x^2}}{81 \sqrt{1+c^2 x^2}}-\frac{b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt{d+c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f^3 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{15 d^2 f g^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac{15}{64} d^2 f g^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{24} d^2 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{5}{16} d^2 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{1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d^2 f^2 g \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^2}-\frac{d^2 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{d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac{5 d^2 f^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt{1+c^2 x^2}}-\frac{15 d^2 f g^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 7.05114, size = 1899, normalized size = 1.55 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

Sqrt[d*(1 + c^2*x^2)]*(-(a*d^2*g*(-27*c^2*f^2 + 2*g^2))/(63*c^4) + (a*d^2*f*(88*c^2*f^2 + 15*g^2)*x)/(128*c^2)
+ (a*d^2*g*(81*c^2*f^2 + g^2)*x^2)/(63*c^2) + (a*d^2*f*(104*c^2*f^2 + 177*g^2)*x^3)/192 + (a*d^2*g*(27*c^2*f^
2 + 5*g^2)*x^4)/21 + (a*c^2*d^2*f*(8*c^2*f^2 + 51*g^2)*x^5)/48 + (a*c^2*d^2*g*(27*c^2*f^2 + 19*g^2)*x^6)/63 +
(3*a*c^4*d^2*f*g^2*x^7)/8 + (a*c^4*d^2*g^3*x^8)/9) + (3*b*d^2*f^2*g*(-(c*x*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2)
)/(9*Sqrt[1 + c^2*x^2]) + ((1 + c^2*x^2)*Sqrt[d*(1 + c^2*x^2)]*ArcSinh[c*x])/3))/c^2 + (6*b*d^2*f^2*g*((2*c*x*
Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/(45*Sqrt[1 + c^2*x^2]) - (c^3*x^3*Sqrt[d*(1 + c^2*x^2)]*(5 + 3*c^2*x^2))/
(75*Sqrt[1 + c^2*x^2]) + ((d*(1 + c^2*x^2))^(3/2)*(-2 + 3*c^2*x^2)*ArcSinh[c*x])/(15*d)))/c^2 + (b*d^2*g^3*((2
*c*x*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/(45*Sqrt[1 + c^2*x^2]) - (c^3*x^3*Sqrt[d*(1 + c^2*x^2)]*(5 + 3*c^2*x
^2))/(75*Sqrt[1 + c^2*x^2]) + ((d*(1 + c^2*x^2))^(3/2)*(-2 + 3*c^2*x^2)*ArcSinh[c*x])/(15*d)))/c^4 + (3*b*d^2*
f^2*g*((-8*c*x*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/(315*Sqrt[1 + c^2*x^2]) + (4*c^3*x^3*Sqrt[d*(1 + c^2*x^2)]
*(5 + 3*c^2*x^2))/(525*Sqrt[1 + c^2*x^2]) - (c^5*x^5*Sqrt[d*(1 + c^2*x^2)]*(7 + 5*c^2*x^2))/(245*Sqrt[1 + c^2*
x^2]) + ((d*(1 + c^2*x^2))^(3/2)*(8 - 12*c^2*x^2 + 15*c^4*x^4)*ArcSinh[c*x])/(105*d)))/c^2 + (2*b*d^2*g^3*((-8
*c*x*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/(315*Sqrt[1 + c^2*x^2]) + (4*c^3*x^3*Sqrt[d*(1 + c^2*x^2)]*(5 + 3*c^
2*x^2))/(525*Sqrt[1 + c^2*x^2]) - (c^5*x^5*Sqrt[d*(1 + c^2*x^2)]*(7 + 5*c^2*x^2))/(245*Sqrt[1 + c^2*x^2]) + ((
d*(1 + c^2*x^2))^(3/2)*(8 - 12*c^2*x^2 + 15*c^4*x^4)*ArcSinh[c*x])/(105*d)))/c^4 + (b*d^2*g^3*((16*c*x*Sqrt[d*
(1 + c^2*x^2)]*(3 + c^2*x^2))/(945*Sqrt[1 + c^2*x^2]) - (8*c^3*x^3*Sqrt[d*(1 + c^2*x^2)]*(5 + 3*c^2*x^2))/(157
5*Sqrt[1 + c^2*x^2]) + (2*c^5*x^5*Sqrt[d*(1 + c^2*x^2)]*(7 + 5*c^2*x^2))/(735*Sqrt[1 + c^2*x^2]) - (c^7*x^7*Sq
rt[d*(1 + c^2*x^2)]*(9 + 7*c^2*x^2))/(567*Sqrt[1 + c^2*x^2]) + ((d*(1 + c^2*x^2))^(3/2)*(-16 + 24*c^2*x^2 - 30
*c^4*x^4 + 35*c^6*x^6)*ArcSinh[c*x])/(315*d)))/c^4 + (5*a*d^(5/2)*f*(8*c^2*f^2 - 3*g^2)*Log[c*d*x + Sqrt[d]*Sq
rt[d*(1 + c^2*x^2)]])/(128*c^3) + (b*d^2*f^3*Sqrt[d*(1 + c^2*x^2)]*(-Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(Ar
cSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/(8*c*Sqrt[1 + c^2*x^2]) - (b*d^2*f^3*Sqrt[d*(1 + c^2*x^2)]*(8*ArcSinh[c*x
]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/(64*c*Sqrt[1 + c^2*x^2]) - (3*b*d^2*f*g^2*S
qrt[d*(1 + c^2*x^2)]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/(128*c^3
*Sqrt[1 + c^2*x^2]) + (b*d^2*f^3*Sqrt[d*(1 + c^2*x^2)]*(72*ArcSinh[c*x]^2 + 18*Cosh[2*ArcSinh[c*x]] + 9*Cosh[4
*ArcSinh[c*x]] - 2*Cosh[6*ArcSinh[c*x]] - 36*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]] - 36*ArcSinh[c*x]*Sinh[4*ArcSin
h[c*x]] + 12*ArcSinh[c*x]*Sinh[6*ArcSinh[c*x]]))/(2304*c*Sqrt[1 + c^2*x^2]) + (b*d^2*f*g^2*Sqrt[d*(1 + c^2*x^2
)]*(72*ArcSinh[c*x]^2 + 18*Cosh[2*ArcSinh[c*x]] + 9*Cosh[4*ArcSinh[c*x]] - 2*Cosh[6*ArcSinh[c*x]] - 36*ArcSinh
[c*x]*Sinh[2*ArcSinh[c*x]] - 36*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]] + 12*ArcSinh[c*x]*Sinh[6*ArcSinh[c*x]]))/(38
4*c^3*Sqrt[1 + c^2*x^2]) - (b*d^2*f*g^2*Sqrt[d*(1 + c^2*x^2)]*(1440*ArcSinh[c*x]^2 + 576*Cosh[2*ArcSinh[c*x]]
+ 144*Cosh[4*ArcSinh[c*x]] - 64*Cosh[6*ArcSinh[c*x]] + 9*Cosh[8*ArcSinh[c*x]] - 1152*ArcSinh[c*x]*Sinh[2*ArcSi
nh[c*x]] - 576*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]] + 384*ArcSinh[c*x]*Sinh[6*ArcSinh[c*x]] - 72*ArcSinh[c*x]*Sin
h[8*ArcSinh[c*x]]))/(24576*c^3*Sqrt[1 + c^2*x^2])

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Maple [A]  time = 0.575, size = 1954, 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)^(5/2)*(a+b*arcsinh(c*x)),x)

[Out]

359/24576*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d^2/c^3/(c^2*x^2+1)^(1/2)-1/36*b*(d*(c^2*x^2+1))^(1/2)*f^3*d^2*c^5/(c^
2*x^2+1)^(1/2)*x^6-13/96*b*(d*(c^2*x^2+1))^(1/2)*f^3*d^2*c^3/(c^2*x^2+1)^(1/2)*x^4-11/32*b*(d*(c^2*x^2+1))^(1/
2)*f^3*d^2*c/(c^2*x^2+1)^(1/2)*x^2-1/81*b*(d*(c^2*x^2+1))^(1/2)*g^3*d^2*c^5/(c^2*x^2+1)^(1/2)*x^9-19/441*b*(d*
(c^2*x^2+1))^(1/2)*g^3*d^2*c^3/(c^2*x^2+1)^(1/2)*x^7-1/21*b*(d*(c^2*x^2+1))^(1/2)*g^3*d^2*c/(c^2*x^2+1)^(1/2)*
x^5-1/189*b*(d*(c^2*x^2+1))^(1/2)*g^3*d^2/c/(c^2*x^2+1)^(1/2)*x^3+2/63*b*(d*(c^2*x^2+1))^(1/2)*g^3*d^2/c^3/(c^
2*x^2+1)^(1/2)*x+5/32*b*(d*(c^2*x^2+1))^(1/2)*f^3*arcsinh(c*x)^2*d^2/(c^2*x^2+1)^(1/2)/c-2/63*b*(d*(c^2*x^2+1)
)^(1/2)*g^3*d^2/c^4/(c^2*x^2+1)*arcsinh(c*x)+11/16*b*(d*(c^2*x^2+1))^(1/2)*f^3*d^2/(c^2*x^2+1)*arcsinh(c*x)*x+
16/63*b*(d*(c^2*x^2+1))^(1/2)*g^3*d^2/(c^2*x^2+1)*arcsinh(c*x)*x^4+3/8*a*f*g^2*x*(c^2*d*x^2+d)^(7/2)/c^2/d-5/6
4*a*f*g^2/c^2*d*x*(c^2*d*x^2+d)^(3/2)-15/128*a*f*g^2/c^2*d^2*x*(c^2*d*x^2+d)^(1/2)-15/128*a*f*g^2/c^2*d^3*ln(x
*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/9*a*g^3*x^2*(c^2*d*x^2+d)^(7/2)/c^2/d-1/16*a*f*g^2/c
^2*x*(c^2*d*x^2+d)^(5/2)+3/7*a*f^2*g/c^2/d*(c^2*d*x^2+d)^(7/2)-299/2304*b*(d*(c^2*x^2+1))^(1/2)*f^3*d^2/c/(c^2
*x^2+1)^(1/2)+3/7*b*(d*(c^2*x^2+1))^(1/2)*g*d^2*c^6/(c^2*x^2+1)*arcsinh(c*x)*x^8*f^2+12/7*b*(d*(c^2*x^2+1))^(1
/2)*g*d^2*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^6*f^2+18/7*b*(d*(c^2*x^2+1))^(1/2)*g*d^2*c^2/(c^2*x^2+1)*arcsinh(c*x)
*x^4*f^2+3/8*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d^2*c^6/(c^2*x^2+1)*arcsinh(c*x)*x^9+23/16*b*(d*(c^2*x^2+1))^(1/2)*
f*g^2*d^2*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^7+127/64*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d^2*c^2/(c^2*x^2+1)*arcsinh(c*
x)*x^5+15/128*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d^2/c^2/(c^2*x^2+1)*arcsinh(c*x)*x-2/63*a*g^3/d/c^4*(c^2*d*x^2+d)^
(7/2)-17/96*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d^2*c^3/(c^2*x^2+1)^(1/2)*x^6-59/256*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d
^2*c/(c^2*x^2+1)^(1/2)*x^4-15/256*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d^2/c/(c^2*x^2+1)^(1/2)*x^2+3/7*b*(d*(c^2*x^2+
1))^(1/2)*g*d^2/c^2/(c^2*x^2+1)*arcsinh(c*x)*f^2+1/6*b*(d*(c^2*x^2+1))^(1/2)*f^3*d^2*c^6/(c^2*x^2+1)*arcsinh(c
*x)*x^7+17/24*b*(d*(c^2*x^2+1))^(1/2)*f^3*d^2*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^5+59/48*b*(d*(c^2*x^2+1))^(1/2)*f
^3*d^2*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^3+12/7*b*(d*(c^2*x^2+1))^(1/2)*g*d^2/(c^2*x^2+1)*arcsinh(c*x)*x^2*f^2+13
3/128*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d^2/(c^2*x^2+1)*arcsinh(c*x)*x^3-15/256*b*(d*(c^2*x^2+1))^(1/2)*f*arcsinh(
c*x)^2*d^2/(c^2*x^2+1)^(1/2)/c^3*g^2+1/9*b*(d*(c^2*x^2+1))^(1/2)*g^3*d^2*c^6/(c^2*x^2+1)*arcsinh(c*x)*x^10+26/
63*b*(d*(c^2*x^2+1))^(1/2)*g^3*d^2*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^8+34/63*b*(d*(c^2*x^2+1))^(1/2)*g^3*d^2*c^2/
(c^2*x^2+1)*arcsinh(c*x)*x^6-1/63*b*(d*(c^2*x^2+1))^(1/2)*g^3*d^2/c^2/(c^2*x^2+1)*arcsinh(c*x)*x^2-3/49*b*(d*(
c^2*x^2+1))^(1/2)*g*d^2*c^5/(c^2*x^2+1)^(1/2)*x^7*f^2-9/35*b*(d*(c^2*x^2+1))^(1/2)*g*d^2*c^3/(c^2*x^2+1)^(1/2)
*x^5*f^2-3/7*b*(d*(c^2*x^2+1))^(1/2)*g*d^2*c/(c^2*x^2+1)^(1/2)*x^3*f^2-3/7*b*(d*(c^2*x^2+1))^(1/2)*g*d^2/c/(c^
2*x^2+1)^(1/2)*x*f^2-3/64*b*(d*(c^2*x^2+1))^(1/2)*f*g^2*d^2*c^5/(c^2*x^2+1)^(1/2)*x^8+1/6*a*f^3*x*(c^2*d*x^2+d
)^(5/2)+5/16*a*f^3*d^2*x*(c^2*d*x^2+d)^(1/2)+5/16*a*f^3*d^3*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2
*d)^(1/2)+5/24*a*f^3*d*x*(c^2*d*x^2+d)^(3/2)

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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)^(5/2)*(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^{4} d^{2} g^{3} x^{7} + 3 \, a c^{4} d^{2} f g^{2} x^{6} + 3 \, a d^{2} f^{2} g x + a d^{2} f^{3} +{\left (3 \, a c^{4} d^{2} f^{2} g + 2 \, a c^{2} d^{2} g^{3}\right )} x^{5} +{\left (a c^{4} d^{2} f^{3} + 6 \, a c^{2} d^{2} f g^{2}\right )} x^{4} +{\left (6 \, a c^{2} d^{2} f^{2} g + a d^{2} g^{3}\right )} x^{3} +{\left (2 \, a c^{2} d^{2} f^{3} + 3 \, a d^{2} f g^{2}\right )} x^{2} +{\left (b c^{4} d^{2} g^{3} x^{7} + 3 \, b c^{4} d^{2} f g^{2} x^{6} + 3 \, b d^{2} f^{2} g x + b d^{2} f^{3} +{\left (3 \, b c^{4} d^{2} f^{2} g + 2 \, b c^{2} d^{2} g^{3}\right )} x^{5} +{\left (b c^{4} d^{2} f^{3} + 6 \, b c^{2} d^{2} f g^{2}\right )} x^{4} +{\left (6 \, b c^{2} d^{2} f^{2} g + b d^{2} g^{3}\right )} x^{3} +{\left (2 \, b c^{2} d^{2} f^{3} + 3 \, b d^{2} 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)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*g^3*x^7 + 3*a*c^4*d^2*f*g^2*x^6 + 3*a*d^2*f^2*g*x + a*d^2*f^3 + (3*a*c^4*d^2*f^2*g + 2*a*c
^2*d^2*g^3)*x^5 + (a*c^4*d^2*f^3 + 6*a*c^2*d^2*f*g^2)*x^4 + (6*a*c^2*d^2*f^2*g + a*d^2*g^3)*x^3 + (2*a*c^2*d^2
*f^3 + 3*a*d^2*f*g^2)*x^2 + (b*c^4*d^2*g^3*x^7 + 3*b*c^4*d^2*f*g^2*x^6 + 3*b*d^2*f^2*g*x + b*d^2*f^3 + (3*b*c^
4*d^2*f^2*g + 2*b*c^2*d^2*g^3)*x^5 + (b*c^4*d^2*f^3 + 6*b*c^2*d^2*f*g^2)*x^4 + (6*b*c^2*d^2*f^2*g + b*d^2*g^3)
*x^3 + (2*b*c^2*d^2*f^3 + 3*b*d^2*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)**(5/2)*(a+b*asinh(c*x)),x)

[Out]

Timed out

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Giac [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)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

Timed out