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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.925864, antiderivative size = 901, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 15, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5835, 5821, 5684, 5682, 5675, 30, 14, 261, 5717, 194, 5744, 5742, 5758, 266, 43} \[ -\frac{b c^5 d^2 g^2 \sqrt{c^2 d x^2+d} x^8}{64 \sqrt{c^2 x^2+1}}-\frac{2 b c^5 d^2 f g \sqrt{c^2 d x^2+d} x^7}{49 \sqrt{c^2 x^2+1}}-\frac{17 b c^3 d^2 g^2 \sqrt{c^2 d x^2+d} x^6}{288 \sqrt{c^2 x^2+1}}-\frac{6 b c^3 d^2 f g \sqrt{c^2 d x^2+d} x^5}{35 \sqrt{c^2 x^2+1}}-\frac{5 b c^3 d^2 f^2 \sqrt{c^2 d x^2+d} x^4}{96 \sqrt{c^2 x^2+1}}-\frac{59 b c d^2 g^2 \sqrt{c^2 d x^2+d} x^4}{768 \sqrt{c^2 x^2+1}}+\frac{5}{64} d^2 g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3+\frac{1}{8} d^2 g^2 \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3+\frac{5}{48} d^2 g^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3-\frac{2 b c d^2 f g \sqrt{c^2 d x^2+d} x^3}{7 \sqrt{c^2 x^2+1}}-\frac{25 b c d^2 f^2 \sqrt{c^2 d x^2+d} x^2}{96 \sqrt{c^2 x^2+1}}-\frac{5 b d^2 g^2 \sqrt{c^2 d x^2+d} x^2}{256 c \sqrt{c^2 x^2+1}}+\frac{5}{16} d^2 f^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac{5 d^2 g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x}{128 c^2}+\frac{1}{6} d^2 f^2 \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac{5}{24} d^2 f^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x-\frac{2 b d^2 f g \sqrt{c^2 d x^2+d} x}{7 c \sqrt{c^2 x^2+1}}+\frac{5 d^2 f^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt{c^2 x^2+1}}-\frac{5 d^2 g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{c^2 x^2+1}}+\frac{2 d^2 f g \left (c^2 x^2+1\right )^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac{b d^2 f^2 \left (c^2 x^2+1\right )^{5/2} \sqrt{c^2 d x^2+d}}{36 c} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^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 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])

Rubi steps

\begin{align*} \int (f+g x)^2 \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)^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 \sqrt{d+c^2 d x^2}\right ) \int \left (f^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+2 f g x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+g^2 x^2 \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^2 \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 (2 d^2 f 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 (d^2 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{1}{6} d^2 f^2 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{1}{8} d^2 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{2 d^2 f 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{\left (5 d^2 f^2 \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^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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{48} d^2 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^2 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{1}{8} d^2 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{2 d^2 f 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{\left (5 d^2 f^2 \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^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} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{16 \sqrt{1+c^2 x^2}}-\frac{\left (b c d^2 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 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} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{64} d^2 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^2 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}{48} d^2 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^2 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{1}{8} d^2 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{2 d^2 f 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{\left (5 d^2 f^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 \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 \left (a+b \sinh ^{-1}(c x)\right )}{\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 ) \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 (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} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5 d^2 g^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac{5}{64} d^2 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^2 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}{48} d^2 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^2 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{1}{8} d^2 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{2 d^2 f 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{5 d^2 f^2 \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 (5 d^2 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 (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} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5 d^2 g^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac{5}{64} d^2 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^2 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}{48} d^2 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^2 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{1}{8} d^2 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{2 d^2 f 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{5 d^2 f^2 \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{5 d^2 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 = 2.71485, size = 1047, normalized size = 1.16 \[ \frac{d^2 \left (-737280 b f g x^7 \sqrt{c^2 d x^2+d} c^8+2257920 a g^2 x^7 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^7+5160960 a f g x^6 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^7+3010560 a f^2 x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^7-3096576 b f g x^5 \sqrt{c^2 d x^2+d} c^6+6397440 a g^2 x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^5+15482880 a f g x^4 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^5+9784320 a f^2 x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^5-5160960 b f g x^3 \sqrt{c^2 d x^2+d} c^4+5550720 a g^2 x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^3+15482880 a f g x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^3+12418560 a f^2 x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^3-211680 b f^2 \sqrt{c^2 d x^2+d} \cosh \left (4 \sinh ^{-1}(c x)\right ) c^2-15680 b f^2 \sqrt{c^2 d x^2+d} \cosh \left (6 \sinh ^{-1}(c x)\right ) c^2+5644800 a \sqrt{d} f^2 \sqrt{c^2 x^2+1} \log \left (c d x+\sqrt{d} \sqrt{c^2 d x^2+d}\right ) c^2-5160960 b f g x \sqrt{c^2 d x^2+d} c^2+5160960 a f g \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c+705600 a g^2 x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c+352800 b \left (8 c^2 f^2-g^2\right ) \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2-141120 b \left (15 c^2 f^2-g^2\right ) \sqrt{c^2 d x^2+d} \cosh \left (2 \sinh ^{-1}(c x)\right )-35280 b g^2 \sqrt{c^2 d x^2+d} \cosh \left (4 \sinh ^{-1}(c x)\right )-15680 b g^2 \sqrt{c^2 d x^2+d} \cosh \left (6 \sinh ^{-1}(c x)\right )-2205 b g^2 \sqrt{c^2 d x^2+d} \cosh \left (8 \sinh ^{-1}(c x)\right )-705600 a \sqrt{d} g^2 \sqrt{c^2 x^2+1} \log \left (c d x+\sqrt{d} \sqrt{c^2 d x^2+d}\right )+840 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left (6144 f g x^6 \sqrt{c^2 x^2+1} c^7+18432 f g x^4 \sqrt{c^2 x^2+1} c^5+18432 f g x^2 \sqrt{c^2 x^2+1} c^3+112 f^2 \sinh \left (6 \sinh ^{-1}(c x)\right ) c^2+6144 f g \sqrt{c^2 x^2+1} c+336 \left (15 c^2 f^2-g^2\right ) \sinh \left (2 \sinh ^{-1}(c x)\right )+168 \left (6 c^2 f^2+g^2\right ) \sinh \left (4 \sinh ^{-1}(c x)\right )+112 g^2 \sinh \left (6 \sinh ^{-1}(c x)\right )+21 g^2 \sinh \left (8 \sinh ^{-1}(c x)\right )\right )\right )}{18063360 c^3 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(-5160960*b*c^2*f*g*x*Sqrt[d + c^2*d*x^2] - 5160960*b*c^4*f*g*x^3*Sqrt[d + c^2*d*x^2] - 3096576*b*c^6*f*g
*x^5*Sqrt[d + c^2*d*x^2] - 737280*b*c^8*f*g*x^7*Sqrt[d + c^2*d*x^2] + 5160960*a*c*f*g*Sqrt[1 + c^2*x^2]*Sqrt[d
+ c^2*d*x^2] + 12418560*a*c^3*f^2*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 705600*a*c*g^2*x*Sqrt[1 + c^2*x^2
]*Sqrt[d + c^2*d*x^2] + 15482880*a*c^3*f*g*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 9784320*a*c^5*f^2*x^3*S
qrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 5550720*a*c^3*g^2*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 15482880*
a*c^5*f*g*x^4*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 3010560*a*c^7*f^2*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x
^2] + 6397440*a*c^5*g^2*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 5160960*a*c^7*f*g*x^6*Sqrt[1 + c^2*x^2]*Sq
rt[d + c^2*d*x^2] + 2257920*a*c^7*g^2*x^7*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 352800*b*(8*c^2*f^2 - g^2)*S
qrt[d + c^2*d*x^2]*ArcSinh[c*x]^2 - 141120*b*(15*c^2*f^2 - g^2)*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 211
680*b*c^2*f^2*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 35280*b*g^2*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]]
- 15680*b*c^2*f^2*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] - 15680*b*g^2*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*
x]] - 2205*b*g^2*Sqrt[d + c^2*d*x^2]*Cosh[8*ArcSinh[c*x]] + 5644800*a*c^2*Sqrt[d]*f^2*Sqrt[1 + c^2*x^2]*Log[c*
d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 705600*a*Sqrt[d]*g^2*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d
*x^2]] + 840*b*Sqrt[d + c^2*d*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*Sin
h[6*ArcSinh[c*x]] + 21*g^2*Sinh[8*ArcSinh[c*x]])))/(18063360*c^3*Sqrt[1 + c^2*x^2])

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Maple [A]  time = 0.475, size = 1424, 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)^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x)

[Out]

2/7*b*(d*(c^2*x^2+1))^(1/2)*f*g*d^2/c^2/(c^2*x^2+1)*arcsinh(c*x)+1/6*b*(d*(c^2*x^2+1))^(1/2)*d^2*c^6/(c^2*x^2+
1)*arcsinh(c*x)*x^7*f^2+17/24*b*(d*(c^2*x^2+1))^(1/2)*d^2*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^5*f^2+59/48*b*(d*(c^2
*x^2+1))^(1/2)*d^2*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^3*f^2+8/7*b*(d*(c^2*x^2+1))^(1/2)*f*g*d^2/(c^2*x^2+1)*arcsin
h(c*x)*x^2+1/8*b*(d*(c^2*x^2+1))^(1/2)*g^2*d^2*c^6/(c^2*x^2+1)*arcsinh(c*x)*x^9+23/48*b*(d*(c^2*x^2+1))^(1/2)*
g^2*d^2*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^7+127/192*b*(d*(c^2*x^2+1))^(1/2)*g^2*d^2*c^2/(c^2*x^2+1)*arcsinh(c*x)*
x^5+2/7*b*(d*(c^2*x^2+1))^(1/2)*f*g*d^2*c^6/(c^2*x^2+1)*arcsinh(c*x)*x^8+8/7*b*(d*(c^2*x^2+1))^(1/2)*f*g*d^2*c
^4/(c^2*x^2+1)*arcsinh(c*x)*x^6+12/7*b*(d*(c^2*x^2+1))^(1/2)*f*g*d^2*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^4-1/48*a*g
^2/c^2*x*(c^2*d*x^2+d)^(5/2)-2/49*b*(d*(c^2*x^2+1))^(1/2)*f*g*d^2*c^5/(c^2*x^2+1)^(1/2)*x^7-6/35*b*(d*(c^2*x^2
+1))^(1/2)*f*g*d^2*c^3/(c^2*x^2+1)^(1/2)*x^5-2/7*b*(d*(c^2*x^2+1))^(1/2)*f*g*d^2*c/(c^2*x^2+1)^(1/2)*x^3-2/7*b
*(d*(c^2*x^2+1))^(1/2)*f*g*d^2/c/(c^2*x^2+1)^(1/2)*x+5/128*b*(d*(c^2*x^2+1))^(1/2)*g^2*d^2/c^2/(c^2*x^2+1)*arc
sinh(c*x)*x+5/24*a*f^2*d*x*(c^2*d*x^2+d)^(3/2)+5/16*a*f^2*d^2*x*(c^2*d*x^2+d)^(1/2)+5/16*a*f^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)+1/6*a*f^2*x*(c^2*d*x^2+d)^(5/2)-5/192*a*g^2/c^2*d*x*(c^2*d*x^
2+d)^(3/2)-5/128*a*g^2/c^2*d^2*x*(c^2*d*x^2+d)^(1/2)-5/128*a*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)+2/7*a*f*g/c^2/d*(c^2*d*x^2+d)^(7/2)+1/8*a*g^2*x*(c^2*d*x^2+d)^(7/2)/c^2/d+359/73728*b*(
d*(c^2*x^2+1))^(1/2)*g^2*d^2/c^3/(c^2*x^2+1)^(1/2)-299/2304*b*(d*(c^2*x^2+1))^(1/2)*d^2/c/(c^2*x^2+1)^(1/2)*f^
2-1/36*b*(d*(c^2*x^2+1))^(1/2)*d^2*c^5/(c^2*x^2+1)^(1/2)*x^6*f^2-13/96*b*(d*(c^2*x^2+1))^(1/2)*d^2*c^3/(c^2*x^
2+1)^(1/2)*x^4*f^2-11/32*b*(d*(c^2*x^2+1))^(1/2)*d^2*c/(c^2*x^2+1)^(1/2)*x^2*f^2-1/64*b*(d*(c^2*x^2+1))^(1/2)*
g^2*d^2*c^5/(c^2*x^2+1)^(1/2)*x^8-17/288*b*(d*(c^2*x^2+1))^(1/2)*g^2*d^2*c^3/(c^2*x^2+1)^(1/2)*x^6-59/768*b*(d
*(c^2*x^2+1))^(1/2)*g^2*d^2*c/(c^2*x^2+1)^(1/2)*x^4-5/256*b*(d*(c^2*x^2+1))^(1/2)*g^2*d^2/c/(c^2*x^2+1)^(1/2)*
x^2+11/16*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)*arcsinh(c*x)*x*f^2+5/32*b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)
^2*d^2/(c^2*x^2+1)^(1/2)/c*f^2-5/256*b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)^2*d^2/(c^2*x^2+1)^(1/2)/c^3*g^2+133/
384*b*(d*(c^2*x^2+1))^(1/2)*g^2*d^2/(c^2*x^2+1)*arcsinh(c*x)*x^3

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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)^2*(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^{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} +{\left (a c^{4} d^{2} f^{2} + 2 \, a c^{2} d^{2} g^{2}\right )} x^{4} +{\left (2 \, a c^{2} d^{2} f^{2} + a d^{2} g^{2}\right )} x^{2} +{\left (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} +{\left (b c^{4} d^{2} f^{2} + 2 \, b c^{2} d^{2} g^{2}\right )} x^{4} +{\left (2 \, b c^{2} d^{2} f^{2} + b d^{2} 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)^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)

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

[Out]

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