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

Optimal. Leaf size=984 \[ -\frac{b d x^3 \sqrt{c^2 d x^2+d} c^3}{9 g \sqrt{c^2 x^2+1}}+\frac{b d f x^2 \sqrt{c^2 d x^2+d} c^3}{4 g^2 \sqrt{c^2 x^2+1}}-\frac{d f x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) c^2}{2 g^2}-\frac{d \left (c^2 f^2+g^2\right ) x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{2 b g^3 \sqrt{c^2 x^2+1}}-\frac{d f \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{4 b g^2 \sqrt{c^2 x^2+1}}-\frac{b d \left (c^2 f^2+g^2\right ) x \sqrt{c^2 d x^2+d} c}{g^3 \sqrt{c^2 x^2+1}}-\frac{b d x \sqrt{c^2 d x^2+d} c}{3 g \sqrt{c^2 x^2+1}}+\frac{b d \left (c^2 f^2+g^2\right ) \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g^3}+\frac{d \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{c^2 x^2+1}}\right )}{g^4 \sqrt{c^2 x^2+1}}+\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}+1\right )}{g^4 \sqrt{c^2 x^2+1}}-\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}+1\right )}{g^4 \sqrt{c^2 x^2+1}}+\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^4 \sqrt{c^2 x^2+1}}-\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{g^4 \sqrt{c^2 x^2+1}}+\frac{a d \left (c^2 f^2+g^2\right ) \sqrt{c^2 d x^2+d}}{g^3}+\frac{d \left (c^2 f^2+g^2\right ) \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^2 (f+g x) c}-\frac{d \left (c^2 f^2+g^2\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) \sqrt{c^2 x^2+1} c} \]

[Out]

(a*d*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2])/g^3 - (b*c*d*x*Sqrt[d + c^2*d*x^2])/(3*g*Sqrt[1 + c^2*x^2]) - (b*c*d
*(c^2*f^2 + g^2)*x*Sqrt[d + c^2*d*x^2])/(g^3*Sqrt[1 + c^2*x^2]) + (b*c^3*d*f*x^2*Sqrt[d + c^2*d*x^2])/(4*g^2*S
qrt[1 + c^2*x^2]) - (b*c^3*d*x^3*Sqrt[d + c^2*d*x^2])/(9*g*Sqrt[1 + c^2*x^2]) + (b*d*(c^2*f^2 + g^2)*Sqrt[d +
c^2*d*x^2]*ArcSinh[c*x])/g^3 - (c^2*d*f*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*g^2) + (d*(1 + c^2*x^2)
*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*g) - (c*d*f*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*g^2
*Sqrt[1 + c^2*x^2]) - (c*d*(c^2*f^2 + g^2)*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*g^3*Sqrt[1 + c^2
*x^2]) - (d*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^4*(f + g*x)*Sqrt[1 + c^2*x^
2]) + (d*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^2*(f + g*x)) -
(a*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])]
)/(g^4*Sqrt[1 + c^2*x^2]) + (b*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x
]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(g^4*Sqrt[1 + c^2*x^2]) - (b*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*Ar
cSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(g^4*Sqrt[1 + c^2*x^2]) + (b*d*(c^2*f^2 +
g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(g^4*Sqrt[1 + c^
2*x^2]) - (b*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 +
g^2]))])/(g^4*Sqrt[1 + c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.88879, antiderivative size = 984, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 24, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {5835, 5825, 5682, 5675, 30, 5717, 5823, 683, 5815, 6742, 261, 725, 206, 5859, 1654, 12, 5857, 8, 5831, 3322, 2264, 2190, 2279, 2391} \[ -\frac{b d x^3 \sqrt{c^2 d x^2+d} c^3}{9 g \sqrt{c^2 x^2+1}}+\frac{b d f x^2 \sqrt{c^2 d x^2+d} c^3}{4 g^2 \sqrt{c^2 x^2+1}}-\frac{d f x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) c^2}{2 g^2}-\frac{d \left (c^2 f^2+g^2\right ) x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{2 b g^3 \sqrt{c^2 x^2+1}}-\frac{d f \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{4 b g^2 \sqrt{c^2 x^2+1}}-\frac{b d \left (c^2 f^2+g^2\right ) x \sqrt{c^2 d x^2+d} c}{g^3 \sqrt{c^2 x^2+1}}-\frac{b d x \sqrt{c^2 d x^2+d} c}{3 g \sqrt{c^2 x^2+1}}+\frac{b d \left (c^2 f^2+g^2\right ) \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g^3}+\frac{d \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{c^2 x^2+1}}\right )}{g^4 \sqrt{c^2 x^2+1}}+\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}+1\right )}{g^4 \sqrt{c^2 x^2+1}}-\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}+1\right )}{g^4 \sqrt{c^2 x^2+1}}+\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^4 \sqrt{c^2 x^2+1}}-\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{g^4 \sqrt{c^2 x^2+1}}+\frac{a d \left (c^2 f^2+g^2\right ) \sqrt{c^2 d x^2+d}}{g^3}+\frac{d \left (c^2 f^2+g^2\right ) \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^2 (f+g x) c}-\frac{d \left (c^2 f^2+g^2\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) \sqrt{c^2 x^2+1} c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*d*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2])/g^3 - (b*c*d*x*Sqrt[d + c^2*d*x^2])/(3*g*Sqrt[1 + c^2*x^2]) - (b*c*d
*(c^2*f^2 + g^2)*x*Sqrt[d + c^2*d*x^2])/(g^3*Sqrt[1 + c^2*x^2]) + (b*c^3*d*f*x^2*Sqrt[d + c^2*d*x^2])/(4*g^2*S
qrt[1 + c^2*x^2]) - (b*c^3*d*x^3*Sqrt[d + c^2*d*x^2])/(9*g*Sqrt[1 + c^2*x^2]) + (b*d*(c^2*f^2 + g^2)*Sqrt[d +
c^2*d*x^2]*ArcSinh[c*x])/g^3 - (c^2*d*f*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*g^2) + (d*(1 + c^2*x^2)
*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*g) - (c*d*f*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*g^2
*Sqrt[1 + c^2*x^2]) - (c*d*(c^2*f^2 + g^2)*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*g^3*Sqrt[1 + c^2
*x^2]) - (d*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^4*(f + g*x)*Sqrt[1 + c^2*x^
2]) + (d*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^2*(f + g*x)) -
(a*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])]
)/(g^4*Sqrt[1 + c^2*x^2]) + (b*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x
]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(g^4*Sqrt[1 + c^2*x^2]) - (b*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*Ar
cSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(g^4*Sqrt[1 + c^2*x^2]) + (b*d*(c^2*f^2 +
g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(g^4*Sqrt[1 + c^
2*x^2]) - (b*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 +
g^2]))])/(g^4*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 5825

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Int[ExpandIntegrand[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n, (f + g*x)^m*(d + e*x^2)^(p - 1/2), x], x] /; Fr
eeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 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 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 5823

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

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 5815

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*(x_)^2)^(p_.))/((d_) + (e_.)*(x_))^2
, x_Symbol] :> With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcSinh[c*x])^n, u, x] - Di
st[b*c*n, Int[SimplifyIntegrand[(u*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b
, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[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 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 5859

Int[(ArcSinh[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x^2)^p, RFx*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x]
&& IGtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]

Rule 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 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 5857

Int[ArcSinh[(c_.)*(x_)]^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = ExpandIntegrand[(d + e
*x^2)^p*ArcSinh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] &&
IGtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 5831

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

Rule 3322

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(-(I*b) + 2*a*E^(-(I*e) + f*fz*x) + I*b*E^(2*(-(I*e) + f*fz*x))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{f+g x} \, dx &=\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \int \left (-\frac{c^2 f \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g^2}+\frac{c^2 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g}+\frac{\left (c^2 f^2+g^2\right ) \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g^2 (f+g x)}\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\left (d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2}\right ) \int \frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt{1+c^2 x^2}}-\frac{\left (c^2 d f \sqrt{d+c^2 d x^2}\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (c^2 d \sqrt{d+c^2 d x^2}\right ) \int x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{g \sqrt{1+c^2 x^2}}\\ &=-\frac{c^2 d f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac{d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{\left (d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (-g+2 c^2 f x+c^2 g x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c \sqrt{1+c^2 x^2}}-\frac{\left (c^2 d f \sqrt{d+c^2 d x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{2 g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b c^3 d f \sqrt{d+c^2 d x^2}\right ) \int x \, dx}{2 g^2 \sqrt{1+c^2 x^2}}-\frac{\left (b c d \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 g \sqrt{1+c^2 x^2}}\\ &=-\frac{b c d x \sqrt{d+c^2 d x^2}}{3 g \sqrt{1+c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d+c^2 d x^2}}{4 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d+c^2 d x^2}}{9 g \sqrt{1+c^2 x^2}}-\frac{c^2 d f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{1+c^2 x^2}}-\frac{d \left (1+\frac{c^2 f^2}{g^2}\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac{\left (d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (\frac{c^2 x}{g}+\frac{1+\frac{c^2 f^2}{g^2}}{f+g x}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c d x \sqrt{d+c^2 d x^2}}{3 g \sqrt{1+c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d+c^2 d x^2}}{4 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d+c^2 d x^2}}{9 g \sqrt{1+c^2 x^2}}-\frac{c^2 d f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{1+c^2 x^2}}-\frac{d \left (1+\frac{c^2 f^2}{g^2}\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac{\left (d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2}\right ) \int \left (\frac{a \left (c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt{1+c^2 x^2}}+\frac{b \left (c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2\right ) \sinh ^{-1}(c x)}{g^2 (f+g x) \sqrt{1+c^2 x^2}}\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c d x \sqrt{d+c^2 d x^2}}{3 g \sqrt{1+c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d+c^2 d x^2}}{4 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d+c^2 d x^2}}{9 g \sqrt{1+c^2 x^2}}-\frac{c^2 d f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{1+c^2 x^2}}-\frac{d \left (1+\frac{c^2 f^2}{g^2}\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac{\left (a d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2}\right ) \int \frac{c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt{1+c^2 x^2}} \, dx}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2\right ) \sinh ^{-1}(c x)}{(f+g x) \sqrt{1+c^2 x^2}} \, dx}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}{g^3}-\frac{b c d x \sqrt{d+c^2 d x^2}}{3 g \sqrt{1+c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d+c^2 d x^2}}{4 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d+c^2 d x^2}}{9 g \sqrt{1+c^2 x^2}}-\frac{c^2 d f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{1+c^2 x^2}}-\frac{d \left (1+\frac{c^2 f^2}{g^2}\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac{\left (a d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2}\right ) \int \frac{c^2 g^2 \left (c^2 f^2+g^2\right )}{(f+g x) \sqrt{1+c^2 x^2}} \, dx}{c^2 g^4 \sqrt{1+c^2 x^2}}+\frac{\left (b d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2}\right ) \int \left (\frac{c^2 g x \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}+\frac{\left (c^2 f^2+g^2\right ) \sinh ^{-1}(c x)}{(f+g x) \sqrt{1+c^2 x^2}}\right ) \, dx}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}{g^3}-\frac{b c d x \sqrt{d+c^2 d x^2}}{3 g \sqrt{1+c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d+c^2 d x^2}}{4 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d+c^2 d x^2}}{9 g \sqrt{1+c^2 x^2}}-\frac{c^2 d f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{1+c^2 x^2}}-\frac{d \left (1+\frac{c^2 f^2}{g^2}\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac{\left (b c^2 d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2}\right ) \int \frac{x \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{g \sqrt{1+c^2 x^2}}+\frac{\left (a d \left (1+\frac{c^2 f^2}{g^2}\right ) \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{(f+g x) \sqrt{1+c^2 x^2}} \, dx}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b d \left (1+\frac{c^2 f^2}{g^2}\right ) \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}\right ) \int \frac{\sinh ^{-1}(c x)}{(f+g x) \sqrt{1+c^2 x^2}} \, dx}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}{g^3}-\frac{b c d x \sqrt{d+c^2 d x^2}}{3 g \sqrt{1+c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d+c^2 d x^2}}{4 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d+c^2 d x^2}}{9 g \sqrt{1+c^2 x^2}}+\frac{b d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g^3}-\frac{c^2 d f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{1+c^2 x^2}}-\frac{d \left (1+\frac{c^2 f^2}{g^2}\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{\left (b c d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt{1+c^2 x^2}}-\frac{\left (a d \left (1+\frac{c^2 f^2}{g^2}\right ) \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c^2 f^2+g^2-x^2} \, dx,x,\frac{g-c^2 f x}{\sqrt{1+c^2 x^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b d \left (1+\frac{c^2 f^2}{g^2}\right ) \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}{g^3}-\frac{b c d x \sqrt{d+c^2 d x^2}}{3 g \sqrt{1+c^2 x^2}}-\frac{b c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2}}{g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d+c^2 d x^2}}{4 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d+c^2 d x^2}}{9 g \sqrt{1+c^2 x^2}}+\frac{b d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g^3}-\frac{c^2 d f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{1+c^2 x^2}}-\frac{d \left (1+\frac{c^2 f^2}{g^2}\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{1+c^2 x^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}+\frac{\left (2 b d \left (1+\frac{c^2 f^2}{g^2}\right ) \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c e^x f-g+e^{2 x} g} \, dx,x,\sinh ^{-1}(c x)\right )}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}{g^3}-\frac{b c d x \sqrt{d+c^2 d x^2}}{3 g \sqrt{1+c^2 x^2}}-\frac{b c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2}}{g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d+c^2 d x^2}}{4 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d+c^2 d x^2}}{9 g \sqrt{1+c^2 x^2}}+\frac{b d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g^3}-\frac{c^2 d f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{1+c^2 x^2}}-\frac{d \left (1+\frac{c^2 f^2}{g^2}\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{1+c^2 x^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}+\frac{\left (2 b d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g-2 \sqrt{c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(c x)\right )}{g \sqrt{1+c^2 x^2}}-\frac{\left (2 b d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g+2 \sqrt{c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(c x)\right )}{g \sqrt{1+c^2 x^2}}\\ &=\frac{a d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}{g^3}-\frac{b c d x \sqrt{d+c^2 d x^2}}{3 g \sqrt{1+c^2 x^2}}-\frac{b c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2}}{g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d+c^2 d x^2}}{4 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d+c^2 d x^2}}{9 g \sqrt{1+c^2 x^2}}+\frac{b d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g^3}-\frac{c^2 d f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{1+c^2 x^2}}-\frac{d \left (1+\frac{c^2 f^2}{g^2}\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{1+c^2 x^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}+\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}-\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}-\frac{\left (b d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x g}{2 c f-2 \sqrt{c^2 f^2+g^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x g}{2 c f+2 \sqrt{c^2 f^2+g^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}{g^3}-\frac{b c d x \sqrt{d+c^2 d x^2}}{3 g \sqrt{1+c^2 x^2}}-\frac{b c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2}}{g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d+c^2 d x^2}}{4 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d+c^2 d x^2}}{9 g \sqrt{1+c^2 x^2}}+\frac{b d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g^3}-\frac{c^2 d f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{1+c^2 x^2}}-\frac{d \left (1+\frac{c^2 f^2}{g^2}\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{1+c^2 x^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}+\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}-\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}-\frac{\left (b d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f-2 \sqrt{c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f+2 \sqrt{c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}{g^3}-\frac{b c d x \sqrt{d+c^2 d x^2}}{3 g \sqrt{1+c^2 x^2}}-\frac{b c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2}}{g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d+c^2 d x^2}}{4 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d+c^2 d x^2}}{9 g \sqrt{1+c^2 x^2}}+\frac{b d \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g^3}-\frac{c^2 d f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d \left (c^2 f^2+g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{1+c^2 x^2}}-\frac{d \left (1+\frac{c^2 f^2}{g^2}\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{d \left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{1+c^2 x^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}+\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}-\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}+\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \text{Li}_2\left (-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}-\frac{b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2} \text{Li}_2\left (-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{g^4 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 13.5387, size = 2889, normalized size = 2.94 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[d*(1 + c^2*x^2)]*((a*d*(3*c^2*f^2 + 4*g^2))/(3*g^3) - (a*c^2*d*f*x)/(2*g^2) + (a*c^2*d*x^2)/(3*g)) + (a*d
^(3/2)*(c^2*f^2 + g^2)^(3/2)*Log[f + g*x])/g^4 - (a*c*d^(3/2)*f*(2*c^2*f^2 + 3*g^2)*Log[c*d*x + Sqrt[d]*Sqrt[d
*(1 + c^2*x^2)]])/(2*g^4) - (a*d^(3/2)*(c^2*f^2 + g^2)^(3/2)*Log[d*g - c^2*d*f*x + Sqrt[d]*Sqrt[c^2*f^2 + g^2]
*Sqrt[d*(1 + c^2*x^2)]])/g^4 + (b*d*Sqrt[d*(1 + c^2*x^2)]*((-2*c*g*x)/Sqrt[1 + c^2*x^2] + 2*g*ArcSinh[c*x] - (
c*f*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] + (2*(c^2*f^2 + g^2)*(((-I)*Pi*ArcTanh[(-g + c*f*Tanh[ArcSinh[c*x]/2])/S
qrt[c^2*f^2 + g^2]])/Sqrt[c^2*f^2 + g^2] - (2*ArcCos[((-I)*c*f)/g]*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSin
h[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (Pi - (2*I)*ArcSinh[c*x])*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c
*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[
c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] - (2*I)*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^
2) - g^2]])*Log[((1/2 - I/2)*Sqrt[-(c^2*f^2) - g^2])/(E^(ArcSinh[c*x]/2)*Sqrt[(-I)*g]*Sqrt[c*f + c*g*x])] + (A
rcCos[((-I)*c*f)/g] + (2*I)*(ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] +
ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]]))*Log[((1/2 + I/2)*E^(ArcSinh[c
*x]/2)*Sqrt[-(c^2*f^2) - g^2])/(Sqrt[(-I)*g]*Sqrt[c*f + c*g*x])] - (ArcCos[((-I)*c*f)/g] + (2*I)*ArcTanh[((c*f
+ I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*((-I)*c*f + g + Sqrt[-(c^2
*f^2) - g^2])*(1 + I*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*
I)*ArcSinh[c*x])/4]))] - (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/
Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*(I*c*f - g + Sqrt[-(c^2*f^2) - g^2])*(I + Cot[(Pi + (2*I)*ArcSinh[c*
x])/4]))/(g*(c*f - I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] + I*(PolyLog[2, ((I*c*f +
Sqrt[-(c^2*f^2) - g^2])*(I*c*f + g - I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g
+ I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) - g^2])
*(-(c*f) + I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) -
g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))]))/Sqrt[-(c^2*f^2) - g^2]))/Sqrt[1 + c^2*x^2]))/(2*g^2) + (b*d*Sqrt[d
*(1 + c^2*x^2)]*((-9*(ArcSinh[c*x]*(Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])] - Log[1 + (E^ArcSi
nh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])]) + PolyLog[2, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])] - Pol
yLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))]))/Sqrt[c^2*f^2 + g^2] + (-18*c*g*(4*c^2*f^2 + g^2)*
x + 18*g*(4*c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 18*c*f*(2*c^2*f^2 + g^2)*ArcSinh[c*x]^2 + 9*c*f*g^
2*Cosh[2*ArcSinh[c*x]] + 6*g^3*ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]] + 9*(8*c^4*f^4 + 8*c^2*f^2*g^2 + g^4)*(((-I)*
Pi*ArcTanh[(-g + c*f*Tanh[ArcSinh[c*x]/2])/Sqrt[c^2*f^2 + g^2]])/Sqrt[c^2*f^2 + g^2] - (2*ArcCos[((-I)*c*f)/g]
*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (Pi - (2*I)*ArcSinh[c*x])*Ar
cTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (ArcCos[((-I)*c*f)/g] - (2*I)*A
rcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] - (2*I)*ArcTanh[((c*f - I*g)*Tan
[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((1/2 - I/2)*Sqrt[-(c^2*f^2) - g^2])/(E^(ArcSinh[c
*x]/2)*Sqrt[(-I)*g]*Sqrt[c*f + c*g*x])] + (ArcCos[((-I)*c*f)/g] + (2*I)*(ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*
ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*
f^2) - g^2]]))*Log[((1/2 + I/2)*E^(ArcSinh[c*x]/2)*Sqrt[-(c^2*f^2) - g^2])/(Sqrt[(-I)*g]*Sqrt[c*f + c*g*x])] -
(ArcCos[((-I)*c*f)/g] + (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])
*Log[((I*c*f + g)*((-I)*c*f + g + Sqrt[-(c^2*f^2) - g^2])*(1 + I*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f
+ g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] - (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c
*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*(I*c*f - g + Sqrt[-(c^2*
f^2) - g^2])*(I + Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(c*f - I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*Ar
cSinh[c*x])/4]))] + I*(PolyLog[2, ((I*c*f + Sqrt[-(c^2*f^2) - g^2])*(I*c*f + g - I*Sqrt[-(c^2*f^2) - g^2]*Cot[
(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] -
PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) - g^2])*(-(c*f) + I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*
x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))]))/Sqrt[-(c^2*f^2) - g^2]
) - 18*c*f*g^2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]] - 2*g^3*Sinh[3*ArcSinh[c*x]])/g^4))/(72*Sqrt[1 + c^2*x^2])

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Maple [A]  time = 0.22, size = 1838, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a/g^2*c^2*d*f*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2)*x-4/3*b*(d*(c^2*x^2+1))^(1/
2)*d/(c^2*x^2+1)^(1/2)/g*c*x+1/8*b*(d*(c^2*x^2+1))^(1/2)*f*c*d/(c^2*x^2+1)^(1/2)/g^2-1/9*b*(d*(c^2*x^2+1))^(1/
2)*d/(c^2*x^2+1)^(1/2)/g*c^3*x^3+b*(c^2*f^2+g^2)^(3/2)*d*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/g^4*dilog((-(
c*x+(c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))-b*(c^2*f^2+g^2)^(3/2)*d*(d*(c^2*
x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/g^4*dilog(((c*x+(c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2+g^2)^(1/2))/(c*f+(c^2*f^2+g^
2)^(1/2)))-1/2*b*(d*(c^2*x^2+1))^(1/2)*f*c^4*d/(c^2*x^2+1)/g^2*arcsinh(c*x)*x^3-1/2*b*(d*(c^2*x^2+1))^(1/2)*f*
c^2*d/(c^2*x^2+1)/g^2*arcsinh(c*x)*x+b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)/g^3*arcsinh(c*x)*x^2*c^4*f^2+a/g*d*
((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2)+1/3*a/g*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d
*(c^2*f^2+g^2)/g^2)^(3/2)+a/g^3*d*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2)*c^2*f^2+4/3*
b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)/g*arcsinh(c*x)+1/4*b*(d*(c^2*x^2+1))^(1/2)*f*c^3*d/(c^2*x^2+1)^(1/2)/g^2
*x^2-b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)^(1/2)/g^3*x*c^3*f^2+b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)/g^3*arcsi
nh(c*x)*c^2*f^2+b*(c^2*f^2+g^2)^(3/2)*d*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/g^4*arcsinh(c*x)*ln((-(c*x+(c^
2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))-b*(c^2*f^2+g^2)^(3/2)*d*(d*(c^2*x^2+1))
^(1/2)/(c^2*x^2+1)^(1/2)/g^4*arcsinh(c*x)*ln(((c*x+(c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2+g^2)^(1/2))/(c*f+(c^2*f^2
+g^2)^(1/2)))-1/2*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*f^3*arcsinh(c*x)^2*c^3*d/g^4-3/4*b*(d*(c^2*x^2+1))
^(1/2)/(c^2*x^2+1)^(1/2)*f*arcsinh(c*x)^2*c*d/g^2+1/3*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)/g*arcsinh(c*x)*x^4
*c^4+5/3*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)/g*arcsinh(c*x)*x^2*c^2-a/g*d^2/(d*(c^2*f^2+g^2)/g^2)^(1/2)*ln((
2*d*(c^2*f^2+g^2)/g^2-2*c^2*d*f/g*(x+f/g)+2*(d*(c^2*f^2+g^2)/g^2)^(1/2)*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d
*(c^2*f^2+g^2)/g^2)^(1/2))/(x+f/g))-3/2*a/g^2*c^2*d^2*f*ln((-c^2*d*f/g+c^2*d*(x+f/g))/(c^2*d)^(1/2)+((x+f/g)^2
*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(c^2*d)^(1/2)-a/g^4*d^2*c^4*f^3*ln((-c^2*d*f/g+c^2*d*(x
+f/g))/(c^2*d)^(1/2)+((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(c^2*d)^(1/2)-a/g^5*d^2/
(d*(c^2*f^2+g^2)/g^2)^(1/2)*ln((2*d*(c^2*f^2+g^2)/g^2-2*c^2*d*f/g*(x+f/g)+2*(d*(c^2*f^2+g^2)/g^2)^(1/2)*((x+f/
g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(x+f/g))*c^4*f^4-2*a/g^3*d^2/(d*(c^2*f^2+g^2)/g^2)^
(1/2)*ln((2*d*(c^2*f^2+g^2)/g^2-2*c^2*d*f/g*(x+f/g)+2*(d*(c^2*f^2+g^2)/g^2)^(1/2)*((x+f/g)^2*c^2*d-2*c^2*d*f/g
*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(x+f/g))*c^2*f^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((c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))/(g*x+f),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 (\frac{{\left (a c^{2} d x^{2} + a d +{\left (b c^{2} d x^{2} + b d\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d}}{g x + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )}{f + g x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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