\(\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx\) [37]

Optimal result

Integrand size = 30, antiderivative size = 664 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\frac {a \sqrt {d+c^2 d x^2}}{g}-\frac {b c x \sqrt {d+c^2 d x^2}}{g \sqrt {1+c^2 x^2}}+\frac {b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g}-\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {a \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}-\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}-\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}} \]

[Out]

a*(c^2*d*x^2+d)^(1/2)/g+b*arcsinh(c*x)*(c^2*d*x^2+d)^(1/2)/g-b*c*x*(c^2*d*x^2+d)^(1/2)/g/(c^2*x^2+1)^(1/2)-1/2
*c*x*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/g/(c^2*x^2+1)^(1/2)-1/2*(1+c^2*f^2/g^2)*(a+b*arcsinh(c*x))^2*(
c^2*d*x^2+d)^(1/2)/b/c/(g*x+f)/(c^2*x^2+1)^(1/2)-a*arctanh((-c^2*f*x+g)/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2))
*(c^2*f^2+g^2)^(1/2)*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*x^2+1)^(1/2)+b*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(
c*f-(c^2*f^2+g^2)^(1/2)))*(c^2*f^2+g^2)^(1/2)*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*x^2+1)^(1/2)-b*arcsinh(c*x)*ln(1+(c
*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*f^2+g^2)^(1/2)*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*x^2+1)^(1/
2)+b*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2)))*(c^2*f^2+g^2)^(1/2)*(c^2*d*x^2+d)^(1/2)/g
^2/(c^2*x^2+1)^(1/2)-b*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*f^2+g^2)^(1/2)*(c^
2*d*x^2+d)^(1/2)/g^2/(c^2*x^2+1)^(1/2)+1/2*(a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)/b/c/(g*x
+f)

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5845, 5839, 697, 5835, 6874, 267, 739, 212, 5856, 1668, 12, 5855, 5798, 8, 5843, 3403, 2296, 2221, 2317, 2438} \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=-\frac {\sqrt {c^2 d x^2+d} \left (\frac {c^2 f^2}{g^2}+1\right ) (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {c^2 x^2+1} (f+g x)}+\frac {\sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {c x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {c^2 x^2+1}}-\frac {a \sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1}}+\frac {a \sqrt {c^2 d x^2+d}}{g}+\frac {b \sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1}}-\frac {b \sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1}}+\frac {b \text {arcsinh}(c x) \sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2} \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g^2 \sqrt {c^2 x^2+1}}-\frac {b \text {arcsinh}(c x) \sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2} \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g^2 \sqrt {c^2 x^2+1}}+\frac {b \text {arcsinh}(c x) \sqrt {c^2 d x^2+d}}{g}-\frac {b c x \sqrt {c^2 d x^2+d}}{g \sqrt {c^2 x^2+1}} \]

[In]

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

[Out]

(a*Sqrt[d + c^2*d*x^2])/g - (b*c*x*Sqrt[d + c^2*d*x^2])/(g*Sqrt[1 + c^2*x^2]) + (b*Sqrt[d + c^2*d*x^2]*ArcSinh
[c*x])/g - (c*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*g*Sqrt[1 + c^2*x^2]) - ((1 + (c^2*f^2)/g^2)*S
qrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*(f + g*x)*Sqrt[1 + c^2*x^2]) + (Sqrt[1 + c^2*x^2]*Sqrt[d + c
^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*(f + g*x)) - (a*Sqrt[c^2*f^2 + g^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^2*Sqrt[1 + c^2*x^2]) + (b*Sqrt[c^2*f^2 + g^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^2*Sqrt[1 + c^2*x^2]) - (b*S
qrt[c^2*f^2 + g^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^2*Sqrt[1 + c^2*x^2]) + (b*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqr
t[c^2*f^2 + g^2]))])/(g^2*Sqrt[1 + c^2*x^2]) - (b*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcS
inh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(g^2*Sqrt[1 + c^2*x^2])

Rule 8

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

Rule 12

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

Rule 212

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

Rule 267

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

Rule 697

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 739

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 1668

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2317

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 2438

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]

Rule 3403

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 5798

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

Rule 5835

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 5839

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 5843

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 5845

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

Rule 5855

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 5856

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d+c^2 d x^2} \int \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {\sqrt {d+c^2 d x^2} \int \frac {\left (-g+2 c^2 f x+c^2 g x^2\right ) (a+b \text {arcsinh}(c x))^2}{(f+g x)^2} \, dx}{2 b c \sqrt {1+c^2 x^2}} \\ & = -\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}+\frac {\sqrt {d+c^2 d x^2} \int \frac {\left (\frac {c^2 x}{g}+\frac {1+\frac {c^2 f^2}{g^2}}{f+g x}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}+\frac {\sqrt {d+c^2 d x^2} \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 ) \text {arcsinh}(c x)}{g^2 (f+g x) \sqrt {1+c^2 x^2}}\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}+\frac {\left (a \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 \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 ) \text {arcsinh}(c x)}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a \sqrt {d+c^2 d x^2}}{g}-\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}+\frac {\left (a \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 \sqrt {d+c^2 d x^2}\right ) \int \left (\frac {c^2 g x \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}+\frac {\left (c^2 f^2+g^2\right ) \text {arcsinh}(c x)}{(f+g x) \sqrt {1+c^2 x^2}}\right ) \, dx}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a \sqrt {d+c^2 d x^2}}{g}-\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}+\frac {\left (b c^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{g \sqrt {1+c^2 x^2}}+\frac {\left (a \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 \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}\right ) \int \frac {\text {arcsinh}(c x)}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a \sqrt {d+c^2 d x^2}}{g}+\frac {b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g}-\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {\left (b c \sqrt {d+c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt {1+c^2 x^2}}-\frac {\left (a \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}\right ) \text {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 \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{c f+g \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right )}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a \sqrt {d+c^2 d x^2}}{g}-\frac {b c x \sqrt {d+c^2 d x^2}}{g \sqrt {1+c^2 x^2}}+\frac {b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g}-\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {a \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (2 b \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c e^x f-g+e^{2 x} g} \, dx,x,\text {arcsinh}(c x)\right )}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a \sqrt {d+c^2 d x^2}}{g}-\frac {b c x \sqrt {d+c^2 d x^2}}{g \sqrt {1+c^2 x^2}}+\frac {b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g}-\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {a \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (2 b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g-2 \sqrt {c^2 f^2+g^2}} \, dx,x,\text {arcsinh}(c x)\right )}{g \sqrt {1+c^2 x^2}}-\frac {\left (2 b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g+2 \sqrt {c^2 f^2+g^2}} \, dx,x,\text {arcsinh}(c x)\right )}{g \sqrt {1+c^2 x^2}} \\ & = \frac {a \sqrt {d+c^2 d x^2}}{g}-\frac {b c x \sqrt {d+c^2 d x^2}}{g \sqrt {1+c^2 x^2}}+\frac {b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g}-\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {a \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}-\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}-\frac {\left (b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right ) \text {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,\text {arcsinh}(c x)\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right ) \text {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,\text {arcsinh}(c x)\right )}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a \sqrt {d+c^2 d x^2}}{g}-\frac {b c x \sqrt {d+c^2 d x^2}}{g \sqrt {1+c^2 x^2}}+\frac {b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g}-\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {a \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}-\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}-\frac {\left (b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right ) \text {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^{\text {arcsinh}(c x)}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right ) \text {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^{\text {arcsinh}(c x)}\right )}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a \sqrt {d+c^2 d x^2}}{g}-\frac {b c x \sqrt {d+c^2 d x^2}}{g \sqrt {1+c^2 x^2}}+\frac {b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g}-\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {a \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}-\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}-\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 4.93 (sec) , antiderivative size = 1358, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\frac {2 a g \sqrt {d+c^2 d x^2}+2 a \sqrt {d} \sqrt {c^2 f^2+g^2} \log (f+g x)-2 a c \sqrt {d} f \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-2 a \sqrt {d} \sqrt {c^2 f^2+g^2} \log \left (d \left (g-c^2 f x\right )+\sqrt {d} \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right )+b \sqrt {d+c^2 d x^2} \left (-\frac {2 c g x}{\sqrt {1+c^2 x^2}}+2 g \text {arcsinh}(c x)-\frac {c f \text {arcsinh}(c x)^2}{\sqrt {1+c^2 x^2}}+\frac {2 (-i c f+g) (i c f+g) \left (-\frac {i \pi \text {arctanh}\left (\frac {-g+c f \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 f^2+g^2}}-\frac {2 \arccos \left (-\frac {i c f}{g}\right ) \text {arctanh}\left (\frac {(c f+i g) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )+(\pi -2 i \text {arcsinh}(c x)) \text {arctanh}\left (\frac {(c f-i g) \tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )+\left (\arccos \left (-\frac {i c f}{g}\right )-2 i \text {arctanh}\left (\frac {(c f+i g) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )-2 i \text {arctanh}\left (\frac {(c f-i g) \tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )\right ) \log \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) e^{-\frac {1}{2} \text {arcsinh}(c x)} \sqrt {-c^2 f^2-g^2}}{\sqrt {-i g} \sqrt {c (f+g x)}}\right )+\left (\arccos \left (-\frac {i c f}{g}\right )+2 i \left (\text {arctanh}\left (\frac {(c f+i g) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )+\text {arctanh}\left (\frac {(c f-i g) \tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )\right )\right ) \log \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) e^{\frac {1}{2} \text {arcsinh}(c x)} \sqrt {-c^2 f^2-g^2}}{\sqrt {-i g} \sqrt {c (f+g x)}}\right )-\left (\arccos \left (-\frac {i c f}{g}\right )+2 i \text {arctanh}\left (\frac {(c f+i g) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )\right ) \log \left (\frac {(i c f+g) \left (-i c f+g+\sqrt {-c^2 f^2-g^2}\right ) \left (1+i \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}{g \left (i c f+g+i \sqrt {-c^2 f^2-g^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}\right )-\left (\arccos \left (-\frac {i c f}{g}\right )-2 i \text {arctanh}\left (\frac {(c f+i g) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )\right ) \log \left (\frac {(i c f+g) \left (i c f-g+\sqrt {-c^2 f^2-g^2}\right ) \left (i+\cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}{g \left (c f-i g+\sqrt {-c^2 f^2-g^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (i c f+\sqrt {-c^2 f^2-g^2}\right ) \left (i c f+g-i \sqrt {-c^2 f^2-g^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}{g \left (i c f+g+i \sqrt {-c^2 f^2-g^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c f+i \sqrt {-c^2 f^2-g^2}\right ) \left (-c f+i g+\sqrt {-c^2 f^2-g^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}{g \left (i c f+g+i \sqrt {-c^2 f^2-g^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}\right )\right )}{\sqrt {-c^2 f^2-g^2}}\right )}{\sqrt {1+c^2 x^2}}\right )}{2 g^2} \]

[In]

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

[Out]

(2*a*g*Sqrt[d + c^2*d*x^2] + 2*a*Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Log[f + g*x] - 2*a*c*Sqrt[d]*f*Log[c*d*x + Sqrt[d
]*Sqrt[d + c^2*d*x^2]] - 2*a*Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Log[d*(g - c^2*f*x) + Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Sqr
t[d + c^2*d*x^2]] + b*Sqrt[d + c^2*d*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*((-I)*c*f + g)*(I*c*f + g)*(((-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])*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 + 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]] + ArcTa
nh[((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 + 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)*Ar
cSinh[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)

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 747, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {a \left (\sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}-\frac {c^{2} d f \ln \left (\frac {-\frac {c^{2} d f}{g}+c^{2} d \left (x +\frac {f}{g}\right )}{\sqrt {c^{2} d}}+\sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\right )}{g \sqrt {c^{2} d}}-\frac {d \left (c^{2} f^{2}+g^{2}\right ) \ln \left (\frac {\frac {2 d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}-\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g^{2} \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}\right )}{g}+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (c x \right )^{2} c}{2 \sqrt {c^{2} x^{2}+1}\, g^{2}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{2 \left (c^{2} x^{2}+1\right ) g}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{2 \left (c^{2} x^{2}+1\right ) g}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )+\operatorname {dilog}\left (\frac {-\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )\right )}{\sqrt {c^{2} x^{2}+1}\, g^{2}}\right )\)	\(747\)
parts	\(\frac {a \left (\sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}-\frac {c^{2} d f \ln \left (\frac {-\frac {c^{2} d f}{g}+c^{2} d \left (x +\frac {f}{g}\right )}{\sqrt {c^{2} d}}+\sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\right )}{g \sqrt {c^{2} d}}-\frac {d \left (c^{2} f^{2}+g^{2}\right ) \ln \left (\frac {\frac {2 d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}-\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g^{2} \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}\right )}{g}+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (c x \right )^{2} c}{2 \sqrt {c^{2} x^{2}+1}\, g^{2}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{2 \left (c^{2} x^{2}+1\right ) g}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{2 \left (c^{2} x^{2}+1\right ) g}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )+\operatorname {dilog}\left (\frac {-\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )\right )}{\sqrt {c^{2} x^{2}+1}\, g^{2}}\right )\)	\(747\)

[In]

int((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(g*x+f),x,method=_RETURNVERBOSE)

[Out]

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)^(1/2)-c^2*d*f/g*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)-d*(c^2*f^2+g^2)/g^
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)))+b*(-1/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+
1)^(1/2)*f*arcsinh(c*x)^2*c/g^2+1/2*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*(-1+arcsinh(c*x))/
(c^2*x^2+1)/g+1/2*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*x)+1)/(c^2*x^2+1)/g+(d*(c
^2*x^2+1))^(1/2)*(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)*(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)))-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)))+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)))-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))))/g^2)

Fricas [F]

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

[In]

integrate((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(g*x+f),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(g*x+f),x, algorithm="maxima")

[Out]

-(c*sqrt(d)*f*arcsinh(c*x)/g^2 - sqrt(c^2*d*f^2/g^2 + d)*arcsinh(c*f*x/abs(g*x + f) - g/(c*abs(g*x + f)))/g -
sqrt(c^2*d*x^2 + d)/g)*a + b*integrate(sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1))/(g*x + f), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(g*x+f),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d}}{f+g\,x} \,d x \]

[In]

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

[Out]

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