\(\int \frac {(d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx\) [280]

Optimal result

Integrand size = 28, antiderivative size = 687 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\frac {40}{9} b^2 c^2 d^2 \sqrt {d+c^2 d x^2}-\frac {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {2}{27} b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {5 b^2 c^3 d^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \]

[Out]

5/6*c^2*d*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2-1/2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^2+40/9*b^2*c
^2*d^2*(c^2*d*x^2+d)^(1/2)+2/27*b^2*c^2*d^2*(c^2*x^2+1)*(c^2*d*x^2+d)^(1/2)+5/2*c^2*d^2*(a+b*arcsinh(c*x))^2*(
c^2*d*x^2+d)^(1/2)-5*a*b*c^3*d^2*x*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-5*b^2*c^3*d^2*x*arcsinh(c*x)*(c^2*d*x
^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-b*c*d^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/x/(c^2*x^2+1)^(1/2)+1/3*b*c^3*d^2
*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-2/9*b*c^5*d^2*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)
^(1/2)/(c^2*x^2+1)^(1/2)-5*c^2*d^2*(a+b*arcsinh(c*x))^2*arctanh(c*x+(c^2*x^2+1)^(1/2))*(c^2*d*x^2+d)^(1/2)/(c^
2*x^2+1)^(1/2)-b^2*c^2*d^2*arctanh((c^2*x^2+1)^(1/2))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-5*b*c^2*d^2*(a+b*a
rcsinh(c*x))*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+5*b*c^2*d^2*(a+b*arcsinh(
c*x))*polylog(2,c*x+(c^2*x^2+1)^(1/2))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+5*b^2*c^2*d^2*polylog(3,-c*x-(c^2
*x^2+1)^(1/2))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-5*b^2*c^2*d^2*polylog(3,c*x+(c^2*x^2+1)^(1/2))*(c^2*d*x^2
+d)^(1/2)/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5807, 5808, 5806, 5816, 4267, 2611, 2320, 6724, 5772, 267, 5784, 455, 45, 276, 5803, 12, 1265, 911, 1167, 214} \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=-\frac {5 c^2 d^2 \sqrt {c^2 d x^2+d} \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {5 b c^2 d^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}+\frac {5 b c^2 d^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}+\frac {5}{2} c^2 d^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{x \sqrt {c^2 x^2+1}}+\frac {5}{6} c^2 d \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {2 b c^5 d^2 x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{9 \sqrt {c^2 x^2+1}}+\frac {b c^3 d^2 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {5 a b c^3 d^2 x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}+\frac {5 b^2 c^2 d^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {5 b^2 c^2 d^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {5 b^2 c^3 d^2 x \text {arcsinh}(c x) \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}-\frac {b^2 c^2 d^2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right ) \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}+\frac {40}{9} b^2 c^2 d^2 \sqrt {c^2 d x^2+d}+\frac {2}{27} b^2 c^2 d^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \]

[In]

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

[Out]

(40*b^2*c^2*d^2*Sqrt[d + c^2*d*x^2])/9 - (5*a*b*c^3*d^2*x*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (2*b^2*c^2*
d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/27 - (5*b^2*c^3*d^2*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/Sqrt[1 + c^2*x^
2] - (b*c*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(x*Sqrt[1 + c^2*x^2]) + (b*c^3*d^2*x*Sqrt[d + c^2*d*x^
2]*(a + b*ArcSinh[c*x]))/(3*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(9
*Sqrt[1 + c^2*x^2]) + (5*c^2*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (5*c^2*d*(d + c^2*d*x^2)^(3/2
)*(a + b*ArcSinh[c*x])^2)/6 - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(2*x^2) - (5*c^2*d^2*Sqrt[d + c^2
*d*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (b^2*c^2*d^2*Sqrt[d + c^2*d*x^2]*A
rcTanh[Sqrt[1 + c^2*x^2]])/Sqrt[1 + c^2*x^2] - (5*b*c^2*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2
, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (5*b*c^2*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^Arc
Sinh[c*x]])/Sqrt[1 + c^2*x^2] + (5*b^2*c^2*d^2*Sqrt[d + c^2*d*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x
^2] - (5*b^2*c^2*d^2*Sqrt[d + c^2*d*x^2]*PolyLog[3, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2]

Rule 12

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 214

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

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 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1167

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

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5772

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

Rule 5784

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

Rule 5803

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

Rule 5806

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[(1/(m + 2))*Simp[Sqrt[d + e*x^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/(f*(m + 2)))
*Simp[Sqrt[d + e*x^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] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])

Rule 5807

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 + 1))), x] + (-Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x
)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1
+ c^2*x^2)^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}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rule 5808

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/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^
p/(1 + c^2*x^2)^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]

Rule 5816

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

Rule 6724

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {1}{2} \left (5 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x} \, dx+\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^2} \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x \sqrt {1+c^2 x^2}}+\frac {2 b c^3 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {1}{2} \left (5 c^2 d^2\right ) \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x} \, dx-\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {-3+6 c^2 x^2+c^4 x^4}{3 x \sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (5 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x)) \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {\left (5 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {-3+6 c^2 x^2+c^4 x^4}{x \sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x \left (1+\frac {c^2 x^2}{3}\right )}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {\left (5 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \text {csch}(x) \, dx,x,\text {arcsinh}(c x)\right )}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {-3+6 c^2 x+c^4 x^2}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \text {arcsinh}(c x) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {c^2 x}{3}}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {1+c^2 x^2}} \\ & = -\frac {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {5 b^2 c^3 d^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {-8+4 x^2+x^4}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (5 b c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1+c^2 x}}+\frac {1}{3} \sqrt {1+c^2 x}\right ) \, dx,x,x^2\right )}{6 \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {55}{9} b^2 c^2 d^2 \sqrt {d+c^2 d x^2}-\frac {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {5}{27} b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {5 b^2 c^3 d^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (5 c^2+c^2 x^2-\frac {3}{-\frac {1}{c^2}+\frac {x^2}{c^2}}\right ) \, dx,x,\sqrt {1+c^2 x^2}\right )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}} \\ & = \frac {40}{9} b^2 c^2 d^2 \sqrt {d+c^2 d x^2}-\frac {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {2}{27} b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {5 b^2 c^3 d^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \\ & = \frac {40}{9} b^2 c^2 d^2 \sqrt {d+c^2 d x^2}-\frac {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {2}{27} b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {5 b^2 c^3 d^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.04 (sec) , antiderivative size = 990, normalized size of antiderivative = 1.44 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\sqrt {d \left (1+c^2 x^2\right )} \left (\frac {7}{3} a^2 c^2 d^2-\frac {a^2 d^2}{2 x^2}+\frac {1}{3} a^2 c^4 d^2 x^2\right )+2 a b c^2 d^2 \left (-\frac {c x \sqrt {d \left (1+c^2 x^2\right )} \left (3+c^2 x^2\right )}{9 \sqrt {1+c^2 x^2}}+\frac {1}{3} \left (1+c^2 x^2\right ) \sqrt {d \left (1+c^2 x^2\right )} \text {arcsinh}(c x)\right )+\frac {5}{2} a^2 c^2 d^{5/2} \log (x)-\frac {5}{2} a^2 c^2 d^{5/2} \log \left (d+\sqrt {d} \sqrt {d \left (1+c^2 x^2\right )}\right )+\frac {4 a b c^2 d^2 \sqrt {d \left (1+c^2 x^2\right )} \left (-c x+\sqrt {1+c^2 x^2} \text {arcsinh}(c x)+\text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {1+c^2 x^2}}+2 b^2 c^2 d^2 \sqrt {d \left (1+c^2 x^2\right )} \left (2-\frac {2 c x \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}+\text {arcsinh}(c x)^2+\frac {\text {arcsinh}(c x)^2 \left (\log \left (1-e^{-\text {arcsinh}(c x)}\right )-\log \left (1+e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {1+c^2 x^2}}+\frac {2 \text {arcsinh}(c x) \left (\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {1+c^2 x^2}}+\frac {2 \left (\operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {1+c^2 x^2}}\right )+\frac {b^2 c^2 d^2 \sqrt {d \left (1+c^2 x^2\right )} \left (27 \sqrt {1+c^2 x^2} \left (2+\text {arcsinh}(c x)^2\right )+\left (2+9 \text {arcsinh}(c x)^2\right ) \cosh (3 \text {arcsinh}(c x))-6 \text {arcsinh}(c x) (9 c x+\sinh (3 \text {arcsinh}(c x)))\right )}{108 \sqrt {1+c^2 x^2}}+\frac {a b c^2 d^2 \sqrt {d \left (1+c^2 x^2\right )} \left (-2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-4 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{4 \sqrt {1+c^2 x^2}}+\frac {b^2 c^2 d^2 \sqrt {d \left (1+c^2 x^2\right )} \left (-4 \text {arcsinh}(c x) \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\text {arcsinh}(c x)^2 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x)^2 \log \left (1-e^{-\text {arcsinh}(c x)}\right )-4 \text {arcsinh}(c x)^2 \log \left (1+e^{-\text {arcsinh}(c x)}\right )+8 \log \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+8 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-8 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+8 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c x)}\right )-8 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x)^2 \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x) \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{8 \sqrt {1+c^2 x^2}} \]

[In]

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

[Out]

Sqrt[d*(1 + c^2*x^2)]*((7*a^2*c^2*d^2)/3 - (a^2*d^2)/(2*x^2) + (a^2*c^4*d^2*x^2)/3) + 2*a*b*c^2*d^2*(-1/9*(c*x
*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/Sqrt[1 + c^2*x^2] + ((1 + c^2*x^2)*Sqrt[d*(1 + c^2*x^2)]*ArcSinh[c*x])/3
) + (5*a^2*c^2*d^(5/2)*Log[x])/2 - (5*a^2*c^2*d^(5/2)*Log[d + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/2 + (4*a*b*c^2*d
^2*Sqrt[d*(1 + c^2*x^2)]*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] -
ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqr
t[1 + c^2*x^2] + 2*b^2*c^2*d^2*Sqrt[d*(1 + c^2*x^2)]*(2 - (2*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ArcSinh[c*x
]^2 + (ArcSinh[c*x]^2*(Log[1 - E^(-ArcSinh[c*x])] - Log[1 + E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (2*ArcSin
h[c*x]*(PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (2*(PolyLog[3, -E
^(-ArcSinh[c*x])] - PolyLog[3, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2]) + (b^2*c^2*d^2*Sqrt[d*(1 + c^2*x^2)]*(2
7*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x]^2) + (2 + 9*ArcSinh[c*x]^2)*Cosh[3*ArcSinh[c*x]] - 6*ArcSinh[c*x]*(9*c*x
+ Sinh[3*ArcSinh[c*x]])))/(108*Sqrt[1 + c^2*x^2]) + (a*b*c^2*d^2*Sqrt[d*(1 + c^2*x^2)]*(-2*Coth[ArcSinh[c*x]/
2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 +
E^(-ArcSinh[c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[Ar
cSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/(4*Sqrt[1 + c^2*x^2]) + (b^2*c^2*d^2*Sqrt[d*(1 + c^2*x^2)]*(-4*ArcS
inh[c*x]*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c
*x])] - 4*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 8*Log[Tanh[ArcSinh[c*x]/2]] + 8*ArcSinh[c*x]*PolyLog[2,
-E^(-ArcSinh[c*x])] - 8*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] + 8*PolyLog[3, -E^(-ArcSinh[c*x])] - 8*Poly
Log[3, E^(-ArcSinh[c*x])] - ArcSinh[c*x]^2*Sech[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2]))/(8*S
qrt[1 + c^2*x^2])

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 720, normalized size of antiderivative = 1.05

method	result	size
default	\(a^{2} \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}+\frac {5 c^{2} \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5}+d \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )\right )\right )}{2}\right )+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (18 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-12 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}+4 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+126 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+135 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-135 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-252 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}-270 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+270 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+244 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-108 \,\operatorname {arctanh}\left (c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+270 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-270 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-27 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )^{2}-54 \,\operatorname {arcsinh}\left (c x \right ) c x \right ) d^{2}}{54 \sqrt {c^{2} x^{2}+1}\, x^{2}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-2 c^{5} x^{5}+42 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+45 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-45 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-42 c^{3} x^{3}+45 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-45 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-9 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-9 c x \right ) d^{2}}{9 \sqrt {c^{2} x^{2}+1}\, x^{2}}\)	\(720\)
parts	\(a^{2} \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}+\frac {5 c^{2} \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5}+d \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )\right )\right )}{2}\right )+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (18 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-12 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}+4 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+126 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+135 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-135 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-252 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}-270 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+270 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+244 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-108 \,\operatorname {arctanh}\left (c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+270 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-270 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-27 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )^{2}-54 \,\operatorname {arcsinh}\left (c x \right ) c x \right ) d^{2}}{54 \sqrt {c^{2} x^{2}+1}\, x^{2}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-2 c^{5} x^{5}+42 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+45 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-45 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-42 c^{3} x^{3}+45 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-45 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-9 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-9 c x \right ) d^{2}}{9 \sqrt {c^{2} x^{2}+1}\, x^{2}}\)	\(720\)

[In]

int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^3,x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/2/d/x^2*(c^2*d*x^2+d)^(7/2)+5/2*c^2*(1/5*(c^2*d*x^2+d)^(5/2)+d*(1/3*(c^2*d*x^2+d)^(3/2)+d*((c^2*d*x^2+
d)^(1/2)-d^(1/2)*ln((2*d+2*d^(1/2)*(c^2*d*x^2+d)^(1/2))/x)))))+1/54*b^2*(d*(c^2*x^2+1))^(1/2)*(18*(c^2*x^2+1)^
(1/2)*arcsinh(c*x)^2*x^4*c^4-12*arcsinh(c*x)*c^5*x^5+4*c^4*x^4*(c^2*x^2+1)^(1/2)+126*arcsinh(c*x)^2*(c^2*x^2+1
)^(1/2)*x^2*c^2+135*arcsinh(c*x)^2*ln(1-c*x-(c^2*x^2+1)^(1/2))*x^2*c^2-135*arcsinh(c*x)^2*ln(1+c*x+(c^2*x^2+1)
^(1/2))*x^2*c^2-252*arcsinh(c*x)*c^3*x^3-270*arcsinh(c*x)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*x^2*c^2+270*arcsin
h(c*x)*polylog(2,c*x+(c^2*x^2+1)^(1/2))*x^2*c^2+244*c^2*x^2*(c^2*x^2+1)^(1/2)-108*arctanh(c*x+(c^2*x^2+1)^(1/2
))*x^2*c^2+270*polylog(3,-c*x-(c^2*x^2+1)^(1/2))*x^2*c^2-270*polylog(3,c*x+(c^2*x^2+1)^(1/2))*x^2*c^2-27*(c^2*
x^2+1)^(1/2)*arcsinh(c*x)^2-54*arcsinh(c*x)*c*x)*d^2/(c^2*x^2+1)^(1/2)/x^2+1/9*a*b*(d*(c^2*x^2+1))^(1/2)*(6*ar
csinh(c*x)*(c^2*x^2+1)^(1/2)*x^4*c^4-2*c^5*x^5+42*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^2*c^2+45*arcsinh(c*x)*ln(1-
c*x-(c^2*x^2+1)^(1/2))*x^2*c^2-45*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*x^2*c^2-42*c^3*x^3+45*polylog(2,c*x
+(c^2*x^2+1)^(1/2))*x^2*c^2-45*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*x^2*c^2-9*arcsinh(c*x)*(c^2*x^2+1)^(1/2)-9*c*
x)*d^2/(c^2*x^2+1)^(1/2)/x^2

Fricas [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="fricas")

[Out]

integral((a^2*c^4*d^2*x^4 + 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 + 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcs
inh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 + 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d)/x^3, x)

Sympy [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]

[In]

integrate((c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**2/x**3,x)

[Out]

Integral((d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))**2/x**3, x)

Maxima [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="maxima")

[Out]

-1/6*(15*c^2*d^(5/2)*arcsinh(1/(c*abs(x))) - 3*(c^2*d*x^2 + d)^(5/2)*c^2 - 5*(c^2*d*x^2 + d)^(3/2)*c^2*d - 15*
sqrt(c^2*d*x^2 + d)*c^2*d^2 + 3*(c^2*d*x^2 + d)^(7/2)/(d*x^2))*a^2 + integrate((c^2*d*x^2 + d)^(5/2)*b^2*log(c
*x + sqrt(c^2*x^2 + 1))^2/x^3 + 2*(c^2*d*x^2 + d)^(5/2)*a*b*log(c*x + sqrt(c^2*x^2 + 1))/x^3, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^3,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. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^3} \,d x \]

[In]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2))/x^3,x)

[Out]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2))/x^3, x)