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

Optimal. Leaf size=515 \[ -\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt{c^2 d x^2+d}}+\frac{4 x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}-\frac{8 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac{16 a b x \sqrt{c^2 x^2+1}}{3 c^5 d \sqrt{c^2 d x^2+d}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{c^2 d x^2+d}}-\frac{2 b x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt{c^2 d x^2+d}}-\frac{2 b x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt{c^2 d x^2+d}}+\frac{4 b \sqrt{c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^6 d \sqrt{c^2 d x^2+d}}+\frac{2 b^2 \left (c^2 x^2+1\right )^2}{27 c^6 d \sqrt{c^2 d x^2+d}}-\frac{32 b^2 \left (c^2 x^2+1\right )}{9 c^6 d \sqrt{c^2 d x^2+d}}+\frac{16 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d \sqrt{c^2 d x^2+d}} \]

[Out]

(16*a*b*x*Sqrt[1 + c^2*x^2])/(3*c^5*d*Sqrt[d + c^2*d*x^2]) - (32*b^2*(1 + c^2*x^2))/(9*c^6*d*Sqrt[d + c^2*d*x^
2]) + (2*b^2*(1 + c^2*x^2)^2)/(27*c^6*d*Sqrt[d + c^2*d*x^2]) + (16*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^
5*d*Sqrt[d + c^2*d*x^2]) - (2*b*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(c^5*d*Sqrt[d + c^2*d*x^2]) - (2*b*x
^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c^3*d*Sqrt[d + c^2*d*x^2]) - (x^4*(a + b*ArcSinh[c*x])^2)/(c^2*d
*Sqrt[d + c^2*d*x^2]) - (8*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^6*d^2) + (4*x^2*Sqrt[d + c^2*d*x^2
]*(a + b*ArcSinh[c*x])^2)/(3*c^4*d^2) + (4*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c
^6*d*Sqrt[d + c^2*d*x^2]) - ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^6*d*Sqrt[d + c^2*
d*x^2]) + ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^6*d*Sqrt[d + c^2*d*x^2])

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Rubi [A]  time = 0.792561, antiderivative size = 515, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules used = {5751, 5758, 5717, 5653, 261, 5661, 266, 43, 5767, 5693, 4180, 2279, 2391} \[ -\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt{c^2 d x^2+d}}+\frac{4 x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}-\frac{8 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac{16 a b x \sqrt{c^2 x^2+1}}{3 c^5 d \sqrt{c^2 d x^2+d}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{c^2 d x^2+d}}-\frac{2 b x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt{c^2 d x^2+d}}-\frac{2 b x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt{c^2 d x^2+d}}+\frac{4 b \sqrt{c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^6 d \sqrt{c^2 d x^2+d}}+\frac{2 b^2 \left (c^2 x^2+1\right )^2}{27 c^6 d \sqrt{c^2 d x^2+d}}-\frac{32 b^2 \left (c^2 x^2+1\right )}{9 c^6 d \sqrt{c^2 d x^2+d}}+\frac{16 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*a*b*x*Sqrt[1 + c^2*x^2])/(3*c^5*d*Sqrt[d + c^2*d*x^2]) - (32*b^2*(1 + c^2*x^2))/(9*c^6*d*Sqrt[d + c^2*d*x^
2]) + (2*b^2*(1 + c^2*x^2)^2)/(27*c^6*d*Sqrt[d + c^2*d*x^2]) + (16*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^
5*d*Sqrt[d + c^2*d*x^2]) - (2*b*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(c^5*d*Sqrt[d + c^2*d*x^2]) - (2*b*x
^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c^3*d*Sqrt[d + c^2*d*x^2]) - (x^4*(a + b*ArcSinh[c*x])^2)/(c^2*d
*Sqrt[d + c^2*d*x^2]) - (8*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^6*d^2) + (4*x^2*Sqrt[d + c^2*d*x^2
]*(a + b*ArcSinh[c*x])^2)/(3*c^4*d^2) + (4*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c
^6*d*Sqrt[d + c^2*d*x^2]) - ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^6*d*Sqrt[d + c^2*
d*x^2]) + ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^6*d*Sqrt[d + c^2*d*x^2])

Rule 5751

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

Rule 5758

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

Rule 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 5653

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 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 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt
[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 266

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

Rule 43

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

Rule 5767

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

Rule 5693

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

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && 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{x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d+c^2 d x^2}}+\frac{4 \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{d+c^2 d x^2}} \, dx}{c^2 d}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c d \sqrt{d+c^2 d x^2}}\\ &=\frac{2 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d \sqrt{d+c^2 d x^2}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d+c^2 d x^2}}+\frac{4 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}-\frac{8 \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{d+c^2 d x^2}} \, dx}{3 c^4 d}-\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^3 d \sqrt{d+c^2 d x^2}}-\frac{\left (8 b \sqrt{1+c^2 x^2}\right ) \int x^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 c^3 d \sqrt{d+c^2 d x^2}}-\frac{\left (2 b^2 \sqrt{1+c^2 x^2}\right ) \int \frac{x^3}{\sqrt{1+c^2 x^2}} \, dx}{3 c^2 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{2 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt{d+c^2 d x^2}}-\frac{2 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt{d+c^2 d x^2}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d+c^2 d x^2}}-\frac{8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac{4 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^5 d \sqrt{d+c^2 d x^2}}+\frac{\left (16 b \sqrt{1+c^2 x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 c^5 d \sqrt{d+c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{1+c^2 x^2}\right ) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{c^4 d \sqrt{d+c^2 d x^2}}-\frac{\left (b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{3 c^2 d \sqrt{d+c^2 d x^2}}+\frac{\left (8 b^2 \sqrt{1+c^2 x^2}\right ) \int \frac{x^3}{\sqrt{1+c^2 x^2}} \, dx}{9 c^2 d \sqrt{d+c^2 d x^2}}\\ &=\frac{16 a b x \sqrt{1+c^2 x^2}}{3 c^5 d \sqrt{d+c^2 d x^2}}+\frac{2 b^2 \left (1+c^2 x^2\right )}{c^6 d \sqrt{d+c^2 d x^2}}-\frac{2 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt{d+c^2 d x^2}}-\frac{2 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt{d+c^2 d x^2}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d+c^2 d x^2}}-\frac{8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac{4 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^6 d \sqrt{d+c^2 d x^2}}+\frac{\left (16 b^2 \sqrt{1+c^2 x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{3 c^5 d \sqrt{d+c^2 d x^2}}-\frac{\left (b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2 \sqrt{1+c^2 x}}+\frac{\sqrt{1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{3 c^2 d \sqrt{d+c^2 d x^2}}+\frac{\left (4 b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{9 c^2 d \sqrt{d+c^2 d x^2}}\\ &=\frac{16 a b x \sqrt{1+c^2 x^2}}{3 c^5 d \sqrt{d+c^2 d x^2}}+\frac{8 b^2 \left (1+c^2 x^2\right )}{3 c^6 d \sqrt{d+c^2 d x^2}}-\frac{2 b^2 \left (1+c^2 x^2\right )^2}{9 c^6 d \sqrt{d+c^2 d x^2}}+\frac{16 b^2 x \sqrt{1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d \sqrt{d+c^2 d x^2}}-\frac{2 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt{d+c^2 d x^2}}-\frac{2 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt{d+c^2 d x^2}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d+c^2 d x^2}}-\frac{8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac{4 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac{4 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt{d+c^2 d x^2}}-\frac{\left (2 i b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^6 d \sqrt{d+c^2 d x^2}}+\frac{\left (2 i b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^6 d \sqrt{d+c^2 d x^2}}-\frac{\left (16 b^2 \sqrt{1+c^2 x^2}\right ) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{3 c^4 d \sqrt{d+c^2 d x^2}}+\frac{\left (4 b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2 \sqrt{1+c^2 x}}+\frac{\sqrt{1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{9 c^2 d \sqrt{d+c^2 d x^2}}\\ &=\frac{16 a b x \sqrt{1+c^2 x^2}}{3 c^5 d \sqrt{d+c^2 d x^2}}-\frac{32 b^2 \left (1+c^2 x^2\right )}{9 c^6 d \sqrt{d+c^2 d x^2}}+\frac{2 b^2 \left (1+c^2 x^2\right )^2}{27 c^6 d \sqrt{d+c^2 d x^2}}+\frac{16 b^2 x \sqrt{1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d \sqrt{d+c^2 d x^2}}-\frac{2 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt{d+c^2 d x^2}}-\frac{2 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt{d+c^2 d x^2}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d+c^2 d x^2}}-\frac{8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac{4 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac{4 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt{d+c^2 d x^2}}-\frac{\left (2 i b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt{d+c^2 d x^2}}+\frac{\left (2 i b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt{d+c^2 d x^2}}\\ &=\frac{16 a b x \sqrt{1+c^2 x^2}}{3 c^5 d \sqrt{d+c^2 d x^2}}-\frac{32 b^2 \left (1+c^2 x^2\right )}{9 c^6 d \sqrt{d+c^2 d x^2}}+\frac{2 b^2 \left (1+c^2 x^2\right )^2}{27 c^6 d \sqrt{d+c^2 d x^2}}+\frac{16 b^2 x \sqrt{1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d \sqrt{d+c^2 d x^2}}-\frac{2 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt{d+c^2 d x^2}}-\frac{2 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt{d+c^2 d x^2}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d+c^2 d x^2}}-\frac{8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac{4 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac{4 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt{d+c^2 d x^2}}-\frac{2 i b^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt{d+c^2 d x^2}}+\frac{2 i b^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt{d+c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.605713, size = 427, normalized size = 0.83 \[ \frac{-54 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )+54 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )+9 a^2 c^4 x^4-36 a^2 c^2 x^2-72 a^2-6 a b c^3 x^3 \sqrt{c^2 x^2+1}+90 a b c x \sqrt{c^2 x^2+1}+18 a b c^4 x^4 \sinh ^{-1}(c x)-72 a b c^2 x^2 \sinh ^{-1}(c x)+108 a b \sqrt{c^2 x^2+1} \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )-144 a b \sinh ^{-1}(c x)+2 b^2 c^4 x^4-92 b^2 c^2 x^2+9 b^2 c^4 x^4 \sinh ^{-1}(c x)^2-6 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-36 b^2 c^2 x^2 \sinh ^{-1}(c x)^2+90 b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-54 i b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )+54 i b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )-72 b^2 \sinh ^{-1}(c x)^2-94 b^2}{27 c^6 d \sqrt{c^2 d x^2+d}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-72*a^2 - 94*b^2 - 36*a^2*c^2*x^2 - 92*b^2*c^2*x^2 + 9*a^2*c^4*x^4 + 2*b^2*c^4*x^4 + 90*a*b*c*x*Sqrt[1 + c^2*
x^2] - 6*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 144*a*b*ArcSinh[c*x] - 72*a*b*c^2*x^2*ArcSinh[c*x] + 18*a*b*c^4*x^4*A
rcSinh[c*x] + 90*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 6*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 72*b^
2*ArcSinh[c*x]^2 - 36*b^2*c^2*x^2*ArcSinh[c*x]^2 + 9*b^2*c^4*x^4*ArcSinh[c*x]^2 + 108*a*b*Sqrt[1 + c^2*x^2]*Ar
cTan[Tanh[ArcSinh[c*x]/2]] - (54*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + (54*I)*b^2*
Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] - (54*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcS
inh[c*x]] + (54*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I/E^ArcSinh[c*x]])/(27*c^6*d*Sqrt[d + c^2*d*x^2])

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Maple [A]  time = 0.369, size = 933, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*x^4/c^2/d/(c^2*d*x^2+d)^(1/2)-4/3*a^2/c^4*x^2/d/(c^2*d*x^2+d)^(1/2)-8/3*a^2/c^6/d/(c^2*d*x^2+d)^(1/2)-
94/27*b^2*(d*(c^2*x^2+1))^(1/2)/c^6/d^2/(c^2*x^2+1)+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/c^2/d^2/(c^2*x^2+1)*arcsinh(
c*x)^2*x^4-4/3*b^2*(d*(c^2*x^2+1))^(1/2)/c^4/d^2/(c^2*x^2+1)*arcsinh(c*x)^2*x^2+2*I*b^2*(d*(c^2*x^2+1))^(1/2)/
(c^2*x^2+1)^(1/2)/c^6/d^2*arcsinh(c*x)*ln(1-I*(c*x+(c^2*x^2+1)^(1/2)))-2*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+
1)^(1/2)/c^6/d^2*dilog(1+I*(c*x+(c^2*x^2+1)^(1/2)))+2/27*b^2*(d*(c^2*x^2+1))^(1/2)/c^2/d^2/(c^2*x^2+1)*x^4-92/
27*b^2*(d*(c^2*x^2+1))^(1/2)/c^4/d^2/(c^2*x^2+1)*x^2+2*I*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^2*l
n(c*x+(c^2*x^2+1)^(1/2)+I)-2/9*b^2*(d*(c^2*x^2+1))^(1/2)/c^3/d^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*x^3+10/3*b^2*(
d*(c^2*x^2+1))^(1/2)/c^5/d^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*x+2*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/
c^6/d^2*dilog(1-I*(c*x+(c^2*x^2+1)^(1/2)))-8/3*b^2*(d*(c^2*x^2+1))^(1/2)/c^6/d^2/(c^2*x^2+1)*arcsinh(c*x)^2+2/
3*a*b*(d*(c^2*x^2+1))^(1/2)/c^2/d^2/(c^2*x^2+1)*arcsinh(c*x)*x^4-2/9*a*b*(d*(c^2*x^2+1))^(1/2)/c^3/d^2/(c^2*x^
2+1)^(1/2)*x^3-8/3*a*b*(d*(c^2*x^2+1))^(1/2)/c^4/d^2/(c^2*x^2+1)*arcsinh(c*x)*x^2+10/3*a*b*(d*(c^2*x^2+1))^(1/
2)/c^5/d^2/(c^2*x^2+1)^(1/2)*x-16/3*a*b*(d*(c^2*x^2+1))^(1/2)/c^6/d^2/(c^2*x^2+1)*arcsinh(c*x)-2*I*a*b*(d*(c^2
*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^2*ln(c*x+(c^2*x^2+1)^(1/2)-I)-2*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)
^(1/2)/c^6/d^2*arcsinh(c*x)*ln(1+I*(c*x+(c^2*x^2+1)^(1/2)))

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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(x^5*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(3/2),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 (b^{2} x^{5} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b x^{5} \operatorname{arsinh}\left (c x\right ) + a^{2} x^{5}\right )} \sqrt{c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*x^5*arcsinh(c*x)^2 + 2*a*b*x^5*arcsinh(c*x) + a^2*x^5)*sqrt(c^2*d*x^2 + d)/(c^4*d^2*x^4 + 2*c^2*
d^2*x^2 + d^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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