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

Optimal. Leaf size=512 \[ \frac{11 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{c^2 d x^2+d}}-\frac{11 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{c^2 d x^2+d}}-\frac{16 a b x \sqrt{c^2 x^2+1}}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}+\frac{11 b x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}+\frac{8 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac{22 b \sqrt{c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt{c^2 d x^2+d}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac{2 b^2 \left (c^2 x^2+1\right )}{c^6 d^2 \sqrt{c^2 d x^2+d}}+\frac{b^2}{3 c^6 d^2 \sqrt{c^2 d x^2+d}}-\frac{16 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt{c^2 d x^2+d}} \]

[Out]

b^2/(3*c^6*d^2*Sqrt[d + c^2*d*x^2]) - (16*a*b*x*Sqrt[1 + c^2*x^2])/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (2*b^2*(1
+ c^2*x^2))/(c^6*d^2*Sqrt[d + c^2*d*x^2]) - (16*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^5*d^2*Sqrt[d + c^2
*d*x^2]) - (b*x^3*(a + b*ArcSinh[c*x]))/(3*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) + (11*b*x*Sqrt[1 + c
^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) - (x^4*(a + b*ArcSinh[c*x])^2)/(3*c^2*d*(d + c^2
*d*x^2)^(3/2)) - (4*x^2*(a + b*ArcSinh[c*x])^2)/(3*c^4*d^2*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^3) - (22*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(3*c^6*d
^2*Sqrt[d + c^2*d*x^2]) + (((11*I)/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^6*d^2*Sqrt[d +
c^2*d*x^2]) - (((11*I)/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^6*d^2*Sqrt[d + c^2*d*x^2])

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Rubi [A]  time = 0.881223, antiderivative size = 512, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {5751, 5717, 5653, 261, 5767, 5693, 4180, 2279, 2391, 266, 43} \[ \frac{11 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{c^2 d x^2+d}}-\frac{11 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{c^2 d x^2+d}}-\frac{16 a b x \sqrt{c^2 x^2+1}}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}+\frac{11 b x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}+\frac{8 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac{22 b \sqrt{c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt{c^2 d x^2+d}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac{2 b^2 \left (c^2 x^2+1\right )}{c^6 d^2 \sqrt{c^2 d x^2+d}}+\frac{b^2}{3 c^6 d^2 \sqrt{c^2 d x^2+d}}-\frac{16 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

b^2/(3*c^6*d^2*Sqrt[d + c^2*d*x^2]) - (16*a*b*x*Sqrt[1 + c^2*x^2])/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (2*b^2*(1
+ c^2*x^2))/(c^6*d^2*Sqrt[d + c^2*d*x^2]) - (16*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^5*d^2*Sqrt[d + c^2
*d*x^2]) - (b*x^3*(a + b*ArcSinh[c*x]))/(3*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) + (11*b*x*Sqrt[1 + c
^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) - (x^4*(a + b*ArcSinh[c*x])^2)/(3*c^2*d*(d + c^2
*d*x^2)^(3/2)) - (4*x^2*(a + b*ArcSinh[c*x])^2)/(3*c^4*d^2*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^3) - (22*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(3*c^6*d
^2*Sqrt[d + c^2*d*x^2]) + (((11*I)/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^6*d^2*Sqrt[d +
c^2*d*x^2]) - (((11*I)/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^6*d^2*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 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 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]

Rule 266

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

Rule 43

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

Rubi steps

\begin{align*} \int \frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx &=-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{4 \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{3 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 )}{\left (1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d+c^2 d x^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^2}+\frac{\left (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^2 \sqrt{d+c^2 d x^2}}+\frac{\left (8 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}{3 c^3 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (b^2 \sqrt{1+c^2 x^2}\right ) \int \frac{x^3}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 c^2 d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}+\frac{11 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d+c^2 d x^2}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \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^3}-\frac{\left (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^2 \sqrt{d+c^2 d x^2}}-\frac{\left (8 b \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^5 d^2 \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^2 \sqrt{d+c^2 d x^2}}-\frac{\left (b^2 \sqrt{1+c^2 x^2}\right ) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{c^4 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (8 b^2 \sqrt{1+c^2 x^2}\right ) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{3 c^4 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{6 c^2 d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{16 a b x \sqrt{1+c^2 x^2}}{3 c^5 d^2 \sqrt{d+c^2 d x^2}}-\frac{11 b^2 \left (1+c^2 x^2\right )}{3 c^6 d^2 \sqrt{d+c^2 d x^2}}-\frac{b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}+\frac{11 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d+c^2 d x^2}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \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^3}-\frac{\left (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^2 \sqrt{d+c^2 d x^2}}-\frac{\left (8 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 )}{3 c^6 d^2 \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^2 \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 \left (1+c^2 x\right )^{3/2}}+\frac{1}{c^2 \sqrt{1+c^2 x}}\right ) \, dx,x,x^2\right )}{6 c^2 d^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{b^2}{3 c^6 d^2 \sqrt{d+c^2 d x^2}}-\frac{16 a b x \sqrt{1+c^2 x^2}}{3 c^5 d^2 \sqrt{d+c^2 d x^2}}-\frac{10 b^2 \left (1+c^2 x^2\right )}{3 c^6 d^2 \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^2 \sqrt{d+c^2 d x^2}}-\frac{b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}+\frac{11 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d+c^2 d x^2}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \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^3}-\frac{22 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (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^2 \sqrt{d+c^2 d x^2}}-\frac{\left (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^2 \sqrt{d+c^2 d x^2}}+\frac{\left (8 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 )}{3 c^6 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (8 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 )}{3 c^6 d^2 \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^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{b^2}{3 c^6 d^2 \sqrt{d+c^2 d x^2}}-\frac{16 a b x \sqrt{1+c^2 x^2}}{3 c^5 d^2 \sqrt{d+c^2 d x^2}}+\frac{2 b^2 \left (1+c^2 x^2\right )}{c^6 d^2 \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^2 \sqrt{d+c^2 d x^2}}-\frac{b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}+\frac{11 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d+c^2 d x^2}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \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^3}-\frac{22 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (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^2 \sqrt{d+c^2 d x^2}}-\frac{\left (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^2 \sqrt{d+c^2 d x^2}}+\frac{\left (8 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 )}{3 c^6 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (8 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 )}{3 c^6 d^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{b^2}{3 c^6 d^2 \sqrt{d+c^2 d x^2}}-\frac{16 a b x \sqrt{1+c^2 x^2}}{3 c^5 d^2 \sqrt{d+c^2 d x^2}}+\frac{2 b^2 \left (1+c^2 x^2\right )}{c^6 d^2 \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^2 \sqrt{d+c^2 d x^2}}-\frac{b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}+\frac{11 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d+c^2 d x^2}}-\frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \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^3}-\frac{22 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{d+c^2 d x^2}}+\frac{11 i b^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{d+c^2 d x^2}}-\frac{11 i b^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{d+c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 1.73931, size = 333, normalized size = 0.65 \[ \frac{\sqrt{c^2 d x^2+d} \left (b^2 \left (11 i \left (c^2 x^2+1\right )^{3/2} \left (\text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )\right )+3 \left (c^2 x^2+1\right )^2 \left (\sinh ^{-1}(c x)^2+2\right )-6 c x \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)+\left (c^2 x^2+1\right ) \left (6 \sinh ^{-1}(c x)^2+1\right )+c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+11 i \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x) \left (\log \left (1-i e^{-\sinh ^{-1}(c x)}\right )-\log \left (1+i e^{-\sinh ^{-1}(c x)}\right )\right )-\sinh ^{-1}(c x)^2\right )+a^2 \left (3 c^4 x^4+12 c^2 x^2+8\right )+a b \left (2 \left (3 c^4 x^4+12 c^2 x^2+8\right ) \sinh ^{-1}(c x)-\sqrt{c^2 x^2+1} \left (c x \left (6 c^2 x^2+5\right )+22 \left (c^2 x^2+1\right ) \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )\right )\right )}{3 c^6 d^3 \left (c^2 x^2+1\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[d + c^2*d*x^2]*(a^2*(8 + 12*c^2*x^2 + 3*c^4*x^4) + a*b*(2*(8 + 12*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x] - Sq
rt[1 + c^2*x^2]*(c*x*(5 + 6*c^2*x^2) + 22*(1 + c^2*x^2)*ArcTan[Tanh[ArcSinh[c*x]/2]])) + b^2*(c*x*Sqrt[1 + c^2
*x^2]*ArcSinh[c*x] - 6*c*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x] - ArcSinh[c*x]^2 + 3*(1 + c^2*x^2)^2*(2 + ArcSinh[
c*x]^2) + (1 + c^2*x^2)*(1 + 6*ArcSinh[c*x]^2) + (11*I)*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*(Log[1 - I/E^ArcSinh[
c*x]] - Log[1 + I/E^ArcSinh[c*x]]) + (11*I)*(1 + c^2*x^2)^(3/2)*(PolyLog[2, (-I)/E^ArcSinh[c*x]] - PolyLog[2,
I/E^ArcSinh[c*x]]))))/(3*c^6*d^3*(1 + c^2*x^2)^2)

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Maple [B]  time = 0.342, size = 1040, normalized size = 2. \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)^(5/2),x)

[Out]

4*a^2/c^4*x^2/d/(c^2*d*x^2+d)^(3/2)+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/d^3/(c^2*x^2+1)^2/c^6+2*b^2*(d*(c^2*x^2+1))^
(1/2)/c^6/d^3/(c^2*x^2+1)-11/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^3*arcsinh(c*x)*ln(1-I*(c*x+
(c^2*x^2+1)^(1/2)))+11/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^3*arcsinh(c*x)*ln(1+I*(c*x+(c^2*x
^2+1)^(1/2)))+11/3*I*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^3*ln(c*x+(c^2*x^2+1)^(1/2)-I)-11/3*I*a*
b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^3*ln(c*x+(c^2*x^2+1)^(1/2)+I)+2*a*b*(d*(c^2*x^2+1))^(1/2)/c^4/
d^3/(c^2*x^2+1)*arcsinh(c*x)*x^2+4*a*b*(d*(c^2*x^2+1))^(1/2)/d^3/(c^2*x^2+1)^2/c^4*arcsinh(c*x)*x^2+8/3*a^2/c^
6/d/(c^2*d*x^2+d)^(3/2)+a^2*x^4/c^2/d/(c^2*d*x^2+d)^(3/2)-2*b^2*(d*(c^2*x^2+1))^(1/2)/c^5/d^3/(c^2*x^2+1)^(1/2
)*arcsinh(c*x)*x+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/d^3/(c^2*x^2+1)^(3/2)/c^5*arcsinh(c*x)*x+1/3*b^2*(d*(c^2*x^2+1)
)^(1/2)/d^3/(c^2*x^2+1)^2/c^4*x^2+5/3*b^2*(d*(c^2*x^2+1))^(1/2)/d^3/(c^2*x^2+1)^2/c^6*arcsinh(c*x)^2+2*b^2*(d*
(c^2*x^2+1))^(1/2)/c^4/d^3/(c^2*x^2+1)*x^2+b^2*(d*(c^2*x^2+1))^(1/2)/c^6/d^3/(c^2*x^2+1)*arcsinh(c*x)^2+1/3*a*
b*(d*(c^2*x^2+1))^(1/2)/d^3/(c^2*x^2+1)^(3/2)/c^5*x-2*a*b*(d*(c^2*x^2+1))^(1/2)/c^5/d^3/(c^2*x^2+1)^(1/2)*x+11
/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^3*dilog(1+I*(c*x+(c^2*x^2+1)^(1/2)))+b^2*(d*(c^2*x^2+1)
)^(1/2)/c^4/d^3/(c^2*x^2+1)*arcsinh(c*x)^2*x^2+2*b^2*(d*(c^2*x^2+1))^(1/2)/d^3/(c^2*x^2+1)^2/c^4*arcsinh(c*x)^
2*x^2-11/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^3*dilog(1-I*(c*x+(c^2*x^2+1)^(1/2)))+2*a*b*(d*(
c^2*x^2+1))^(1/2)/c^6/d^3/(c^2*x^2+1)*arcsinh(c*x)+10/3*a*b*(d*(c^2*x^2+1))^(1/2)/d^3/(c^2*x^2+1)^2/c^6*arcsin
h(c*x)

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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)^(5/2),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^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}}, 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)^(5/2),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^6*d^3*x^6 + 3*c^4*
d^3*x^4 + 3*c^2*d^3*x^2 + d^3), 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)**(5/2),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{5}{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)^(5/2),x, algorithm="giac")

[Out]

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