3.311 \(\int \frac{x^4 (a+b \sinh ^{-1}(c x))^2}{(d+c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=398 \[ \frac{4 b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{b x^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt{c^2 d x^2+d}}+\frac{\sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{4 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}+\frac{8 b \sqrt{c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}-\frac{b^2 x}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}+\frac{b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt{c^2 d x^2+d}} \]

[Out]

-(b^2*x)/(3*c^4*d^2*Sqrt[d + c^2*d*x^2]) + (b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]
) - (b*x^2*(a + b*ArcSinh[c*x]))/(3*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (x^3*(a + b*ArcSinh[c*x])
^2)/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) - (x*(a + b*ArcSinh[c*x])^2)/(c^4*d^2*Sqrt[d + c^2*d*x^2]) - (4*Sqrt[1 + c
^2*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(
3*b*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (8*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3
*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (4*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c^5*d^2*Sqrt[d +
c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.761133, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5751, 5677, 5675, 5714, 3718, 2190, 2279, 2391, 288, 215} \[ \frac{4 b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{b x^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt{c^2 d x^2+d}}+\frac{\sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{4 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}+\frac{8 b \sqrt{c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}-\frac{b^2 x}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}+\frac{b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]

[Out]

-(b^2*x)/(3*c^4*d^2*Sqrt[d + c^2*d*x^2]) + (b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]
) - (b*x^2*(a + b*ArcSinh[c*x]))/(3*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (x^3*(a + b*ArcSinh[c*x])
^2)/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) - (x*(a + b*ArcSinh[c*x])^2)/(c^4*d^2*Sqrt[d + c^2*d*x^2]) - (4*Sqrt[1 + c
^2*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(
3*b*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (8*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3
*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (4*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c^5*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 5677

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

Rule 5675

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

Rule 5714

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

Rule 3718

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

Rule 2190

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

Rule 2279

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

Rule 2391

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

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 1.24282, size = 359, normalized size = 0.9 \[ \frac{-b^2 \sqrt{d} \left (4 \left (c^2 x^2+1\right )^{3/2} \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )+c^3 x^3+4 c^3 x^3 \sinh ^{-1}(c x)^2-\left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)^3-4 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)^2-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-8 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x) \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )+c x+3 c x \sinh ^{-1}(c x)^2\right )+a^2 (-c) \sqrt{d} x \left (4 c^2 x^2+3\right )+3 a^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )+a b \sqrt{d} \left (\sqrt{c^2 x^2+1}-8 c x \left (c^2 x^2+1\right ) \sinh ^{-1}(c x)+\left (c^2 x^2+1\right )^{3/2} \left (4 \log \left (c^2 x^2+1\right )+3 \sinh ^{-1}(c x)^2\right )+2 c x \sinh ^{-1}(c x)\right )}{3 c^5 d^{5/2} \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]

[Out]

(-(a^2*c*Sqrt[d]*x*(3 + 4*c^2*x^2)) + a*b*Sqrt[d]*(Sqrt[1 + c^2*x^2] + 2*c*x*ArcSinh[c*x] - 8*c*x*(1 + c^2*x^2
)*ArcSinh[c*x] + (1 + c^2*x^2)^(3/2)*(3*ArcSinh[c*x]^2 + 4*Log[1 + c^2*x^2])) + 3*a^2*(1 + c^2*x^2)*Sqrt[d + c
^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - b^2*Sqrt[d]*(c*x + c^3*x^3 - Sqrt[1 + c^2*x^2]*ArcSinh[c*
x] + 3*c*x*ArcSinh[c*x]^2 + 4*c^3*x^3*ArcSinh[c*x]^2 - 4*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]^2 - (1 + c^2*x^2)^(3
/2)*ArcSinh[c*x]^3 - 8*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[c*x])] + 4*(1 + c^2*x^2)^(3/2)*P
olyLog[2, -E^(-2*ArcSinh[c*x])]))/(3*c^5*d^(5/2)*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])

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Maple [B]  time = 0.378, size = 3705, normalized size = 9.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2/c^4/d^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)-1/3*a^2*x^3/c^2/d/(c^2*d*x^2+d)^(3/2)-
17*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/d^3*x^5+84*b^2*(d*(c^2*x^2+1))^
(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c/d^3*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)*x^4+8*b^2*(d*(c
^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c/d^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^4+4*
b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^4/d^3*arcsinh(c*x)*(c^2*x^2+1)*x
+28/3*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^2/d^3*arcsinh(c*x)*(c^2*x^
2+1)*x^3+220/3*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^3/d^3*arcsinh(c*x
)^2*(c^2*x^2+1)^(1/2)*x^2+13*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^3/d
^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^2+32*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x
^2+16)*c/d^3*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)*x^6-64*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*
x^4+71*c^2*x^2+16)*c^2/d^3*arcsinh(c*x)*x^7+8*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*
c^2*x^2+16)/c/d^3*(c^2*x^2+1)^(1/2)*x^4+28/3*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c
^2*x^2+16)/c^2/d^3*(c^2*x^2+1)*x^3-362/3*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x
^2+16)/c^2/d^3*arcsinh(c*x)*x^3+13*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)
/c^3/d^3*x^2*(c^2*x^2+1)^(1/2)+4*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c
^4/d^3*(c^2*x^2+1)*x-32*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^4/d^3*ar
csinh(c*x)*x+128/3*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^5/d^3*arcsinh
(c*x)*(c^2*x^2+1)^(1/2)+a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^5/d^3*arcsinh(c*x)^2+64*a*b*(d*(c^2*x^2+
1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)*c/d^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^6+168*a*b*(
d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c/d^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^
4+440/3*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^3/d^3*arcsinh(c*x)*(c^2*
x^2+1)^(1/2)*x^2-16/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^5/d^3*arcsinh(c*x)-16/3*a*b*(d*(c^2*x^2+1)
)^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)*c^2/d^3*x^7+16/3*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x
^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/d^3*(c^2*x^2+1)*x^5-152*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*
x^6+118*c^4*x^4+71*c^2*x^2+16)/d^3*arcsinh(c*x)*x^5-40/3*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*
c^4*x^4+71*c^2*x^2+16)/c^2/d^3*x^3-4*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+1
6)/c^4/d^3*x+16/3*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^5/d^3*(c^2*x^2
+1)^(1/2)+8/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^5/d^3*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)-8/3*b^2*(d*(
c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^2/d^3*(c^2*x^2+1)*x^3-40/3*b^2*(d*(c^2*x
^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^2/d^3*arcsinh(c*x)*x^3+55/3*b^2*(d*(c^2*x^2+1
))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^3/d^3*x^2*(c^2*x^2+1)^(1/2)-16*b^2*(d*(c^2*x^2+1)
)^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^4/d^3*arcsinh(c*x)^2*x-4/3*b^2*(d*(c^2*x^2+1))^(1/
2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^4/d^3*(c^2*x^2+1)*x+64/3*b^2*(d*(c^2*x^2+1))^(1/2)/(24*
c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^5/d^3*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)-4*b^2*(d*(c^2*x^2+1))^(
1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^4/d^3*arcsinh(c*x)*x+16/3*b^2*(d*(c^2*x^2+1))^(1/2)/(
24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^5/d^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+16/3*b^2*(d*(c^2*x^2+1
))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/d^3*arcsinh(c*x)*(c^2*x^2+1)*x^5+8/3*b^2*(d*(c^2*x^
2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^5/d^3*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)-32*b^2*(d*(c^2*x^2+1))^(1/2
)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)*c^2/d^3*arcsinh(c*x)^2*x^7-16/3*b^2*(d*(c^2*x^2+1))^(1/2)/
(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)*c^2/d^3*arcsinh(c*x)*x^7+8*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8
*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)*c/d^3*(c^2*x^2+1)^(1/2)*x^6+21*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^
8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c/d^3*(c^2*x^2+1)^(1/2)*x^4-181/3*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^
8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^2/d^3*arcsinh(c*x)^2*x^3+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(
1/2)/c^5/d^3*arcsinh(c*x)^3-76*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/d^3
*arcsinh(c*x)^2*x^5-4/3*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/d^3*(c^2*x
^2+1)*x^5-44/3*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/d^3*arcsinh(c*x)*x^
5-20/3*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)*c^2/d^3*x^7-43/3*b^2*(d*(c^
2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^2/d^3*x^3-4*b^2*(d*(c^2*x^2+1))^(1/2)/(24*
c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/c^4/d^3*x+16/3*b^2*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+
118*c^4*x^4+71*c^2*x^2+16)/c^5/d^3*(c^2*x^2+1)^(1/2)-8/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^5/d^3*a
rcsinh(c*x)^2+4/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^5/d^3*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)-44
/3*a*b*(d*(c^2*x^2+1))^(1/2)/(24*c^8*x^8+87*c^6*x^6+118*c^4*x^4+71*c^2*x^2+16)/d^3*x^5-a^2/c^4/d^2*x/(c^2*d*x^
2+d)^(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^4*(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^{4} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname{arsinh}\left (c x\right ) + a^{2} x^{4}\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^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

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