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

Optimal. Leaf size=745 \[ \frac{d^2 7^{-n-1} e^{-\frac{7 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 5^{-n} e^{-\frac{5 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 3^{1-n} e^{-\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{5 d^2 e^{-\frac{a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{5 d^2 e^{a/b} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 3^{1-n} e^{\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 5^{-n} e^{\frac{5 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 7^{-n-1} e^{\frac{7 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}} \]

[Out]

(7^(-1 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-7*(a + b*ArcSinh[c*x]))/b])/(128*c^2
*E^((7*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^
n*Gamma[1 + n, (-5*(a + b*ArcSinh[c*x]))/b])/(128*5^n*c^2*E^((5*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x]
)/b))^n) + (3^(1 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b]
)/(128*c^2*E^((3*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*Ar
cSinh[c*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(128*c^2*E^(a/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*
x])/b))^n) + (5*d^2*E^(a/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b*ArcSinh[c*x])/b])/(
128*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (3^(1 - n)*d^2*E^((3*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*
ArcSinh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/(128*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^
n) + (d^2*E^((5*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (5*(a + b*ArcSinh[c*x]))/b])/(12
8*5^n*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (7^(-1 - n)*d^2*E^((7*a)/b)*Sqrt[d + c^2*d*x^2]*(a +
b*ArcSinh[c*x])^n*Gamma[1 + n, (7*(a + b*ArcSinh[c*x]))/b])/(128*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/
b)^n)

________________________________________________________________________________________

Rubi [A]  time = 0.785146, antiderivative size = 745, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {5782, 5779, 5448, 3308, 2181} \[ \frac{d^2 7^{-n-1} e^{-\frac{7 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 5^{-n} e^{-\frac{5 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 3^{1-n} e^{-\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{5 d^2 e^{-\frac{a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{5 d^2 e^{a/b} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 3^{1-n} e^{\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 5^{-n} e^{\frac{5 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 7^{-n-1} e^{\frac{7 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7^(-1 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-7*(a + b*ArcSinh[c*x]))/b])/(128*c^2
*E^((7*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^
n*Gamma[1 + n, (-5*(a + b*ArcSinh[c*x]))/b])/(128*5^n*c^2*E^((5*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x]
)/b))^n) + (3^(1 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b]
)/(128*c^2*E^((3*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*Ar
cSinh[c*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(128*c^2*E^(a/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*
x])/b))^n) + (5*d^2*E^(a/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b*ArcSinh[c*x])/b])/(
128*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (3^(1 - n)*d^2*E^((3*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*
ArcSinh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/(128*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^
n) + (d^2*E^((5*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (5*(a + b*ArcSinh[c*x]))/b])/(12
8*5^n*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (7^(-1 - n)*d^2*E^((7*a)/b)*Sqrt[d + c^2*d*x^2]*(a +
b*ArcSinh[c*x])^n*Gamma[1 + n, (7*(a + b*ArcSinh[c*x]))/b])/(128*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/
b)^n)

Rule 5782

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(d^IntPa
rt[p]*(d + e*x^2)^FracPart[p])/(1 + c^2*x^2)^FracPart[p], Int[x^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x],
x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] &&  !(Integer
Q[p] || GtQ[d, 0])

Rule 5779

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

Rule 5448

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

Rule 3308

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx &=\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh ^6(x) \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 \sqrt{1+c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{5}{64} (a+b x)^n \sinh (x)+\frac{9}{64} (a+b x)^n \sinh (3 x)+\frac{5}{64} (a+b x)^n \sinh (5 x)+\frac{1}{64} (a+b x)^n \sinh (7 x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 \sqrt{1+c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \sinh (7 x) \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^2 \sqrt{1+c^2 x^2}}+\frac{\left (5 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^2 \sqrt{1+c^2 x^2}}+\frac{\left (5 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \sinh (5 x) \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^2 \sqrt{1+c^2 x^2}}+\frac{\left (9 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \sinh (3 x) \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^2 \sqrt{1+c^2 x^2}}\\ &=-\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-7 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt{1+c^2 x^2}}+\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{7 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (5 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-5 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (5 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt{1+c^2 x^2}}+\frac{\left (5 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^x (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt{1+c^2 x^2}}+\frac{\left (5 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{5 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (9 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt{1+c^2 x^2}}+\frac{\left (9 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt{1+c^2 x^2}}\\ &=\frac{7^{-1-n} d^2 e^{-\frac{7 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1+c^2 x^2}}+\frac{5^{-n} d^2 e^{-\frac{5 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1+c^2 x^2}}+\frac{3^{1-n} d^2 e^{-\frac{3 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1+c^2 x^2}}+\frac{5 d^2 e^{-\frac{a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt{1+c^2 x^2}}+\frac{5 d^2 e^{a/b} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt{1+c^2 x^2}}+\frac{3^{1-n} d^2 e^{\frac{3 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1+c^2 x^2}}+\frac{5^{-n} d^2 e^{\frac{5 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1+c^2 x^2}}+\frac{7^{-1-n} d^2 e^{\frac{7 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 2.92995, size = 685, normalized size = 0.92 \[ \frac{d^3 105^{-n-1} e^{-\frac{7 a}{b}} \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \left (-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{-2 n} \left (5^{n+2} 21^{n+1} e^{\frac{8 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{2 n} \left (-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \text{Gamma}\left (n+1,\frac{a}{b}+\sinh ^{-1}(c x)\right )+15^{n+1} \left (-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+e^{\frac{2 a}{b}} \left (5\ 21^{n+1} \left (-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+9\ 35^{n+1} e^{\frac{2 a}{b}} \left (-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+5^{n+2} 21^{n+1} e^{\frac{4 a}{b}} \left (-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right )+8\ 35^{n+1} e^{\frac{8 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{2 n} \left (-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \text{Gamma}\left (n+1,\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+8\ 21^{n+1} e^{\frac{10 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{2 n} \left (-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \text{Gamma}\left (n+1,\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+35^{n+1} e^{\frac{8 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{3 n} \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-3^{n+2} 7^{n+1} e^{\frac{10 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{3 n} \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+15^{n+1} e^{\frac{12 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{3 n} \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )\right )}{128 c^2 \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(105^(-1 - n)*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*(5^(2 + n)*21^(1 + n)*E^((8*a)/b)*(-((a + b*ArcSinh
[c*x])/b))^(2*n)*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, a/b + ArcSinh[c*x]] + 15^(1 + n)*(-((a + b*Arc
Sinh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-7*(a + b*ArcSinh[c*x]))/b] + E^((2*a)/b)*(5*21^(1 + n)*(-((a + b*ArcSi
nh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-5*(a + b*ArcSinh[c*x]))/b] + 9*35^(1 + n)*E^((2*a)/b)*(-((a + b*ArcSinh[
c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b] + 5^(2 + n)*21^(1 + n)*E^((4*a)/b)*(-((a + b*Arc
Sinh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)] + 35^(1 + n)*E^((8*a)/b)*(a/b + ArcSinh[c*x])
^n*(-((a + b*ArcSinh[c*x])/b))^(3*n)*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b] + 8*35^(1 + n)*E^((8*a)/b)*(-((a
+ b*ArcSinh[c*x])/b))^(2*n)*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b] - 3^(2
+ n)*7^(1 + n)*E^((10*a)/b)*(a/b + ArcSinh[c*x])^n*(-((a + b*ArcSinh[c*x])/b))^(3*n)*Gamma[1 + n, (5*(a + b*A
rcSinh[c*x]))/b] + 8*21^(1 + n)*E^((10*a)/b)*(-((a + b*ArcSinh[c*x])/b))^(2*n)*(-((a + b*ArcSinh[c*x])^2/b^2))
^n*Gamma[1 + n, (5*(a + b*ArcSinh[c*x]))/b] + 15^(1 + n)*E^((12*a)/b)*(a/b + ArcSinh[c*x])^n*(-((a + b*ArcSinh
[c*x])/b))^(3*n)*Gamma[1 + n, (7*(a + b*ArcSinh[c*x]))/b])))/(128*c^2*E^((7*a)/b)*Sqrt[d + c^2*d*x^2]*(-((a +
b*ArcSinh[c*x])/b))^n*(-((a + b*ArcSinh[c*x])^2/b^2))^(2*n))

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Maple [F]  time = 0.161, size = 0, normalized size = 0. \begin{align*} \int x \left ({c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c^{4} d^{2} x^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x\right )} \sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x)*sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^n, 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*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**n,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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