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

Optimal. Leaf size=793 \[ \frac{d^2 7^{-n-1} e^{-\frac{7 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 5^{-n} e^{-\frac{5 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d^2 3^{1-n} e^{-\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 e^{-\frac{a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 d^2 e^{a/b} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 3^{1-n} e^{\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d^2 5^{-n} e^{\frac{5 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 7^{-n-1} e^{\frac{7 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(7^(-1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-7*(a + b*ArcCosh[c*x]))/b])/(128*c^2
*E^((7*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (d^2*Sqrt[d - c^2*d*x^2]*(a + b*Arc
Cosh[c*x])^n*Gamma[1 + n, (-5*(a + b*ArcCosh[c*x]))/b])/(128*5^n*c^2*E^((5*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*
(-((a + b*ArcCosh[c*x])/b))^n) + (3^(1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-3*(a
+ b*ArcCosh[c*x]))/b])/(128*c^2*E^((3*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (5*
d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(128*c^2*E^(a/b)*Sqrt[
-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (5*d^2*E^(a/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x
])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(128*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) -
(3^(1 - n)*d^2*E^((3*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b
])/(128*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) + (d^2*E^((5*a)/b)*Sqrt[d - c^2*d*x^2]*(a
+ b*ArcCosh[c*x])^n*Gamma[1 + n, (5*(a + b*ArcCosh[c*x]))/b])/(128*5^n*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a +
b*ArcCosh[c*x])/b)^n) - (7^(-1 - n)*d^2*E^((7*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (
7*(a + b*ArcCosh[c*x]))/b])/(128*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)

________________________________________________________________________________________

Rubi [A]  time = 1.05402, antiderivative size = 793, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {5798, 5781, 5448, 3307, 2181} \[ \frac{d^2 7^{-n-1} e^{-\frac{7 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 5^{-n} e^{-\frac{5 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d^2 3^{1-n} e^{-\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 e^{-\frac{a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 d^2 e^{a/b} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 3^{1-n} e^{\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d^2 5^{-n} e^{\frac{5 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 7^{-n-1} e^{\frac{7 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7^(-1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-7*(a + b*ArcCosh[c*x]))/b])/(128*c^2
*E^((7*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (d^2*Sqrt[d - c^2*d*x^2]*(a + b*Arc
Cosh[c*x])^n*Gamma[1 + n, (-5*(a + b*ArcCosh[c*x]))/b])/(128*5^n*c^2*E^((5*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*
(-((a + b*ArcCosh[c*x])/b))^n) + (3^(1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-3*(a
+ b*ArcCosh[c*x]))/b])/(128*c^2*E^((3*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (5*
d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(128*c^2*E^(a/b)*Sqrt[
-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (5*d^2*E^(a/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x
])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(128*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) -
(3^(1 - n)*d^2*E^((3*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b
])/(128*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) + (d^2*E^((5*a)/b)*Sqrt[d - c^2*d*x^2]*(a
+ b*ArcCosh[c*x])^n*Gamma[1 + n, (5*(a + b*ArcCosh[c*x]))/b])/(128*5^n*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a +
b*ArcCosh[c*x])/b)^n) - (7^(-1 - n)*d^2*E^((7*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (
7*(a + b*ArcCosh[c*x]))/b])/(128*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)

Rule 5798

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

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 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 3307

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

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

Mathematica [A]  time = 3.80858, size = 633, normalized size = 0.8 \[ \frac{d^3 5^{-n} 21^{-n-1} e^{-\frac{7 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-3 n} \left (\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \left (e^{\frac{2 a}{b}} \left (21^{n+1} \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-9\ 5^n 7^{n+1} e^{\frac{2 a}{b}} \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+105^{n+1} e^{\frac{4 a}{b}} \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )+16\ 5^n 7^{n+1} e^{\frac{8 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{2 n} \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n \text{Gamma}\left (n+1,\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-5^n 7^{n+2} e^{\frac{8 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{3 n} \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-21^{n+1} e^{\frac{10 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{3 n} \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3^{n+1} 5^n e^{\frac{12 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{3 n} \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,\frac{7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )-3^{n+1} 5^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )-105^{n+1} e^{\frac{8 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )}{128 c^2 \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(21^(-1 - n)*d^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(-(105^(1 + n)*E^((8*a)/b)*(-((a
+ b*ArcCosh[c*x])/b))^n*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, a/b + ArcCosh[c*x]]) + (a/b + ArcCo
sh[c*x])^n*(-(3^(1 + n)*5^n*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-7*(a + b*ArcCosh[c*x]))/b]) +
E^((2*a)/b)*(21^(1 + n)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-5*(a + b*ArcCosh[c*x]))/b] - 9*5
^n*7^(1 + n)*E^((2*a)/b)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b] + 105
^(1 + n)*E^((4*a)/b)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)] - 5^n*7^(2
+ n)*E^((8*a)/b)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(3*n)*Gamma[1 + n, (3*(a + b*ArcCosh[c*x])
)/b] + 16*5^n*7^(1 + n)*E^((8*a)/b)*(-((a + b*ArcCosh[c*x])/b))^(2*n)*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[
1 + n, (3*(a + b*ArcCosh[c*x]))/b] - 21^(1 + n)*E^((10*a)/b)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b)
)^(3*n)*Gamma[1 + n, (5*(a + b*ArcCosh[c*x]))/b] + 3^(1 + n)*5^n*E^((12*a)/b)*(a/b + ArcCosh[c*x])^n*(-((a + b
*ArcCosh[c*x])/b))^(3*n)*Gamma[1 + n, (7*(a + b*ArcCosh[c*x]))/b]))))/(128*5^n*c^2*E^((7*a)/b)*Sqrt[d - c^2*d*
x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^(3*n))

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Maple [F]  time = 0.286, size = 0, normalized size = 0. \begin{align*} \int x \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}} \left ( a+b{\rm arccosh} \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*arccosh(c*x))^n,x)

[Out]

int(x*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(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{arcosh}\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*arccosh(c*x))^n,x, algorithm="maxima")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccosh(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{arcosh}\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*arccosh(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*arccosh(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*acosh(c*x))**n,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} 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*arccosh(c*x))^n,x, algorithm="giac")

[Out]

sage0*x