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

Optimal. Leaf size=870 \[ \text{result too large to display} \]

[Out]

(-5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(128*b*c^3*(1 + n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (
2^(-11 - 3*n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcCosh[c*x]))/b])/(c^3*E
^((8*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-7 - n)*3^(-1 - n)*d^2*Sqrt[d - c
^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcCosh[c*x]))/b])/(c^3*E^((6*a)/b)*Sqrt[-1 + c*x]*Sq
rt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-4*
(a + b*ArcCosh[c*x]))/b])/(2^(2*(4 + n))*c^3*E^((4*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/
b))^n) + (2^(-7 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])
/(c^3*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-7 - n)*d^2*E^((2*a)/b)*Sq
rt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(c^3*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]*((a + b*ArcCosh[c*x])/b)^n) - (d^2*E^((4*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n,
(4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(4 + n))*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) + (2^
(-7 - n)*3^(-1 - n)*d^2*E^((6*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcCosh[
c*x]))/b])/(c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) - (2^(-11 - 3*n)*d^2*E^((8*a)/b)*Sqrt
[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (8*(a + b*ArcCosh[c*x]))/b])/(c^3*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]*((a + b*ArcCosh[c*x])/b)^n)

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Rubi [A]  time = 1.24155, antiderivative size = 870, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {5798, 5781, 5448, 3307, 2181} \[ \frac{2^{-3 n-11} d^2 e^{-\frac{8 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2^{-n-7} 3^{-n-1} d^2 e^{-\frac{6 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2^{-2 (n+4)} d^2 e^{-\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2^{-n-7} d^2 e^{-\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{128 b c^3 (n+1) \sqrt{c x-1} \sqrt{c x+1}}-\frac{2^{-n-7} d^2 e^{\frac{2 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{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2^{-2 (n+4)} d^2 e^{\frac{4 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{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2^{-n-7} 3^{-n-1} d^2 e^{\frac{6 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{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2^{-3 n-11} d^2 e^{\frac{8 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{8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(128*b*c^3*(1 + n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (
2^(-11 - 3*n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcCosh[c*x]))/b])/(c^3*E
^((8*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-7 - n)*3^(-1 - n)*d^2*Sqrt[d - c
^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcCosh[c*x]))/b])/(c^3*E^((6*a)/b)*Sqrt[-1 + c*x]*Sq
rt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-4*
(a + b*ArcCosh[c*x]))/b])/(2^(2*(4 + n))*c^3*E^((4*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/
b))^n) + (2^(-7 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])
/(c^3*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-7 - n)*d^2*E^((2*a)/b)*Sq
rt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(c^3*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]*((a + b*ArcCosh[c*x])/b)^n) - (d^2*E^((4*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n,
(4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(4 + n))*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) + (2^
(-7 - n)*3^(-1 - n)*d^2*E^((6*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcCosh[
c*x]))/b])/(c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) - (2^(-11 - 3*n)*d^2*E^((8*a)/b)*Sqrt
[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (8*(a + b*ArcCosh[c*x]))/b])/(c^3*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^2 \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^2 (-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 ^2(x) \sinh ^6(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 \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}{128} (a+b x)^n+\frac{1}{32} (a+b x)^n \cosh (2 x)+\frac{1}{32} (a+b x)^n \cosh (4 x)-\frac{1}{32} (a+b x)^n \cosh (6 x)+\frac{1}{128} (a+b x)^n \cosh (8 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \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 (8 x) \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^3 \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 (2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \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 (4 x) \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \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 (6 x) \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \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^{-8 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{256 c^3 \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^{8 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{256 c^3 \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^{-6 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^3 \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^{-4 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^3 \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^{-2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^3 \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^{2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^3 \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^{4 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^3 \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^{6 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2^{-11-3 n} d^2 e^{-\frac{8 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{8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2^{-7-n} 3^{-1-n} d^2 e^{-\frac{6 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{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4^{-4-n} d^2 e^{-\frac{4 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{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2^{-7-n} d^2 e^{-\frac{2 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{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2^{-7-n} d^2 e^{\frac{2 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{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{4^{-4-n} d^2 e^{\frac{4 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{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2^{-7-n} 3^{-1-n} d^2 e^{\frac{6 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{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2^{-11-3 n} d^2 e^{\frac{8 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{8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 6.88134, size = 677, normalized size = 0.78 \[ \frac{d^3 2^{-3 n-11} 3^{-n-1} e^{-\frac{8 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 )^{-n} \left (e^{\frac{8 a}{b}} \left (b 3^{n+1} 4^{n+2} (n+1) e^{\frac{2 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b 2^{n+3} 3^{n+1} (n+1) e^{\frac{4 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-b 4^{n+2} e^{\frac{6 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-b 4^{n+2} n e^{\frac{6 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b 3^{n+1} e^{\frac{8 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b 3^{n+1} n e^{\frac{8 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+5 a 2^{3 n+4} 3^{n+1} \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n+5 b 2^{3 n+4} 3^{n+1} \cosh ^{-1}(c x) \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n\right )+b \left (-3^{n+1}\right ) (n+1) \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b 4^{n+2} (n+1) e^{\frac{2 a}{b}} \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-b 2^{n+3} 3^{n+1} (n+1) e^{\frac{4 a}{b}} \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-b 3^{n+1} 4^{n+2} (n+1) e^{\frac{6 a}{b}} \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{b c^3 (n+1) \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2^(-11 - 3*n)*3^(-1 - n)*d^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(-(3^(1 + n)*b*(1 +
n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcCosh[c*x]))/b]) + 4^(2 + n)*b*E^((2*a)/b)*(1 + n)*(a/b +
ArcCosh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcCosh[c*x]))/b] - 2^(3 + n)*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(a/b + A
rcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b] - 3^(1 + n)*4^(2 + n)*b*E^((6*a)/b)*(1 + n)*(a/b + Ar
cCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b] + E^((8*a)/b)*(5*2^(4 + 3*n)*3^(1 + n)*a*(-((a + b*Arc
Cosh[c*x])^2/b^2))^n + 5*2^(4 + 3*n)*3^(1 + n)*b*ArcCosh[c*x]*(-((a + b*ArcCosh[c*x])^2/b^2))^n + 3^(1 + n)*4^
(2 + n)*b*E^((2*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b] + 2^(3 +
n)*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b] - 4^
(2 + n)*b*E^((6*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcCosh[c*x]))/b] - 4^(2 + n)*b*E^(
(6*a)/b)*n*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcCosh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*(-
((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (8*(a + b*ArcCosh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*n*(-((a + b*Arc
Cosh[c*x])/b))^n*Gamma[1 + n, (8*(a + b*ArcCosh[c*x]))/b])))/(b*c^3*E^((8*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*(-
((a + b*ArcCosh[c*x])^2/b^2))^n)

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Maple [F]  time = 0.319, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n,x)

[Out]

int(x^2*(-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^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-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^2, 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^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}\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^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n,x, algorithm="fricas")

[Out]

integral((c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2)*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**2*(-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^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n,x, algorithm="giac")

[Out]

sage0*x