3.5.30 \(\int x (d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x))^n \, dx\) [430]

Optimal. Leaf size=793 \[ \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}} \]

[Out]

1/128*7^(-1-n)*d^2*(a+b*arccosh(c*x))^n*GAMMA(1+n,-7*(a+b*arccosh(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^2/exp(7*a/b)
/(((-a-b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/128*d^2*(a+b*arccosh(c*x))^n*GAMMA(1+n,-5*(a+b*arcc
osh(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/(5^n)/c^2/exp(5*a/b)/(((-a-b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
+1/128*3^(1-n)*d^2*(a+b*arccosh(c*x))^n*GAMMA(1+n,-3*(a+b*arccosh(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^2/exp(3*a/b)
/(((-a-b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/128*d^2*(a+b*arccosh(c*x))^n*GAMMA(1+n,(-a-b*arccos
h(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^2/exp(a/b)/(((-a-b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/128*d^2
*exp(a/b)*(a+b*arccosh(c*x))^n*GAMMA(1+n,(a+b*arccosh(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^2/(((a+b*arccosh(c*x))/b
)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/128*3^(1-n)*d^2*exp(3*a/b)*(a+b*arccosh(c*x))^n*GAMMA(1+n,3*(a+b*arccosh(c*
x))/b)*(-c^2*d*x^2+d)^(1/2)/c^2/(((a+b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/128*d^2*exp(5*a/b)*(a
+b*arccosh(c*x))^n*GAMMA(1+n,5*(a+b*arccosh(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/(5^n)/c^2/(((a+b*arccosh(c*x))/b)^n)
/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/128*7^(-1-n)*d^2*exp(7*a/b)*(a+b*arccosh(c*x))^n*GAMMA(1+n,7*(a+b*arccosh(c*x))
/b)*(-c^2*d*x^2+d)^(1/2)/c^2/(((a+b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]
time = 0.45, antiderivative size = 793, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5952, 5556, 3388, 2212} \begin {gather*} \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}} \end {gather*}

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 2212

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

Rule 3388

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 5556

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 5952

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

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 2.67, size = 633, normalized size = 0.80 \begin {gather*} \frac {5^{-n} 21^{-1-n} d^3 e^{-\frac {7 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \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 (-105^{1+n} 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} \Gamma \left (1+n,\frac {a}{b}+\cosh ^{-1}(c x)\right )+\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \left (-3^{1+n} 5^n \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+e^{\frac {2 a}{b}} \left (21^{1+n} \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-9\ 5^n 7^{1+n} e^{\frac {2 a}{b}} \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+105^{1+n} e^{\frac {4 a}{b}} \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )-5^n 7^{2+n} e^{\frac {8 a}{b}} \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+16\ 5^n 7^{1+n} 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 \Gamma \left (1+n,\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-21^{1+n} e^{\frac {10 a}{b}} \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3^{1+n} 5^n e^{\frac {12 a}{b}} \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )\right )\right )}{128 c^2 \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[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 = 180.00, size = 0, normalized size = 0.00 \[\int x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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