\(\int x (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))^n \, dx\) [329]

Optimal result

Integrand size = 27, antiderivative size = 793 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\frac {7^{-1-n} d^2 e^{-\frac {7 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {7 (a+b \text {arccosh}(c x))}{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} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {5 (a+b \text {arccosh}(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} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{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} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(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} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(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} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{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} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {5 (a+b \text {arccosh}(c x))}{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} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

1/128*7^(-1-n)*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,(-7 
*a-7*b*arccosh(c*x))/b)/c^2/exp(7*a/b)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/((-(a+b 
*arccosh(c*x))/b)^n)-1/128*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*G 
AMMA(1+n,(-5*a-5*b*arccosh(c*x))/b)/(5^n)/c^2/exp(5*a/b)/(c*x-1)^(1/2)/(c* 
x+1)^(1/2)/((-(a+b*arccosh(c*x))/b)^n)+1/128*3^(1-n)*d^2*(-c^2*d*x^2+d)^(1 
/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,(-3*a-3*b*arccosh(c*x))/b)/c^2/exp(3*a/ 
b)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/((-(a+b*arccosh(c*x))/b)^n)-5/128*d^2*(-c^2 
*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,-(a+b*arccosh(c*x))/b)/c^2/ 
exp(a/b)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/((-(a+b*arccosh(c*x))/b)^n)+5/128*d^2 
*exp(a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,(a+b*arccosh 
(c*x))/b)/c^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(((a+b*arccosh(c*x))/b)^n)-1/128 
*3^(1-n)*d^2*exp(3*a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+ 
n,3*(a+b*arccosh(c*x))/b)/c^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(((a+b*arccosh(c 
*x))/b)^n)+1/128*d^2*exp(5*a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n* 
GAMMA(1+n,5*(a+b*arccosh(c*x))/b)/(5^n)/c^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(( 
(a+b*arccosh(c*x))/b)^n)-1/128*7^(-1-n)*d^2*exp(7*a/b)*(-c^2*d*x^2+d)^(1/2 
)*(a+b*arccosh(c*x))^n*GAMMA(1+n,7*(a+b*arccosh(c*x))/b)/c^2/(c*x-1)^(1/2) 
/(c*x+1)^(1/2)/(((a+b*arccosh(c*x))/b)^n)
 

Mathematica [A] (warning: unable to verify)

Time = 2.53 (sec) , antiderivative size = 633, normalized size of antiderivative = 0.80 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\frac {5^{-n} 21^{-1-n} d^3 e^{-\frac {7 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-3 n} \left (-105^{1+n} e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,\frac {a}{b}+\text {arccosh}(c x)\right )+\left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-3^{1+n} 5^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {7 (a+b \text {arccosh}(c x))}{b}\right )+e^{\frac {2 a}{b}} \left (21^{1+n} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-9\ 5^n 7^{1+n} e^{\frac {2 a}{b}} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+105^{1+n} e^{\frac {4 a}{b}} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )-5^n 7^{2+n} e^{\frac {8 a}{b}} \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+16\ 5^n 7^{1+n} e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{2 n} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-21^{1+n} e^{\frac {10 a}{b}} \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+3^{1+n} 5^n e^{\frac {12 a}{b}} \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {7 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )\right )}{128 c^2 \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(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*ArcC 
osh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, a/b + ArcCosh[c*x]]) + (a/b + ArcCosh 
[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*ArcCo 
sh[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))
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 532, normalized size of antiderivative = 0.67, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6367, 5971, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx\)

\(\Big \downarrow \) 6367

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x))^n \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^6\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b c^2 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (\frac {1}{64} \cosh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n-\frac {5}{64} \cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n+\frac {9}{64} \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n-\frac {5}{64} \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) (a+b \text {arccosh}(c x))^n\right )d(a+b \text {arccosh}(c x))}{b c^2 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{128} b 7^{-n-1} e^{-\frac {7 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {7 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{128} b 5^{-n} e^{-\frac {5 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{128} b 3^{1-n} e^{-\frac {3 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{128} b e^{-\frac {a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {5}{128} b e^{a/b} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{128} b 3^{1-n} e^{\frac {3 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{128} b 5^{-n} e^{\frac {5 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{128} b 7^{-n-1} e^{\frac {7 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {7 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c^2 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5971

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 6367

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> Simp[(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] && Eq 
Q[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [F]

\[\int x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{n}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x \,d x } \] Input:

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

Output:

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*arc 
cosh(c*x) + a)^n, x)
 

Sympy [F(-1)]

Timed out. \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\sqrt {d}\, d^{2} \left (\left (\int \left (\mathit {acosh} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x^{5}d x \right ) c^{4}-2 \left (\int \left (\mathit {acosh} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x^{3}d x \right ) c^{2}+\int \left (\mathit {acosh} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x d x \right ) \] Input:

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

Output:

sqrt(d)*d**2*(int((acosh(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x**5,x)*c** 
4 - 2*int((acosh(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x**3,x)*c**2 + int( 
(acosh(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x,x))