\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))^n}{x} \, dx\) [432]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=-\frac {5^{-1-n} d^3 e^{-\frac {5 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{32 \sqrt {d-c^2 d x^2}}-\frac {5\ 3^{-1-n} d^3 e^{-\frac {3 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{32 \sqrt {d-c^2 d x^2}}+\frac {3^{-n} d^3 e^{-\frac {3 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{8 \sqrt {d-c^2 d x^2}}-\frac {11 d^3 e^{-\frac {a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{16 \sqrt {d-c^2 d x^2}}+\frac {11 d^3 e^{a/b} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{16 \sqrt {d-c^2 d x^2}}+\frac {5\ 3^{-1-n} d^3 e^{\frac {3 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{32 \sqrt {d-c^2 d x^2}}-\frac {3^{-n} d^3 e^{\frac {3 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{8 \sqrt {d-c^2 d x^2}}+\frac {5^{-1-n} d^3 e^{\frac {5 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{32 \sqrt {d-c^2 d x^2}}+d^3 \text {Int}\left (\frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}},x\right ) \]

[Out]

-1/32*5^(-1-n)*d^3*(a+b*arccosh(c*x))^n*GAMMA(1+n,-5*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/exp(5*a
/b)/(((-a-b*arccosh(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)-5/32*3^(-1-n)*d^3*(a+b*arccosh(c*x))^n*GAMMA(1+n,-3*(a+b*
arccosh(c*x))/b)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/exp(3*a/b)/(((-a-b*arccosh(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+1/8*d
^3*(a+b*arccosh(c*x))^n*GAMMA(1+n,-3*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(3^n)/exp(3*a/b)/(((-a-
b*arccosh(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)-11/16*d^3*(a+b*arccosh(c*x))^n*GAMMA(1+n,(-a-b*arccosh(c*x))/b)*(c*
x-1)^(1/2)*(c*x+1)^(1/2)/exp(a/b)/(((-a-b*arccosh(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+11/16*d^3*exp(a/b)*(a+b*arc
cosh(c*x))^n*GAMMA(1+n,(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(((a+b*arccosh(c*x))/b)^n)/(-c^2*d*x^
2+d)^(1/2)+5/32*3^(-1-n)*d^3*exp(3*a/b)*(a+b*arccosh(c*x))^n*GAMMA(1+n,3*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)*(
c*x+1)^(1/2)/(((a+b*arccosh(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)-1/8*d^3*exp(3*a/b)*(a+b*arccosh(c*x))^n*GAMMA(1+n
,3*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(3^n)/(((a+b*arccosh(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+1/3
2*5^(-1-n)*d^3*exp(5*a/b)*(a+b*arccosh(c*x))^n*GAMMA(1+n,5*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(
((a+b*arccosh(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+d^3*Unintegrable((a+b*arccosh(c*x))^n/x/(-c^2*d*x^2+d)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 1.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n}{x} \, dx \]

[In]

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

[Out]

-1/32*(5^(-1 - n)*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-5*(a + b*ArcCosh[c*x]
))/b])/(E^((5*a)/b)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])/b))^n) - (5*3^(-1 - n)*d^3*Sqrt[-1 + c*x]*Sqrt
[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b])/(32*E^((3*a)/b)*Sqrt[d - c^2*d*x^2
]*(-((a + b*ArcCosh[c*x])/b))^n) + (d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-3*(
a + b*ArcCosh[c*x]))/b])/(8*3^n*E^((3*a)/b)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])/b))^n) - (11*d^3*Sqrt[
-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(16*E^(a/b)*Sqrt[d - c
^2*d*x^2]*(-((a + b*ArcCosh[c*x])/b))^n) + (11*d^3*E^(a/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n
*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(16*Sqrt[d - c^2*d*x^2]*((a + b*ArcCosh[c*x])/b)^n) + (5*3^(-1 - n)*d^3
*E^((3*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(32
*Sqrt[d - c^2*d*x^2]*((a + b*ArcCosh[c*x])/b)^n) - (d^3*E^((3*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCos
h[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(8*3^n*Sqrt[d - c^2*d*x^2]*((a + b*ArcCosh[c*x])/b)^n) + (
5^(-1 - n)*d^3*E^((5*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (5*(a + b*ArcCosh[
c*x]))/b])/(32*Sqrt[d - c^2*d*x^2]*((a + b*ArcCosh[c*x])/b)^n) + d^3*Defer[Int][(a + b*ArcCosh[c*x])^n/(x*Sqrt
[d - c^2*d*x^2]), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^3 (a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}}-\frac {3 c^2 d^3 x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}}+\frac {3 c^4 d^3 x^3 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}}-\frac {c^6 d^3 x^5 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}}\right ) \, dx \\ & = d^3 \int \frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx-\left (3 c^2 d^3\right ) \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx+\left (3 c^4 d^3\right ) \int \frac {x^3 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx-\left (c^6 d^3\right ) \int \frac {x^5 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx \\ & = d^3 \int \frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx-\frac {\left (d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int x^n \cosh ^5\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {d-c^2 d x^2}}-\frac {\left (3 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {d-c^2 d x^2}}+\frac {\left (3 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int x^n \cosh ^3\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {d-c^2 d x^2}} \\ & = d^3 \int \frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx-\frac {\left (d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \left (\frac {1}{16} x^n \cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )+\frac {5}{16} x^n \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )+\frac {5}{8} x^n \cosh \left (\frac {a}{b}-\frac {x}{b}\right )\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {d-c^2 d x^2}}-\frac {\left (3 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b \sqrt {d-c^2 d x^2}}-\frac {\left (3 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b \sqrt {d-c^2 d x^2}}+\frac {\left (3 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \left (\frac {1}{4} x^n \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )+\frac {3}{4} x^n \cosh \left (\frac {a}{b}-\frac {x}{b}\right )\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {d-c^2 d x^2}} \\ & = -\frac {3 d^3 e^{-\frac {a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{2 \sqrt {d-c^2 d x^2}}+\frac {3 d^3 e^{a/b} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{2 \sqrt {d-c^2 d x^2}}+d^3 \int \frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx-\frac {\left (d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b \sqrt {d-c^2 d x^2}}-\frac {\left (5 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b \sqrt {d-c^2 d x^2}}-\frac {\left (5 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b \sqrt {d-c^2 d x^2}}+\frac {\left (3 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b \sqrt {d-c^2 d x^2}}+\frac {\left (9 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b \sqrt {d-c^2 d x^2}} \\ & = -\frac {3 d^3 e^{-\frac {a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{2 \sqrt {d-c^2 d x^2}}+\frac {3 d^3 e^{a/b} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{2 \sqrt {d-c^2 d x^2}}+d^3 \int \frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx-\frac {\left (d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{-i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b \sqrt {d-c^2 d x^2}}-\frac {\left (d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b \sqrt {d-c^2 d x^2}}-\frac {\left (5 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b \sqrt {d-c^2 d x^2}}-\frac {\left (5 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b \sqrt {d-c^2 d x^2}}-\frac {\left (5 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b \sqrt {d-c^2 d x^2}}-\frac {\left (5 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b \sqrt {d-c^2 d x^2}}+\frac {\left (3 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b \sqrt {d-c^2 d x^2}}+\frac {\left (3 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b \sqrt {d-c^2 d x^2}}+\frac {\left (9 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b \sqrt {d-c^2 d x^2}}+\frac {\left (9 d^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b \sqrt {d-c^2 d x^2}} \\ & = -\frac {5^{-1-n} d^3 e^{-\frac {5 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{32 \sqrt {d-c^2 d x^2}}-\frac {5\ 3^{-1-n} d^3 e^{-\frac {3 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{32 \sqrt {d-c^2 d x^2}}+\frac {3^{-n} d^3 e^{-\frac {3 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{8 \sqrt {d-c^2 d x^2}}-\frac {11 d^3 e^{-\frac {a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{16 \sqrt {d-c^2 d x^2}}+\frac {11 d^3 e^{a/b} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{16 \sqrt {d-c^2 d x^2}}+\frac {5\ 3^{-1-n} d^3 e^{\frac {3 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{32 \sqrt {d-c^2 d x^2}}-\frac {3^{-n} d^3 e^{\frac {3 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{8 \sqrt {d-c^2 d x^2}}+\frac {5^{-1-n} d^3 e^{\frac {5 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (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 )}{32 \sqrt {d-c^2 d x^2}}+d^3 \int \frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.66 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{x} \,d x } \]

[In]

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

[Out]

integral((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)^n/x, x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{x} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

Mupad [N/A]

Not integrable

Time = 3.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x} \,d x \]

[In]

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

[Out]

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