\(\int \frac {\text {arccosh}(a x)^{5/2}}{\sqrt {c-a^2 c x^2}} \, dx\) [390]

Optimal result

Integrand size = 24, antiderivative size = 48 \[ \int \frac {\text {arccosh}(a x)^{5/2}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {c-a^2 c x^2}} \]

[Out]

2/7*arccosh(a*x)^(7/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/(-a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5892} \[ \int \frac {\text {arccosh}(a x)^{5/2}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {c-a^2 c x^2}} \]

[In]

Int[ArcCosh[a*x]^(5/2)/Sqrt[c - a^2*c*x^2],x]

[Out]

(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(7/2))/(7*a*Sqrt[c - a^2*c*x^2])

Rule 5892

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e,
n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {c-a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arccosh}(a x)^{5/2}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {c-a^2 c x^2}} \]

[In]

Integrate[ArcCosh[a*x]^(5/2)/Sqrt[c - a^2*c*x^2],x]

[Out]

(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(7/2))/(7*a*Sqrt[c - a^2*c*x^2])

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {2 \operatorname {arccosh}\left (a x \right )^{\frac {7}{2}} \sqrt {a x -1}\, \sqrt {a x +1}}{7 \sqrt {-c \left (a x -1\right ) \left (a x +1\right )}\, a}\)	\(41\)

[In]

int(arccosh(a*x)^(5/2)/(-a^2*c*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/7*arccosh(a*x)^(7/2)/(-c*(a*x-1)*(a*x+1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

Fricas [F(-2)]

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

[In]

integrate(arccosh(a*x)^(5/2)/(-a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^{5/2}}{\sqrt {c-a^2 c x^2}} \, dx=\text {Timed out} \]

[In]

integrate(acosh(a*x)**(5/2)/(-a**2*c*x**2+c)**(1/2),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate(arccosh(a*x)^(5/2)/(-a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(arccosh(a*x)^(5/2)/sqrt(-a^2*c*x^2 + c), x)

Giac [F]

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

[In]

integrate(arccosh(a*x)^(5/2)/(-a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^(5/2)/sqrt(-a^2*c*x^2 + c), x)

Mupad [F(-1)]

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

[In]

int(acosh(a*x)^(5/2)/(c - a^2*c*x^2)^(1/2),x)

[Out]

int(acosh(a*x)^(5/2)/(c - a^2*c*x^2)^(1/2), x)