\(\int \frac {\text {arccosh}(a x)^{5/2}}{(c-a^2 c x^2)^{3/2}} \, dx\) [391]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\text {arccosh}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {x \text {arccosh}(a x)^{5/2}}{c \sqrt {c-a^2 c x^2}}+\frac {5 a \sqrt {-1+a x} \sqrt {1+a x} \text {Int}\left (\frac {x \text {arccosh}(a x)^{3/2}}{1-a^2 x^2},x\right )}{2 c \sqrt {c-a^2 c x^2}} \]

[Out]

x*arccosh(a*x)^(5/2)/c/(-a^2*c*x^2+c)^(1/2)+5/2*a*(a*x-1)^(1/2)*(a*x+1)^(1/2)*Unintegrable(x*arccosh(a*x)^(3/2
)/(-a^2*x^2+1),x)/c/(-a^2*c*x^2+c)^(1/2)

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {\text {arccosh}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\text {arccosh}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx \]

[In]

Int[ArcCosh[a*x]^(5/2)/(c - a^2*c*x^2)^(3/2),x]

[Out]

(x*ArcCosh[a*x]^(5/2))/(c*Sqrt[c - a^2*c*x^2]) + (5*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Defer[Int][(x*ArcCosh[a*x]^
(3/2))/(1 - a^2*x^2), x])/(2*c*Sqrt[c - a^2*c*x^2])

Rubi steps \begin{align*} \text {integral}& = \frac {x \text {arccosh}(a x)^{5/2}}{c \sqrt {c-a^2 c x^2}}+\frac {\left (5 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \text {arccosh}(a x)^{3/2}}{1-a^2 x^2} \, dx}{2 c \sqrt {c-a^2 c x^2}} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.78 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\text {arccosh}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\text {arccosh}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx \]

[In]

Integrate[ArcCosh[a*x]^(5/2)/(c - a^2*c*x^2)^(3/2),x]

[Out]

Integrate[ArcCosh[a*x]^(5/2)/(c - a^2*c*x^2)^(3/2), x]

Maple [N/A] (verified)

Not integrable

Time = 1.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

\[\int \frac {\operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]

[In]

int(arccosh(a*x)^(5/2)/(-a^2*c*x^2+c)^(3/2),x)

[Out]

int(arccosh(a*x)^(5/2)/(-a^2*c*x^2+c)^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\text {arccosh}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(arccosh(a*x)^(5/2)/(-a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(acosh(a*x)**(5/2)/(-a**2*c*x**2+c)**(3/2),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\text {arccosh}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(arccosh(a*x)^(5/2)/(-a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate(arccosh(a*x)^(5/2)/(-a^2*c*x^2 + c)^(3/2), x)

Giac [N/A]

Not integrable

Time = 3.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\text {arccosh}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(arccosh(a*x)^(5/2)/(-a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^(5/2)/(-a^2*c*x^2 + c)^(3/2), x)

Mupad [N/A]

Not integrable

Time = 2.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\text {arccosh}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^{5/2}}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]

[In]

int(acosh(a*x)^(5/2)/(c - a^2*c*x^2)^(3/2),x)

[Out]

int(acosh(a*x)^(5/2)/(c - a^2*c*x^2)^(3/2), x)