\(\int \frac {\sqrt {\text {arccosh}(a x)}}{(c-a^2 c x^2)^{5/2}} \, dx\) [383]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\sqrt {\text {arccosh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {x \sqrt {\text {arccosh}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sqrt {\text {arccosh}(a x)}}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right ) \sqrt {\text {arccosh}(a x)}},x\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {Int}\left (\frac {x}{\left (-1+a^2 x^2\right )^2 \sqrt {\text {arccosh}(a x)}},x\right )}{6 c^2 \sqrt {c-a^2 c x^2}} \]

[Out]

1/3*x*arccosh(a*x)^(1/2)/c/(-a^2*c*x^2+c)^(3/2)+2/3*x*arccosh(a*x)^(1/2)/c^2/(-a^2*c*x^2+c)^(1/2)+1/3*a*(a*x-1
)^(1/2)*(a*x+1)^(1/2)*Unintegrable(x/(-a^2*x^2+1)/arccosh(a*x)^(1/2),x)/c^2/(-a^2*c*x^2+c)^(1/2)+1/6*a*(a*x-1)
^(1/2)*(a*x+1)^(1/2)*Unintegrable(x/(a^2*x^2-1)^2/arccosh(a*x)^(1/2),x)/c^2/(-a^2*c*x^2+c)^(1/2)

Rubi [N/A]

Not integrable

Time = 0.21 (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 = {} \[ \int \frac {\sqrt {\text {arccosh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\text {arccosh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \]

[In]

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

[Out]

(x*Sqrt[ArcCosh[a*x]])/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*Sqrt[ArcCosh[a*x]])/(3*c^2*Sqrt[c - a^2*c*x^2]) + (a
*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Defer[Int][x/((1 - a^2*x^2)*Sqrt[ArcCosh[a*x]]), x])/(3*c^2*Sqrt[c - a^2*c*x^2])
+ (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Defer[Int][x/((-1 + a^2*x^2)^2*Sqrt[ArcCosh[a*x]]), x])/(6*c^2*Sqrt[c - a^2
*c*x^2])

Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {\text {arccosh}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\sqrt {\text {arccosh}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{3 c}+\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{(-1+a x)^2 (1+a x)^2 \sqrt {\text {arccosh}(a x)}} \, dx}{6 c^2 \sqrt {c-a^2 c x^2}} \\ & = \frac {x \sqrt {\text {arccosh}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sqrt {\text {arccosh}(a x)}}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (-1+a^2 x^2\right )^2 \sqrt {\text {arccosh}(a x)}} \, dx}{6 c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (1-a^2 x^2\right ) \sqrt {\text {arccosh}(a x)}} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\text {arccosh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\text {arccosh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 98.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\text {arccosh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\operatorname {acosh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {\text {arccosh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {\operatorname {arcosh}\left (a x\right )}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.95 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {\text {arccosh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {\operatorname {arcosh}\left (a x\right )}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.75 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {\text {arccosh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\mathrm {acosh}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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