\(\int \frac {(a+b \text {arccosh}(c x))^{3/2}}{d+e x^2} \, dx\) [559]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {(a+b \text {arccosh}(c x))^{3/2}}{d+e x^2} \, dx=\text {Int}\left (\frac {(a+b \text {arccosh}(c x))^{3/2}}{d+e x^2},x\right ) \]

[Out]

Unintegrable((a+b*arccosh(c*x))^(3/2)/(e*x^2+d),x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {(a+b \text {arccosh}(c x))^{3/2}}{d+e x^2} \, dx=\int \frac {(a+b \text {arccosh}(c x))^{3/2}}{d+e x^2} \, dx \]

[In]

Int[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2),x]

[Out]

Defer[Int][(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b \text {arccosh}(c x))^{3/2}}{d+e x^2} \, dx \\ \end{align*}

Mathematica [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^{3/2}}{d+e x^2} \, dx=\text {\$Aborted} \]

[In]

Integrate[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2),x]

[Out]

$Aborted

Maple [N/A] (verified)

Not integrable

Time = 1.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

int((a+b*arccosh(c*x))^(3/2)/(e*x^2+d),x)

[Out]

int((a+b*arccosh(c*x))^(3/2)/(e*x^2+d),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arccosh}(c x))^{3/2}}{d+e x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arccosh(c*x))^(3/2)/(e*x^2+d),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 37.43 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b \text {arccosh}(c x))^{3/2}}{d+e x^2} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {3}{2}}}{d + e x^{2}}\, dx \]

[In]

integrate((a+b*acosh(c*x))**(3/2)/(e*x**2+d),x)

[Out]

Integral((a + b*acosh(c*x))**(3/2)/(d + e*x**2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arccosh}(c x))^{3/2}}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arccosh(c*x))^(3/2)/(e*x^2+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 19.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c x))^{3/2}}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}{e x^{2} + d} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))^(3/2)/(e*x^2+d),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)^(3/2)/(e*x^2 + d), x)

Mupad [N/A]

Not integrable

Time = 3.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c x))^{3/2}}{d+e x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}}{e\,x^2+d} \,d x \]

[In]

int((a + b*acosh(c*x))^(3/2)/(d + e*x^2),x)

[Out]

int((a + b*acosh(c*x))^(3/2)/(d + e*x^2), x)