\(\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{(d+e x^2)^2} \, dx\) [633]

Optimal result

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

[Out]

Unintegrable((a+b*arcsinh(c*x))^(1/2)/(e*x^2+d)^2,x)

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\left (d+e x^2\right )^2} \, dx=\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\left (d+e x^2\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\left (d+e x^2\right )^2} \, dx=\text {\$Aborted} \]

[In]

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

[Out]

$Aborted

Maple [N/A] (verified)

Not integrable

Time = 1.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

int((a+b*arcsinh(c*x))^(1/2)/(e*x^2+d)^2,x)

[Out]

int((a+b*arcsinh(c*x))^(1/2)/(e*x^2+d)^2,x)

Fricas [F(-2)]

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

[In]

integrate((a+b*arcsinh(c*x))^(1/2)/(e*x^2+d)^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 = 14.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\left (d+e x^2\right )^2} \, dx=\int \frac {\sqrt {a + b \operatorname {asinh}{\left (c x \right )}}}{\left (d + e x^{2}\right )^{2}}\, dx \]

[In]

integrate((a+b*asinh(c*x))**(1/2)/(e*x**2+d)**2,x)

[Out]

Integral(sqrt(a + b*asinh(c*x))/(d + e*x**2)**2, x)

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\left (d+e x^2\right )^2} \, dx=\int { \frac {\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

integrate((a+b*arcsinh(c*x))^(1/2)/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

integrate(sqrt(b*arcsinh(c*x) + a)/(e*x^2 + d)^2, x)

Giac [N/A]

Not integrable

Time = 1.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\left (d+e x^2\right )^2} \, dx=\int { \frac {\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

integrate((a+b*arcsinh(c*x))^(1/2)/(e*x^2+d)^2,x, algorithm="giac")

[Out]

integrate(sqrt(b*arcsinh(c*x) + a)/(e*x^2 + d)^2, x)

Mupad [N/A]

Not integrable

Time = 2.70 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\left (d+e x^2\right )^2} \, dx=\int \frac {\sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )}}{{\left (e\,x^2+d\right )}^2} \,d x \]

[In]

int((a + b*asinh(c*x))^(1/2)/(d + e*x^2)^2,x)

[Out]

int((a + b*asinh(c*x))^(1/2)/(d + e*x^2)^2, x)