\(\int \frac {\sqrt {\text {arcsinh}(\frac {x}{a})}}{(a^2+x^2)^{5/2}} \, dx\) [489]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\left (a^2+x^2\right )^{5/2}} \, dx=\frac {x \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{3 a^2 \left (a^2+x^2\right )^{3/2}}+\frac {2 x \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{3 a^4 \sqrt {a^2+x^2}}-\frac {\sqrt {1+\frac {x^2}{a^2}} \text {Int}\left (\frac {x}{\left (1+\frac {x^2}{a^2}\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}},x\right )}{6 a^5 \sqrt {a^2+x^2}}-\frac {\sqrt {1+\frac {x^2}{a^2}} \text {Int}\left (\frac {x}{\left (1+\frac {x^2}{a^2}\right ) \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}},x\right )}{3 a^5 \sqrt {a^2+x^2}} \]

[Out]

1/3*x*arcsinh(x/a)^(1/2)/a^2/(a^2+x^2)^(3/2)+2/3*x*arcsinh(x/a)^(1/2)/a^4/(a^2+x^2)^(1/2)-1/6*(1+x^2/a^2)^(1/2
)*Unintegrable(x/(1+x^2/a^2)^2/arcsinh(x/a)^(1/2),x)/a^5/(a^2+x^2)^(1/2)-1/3*(1+x^2/a^2)^(1/2)*Unintegrable(x/
(1+x^2/a^2)/arcsinh(x/a)^(1/2),x)/a^5/(a^2+x^2)^(1/2)

Rubi [N/A]

Not integrable

Time = 0.13 (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 = {} \[ \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\left (a^2+x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\left (a^2+x^2\right )^{5/2}} \, dx \]

[In]

Int[Sqrt[ArcSinh[x/a]]/(a^2 + x^2)^(5/2),x]

[Out]

(x*Sqrt[ArcSinh[x/a]])/(3*a^2*(a^2 + x^2)^(3/2)) + (2*x*Sqrt[ArcSinh[x/a]])/(3*a^4*Sqrt[a^2 + x^2]) - (Sqrt[1
+ x^2/a^2]*Defer[Int][x/((1 + x^2/a^2)^2*Sqrt[ArcSinh[x/a]]), x])/(6*a^5*Sqrt[a^2 + x^2]) - (Sqrt[1 + x^2/a^2]
*Defer[Int][x/((1 + x^2/a^2)*Sqrt[ArcSinh[x/a]]), x])/(3*a^5*Sqrt[a^2 + x^2])

Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{3 a^2 \left (a^2+x^2\right )^{3/2}}+\frac {2 \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\left (a^2+x^2\right )^{3/2}} \, dx}{3 a^2}-\frac {\sqrt {1+\frac {x^2}{a^2}} \int \frac {x}{\left (1+\frac {x^2}{a^2}\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}} \, dx}{6 a^5 \sqrt {a^2+x^2}} \\ & = \frac {x \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{3 a^2 \left (a^2+x^2\right )^{3/2}}+\frac {2 x \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{3 a^4 \sqrt {a^2+x^2}}-\frac {\sqrt {1+\frac {x^2}{a^2}} \int \frac {x}{\left (1+\frac {x^2}{a^2}\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}} \, dx}{6 a^5 \sqrt {a^2+x^2}}-\frac {\sqrt {1+\frac {x^2}{a^2}} \int \frac {x}{\left (1+\frac {x^2}{a^2}\right ) \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}} \, dx}{3 a^5 \sqrt {a^2+x^2}} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\left (a^2+x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\left (a^2+x^2\right )^{5/2}} \, dx \]

[In]

Integrate[Sqrt[ArcSinh[x/a]]/(a^2 + x^2)^(5/2),x]

[Out]

Integrate[Sqrt[ArcSinh[x/a]]/(a^2 + x^2)^(5/2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.35 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82

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

[In]

int(arcsinh(x/a)^(1/2)/(a^2+x^2)^(5/2),x)

[Out]

int(arcsinh(x/a)^(1/2)/(a^2+x^2)^(5/2),x)

Fricas [F(-2)]

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

[In]

integrate(arcsinh(x/a)^(1/2)/(a^2+x^2)^(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 = 23.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\left (a^2+x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\operatorname {asinh}{\left (\frac {x}{a} \right )}}}{\left (a^{2} + x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(asinh(x/a)**(1/2)/(a**2+x**2)**(5/2),x)

[Out]

Integral(sqrt(asinh(x/a))/(a**2 + x**2)**(5/2), x)

Maxima [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\left (a^2+x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {\operatorname {arsinh}\left (\frac {x}{a}\right )}}{{\left (a^{2} + x^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(arcsinh(x/a)^(1/2)/(a^2+x^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(arcsinh(x/a))/(a^2 + x^2)^(5/2), x)

Giac [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\left (a^2+x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {\operatorname {arsinh}\left (\frac {x}{a}\right )}}{{\left (a^{2} + x^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(arcsinh(x/a)^(1/2)/(a^2+x^2)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(arcsinh(x/a))/(a^2 + x^2)^(5/2), x)

Mupad [N/A]

Not integrable

Time = 2.76 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\left (a^2+x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\mathrm {asinh}\left (\frac {x}{a}\right )}}{{\left (a^2+x^2\right )}^{5/2}} \,d x \]

[In]

int(asinh(x/a)^(1/2)/(a^2 + x^2)^(5/2),x)

[Out]

int(asinh(x/a)^(1/2)/(a^2 + x^2)^(5/2), x)