\(\int \frac {\text {arcsinh}(a x)^{3/2}}{(c+a^2 c x^2)^{3/2}} \, dx\) [480]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {x \text {arcsinh}(a x)^{3/2}}{c \sqrt {c+a^2 c x^2}}-\frac {3 a \sqrt {1+a^2 x^2} \text {Int}\left (\frac {x \sqrt {\text {arcsinh}(a x)}}{1+a^2 x^2},x\right )}{2 c \sqrt {c+a^2 c x^2}} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\text {arcsinh}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

[In]

Int[ArcSinh[a*x]^(3/2)/(c + a^2*c*x^2)^(3/2),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.91 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\text {arcsinh}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

[In]

Integrate[ArcSinh[a*x]^(3/2)/(c + a^2*c*x^2)^(3/2),x]

[Out]

Integrate[ArcSinh[a*x]^(3/2)/(c + a^2*c*x^2)^(3/2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

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

[In]

int(arcsinh(a*x)^(3/2)/(a^2*c*x^2+c)^(3/2),x)

[Out]

int(arcsinh(a*x)^(3/2)/(a^2*c*x^2+c)^(3/2),x)

Fricas [F(-2)]

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

[In]

integrate(arcsinh(a*x)^(3/2)/(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 [N/A]

Not integrable

Time = 9.76 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(asinh(a*x)**(3/2)/(a**2*c*x**2+c)**(3/2),x)

[Out]

Integral(asinh(a*x)**(3/2)/(c*(a**2*x**2 + 1))**(3/2), x)

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate(arcsinh(a*x)^(3/2)/(a^2*c*x^2 + c)^(3/2), x)

Giac [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate(arcsinh(a*x)^(3/2)/(a^2*c*x^2 + c)^(3/2), x)

Mupad [N/A]

Not integrable

Time = 2.79 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^{3/2}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

[In]

int(asinh(a*x)^(3/2)/(c + a^2*c*x^2)^(3/2),x)

[Out]

int(asinh(a*x)^(3/2)/(c + a^2*c*x^2)^(3/2), x)