\(\int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x^2)^{3/2}} \, dx\) [654]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+e x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.76 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+e x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.68 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.45 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)*sqrt(e*x^2 + d)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

Sympy [N/A]

Not integrable

Time = 4.45 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((a+b*arcsinh(c*x))^2/(e*x^2+d)^(3/2),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-c^2*d>0)', see `assume?` for
more detail

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.64 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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