\(\int \frac {\sqrt {d+e x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx\) [660]

Optimal result

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

[Out]

Unintegrable((e*x^2+d)^(1/2)/(a+b*arcsinh(c*x))^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 = {} \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {d+e x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.57 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {d+e x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {d + e x^{2}}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 575, normalized size of antiderivative = 26.14 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-((c^2*x^2 + 1)^(3/2)*sqrt(e*x^2 + d) + (c^3*x^3 + c*x)*sqrt(e*x^2 + d))/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1)*a*b*
c^2*x + a*b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log(c*x + sqrt(c^2*x^2 + 1))) + integrate(
((2*c^3*e*x^4 + c^3*d*x^2 - c*d)*(c^2*x^2 + 1)*sqrt(e*x^2 + d) + (4*c^4*e*x^5 + 2*(c^4*d + 2*c^2*e)*x^3 + (c^2
*d + e)*x)*sqrt(c^2*x^2 + 1)*sqrt(e*x^2 + d) + (2*c^5*e*x^6 + (c^5*d + 4*c^3*e)*x^4 + 2*(c^3*d + c*e)*x^2 + c*
d)*sqrt(e*x^2 + d))/(a*b*c^5*e*x^6 + (c^5*d + 2*c^3*e)*a*b*x^4 + (2*c^3*d + c*e)*a*b*x^2 + a*b*c*d + (a*b*c^3*
e*x^4 + a*b*c^3*d*x^2)*(c^2*x^2 + 1) + (b^2*c^5*e*x^6 + (c^5*d + 2*c^3*e)*b^2*x^4 + (2*c^3*d + c*e)*b^2*x^2 +
b^2*c*d + (b^2*c^3*e*x^4 + b^2*c^3*d*x^2)*(c^2*x^2 + 1) + 2*(b^2*c^4*e*x^5 + b^2*c^2*d*x + (c^4*d + c^2*e)*b^2
*x^3)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*e*x^5 + a*b*c^2*d*x + (c^4*d + c^2*e)*a*b*x
^3)*sqrt(c^2*x^2 + 1)), x)

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\sqrt {e\,x^2+d}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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