\(\int \frac {\sqrt {d+e x^2}}{(a+b \arcsin (c x))^2} \, dx\) [682]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 7.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {\sqrt {d+e x^2}}{(a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {\sqrt {d + e x^{2}}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.22 (sec) , antiderivative size = 236, normalized size of antiderivative = 10.73 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

((b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*integrate((2*c^2*e*x^3 + (c^2*d - e)*x)*sqrt(e*x^2
+ d)*sqrt(c*x + 1)*sqrt(-c*x + 1)/(a*b*c^3*e*x^4 - a*b*c*d + (a*b*c^3*d - a*b*c*e)*x^2 + (b^2*c^3*e*x^4 - b^2
*c*d + (b^2*c^3*d - b^2*c*e)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))), x) - sqrt(e*x^2 + d)*sqrt(c*x +
1)*sqrt(-c*x + 1))/(b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)

Giac [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {\sqrt {e\,x^2+d}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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