\(\int \frac {1}{(d+e x^2) \sqrt {a+b \arcsin (c x)}} \, dx\) [698]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)}} \, dx=\text {Int}\left (\frac {1}{\left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)}} \, dx=\int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)}} \, dx \]

[In]

Int[1/((d + e*x^2)*Sqrt[a + b*ArcSin[c*x]]),x]

[Out]

Defer[Int][1/((d + e*x^2)*Sqrt[a + b*ArcSin[c*x]]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)}} \, dx=\int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)}} \, dx \]

[In]

Integrate[1/((d + e*x^2)*Sqrt[a + b*ArcSin[c*x]]),x]

[Out]

Integrate[1/((d + e*x^2)*Sqrt[a + b*ArcSin[c*x]]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.97 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

\[\int \frac {1}{\left (e \,x^{2}+d \right ) \sqrt {a +b \arcsin \left (c x \right )}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 2.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)}} \, dx=\int \frac {1}{\sqrt {a + b \operatorname {asin}{\left (c x \right )}} \left (d + e x^{2}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x^2+d)/(a+b*arcsin(c*x))^(1/2),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>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 0.67 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)}} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )} \sqrt {b \arcsin \left (c x\right ) + a}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)}} \, dx=\int \frac {1}{\sqrt {a+b\,\mathrm {asin}\left (c\,x\right )}\,\left (e\,x^2+d\right )} \,d x \]

[In]

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

[Out]

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