\(\int \frac {\sqrt {d x}}{a+b \arcsin (c x)} \, dx\) [221]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {\sqrt {d x}}{a+b \arcsin (c x)} \, dx=\text {Int}\left (\frac {\sqrt {d x}}{a+b \arcsin (c x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {\sqrt {d x}}{a+b \arcsin (c x)} \, dx=\int \frac {\sqrt {d x}}{a+b \arcsin (c x)} \, dx \]

[In]

Int[Sqrt[d*x]/(a + b*ArcSin[c*x]),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.60 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {d x}}{a+b \arcsin (c x)} \, dx=\int \frac {\sqrt {d x}}{a+b \arcsin (c x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d x}}{a+b \arcsin (c x)} \, dx=\int { \frac {\sqrt {d x}}{b \arcsin \left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {d x}}{a+b \arcsin (c x)} \, dx=\int \frac {\sqrt {d x}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d x}}{a+b \arcsin (c x)} \, dx=\int { \frac {\sqrt {d x}}{b \arcsin \left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d x}}{a+b \arcsin (c x)} \, dx=\int { \frac {\sqrt {d x}}{b \arcsin \left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d x}}{a+b \arcsin (c x)} \, dx=\int \frac {\sqrt {d\,x}}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]

[In]

int((d*x)^(1/2)/(a + b*asin(c*x)),x)

[Out]

int((d*x)^(1/2)/(a + b*asin(c*x)), x)