\(\int \frac {(a+b \arcsin (c x))^{3/2}}{x} \, dx\) [181]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {(a+b \arcsin (c x))^{3/2}}{x} \, dx=\text {Int}\left (\frac {(a+b \arcsin (c x))^{3/2}}{x},x\right ) \]

[Out]

Unintegrable((a+b*arcsin(c*x))^(3/2)/x,x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {(a+b \arcsin (c x))^{3/2}}{x} \, dx=\int \frac {(a+b \arcsin (c x))^{3/2}}{x} \, dx \]

[In]

Int[(a + b*ArcSin[c*x])^(3/2)/x,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b \arcsin (c x))^{3/2}}{x} \, dx=\int \frac {(a+b \arcsin (c x))^{3/2}}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

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

[In]

int((a+b*arcsin(c*x))^(3/2)/x,x)

[Out]

int((a+b*arcsin(c*x))^(3/2)/x,x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c x))^{3/2}}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 13.99 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \arcsin (c x))^{3/2}}{x} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}}}{x}\, dx \]

[In]

integrate((a+b*asin(c*x))**(3/2)/x,x)

[Out]

Integral((a + b*asin(c*x))**(3/2)/x, x)

Maxima [N/A]

Not integrable

Time = 0.77 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arcsin (c x))^{3/2}}{x} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}}{x} \,d x } \]

[In]

integrate((a+b*arcsin(c*x))^(3/2)/x,x, algorithm="maxima")

[Out]

integrate((b*arcsin(c*x) + a)^(3/2)/x, x)

Giac [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arcsin (c x))^{3/2}}{x} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}}{x} \,d x } \]

[In]

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

[Out]

integrate((b*arcsin(c*x) + a)^(3/2)/x, x)

Mupad [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arcsin (c x))^{3/2}}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}}{x} \,d x \]

[In]

int((a + b*asin(c*x))^(3/2)/x,x)

[Out]

int((a + b*asin(c*x))^(3/2)/x, x)