\(\int \frac {(d x)^{3/2}}{(a+b \arcsin (c x))^2} \, dx\) [224]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {(d x)^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\text {Int}\left (\frac {(d x)^{3/2}}{(a+b \arcsin (c x))^2},x\right ) \]

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 8.69 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(d x)^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {(d x)^{3/2}}{(a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {(d x)^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {\left (d x\right )^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 10.74 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {(d x)^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {\left (d x\right )^{\frac {3}{2}}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.91 (sec) , antiderivative size = 182, normalized size of antiderivative = 10.11 \[ \int \frac {(d x)^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {\left (d x\right )^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-(sqrt(c*x + 1)*sqrt(-c*x + 1)*d^(3/2)*x^(3/2) - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*sq
rt(d)*integrate(1/2*(5*c^2*d*x^2 - 3*d)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(x)/(a*b*c^3*x^2 - a*b*c + (b^2*c^3*x
^2 - b^2*c)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))), x))/(b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))
+ a*b*c)

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(d x)^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {\left (d x\right )^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(d x)^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {{\left (d\,x\right )}^{3/2}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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