\(\int \frac {x^m (1-c^2 x^2)^{3/2}}{(a+b \arcsin (c x))^2} \, dx\) [389]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\text {Int}\left (\frac {x^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2},x\right ) \]

[Out]

Unintegrable(x^m*(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x)

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx \]

[In]

Int[(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2,x]

[Out]

Defer[Int][(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx \]

[In]

Integrate[(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2,x]

[Out]

Integrate[(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.80 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {x^{m} \left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{\left (a +b \arcsin \left (c x \right )\right )^{2}}d x\]

[In]

int(x^m*(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x)

[Out]

int(x^m*(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{m}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^m*(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral(-(c^2*x^2 - 1)*sqrt(-c^2*x^2 + 1)*x^m/(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2), x)

Sympy [N/A]

Not integrable

Time = 36.92 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x^{m} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

integrate(x**m*(-c**2*x**2+1)**(3/2)/(a+b*asin(c*x))**2,x)

[Out]

Integral(x**m*(-(c*x - 1)*(c*x + 1))**(3/2)/(a + b*asin(c*x))**2, x)

Maxima [N/A]

Not integrable

Time = 1.66 (sec) , antiderivative size = 161, normalized size of antiderivative = 5.75 \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{m}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^m*(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^m*(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x^m\,{\left (1-c^2\,x^2\right )}^{3/2}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]

[In]

int((x^m*(1 - c^2*x^2)^(3/2))/(a + b*asin(c*x))^2,x)

[Out]

int((x^m*(1 - c^2*x^2)^(3/2))/(a + b*asin(c*x))^2, x)