\(\int \frac {1}{x^2 (a+b \arcsin (c x))^{5/2}} \, dx\) [202]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^{5/2}} \, dx=\text {Int}\left (\frac {1}{x^2 (a+b \arcsin (c x))^{5/2}},x\right ) \]

[Out]

Unintegrable(1/x^2/(a+b*arcsin(c*x))^(5/2),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 = {} \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^{5/2}} \, dx=\int \frac {1}{x^2 (a+b \arcsin (c x))^{5/2}} \, dx \]

[In]

Int[1/(x^2*(a + b*ArcSin[c*x])^(5/2)),x]

[Out]

Defer[Int][1/(x^2*(a + b*ArcSin[c*x])^(5/2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 (a+b \arcsin (c x))^{5/2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.79 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^{5/2}} \, dx=\int \frac {1}{x^2 (a+b \arcsin (c x))^{5/2}} \, dx \]

[In]

Integrate[1/(x^2*(a + b*ArcSin[c*x])^(5/2)),x]

[Out]

Integrate[1/(x^2*(a + b*ArcSin[c*x])^(5/2)), x]

Maple [N/A] (verified)

Not integrable

Time = 0.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

\[\int \frac {1}{x^{2} \left (a +b \arcsin \left (c x \right )\right )^{\frac {5}{2}}}d x\]

[In]

int(1/x^2/(a+b*arcsin(c*x))^(5/2),x)

[Out]

int(1/x^2/(a+b*arcsin(c*x))^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/x^2/(a+b*arcsin(c*x))^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 16.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^{5/2}} \, dx=\int \frac {1}{x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/x**2/(a+b*asin(c*x))**(5/2),x)

[Out]

Integral(1/(x**2*(a + b*asin(c*x))**(5/2)), x)

Maxima [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*arcsin(c*x))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((b*arcsin(c*x) + a)^(5/2)*x^2), x)

Giac [N/A]

Not integrable

Time = 1.57 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*arcsin(c*x))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((b*arcsin(c*x) + a)^(5/2)*x^2), x)

Mupad [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^{5/2}} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{5/2}} \,d x \]

[In]

int(1/(x^2*(a + b*asin(c*x))^(5/2)),x)

[Out]

int(1/(x^2*(a + b*asin(c*x))^(5/2)), x)