\(\int \frac {1}{x^2 (1-c^2 x^2)^{5/2} (a+b \arcsin (c x))} \, dx\) [366]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))} \, dx=\text {Int}\left (\frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))},x\right ) \]

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 10.60 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))} \, dx=\int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.18 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

integral(-sqrt(-c^2*x^2 + 1)/(a*c^6*x^8 - 3*a*c^4*x^6 + 3*a*c^2*x^4 - a*x^2 + (b*c^6*x^8 - 3*b*c^4*x^6 + 3*b*c
^2*x^4 - b*x^2)*arcsin(c*x)), x)

Sympy [N/A]

Not integrable

Time = 3.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))} \, dx=\int \frac {1}{x^{2} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 9.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))} \, dx=\int \frac {1}{x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (1-c^2\,x^2\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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