\(\int \frac {1}{(d+e x^2)^{3/2} (a+b \arcsin (c x))^2} \, dx\) [684]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\text {Int}\left (\frac {1}{\left (d+e x^2\right )^{3/2} (a+b \arcsin (c x))^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 25.69 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.91 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(sqrt(e*x^2 + d)/(a^2*e^2*x^4 + 2*a^2*d*e*x^2 + a^2*d^2 + (b^2*e^2*x^4 + 2*b^2*d*e*x^2 + b^2*d^2)*arcs
in(c*x)^2 + 2*(a*b*e^2*x^4 + 2*a*b*d*e*x^2 + a*b*d^2)*arcsin(c*x)), x)

Sympy [N/A]

Not integrable

Time = 4.93 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(e*x**2+d)**(3/2)/(a+b*asin(c*x))**2,x)

[Out]

Integral(1/((a + b*asin(c*x))**2*(d + e*x**2)**(3/2)), x)

Maxima [N/A]

Not integrable

Time = 2.77 (sec) , antiderivative size = 457, normalized size of antiderivative = 20.77 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-((a*b*c*e^2*x^4 + 2*a*b*c*d*e*x^2 + a*b*c*d^2 + (b^2*c*e^2*x^4 + 2*b^2*c*d*e*x^2 + b^2*c*d^2)*arctan2(c*x, sq
rt(c*x + 1)*sqrt(-c*x + 1)))*integrate((2*c^2*e*x^3 - (c^2*d + 3*e)*x)*sqrt(e*x^2 + d)*sqrt(c*x + 1)*sqrt(-c*x
+ 1)/(a*b*c^3*e^3*x^8 + (3*a*b*c^3*d*e^2 - a*b*c*e^3)*x^6 - a*b*c*d^3 + 3*(a*b*c^3*d^2*e - a*b*c*d*e^2)*x^4 +
(a*b*c^3*d^3 - 3*a*b*c*d^2*e)*x^2 + (b^2*c^3*e^3*x^8 + (3*b^2*c^3*d*e^2 - b^2*c*e^3)*x^6 - b^2*c*d^3 + 3*(b^2
*c^3*d^2*e - b^2*c*d*e^2)*x^4 + (b^2*c^3*d^3 - 3*b^2*c*d^2*e)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))
, x) + sqrt(e*x^2 + d)*sqrt(c*x + 1)*sqrt(-c*x + 1))/(a*b*c*e^2*x^4 + 2*a*b*c*d*e*x^2 + a*b*c*d^2 + (b^2*c*e^2
*x^4 + 2*b^2*c*d*e*x^2 + b^2*c*d^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))

Giac [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate(1/((e*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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