\(\int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx\) [279]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\text {Int}\left (\frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.60 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m}}{c^{2} d x^{2} - d} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.71 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.44 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a^{2} x^{m}}{c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{m} \operatorname {asin}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.81 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m}}{c^{2} d x^{2} - d} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.93 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m}}{c^{2} d x^{2} - d} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int \frac {x^m\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{d-c^2\,d\,x^2} \,d x \]

[In]

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

[Out]

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