\(\int \frac {x^m (a+b \arcsin (c x))}{(d-c^2 d x^2)^2} \, dx\) [147]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {x^{1+m} (a+b \arcsin (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {b c x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{2 d^2 (2+m)}+\frac {(1-m) \text {Int}\left (\frac {x^m (a+b \arcsin (c x))}{d-c^2 d x^2},x\right )}{2 d} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx \]

[In]

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

[Out]

(x^(1 + m)*(a + b*ArcSin[c*x]))/(2*d^2*(1 - c^2*x^2)) - (b*c*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 +
m)/2, c^2*x^2])/(2*d^2*(2 + m)) + ((1 - m)*Defer[Int][(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2), x])/(2*d)

Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} (a+b \arcsin (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {(b c) \int \frac {x^{1+m}}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {(1-m) \int \frac {x^m (a+b \arcsin (c x))}{d-c^2 d x^2} \, dx}{2 d} \\ & = \frac {x^{1+m} (a+b \arcsin (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {b c x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{2 d^2 (2+m)}+\frac {(1-m) \int \frac {x^m (a+b \arcsin (c x))}{d-c^2 d x^2} \, dx}{2 d} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 5.78 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.40 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral((b*arcsin(c*x) + a)*x^m/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)

Sympy [N/A]

Not integrable

Time = 16.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x^{m}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{m} \operatorname {asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]

[In]

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

[Out]

(Integral(a*x**m/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b*x**m*asin(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),
x))/d**2

Maxima [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^m\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]

[In]

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

[Out]

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