\(\int (c e+d e x)^m (a+b \arcsin (c+d x))^3 \, dx\) [309]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^3 \, dx=\frac {(e (c+d x))^{1+m} (a+b \arcsin (c+d x))^3}{d e (1+m)}-\frac {3 b \text {Int}\left (\frac {(e (c+d x))^{1+m} (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}},x\right )}{e (1+m)} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 23, 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 = {} \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^3 \, dx=\int (c e+d e x)^m (a+b \arcsin (c+d x))^3 \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.81 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^3 \, dx=\int (c e+d e x)^m (a+b \arcsin (c+d x))^3 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

int((d*e*x+c*e)^m*(a+b*arcsin(d*x+c))^3,x)

[Out]

int((d*e*x+c*e)^m*(a+b*arcsin(d*x+c))^3,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^3 \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{3} {\left (d e x + c e\right )}^{m} \,d x } \]

[In]

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

[Out]

integral((b^3*arcsin(d*x + c)^3 + 3*a*b^2*arcsin(d*x + c)^2 + 3*a^2*b*arcsin(d*x + c) + a^3)*(d*e*x + c*e)^m,
x)

Sympy [N/A]

Not integrable

Time = 20.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^3 \, dx=\int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{3}\, dx \]

[In]

integrate((d*e*x+c*e)**m*(a+b*asin(d*x+c))**3,x)

[Out]

Integral((e*(c + d*x))**m*(a + b*asin(c + d*x))**3, x)

Maxima [N/A]

Not integrable

Time = 7.43 (sec) , antiderivative size = 469, normalized size of antiderivative = 20.39 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^3 \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{3} {\left (d e x + c e\right )}^{m} \,d x } \]

[In]

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

[Out]

(d*e*x + c*e)^(m + 1)*a^3/(d*e*(m + 1)) + ((b^3*d*e^m*x + b^3*c*e^m)*(d*x + c)^m*arctan2(d*x + c, sqrt(d*x + c
+ 1)*sqrt(-d*x - c + 1))^3 + (d*m + d)*integrate(3*((b^3*d*e^m*x + b^3*c*e^m)*sqrt(d*x + c + 1)*sqrt(-d*x - c
+ 1)*(d*x + c)^m*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + ((a*b^2*c^2 - a*b^2)*e^m*m + (a*b
^2*d^2*e^m*m + a*b^2*d^2*e^m)*x^2 + (a*b^2*c^2 - a*b^2)*e^m + 2*(a*b^2*c*d*e^m*m + a*b^2*c*d*e^m)*x)*(d*x + c)
^m*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + ((a^2*b*c^2 - a^2*b)*e^m*m + (a^2*b*d^2*e^m*m +
a^2*b*d^2*e^m)*x^2 + (a^2*b*c^2 - a^2*b)*e^m + 2*(a^2*b*c*d*e^m*m + a^2*b*c*d*e^m)*x)*(d*x + c)^m*arctan2(d*x
+ c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/((d^2*m + d^2)*x^2 + c^2 + (c^2 - 1)*m + 2*(c*d*m + c*d)*x - 1), x
))/(d*m + d)

Giac [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^3 \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{3} {\left (d e x + c e\right )}^{m} \,d x } \]

[In]

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

[Out]

integrate((b*arcsin(d*x + c) + a)^3*(d*e*x + c*e)^m, x)

Mupad [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^m\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int((c*e + d*e*x)^m*(a + b*asin(c + d*x))^3,x)

[Out]

int((c*e + d*e*x)^m*(a + b*asin(c + d*x))^3, x)