\(\int (c e+d e x)^2 (a+b \arcsin (c+d x))^3 \, dx\) [199]

Optimal result

Integrand size = 23, antiderivative size = 235 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^3 \, dx=-\frac {4}{3} a b^2 e^2 x-\frac {14 b^3 e^2 \sqrt {1-(c+d x)^2}}{9 d}+\frac {2 b^3 e^2 \left (1-(c+d x)^2\right )^{3/2}}{27 d}-\frac {4 b^3 e^2 (c+d x) \arcsin (c+d x)}{3 d}-\frac {2 b^2 e^2 (c+d x)^3 (a+b \arcsin (c+d x))}{9 d}+\frac {2 b e^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^3}{3 d} \]

[Out]

-4/3*a*b^2*e^2*x+2/27*b^3*e^2*(1-(d*x+c)^2)^(3/2)/d-4/3*b^3*e^2*(d*x+c)*arcsin(d*x+c)/d-2/9*b^2*e^2*(d*x+c)^3*
(a+b*arcsin(d*x+c))/d+1/3*e^2*(d*x+c)^3*(a+b*arcsin(d*x+c))^3/d-14/9*b^3*e^2*(1-(d*x+c)^2)^(1/2)/d+2/3*b*e^2*(
a+b*arcsin(d*x+c))^2*(1-(d*x+c)^2)^(1/2)/d+1/3*b*e^2*(d*x+c)^2*(a+b*arcsin(d*x+c))^2*(1-(d*x+c)^2)^(1/2)/d

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4889, 12, 4723, 4795, 4767, 4715, 267, 272, 45} \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^3 \, dx=-\frac {2 b^2 e^2 (c+d x)^3 (a+b \arcsin (c+d x))}{9 d}+\frac {2 b e^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^3}{3 d}-\frac {4}{3} a b^2 e^2 x-\frac {4 b^3 e^2 (c+d x) \arcsin (c+d x)}{3 d}+\frac {2 b^3 e^2 \left (1-(c+d x)^2\right )^{3/2}}{27 d}-\frac {14 b^3 e^2 \sqrt {1-(c+d x)^2}}{9 d} \]

[In]

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

[Out]

(-4*a*b^2*e^2*x)/3 - (14*b^3*e^2*Sqrt[1 - (c + d*x)^2])/(9*d) + (2*b^3*e^2*(1 - (c + d*x)^2)^(3/2))/(27*d) - (
4*b^3*e^2*(c + d*x)*ArcSin[c + d*x])/(3*d) - (2*b^2*e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x]))/(9*d) + (2*b*e^2*
Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(3*d) + (b*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSi
n[c + d*x])^2)/(3*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^3)/(3*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4715

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*
d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 4889

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^2 x^2 (a+b \arcsin (x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int x^2 (a+b \arcsin (x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^3}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x^3 (a+b \arcsin (x))^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^3}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {x (a+b \arcsin (x))^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d}-\frac {\left (2 b^2 e^2\right ) \text {Subst}\left (\int x^2 (a+b \arcsin (x)) \, dx,x,c+d x\right )}{3 d} \\ & = -\frac {2 b^2 e^2 (c+d x)^3 (a+b \arcsin (c+d x))}{9 d}+\frac {2 b e^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^3}{3 d}-\frac {\left (4 b^2 e^2\right ) \text {Subst}(\int (a+b \arcsin (x)) \, dx,x,c+d x)}{3 d}+\frac {\left (2 b^3 e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d} \\ & = -\frac {4}{3} a b^2 e^2 x-\frac {2 b^2 e^2 (c+d x)^3 (a+b \arcsin (c+d x))}{9 d}+\frac {2 b e^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^3}{3 d}+\frac {\left (b^3 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x}} \, dx,x,(c+d x)^2\right )}{9 d}-\frac {\left (4 b^3 e^2\right ) \text {Subst}(\int \arcsin (x) \, dx,x,c+d x)}{3 d} \\ & = -\frac {4}{3} a b^2 e^2 x-\frac {4 b^3 e^2 (c+d x) \arcsin (c+d x)}{3 d}-\frac {2 b^2 e^2 (c+d x)^3 (a+b \arcsin (c+d x))}{9 d}+\frac {2 b e^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^3}{3 d}+\frac {\left (b^3 e^2\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx,x,(c+d x)^2\right )}{9 d}+\frac {\left (4 b^3 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d} \\ & = -\frac {4}{3} a b^2 e^2 x-\frac {14 b^3 e^2 \sqrt {1-(c+d x)^2}}{9 d}+\frac {2 b^3 e^2 \left (1-(c+d x)^2\right )^{3/2}}{27 d}-\frac {4 b^3 e^2 (c+d x) \arcsin (c+d x)}{3 d}-\frac {2 b^2 e^2 (c+d x)^3 (a+b \arcsin (c+d x))}{9 d}+\frac {2 b e^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^3}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.85 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^3 \, dx=\frac {e^2 \left ((c+d x)^3 (a+b \arcsin (c+d x))^3-b \left (\frac {2}{9} b^2 \left (2+c^2+2 c d x+d^2 x^2\right ) \sqrt {1-(c+d x)^2}+\frac {2}{3} b (c+d x)^3 (a+b \arcsin (c+d x))-2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2-(c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2+4 b \left (a d x+b \sqrt {1-(c+d x)^2}+b (c+d x) \arcsin (c+d x)\right )\right )\right )}{3 d} \]

[In]

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

[Out]

(e^2*((c + d*x)^3*(a + b*ArcSin[c + d*x])^3 - b*((2*b^2*(2 + c^2 + 2*c*d*x + d^2*x^2)*Sqrt[1 - (c + d*x)^2])/9
+ (2*b*(c + d*x)^3*(a + b*ArcSin[c + d*x]))/3 - 2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 - (c + d*x)
^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 + 4*b*(a*d*x + b*Sqrt[1 - (c + d*x)^2] + b*(c + d*x)*ArcSin
[c + d*x]))))/(3*d)

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(\frac {\frac {e^{2} a^{3} \left (d x +c \right )^{3}}{3}+b^{3} e^{2} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{3}}{3}+\frac {\arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \arcsin \left (d x +c \right )}{3}-\frac {2 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{9}-\frac {2 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{27}\right )+3 e^{2} a \,b^{2} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{2}}{3}+\frac {2 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {2 \left (d x +c \right )^{3}}{27}-\frac {4 d x}{9}-\frac {4 c}{9}\right )+3 e^{2} a^{2} b \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{3}+\frac {\left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (d x +c \right )^{2}}}{9}\right )}{d}\)	\(280\)
default	\(\frac {\frac {e^{2} a^{3} \left (d x +c \right )^{3}}{3}+b^{3} e^{2} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{3}}{3}+\frac {\arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \arcsin \left (d x +c \right )}{3}-\frac {2 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{9}-\frac {2 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{27}\right )+3 e^{2} a \,b^{2} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{2}}{3}+\frac {2 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {2 \left (d x +c \right )^{3}}{27}-\frac {4 d x}{9}-\frac {4 c}{9}\right )+3 e^{2} a^{2} b \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{3}+\frac {\left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (d x +c \right )^{2}}}{9}\right )}{d}\)	\(280\)
parts	\(\frac {e^{2} a^{3} \left (d x +c \right )^{3}}{3 d}+\frac {b^{3} e^{2} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{3}}{3}+\frac {\arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \arcsin \left (d x +c \right )}{3}-\frac {2 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{9}-\frac {2 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{27}\right )}{d}+\frac {3 e^{2} a \,b^{2} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{2}}{3}+\frac {2 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {2 \left (d x +c \right )^{3}}{27}-\frac {4 d x}{9}-\frac {4 c}{9}\right )}{d}+\frac {3 e^{2} a^{2} b \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{3}+\frac {\left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (d x +c \right )^{2}}}{9}\right )}{d}\)	\(288\)

[In]

int((d*e*x+c*e)^2*(a+b*arcsin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/3*e^2*a^3*(d*x+c)^3+b^3*e^2*(1/3*(d*x+c)^3*arcsin(d*x+c)^3+1/3*arcsin(d*x+c)^2*((d*x+c)^2+2)*(1-(d*x+c)
^2)^(1/2)-4/3*(1-(d*x+c)^2)^(1/2)-4/3*(d*x+c)*arcsin(d*x+c)-2/9*(d*x+c)^3*arcsin(d*x+c)-2/27*((d*x+c)^2+2)*(1-
(d*x+c)^2)^(1/2))+3*e^2*a*b^2*(1/3*(d*x+c)^3*arcsin(d*x+c)^2+2/9*arcsin(d*x+c)*((d*x+c)^2+2)*(1-(d*x+c)^2)^(1/
2)-2/27*(d*x+c)^3-4/9*d*x-4/9*c)+3*e^2*a^2*b*(1/3*(d*x+c)^3*arcsin(d*x+c)+1/9*(d*x+c)^2*(1-(d*x+c)^2)^(1/2)+2/
9*(1-(d*x+c)^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (211) = 422\).

Time = 0.28 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.26 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^3 \, dx=\frac {3 \, {\left (3 \, a^{3} - 2 \, a b^{2}\right )} d^{3} e^{2} x^{3} + 9 \, {\left (3 \, a^{3} - 2 \, a b^{2}\right )} c d^{2} e^{2} x^{2} - 9 \, {\left (4 \, a b^{2} - {\left (3 \, a^{3} - 2 \, a b^{2}\right )} c^{2}\right )} d e^{2} x + 9 \, {\left (b^{3} d^{3} e^{2} x^{3} + 3 \, b^{3} c d^{2} e^{2} x^{2} + 3 \, b^{3} c^{2} d e^{2} x + b^{3} c^{3} e^{2}\right )} \arcsin \left (d x + c\right )^{3} + 27 \, {\left (a b^{2} d^{3} e^{2} x^{3} + 3 \, a b^{2} c d^{2} e^{2} x^{2} + 3 \, a b^{2} c^{2} d e^{2} x + a b^{2} c^{3} e^{2}\right )} \arcsin \left (d x + c\right )^{2} + 3 \, {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} b - 2 \, b^{3}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (4 \, b^{3} - {\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2}\right )} d e^{2} x - {\left (12 \, b^{3} c - {\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{3}\right )} e^{2}\right )} \arcsin \left (d x + c\right ) + {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (9 \, a^{2} b - 2 \, b^{3}\right )} c d e^{2} x + {\left (18 \, a^{2} b - 40 \, b^{3} + {\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2}\right )} e^{2} + 9 \, {\left (b^{3} d^{2} e^{2} x^{2} + 2 \, b^{3} c d e^{2} x + {\left (b^{3} c^{2} + 2 \, b^{3}\right )} e^{2}\right )} \arcsin \left (d x + c\right )^{2} + 18 \, {\left (a b^{2} d^{2} e^{2} x^{2} + 2 \, a b^{2} c d e^{2} x + {\left (a b^{2} c^{2} + 2 \, a b^{2}\right )} e^{2}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{27 \, d} \]

[In]

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

[Out]

1/27*(3*(3*a^3 - 2*a*b^2)*d^3*e^2*x^3 + 9*(3*a^3 - 2*a*b^2)*c*d^2*e^2*x^2 - 9*(4*a*b^2 - (3*a^3 - 2*a*b^2)*c^2
)*d*e^2*x + 9*(b^3*d^3*e^2*x^3 + 3*b^3*c*d^2*e^2*x^2 + 3*b^3*c^2*d*e^2*x + b^3*c^3*e^2)*arcsin(d*x + c)^3 + 27
*(a*b^2*d^3*e^2*x^3 + 3*a*b^2*c*d^2*e^2*x^2 + 3*a*b^2*c^2*d*e^2*x + a*b^2*c^3*e^2)*arcsin(d*x + c)^2 + 3*((9*a
^2*b - 2*b^3)*d^3*e^2*x^3 + 3*(9*a^2*b - 2*b^3)*c*d^2*e^2*x^2 - 3*(4*b^3 - (9*a^2*b - 2*b^3)*c^2)*d*e^2*x - (1
2*b^3*c - (9*a^2*b - 2*b^3)*c^3)*e^2)*arcsin(d*x + c) + ((9*a^2*b - 2*b^3)*d^2*e^2*x^2 + 2*(9*a^2*b - 2*b^3)*c
*d*e^2*x + (18*a^2*b - 40*b^3 + (9*a^2*b - 2*b^3)*c^2)*e^2 + 9*(b^3*d^2*e^2*x^2 + 2*b^3*c*d*e^2*x + (b^3*c^2 +
2*b^3)*e^2)*arcsin(d*x + c)^2 + 18*(a*b^2*d^2*e^2*x^2 + 2*a*b^2*c*d*e^2*x + (a*b^2*c^2 + 2*a*b^2)*e^2)*arcsin
(d*x + c))*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1173 vs. \(2 (211) = 422\).

Time = 0.49 (sec) , antiderivative size = 1173, normalized size of antiderivative = 4.99 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((a**3*c**2*e**2*x + a**3*c*d*e**2*x**2 + a**3*d**2*e**2*x**3/3 + a**2*b*c**3*e**2*asin(c + d*x)/d +
3*a**2*b*c**2*e**2*x*asin(c + d*x) + a**2*b*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(3*d) + 3*a**2*b*c
*d*e**2*x**2*asin(c + d*x) + 2*a**2*b*c*e**2*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/3 + a**2*b*d**2*e**2*x**3
*asin(c + d*x) + a**2*b*d*e**2*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/3 + 2*a**2*b*e**2*sqrt(-c**2 - 2*c*d
*x - d**2*x**2 + 1)/(3*d) + a*b**2*c**3*e**2*asin(c + d*x)**2/d + 3*a*b**2*c**2*e**2*x*asin(c + d*x)**2 - 2*a*
b**2*c**2*e**2*x/3 + 2*a*b**2*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(3*d) + 3*a*b**2*c
*d*e**2*x**2*asin(c + d*x)**2 - 2*a*b**2*c*d*e**2*x**2/3 + 4*a*b**2*c*e**2*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2
+ 1)*asin(c + d*x)/3 + a*b**2*d**2*e**2*x**3*asin(c + d*x)**2 - 2*a*b**2*d**2*e**2*x**3/9 + 2*a*b**2*d*e**2*x*
*2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/3 - 4*a*b**2*e**2*x/3 + 4*a*b**2*e**2*sqrt(-c**2 - 2*c*
d*x - d**2*x**2 + 1)*asin(c + d*x)/(3*d) + b**3*c**3*e**2*asin(c + d*x)**3/(3*d) - 2*b**3*c**3*e**2*asin(c + d
*x)/(9*d) + b**3*c**2*e**2*x*asin(c + d*x)**3 - 2*b**3*c**2*e**2*x*asin(c + d*x)/3 + b**3*c**2*e**2*sqrt(-c**2
- 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/(3*d) - 2*b**3*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(2
7*d) + b**3*c*d*e**2*x**2*asin(c + d*x)**3 - 2*b**3*c*d*e**2*x**2*asin(c + d*x)/3 + 2*b**3*c*e**2*x*sqrt(-c**2
- 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/3 - 4*b**3*c*e**2*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/27 - 4*
b**3*c*e**2*asin(c + d*x)/(3*d) + b**3*d**2*e**2*x**3*asin(c + d*x)**3/3 - 2*b**3*d**2*e**2*x**3*asin(c + d*x)
/9 + b**3*d*e**2*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/3 - 2*b**3*d*e**2*x**2*sqrt(-c**2
- 2*c*d*x - d**2*x**2 + 1)/27 - 4*b**3*e**2*x*asin(c + d*x)/3 + 2*b**3*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2
+ 1)*asin(c + d*x)**2/(3*d) - 40*b**3*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(27*d), Ne(d, 0)), (c**2*e**2
*x*(a + b*asin(c))**3, True))

Maxima [F]

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

[In]

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

[Out]

1/3*a^3*d^2*e^2*x^3 + a^3*c*d*e^2*x^2 + 3/2*(2*x^2*arcsin(d*x + c) + d*(3*c^2*arcsin(-(d^2*x + c*d)/sqrt(c^2*d
^2 - (c^2 - 1)*d^2))/d^3 + sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x/d^2 - (c^2 - 1)*arcsin(-(d^2*x + c*d)/sqrt(c^2
*d^2 - (c^2 - 1)*d^2))/d^3 - 3*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c/d^3))*a^2*b*c*d*e^2 + 1/6*(6*x^3*arcsin(d*
x + c) + d*(2*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x^2/d^2 - 15*c^3*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 -
1)*d^2))/d^4 - 5*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c*x/d^3 + 9*(c^2 - 1)*c*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^2
- (c^2 - 1)*d^2))/d^4 + 15*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c^2/d^4 - 4*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*
(c^2 - 1)/d^4))*a^2*b*d^2*e^2 + a^3*c^2*e^2*x + 3*((d*x + c)*arcsin(d*x + c) + sqrt(-(d*x + c)^2 + 1))*a^2*b*c
^2*e^2/d + 1/3*(b^3*d^2*e^2*x^3 + 3*b^3*c*d*e^2*x^2 + 3*b^3*c^2*e^2*x)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt
(-d*x - c + 1))^3 + integrate(((b^3*d^3*e^2*x^3 + 3*b^3*c*d^2*e^2*x^2 + 3*b^3*c^2*d*e^2*x)*sqrt(d*x + c + 1)*s
qrt(-d*x - c + 1)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + 3*(a*b^2*d^4*e^2*x^4 + 4*a*b^2*c*
d^3*e^2*x^3 + (6*a*b^2*c^2 - a*b^2)*d^2*e^2*x^2 + 2*(2*a*b^2*c^3 - a*b^2*c)*d*e^2*x + (a*b^2*c^4 - a*b^2*c^2)*
e^2)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2)/(d^2*x^2 + 2*c*d*x + c^2 - 1), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (211) = 422\).

Time = 0.35 (sec) , antiderivative size = 504, normalized size of antiderivative = 2.14 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^3 \, dx=\frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{3} e^{2} \arcsin \left (d x + c\right )^{3}}{3 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a b^{2} e^{2} \arcsin \left (d x + c\right )^{2}}{d} + \frac {{\left (d x + c\right )} b^{3} e^{2} \arcsin \left (d x + c\right )^{3}}{3 \, d} - \frac {{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b^{3} e^{2} \arcsin \left (d x + c\right )^{2}}{3 \, d} + \frac {{\left (d x + c\right )}^{3} a^{3} e^{2}}{3 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a^{2} b e^{2} \arcsin \left (d x + c\right )}{d} - \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{3} e^{2} \arcsin \left (d x + c\right )}{9 \, d} + \frac {{\left (d x + c\right )} a b^{2} e^{2} \arcsin \left (d x + c\right )^{2}}{d} - \frac {2 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a b^{2} e^{2} \arcsin \left (d x + c\right )}{3 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3} e^{2} \arcsin \left (d x + c\right )^{2}}{d} - \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a b^{2} e^{2}}{9 \, d} + \frac {{\left (d x + c\right )} a^{2} b e^{2} \arcsin \left (d x + c\right )}{d} - \frac {14 \, {\left (d x + c\right )} b^{3} e^{2} \arcsin \left (d x + c\right )}{9 \, d} - \frac {{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a^{2} b e^{2}}{3 \, d} + \frac {2 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b^{3} e^{2}}{27 \, d} + \frac {2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{2} e^{2} \arcsin \left (d x + c\right )}{d} - \frac {14 \, {\left (d x + c\right )} a b^{2} e^{2}}{9 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b e^{2}}{d} - \frac {14 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3} e^{2}}{9 \, d} \]

[In]

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

[Out]

1/3*((d*x + c)^2 - 1)*(d*x + c)*b^3*e^2*arcsin(d*x + c)^3/d + ((d*x + c)^2 - 1)*(d*x + c)*a*b^2*e^2*arcsin(d*x
+ c)^2/d + 1/3*(d*x + c)*b^3*e^2*arcsin(d*x + c)^3/d - 1/3*(-(d*x + c)^2 + 1)^(3/2)*b^3*e^2*arcsin(d*x + c)^2
/d + 1/3*(d*x + c)^3*a^3*e^2/d + ((d*x + c)^2 - 1)*(d*x + c)*a^2*b*e^2*arcsin(d*x + c)/d - 2/9*((d*x + c)^2 -
1)*(d*x + c)*b^3*e^2*arcsin(d*x + c)/d + (d*x + c)*a*b^2*e^2*arcsin(d*x + c)^2/d - 2/3*(-(d*x + c)^2 + 1)^(3/2
)*a*b^2*e^2*arcsin(d*x + c)/d + sqrt(-(d*x + c)^2 + 1)*b^3*e^2*arcsin(d*x + c)^2/d - 2/9*((d*x + c)^2 - 1)*(d*
x + c)*a*b^2*e^2/d + (d*x + c)*a^2*b*e^2*arcsin(d*x + c)/d - 14/9*(d*x + c)*b^3*e^2*arcsin(d*x + c)/d - 1/3*(-
(d*x + c)^2 + 1)^(3/2)*a^2*b*e^2/d + 2/27*(-(d*x + c)^2 + 1)^(3/2)*b^3*e^2/d + 2*sqrt(-(d*x + c)^2 + 1)*a*b^2*
e^2*arcsin(d*x + c)/d - 14/9*(d*x + c)*a*b^2*e^2/d + sqrt(-(d*x + c)^2 + 1)*a^2*b*e^2/d - 14/9*sqrt(-(d*x + c)
^2 + 1)*b^3*e^2/d

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3 \,d x \]

[In]

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

[Out]

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