\(\int (c e+d e x) (a+b \arcsin (c+d x))^4 \, dx\) [208]

Optimal result

Integrand size = 21, antiderivative size = 198 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^4 \, dx=\frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{2 d}+\frac {3 b^2 e (a+b \arcsin (c+d x))^2}{4 d}-\frac {3 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^2}{2 d}+\frac {b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{d}-\frac {e (a+b \arcsin (c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^4}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4889, 12, 4723, 4795, 4737, 30} \[ \int (c e+d e x) (a+b \arcsin (c+d x))^4 \, dx=-\frac {3 b^3 e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{2 d}-\frac {3 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^2}{2 d}+\frac {3 b^2 e (a+b \arcsin (c+d x))^2}{4 d}+\frac {b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^4}{2 d}-\frac {e (a+b \arcsin (c+d x))^4}{4 d}+\frac {3 b^4 e (c+d x)^2}{4 d} \]

[In]

Int[(c*e + d*e*x)*(a + b*ArcSin[c + d*x])^4,x]

[Out]

(3*b^4*e*(c + d*x)^2)/(4*d) - (3*b^3*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(2*d) + (3*b^2
*e*(a + b*ArcSin[c + d*x])^2)/(4*d) - (3*b^2*e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2)/(2*d) + (b*e*(c + d*x)*S
qrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/d - (e*(a + b*ArcSin[c + d*x])^4)/(4*d) + (e*(c + d*x)^2*(a +
b*ArcSin[c + d*x])^4)/(2*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -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 x (a+b \arcsin (x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int x (a+b \arcsin (x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {e (c+d x)^2 (a+b \arcsin (c+d x))^4}{2 d}-\frac {(2 b e) \text {Subst}\left (\int \frac {x^2 (a+b \arcsin (x))^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^4}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {(a+b \arcsin (x))^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}-\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int x (a+b \arcsin (x))^2 \, dx,x,c+d x\right )}{d} \\ & = -\frac {3 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^2}{2 d}+\frac {b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{d}-\frac {e (a+b \arcsin (c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^4}{2 d}+\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {x^2 (a+b \arcsin (x))}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {3 b^3 e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{2 d}-\frac {3 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^2}{2 d}+\frac {b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{d}-\frac {e (a+b \arcsin (c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^4}{2 d}+\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {a+b \arcsin (x)}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d}+\frac {\left (3 b^4 e\right ) \text {Subst}(\int x \, dx,x,c+d x)}{2 d} \\ & = \frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{2 d}+\frac {3 b^2 e (a+b \arcsin (c+d x))^2}{4 d}-\frac {3 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^2}{2 d}+\frac {b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{d}-\frac {e (a+b \arcsin (c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^4}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.82 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^4 \, dx=-\frac {e \left (-4 b (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3+(a+b \arcsin (c+d x))^4-2 (c+d x)^2 (a+b \arcsin (c+d x))^4+3 b^2 \left (-b^2 (c+d x)^2+2 b (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))-(a+b \arcsin (c+d x))^2+2 (c+d x)^2 (a+b \arcsin (c+d x))^2\right )\right )}{4 d} \]

[In]

Integrate[(c*e + d*e*x)*(a + b*ArcSin[c + d*x])^4,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(411\) vs. \(2(182)=364\).

Time = 0.88 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.08

method	result	size
derivativedivides	\(\frac {\frac {e \,a^{4} \left (d x +c \right )^{2}}{2}+e \,b^{4} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{4}}{2}+\arcsin \left (d x +c \right )^{3} \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )-\frac {3 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}-\frac {3 \arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}+\frac {3 \arcsin \left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}-\frac {3 \arcsin \left (d x +c \right )^{4}}{4}\right )+4 e a \,b^{3} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{3}}{2}+\frac {3 \arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{4}-\frac {3 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{8}-\frac {3 \arcsin \left (d x +c \right )}{8}-\frac {\arcsin \left (d x +c \right )^{3}}{2}\right )+6 e \,a^{2} b^{2} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}+\frac {\arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}-\frac {\arcsin \left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{2}}{4}\right )+4 e \,a^{3} b \left (\frac {\left (d x +c \right )^{2} \arcsin \left (d x +c \right )}{2}+\frac {\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{4}-\frac {\arcsin \left (d x +c \right )}{4}\right )}{d}\)	\(412\)
default	\(\frac {\frac {e \,a^{4} \left (d x +c \right )^{2}}{2}+e \,b^{4} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{4}}{2}+\arcsin \left (d x +c \right )^{3} \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )-\frac {3 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}-\frac {3 \arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}+\frac {3 \arcsin \left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}-\frac {3 \arcsin \left (d x +c \right )^{4}}{4}\right )+4 e a \,b^{3} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{3}}{2}+\frac {3 \arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{4}-\frac {3 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{8}-\frac {3 \arcsin \left (d x +c \right )}{8}-\frac {\arcsin \left (d x +c \right )^{3}}{2}\right )+6 e \,a^{2} b^{2} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}+\frac {\arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}-\frac {\arcsin \left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{2}}{4}\right )+4 e \,a^{3} b \left (\frac {\left (d x +c \right )^{2} \arcsin \left (d x +c \right )}{2}+\frac {\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{4}-\frac {\arcsin \left (d x +c \right )}{4}\right )}{d}\)	\(412\)
parts	\(e \,a^{4} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e \,b^{4} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{4}}{2}+\arcsin \left (d x +c \right )^{3} \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )-\frac {3 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}-\frac {3 \arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}+\frac {3 \arcsin \left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}-\frac {3 \arcsin \left (d x +c \right )^{4}}{4}\right )}{d}+\frac {4 e a \,b^{3} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{3}}{2}+\frac {3 \arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{4}-\frac {3 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{8}-\frac {3 \arcsin \left (d x +c \right )}{8}-\frac {\arcsin \left (d x +c \right )^{3}}{2}\right )}{d}+\frac {6 e \,a^{2} b^{2} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}+\frac {\arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}-\frac {\arcsin \left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{2}}{4}\right )}{d}+\frac {4 e \,a^{3} b \left (\frac {\left (d x +c \right )^{2} \arcsin \left (d x +c \right )}{2}+\frac {\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{4}-\frac {\arcsin \left (d x +c \right )}{4}\right )}{d}\)	\(422\)

[In]

int((d*e*x+c*e)*(a+b*arcsin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/2*e*a^4*(d*x+c)^2+e*b^4*(1/2*((d*x+c)^2-1)*arcsin(d*x+c)^4+arcsin(d*x+c)^3*((d*x+c)*(1-(d*x+c)^2)^(1/2)
+arcsin(d*x+c))-3/2*((d*x+c)^2-1)*arcsin(d*x+c)^2-3/2*arcsin(d*x+c)*((d*x+c)*(1-(d*x+c)^2)^(1/2)+arcsin(d*x+c)
)+3/4*arcsin(d*x+c)^2+3/4*(d*x+c)^2-3/4*arcsin(d*x+c)^4)+4*e*a*b^3*(1/2*((d*x+c)^2-1)*arcsin(d*x+c)^3+3/4*arcs
in(d*x+c)^2*((d*x+c)*(1-(d*x+c)^2)^(1/2)+arcsin(d*x+c))-3/4*((d*x+c)^2-1)*arcsin(d*x+c)-3/8*(d*x+c)*(1-(d*x+c)
^2)^(1/2)-3/8*arcsin(d*x+c)-1/2*arcsin(d*x+c)^3)+6*e*a^2*b^2*(1/2*((d*x+c)^2-1)*arcsin(d*x+c)^2+1/2*arcsin(d*x
+c)*((d*x+c)*(1-(d*x+c)^2)^(1/2)+arcsin(d*x+c))-1/4*arcsin(d*x+c)^2-1/4*(d*x+c)^2)+4*e*a^3*b*(1/2*(d*x+c)^2*ar
csin(d*x+c)+1/4*(d*x+c)*(1-(d*x+c)^2)^(1/2)-1/4*arcsin(d*x+c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (182) = 364\).

Time = 0.28 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.44 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^4 \, dx=\frac {{\left (2 \, a^{4} - 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{4} - 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d e x + {\left (2 \, b^{4} d^{2} e x^{2} + 4 \, b^{4} c d e x + {\left (2 \, b^{4} c^{2} - b^{4}\right )} e\right )} \arcsin \left (d x + c\right )^{4} + 4 \, {\left (2 \, a b^{3} d^{2} e x^{2} + 4 \, a b^{3} c d e x + {\left (2 \, a b^{3} c^{2} - a b^{3}\right )} e\right )} \arcsin \left (d x + c\right )^{3} + 3 \, {\left (2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} c d e x - {\left (2 \, a^{2} b^{2} - b^{4} - 2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} c^{2}\right )} e\right )} \arcsin \left (d x + c\right )^{2} + 2 \, {\left (2 \, {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} c d e x - {\left (2 \, a^{3} b - 3 \, a b^{3} - 2 \, {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} c^{2}\right )} e\right )} \arcsin \left (d x + c\right ) + 2 \, {\left ({\left (2 \, a^{3} b - 3 \, a b^{3}\right )} d e x + 2 \, {\left (b^{4} d e x + b^{4} c e\right )} \arcsin \left (d x + c\right )^{3} + {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} c e + 6 \, {\left (a b^{3} d e x + a b^{3} c e\right )} \arcsin \left (d x + c\right )^{2} + 3 \, {\left ({\left (2 \, a^{2} b^{2} - b^{4}\right )} d e x + {\left (2 \, a^{2} b^{2} - b^{4}\right )} c e\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{4 \, d} \]

[In]

integrate((d*e*x+c*e)*(a+b*arcsin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/4*((2*a^4 - 6*a^2*b^2 + 3*b^4)*d^2*e*x^2 + 2*(2*a^4 - 6*a^2*b^2 + 3*b^4)*c*d*e*x + (2*b^4*d^2*e*x^2 + 4*b^4*
c*d*e*x + (2*b^4*c^2 - b^4)*e)*arcsin(d*x + c)^4 + 4*(2*a*b^3*d^2*e*x^2 + 4*a*b^3*c*d*e*x + (2*a*b^3*c^2 - a*b
^3)*e)*arcsin(d*x + c)^3 + 3*(2*(2*a^2*b^2 - b^4)*d^2*e*x^2 + 4*(2*a^2*b^2 - b^4)*c*d*e*x - (2*a^2*b^2 - b^4 -
2*(2*a^2*b^2 - b^4)*c^2)*e)*arcsin(d*x + c)^2 + 2*(2*(2*a^3*b - 3*a*b^3)*d^2*e*x^2 + 4*(2*a^3*b - 3*a*b^3)*c*
d*e*x - (2*a^3*b - 3*a*b^3 - 2*(2*a^3*b - 3*a*b^3)*c^2)*e)*arcsin(d*x + c) + 2*((2*a^3*b - 3*a*b^3)*d*e*x + 2*
(b^4*d*e*x + b^4*c*e)*arcsin(d*x + c)^3 + (2*a^3*b - 3*a*b^3)*c*e + 6*(a*b^3*d*e*x + a*b^3*c*e)*arcsin(d*x + c
)^2 + 3*((2*a^2*b^2 - b^4)*d*e*x + (2*a^2*b^2 - b^4)*c*e)*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. 1027 vs. \(2 (178) = 356\).

Time = 0.49 (sec) , antiderivative size = 1027, normalized size of antiderivative = 5.19 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^4 \, dx=\text {Too large to display} \]

[In]

integrate((d*e*x+c*e)*(a+b*asin(d*x+c))**4,x)

[Out]

Piecewise((a**4*c*e*x + a**4*d*e*x**2/2 + 2*a**3*b*c**2*e*asin(c + d*x)/d + 4*a**3*b*c*e*x*asin(c + d*x) + a**
3*b*c*e*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/d + 2*a**3*b*d*e*x**2*asin(c + d*x) + a**3*b*e*x*sqrt(-c**2 - 2*
c*d*x - d**2*x**2 + 1) - a**3*b*e*asin(c + d*x)/d + 3*a**2*b**2*c**2*e*asin(c + d*x)**2/d + 6*a**2*b**2*c*e*x*
asin(c + d*x)**2 - 3*a**2*b**2*c*e*x + 3*a**2*b**2*c*e*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/d +
3*a**2*b**2*d*e*x**2*asin(c + d*x)**2 - 3*a**2*b**2*d*e*x**2/2 + 3*a**2*b**2*e*x*sqrt(-c**2 - 2*c*d*x - d**2*
x**2 + 1)*asin(c + d*x) - 3*a**2*b**2*e*asin(c + d*x)**2/(2*d) + 2*a*b**3*c**2*e*asin(c + d*x)**3/d - 3*a*b**3
*c**2*e*asin(c + d*x)/d + 4*a*b**3*c*e*x*asin(c + d*x)**3 - 6*a*b**3*c*e*x*asin(c + d*x) + 3*a*b**3*c*e*sqrt(-
c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/d - 3*a*b**3*c*e*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(2*d)
+ 2*a*b**3*d*e*x**2*asin(c + d*x)**3 - 3*a*b**3*d*e*x**2*asin(c + d*x) + 3*a*b**3*e*x*sqrt(-c**2 - 2*c*d*x - d
**2*x**2 + 1)*asin(c + d*x)**2 - 3*a*b**3*e*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/2 - a*b**3*e*asin(c + d*x)
**3/d + 3*a*b**3*e*asin(c + d*x)/(2*d) + b**4*c**2*e*asin(c + d*x)**4/(2*d) - 3*b**4*c**2*e*asin(c + d*x)**2/(
2*d) + b**4*c*e*x*asin(c + d*x)**4 - 3*b**4*c*e*x*asin(c + d*x)**2 + 3*b**4*c*e*x/2 + b**4*c*e*sqrt(-c**2 - 2*
c*d*x - d**2*x**2 + 1)*asin(c + d*x)**3/d - 3*b**4*c*e*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(2*
d) + b**4*d*e*x**2*asin(c + d*x)**4/2 - 3*b**4*d*e*x**2*asin(c + d*x)**2/2 + 3*b**4*d*e*x**2/4 + b**4*e*x*sqrt
(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**3 - 3*b**4*e*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c +
d*x)/2 - b**4*e*asin(c + d*x)**4/(4*d) + 3*b**4*e*asin(c + d*x)**2/(4*d), Ne(d, 0)), (c*e*x*(a + b*asin(c))**
4, True))

Maxima [F]

\[ \int (c e+d e x) (a+b \arcsin (c+d x))^4 \, dx=\int { {\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{4} \,d x } \]

[In]

integrate((d*e*x+c*e)*(a+b*arcsin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/2*a^4*d*e*x^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^3*b*d*e + a^4*c*e*x + 4*((d*x + c)*arcsin(d*x + c) + sqrt(-(
d*x + c)^2 + 1))*a^3*b*c*e/d + 1/2*(b^4*d*e*x^2 + 2*b^4*c*e*x)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x -
c + 1))^4 + integrate(2*((b^4*d^2*e*x^2 + 2*b^4*c*d*e*x)*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)*arctan2(d*x + c,
sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 + 2*(a*b^3*d^3*e*x^3 + 3*a*b^3*c*d^2*e*x^2 + (3*a*b^3*c^2 - a*b^3)*d*
e*x + (a*b^3*c^3 - a*b^3*c)*e)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 + 3*(a^2*b^2*d^3*e*x^3
+ 3*a^2*b^2*c*d^2*e*x^2 + (3*a^2*b^2*c^2 - a^2*b^2)*d*e*x + (a^2*b^2*c^3 - a^2*b^2*c)*e)*arctan2(d*x + c, sqr
t(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. 533 vs. \(2 (182) = 364\).

Time = 0.37 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.69 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^4 \, dx=\frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{4} e \arcsin \left (d x + c\right )^{4}}{2 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{4} e \arcsin \left (d x + c\right )^{3}}{d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{3} e \arcsin \left (d x + c\right )^{3}}{d} + \frac {b^{4} e \arcsin \left (d x + c\right )^{4}}{4 \, d} + \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b^{3} e \arcsin \left (d x + c\right )^{2}}{d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} b^{2} e \arcsin \left (d x + c\right )^{2}}{d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} b^{4} e \arcsin \left (d x + c\right )^{2}}{2 \, d} + \frac {a b^{3} e \arcsin \left (d x + c\right )^{3}}{d} + \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a^{2} b^{2} e \arcsin \left (d x + c\right )}{d} - \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{4} e \arcsin \left (d x + c\right )}{2 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a^{3} b e \arcsin \left (d x + c\right )}{d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{3} e \arcsin \left (d x + c\right )}{d} + \frac {3 \, a^{2} b^{2} e \arcsin \left (d x + c\right )^{2}}{2 \, d} - \frac {3 \, b^{4} e \arcsin \left (d x + c\right )^{2}}{4 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a^{3} b e}{d} - \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b^{3} e}{2 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} a^{4} e}{2 \, d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} b^{2} e}{2 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} b^{4} e}{4 \, d} + \frac {a^{3} b e \arcsin \left (d x + c\right )}{d} - \frac {3 \, a b^{3} e \arcsin \left (d x + c\right )}{2 \, d} - \frac {3 \, a^{2} b^{2} e}{4 \, d} + \frac {3 \, b^{4} e}{8 \, d} \]

[In]

integrate((d*e*x+c*e)*(a+b*arcsin(d*x+c))^4,x, algorithm="giac")

[Out]

1/2*((d*x + c)^2 - 1)*b^4*e*arcsin(d*x + c)^4/d + sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^4*e*arcsin(d*x + c)^3/d +
2*((d*x + c)^2 - 1)*a*b^3*e*arcsin(d*x + c)^3/d + 1/4*b^4*e*arcsin(d*x + c)^4/d + 3*sqrt(-(d*x + c)^2 + 1)*(d
*x + c)*a*b^3*e*arcsin(d*x + c)^2/d + 3*((d*x + c)^2 - 1)*a^2*b^2*e*arcsin(d*x + c)^2/d - 3/2*((d*x + c)^2 - 1
)*b^4*e*arcsin(d*x + c)^2/d + a*b^3*e*arcsin(d*x + c)^3/d + 3*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a^2*b^2*e*arcsi
n(d*x + c)/d - 3/2*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^4*e*arcsin(d*x + c)/d + 2*((d*x + c)^2 - 1)*a^3*b*e*arcs
in(d*x + c)/d - 3*((d*x + c)^2 - 1)*a*b^3*e*arcsin(d*x + c)/d + 3/2*a^2*b^2*e*arcsin(d*x + c)^2/d - 3/4*b^4*e*
arcsin(d*x + c)^2/d + sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a^3*b*e/d - 3/2*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a*b^3*
e/d + 1/2*((d*x + c)^2 - 1)*a^4*e/d - 3/2*((d*x + c)^2 - 1)*a^2*b^2*e/d + 3/4*((d*x + c)^2 - 1)*b^4*e/d + a^3*
b*e*arcsin(d*x + c)/d - 3/2*a*b^3*e*arcsin(d*x + c)/d - 3/4*a^2*b^2*e/d + 3/8*b^4*e/d

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x) (a+b \arcsin (c+d x))^4 \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4 \,d x \]

[In]

int((c*e + d*e*x)*(a + b*asin(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)*(a + b*asin(c + d*x))^4, x)