3.198 \(\int (c e+d e x)^3 (a+b \sin ^{-1}(c+d x))^3 \, dx\)

Optimal. Leaf size=287 \[ -\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}-\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}+\frac{3 b e^3 \sqrt{1-(c+d x)^2} (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}+\frac{9 b e^3 \sqrt{1-(c+d x)^2} (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{32 d}-\frac{3 b^3 e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{128 d}-\frac{45 b^3 e^3 \sqrt{1-(c+d x)^2} (c+d x)}{256 d}+\frac{45 b^3 e^3 \sin ^{-1}(c+d x)}{256 d} \]

[Out]

(-45*b^3*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(256*d) - (3*b^3*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(128*d)
+ (45*b^3*e^3*ArcSin[c + d*x])/(256*d) - (9*b^2*e^3*(c + d*x)^2*(a + b*ArcSin[c + d*x]))/(32*d) - (3*b^2*e^3*(
c + d*x)^4*(a + b*ArcSin[c + d*x]))/(32*d) + (9*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^
2)/(32*d) + (3*b*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(16*d) - (3*e^3*(a + b*ArcSi
n[c + d*x])^3)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^3)/(4*d)

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Rubi [A]  time = 0.400394, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4805, 12, 4627, 4707, 4641, 321, 216} \[ -\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}-\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}+\frac{3 b e^3 \sqrt{1-(c+d x)^2} (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}+\frac{9 b e^3 \sqrt{1-(c+d x)^2} (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{32 d}-\frac{3 b^3 e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{128 d}-\frac{45 b^3 e^3 \sqrt{1-(c+d x)^2} (c+d x)}{256 d}+\frac{45 b^3 e^3 \sin ^{-1}(c+d x)}{256 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*b^3*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(256*d) - (3*b^3*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(128*d)
+ (45*b^3*e^3*ArcSin[c + d*x])/(256*d) - (9*b^2*e^3*(c + d*x)^2*(a + b*ArcSin[c + d*x]))/(32*d) - (3*b^2*e^3*(
c + d*x)^4*(a + b*ArcSin[c + d*x]))/(32*d) + (9*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^
2)/(32*d) + (3*b*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(16*d) - (3*e^3*(a + b*ArcSi
n[c + d*x])^3)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^3)/(4*d)

Rule 4805

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]

Rule 12

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

Rule 4627

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 4707

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

Rule 4641

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

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

Rubi steps

\begin{align*} \int (c e+d e x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int e^3 x^3 \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int x^3 \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}-\frac{\left (3 b e^3\right ) \operatorname{Subst}\left (\int \frac{x^4 \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}-\frac{\left (9 b e^3\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{16 d}-\frac{\left (3 b^2 e^3\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{8 d}\\ &=-\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}+\frac{9 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}+\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}-\frac{\left (9 b e^3\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \sin ^{-1}(x)\right )^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{32 d}-\frac{\left (9 b^2 e^3\right ) \operatorname{Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{16 d}+\frac{\left (3 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{32 d}\\ &=-\frac{3 b^3 e^3 (c+d x)^3 \sqrt{1-(c+d x)^2}}{128 d}-\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}-\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}+\frac{9 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}+\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}+\frac{\left (9 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{128 d}+\frac{\left (9 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{32 d}\\ &=-\frac{45 b^3 e^3 (c+d x) \sqrt{1-(c+d x)^2}}{256 d}-\frac{3 b^3 e^3 (c+d x)^3 \sqrt{1-(c+d x)^2}}{128 d}-\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}-\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}+\frac{9 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}+\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}+\frac{\left (9 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{256 d}+\frac{\left (9 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{64 d}\\ &=-\frac{45 b^3 e^3 (c+d x) \sqrt{1-(c+d x)^2}}{256 d}-\frac{3 b^3 e^3 (c+d x)^3 \sqrt{1-(c+d x)^2}}{128 d}+\frac{45 b^3 e^3 \sin ^{-1}(c+d x)}{256 d}-\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}-\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}+\frac{9 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}+\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.498578, size = 232, normalized size = 0.81 \[ \frac{e^3 \left ((c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3-\frac{3}{8} \left (b^2 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )+3 b^2 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )-2 b \sqrt{1-(c+d x)^2} (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2-3 b \sqrt{1-(c+d x)^2} (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2+\left (a+b \sin ^{-1}(c+d x)\right )^3+\frac{1}{4} b^3 \sqrt{1-(c+d x)^2} (c+d x)^3+\frac{15}{8} b^3 \sqrt{1-(c+d x)^2} (c+d x)-\frac{15}{8} b^3 \sin ^{-1}(c+d x)\right )\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^3*((c + d*x)^4*(a + b*ArcSin[c + d*x])^3 - (3*((15*b^3*(c + d*x)*Sqrt[1 - (c + d*x)^2])/8 + (b^3*(c + d*x)^
3*Sqrt[1 - (c + d*x)^2])/4 - (15*b^3*ArcSin[c + d*x])/8 + 3*b^2*(c + d*x)^2*(a + b*ArcSin[c + d*x]) + b^2*(c +
d*x)^4*(a + b*ArcSin[c + d*x]) - 3*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 - 2*b*(c + d*x
)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 + (a + b*ArcSin[c + d*x])^3))/8))/(4*d)

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Maple [A]  time = 0.043, size = 397, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({\frac{{e}^{3} \left ( dx+c \right ) ^{4}{a}^{3}}{4}}+{e}^{3}{b}^{3} \left ({\frac{ \left ( dx+c \right ) ^{4} \left ( \arcsin \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}}{32} \left ( -2\, \left ( dx+c \right ) ^{3}\sqrt{1- \left ( dx+c \right ) ^{2}}-3\, \left ( dx+c \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}+3\,\arcsin \left ( dx+c \right ) \right ) }-{\frac{3\, \left ( dx+c \right ) ^{4}\arcsin \left ( dx+c \right ) }{32}}-{\frac{ \left ( 3\,dx+3\,c \right ) \left ( 2\, \left ( dx+c \right ) ^{2}+3 \right ) }{256}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{27\,\arcsin \left ( dx+c \right ) }{256}}-{\frac{9\,\arcsin \left ( dx+c \right ) \left ( \left ( dx+c \right ) ^{2}-1 \right ) }{32}}-{\frac{9\,dx+9\,c}{64}\sqrt{1- \left ( dx+c \right ) ^{2}}}+{\frac{3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{3}}{16}} \right ) +3\,{e}^{3}a{b}^{2} \left ( 1/4\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) ^{4}-1/16\,\arcsin \left ( dx+c \right ) \left ( -2\, \left ( dx+c \right ) ^{3}\sqrt{1- \left ( dx+c \right ) ^{2}}-3\, \left ( dx+c \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}+3\,\arcsin \left ( dx+c \right ) \right ) +{\frac{3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}}{32}}-1/32\, \left ( dx+c \right ) ^{4}-{\frac{3\, \left ( dx+c \right ) ^{2}}{32}} \right ) +3\,{e}^{3}{a}^{2}b \left ( 1/4\, \left ( dx+c \right ) ^{4}\arcsin \left ( dx+c \right ) +1/16\, \left ( dx+c \right ) ^{3}\sqrt{1- \left ( dx+c \right ) ^{2}}+{\frac{ \left ( 3\,dx+3\,c \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}}{32}}-{\frac{3\,\arcsin \left ( dx+c \right ) }{32}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/4*e^3*(d*x+c)^4*a^3+e^3*b^3*(1/4*(d*x+c)^4*arcsin(d*x+c)^3-3/32*arcsin(d*x+c)^2*(-2*(d*x+c)^3*(1-(d*x+c
)^2)^(1/2)-3*(d*x+c)*(1-(d*x+c)^2)^(1/2)+3*arcsin(d*x+c))-3/32*(d*x+c)^4*arcsin(d*x+c)-3/256*(d*x+c)*(2*(d*x+c
)^2+3)*(1-(d*x+c)^2)^(1/2)-27/256*arcsin(d*x+c)-9/32*arcsin(d*x+c)*((d*x+c)^2-1)-9/64*(d*x+c)*(1-(d*x+c)^2)^(1
/2)+3/16*arcsin(d*x+c)^3)+3*e^3*a*b^2*(1/4*arcsin(d*x+c)^2*(d*x+c)^4-1/16*arcsin(d*x+c)*(-2*(d*x+c)^3*(1-(d*x+
c)^2)^(1/2)-3*(d*x+c)*(1-(d*x+c)^2)^(1/2)+3*arcsin(d*x+c))+3/32*arcsin(d*x+c)^2-1/32*(d*x+c)^4-3/32*(d*x+c)^2)
+3*e^3*a^2*b*(1/4*(d*x+c)^4*arcsin(d*x+c)+1/16*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)+3/32*(d*x+c)*(1-(d*x+c)^2)^(1/2)-
3/32*arcsin(d*x+c)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.84205, size = 1597, normalized size = 5.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/256*(8*(8*a^3 - 3*a*b^2)*d^4*e^3*x^4 + 32*(8*a^3 - 3*a*b^2)*c*d^3*e^3*x^3 - 24*(3*a*b^2 - 2*(8*a^3 - 3*a*b^2
)*c^2)*d^2*e^3*x^2 - 16*(9*a*b^2*c - 2*(8*a^3 - 3*a*b^2)*c^3)*d*e^3*x + 8*(8*b^3*d^4*e^3*x^4 + 32*b^3*c*d^3*e^
3*x^3 + 48*b^3*c^2*d^2*e^3*x^2 + 32*b^3*c^3*d*e^3*x + (8*b^3*c^4 - 3*b^3)*e^3)*arcsin(d*x + c)^3 + 24*(8*a*b^2
*d^4*e^3*x^4 + 32*a*b^2*c*d^3*e^3*x^3 + 48*a*b^2*c^2*d^2*e^3*x^2 + 32*a*b^2*c^3*d*e^3*x + (8*a*b^2*c^4 - 3*a*b
^2)*e^3)*arcsin(d*x + c)^2 + 3*(8*(8*a^2*b - b^3)*d^4*e^3*x^4 + 32*(8*a^2*b - b^3)*c*d^3*e^3*x^3 - 24*(b^3 - 2
*(8*a^2*b - b^3)*c^2)*d^2*e^3*x^2 - 16*(3*b^3*c - 2*(8*a^2*b - b^3)*c^3)*d*e^3*x - (24*b^3*c^2 - 8*(8*a^2*b -
b^3)*c^4 + 24*a^2*b - 15*b^3)*e^3)*arcsin(d*x + c) + 3*(2*(8*a^2*b - b^3)*d^3*e^3*x^3 + 6*(8*a^2*b - b^3)*c*d^
2*e^3*x^2 + 3*(8*a^2*b - 5*b^3 + 2*(8*a^2*b - b^3)*c^2)*d*e^3*x + (2*(8*a^2*b - b^3)*c^3 + 3*(8*a^2*b - 5*b^3)
*c)*e^3 + 8*(2*b^3*d^3*e^3*x^3 + 6*b^3*c*d^2*e^3*x^2 + 3*(2*b^3*c^2 + b^3)*d*e^3*x + (2*b^3*c^3 + 3*b^3*c)*e^3
)*arcsin(d*x + c)^2 + 16*(2*a*b^2*d^3*e^3*x^3 + 6*a*b^2*c*d^2*e^3*x^2 + 3*(2*a*b^2*c^2 + a*b^2)*d*e^3*x + (2*a
*b^2*c^3 + 3*a*b^2*c)*e^3)*arcsin(d*x + c))*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1))/d

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Sympy [A]  time = 14.784, size = 1828, normalized size = 6.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*c**3*e**3*x + 3*a**3*c**2*d*e**3*x**2/2 + a**3*c*d**2*e**3*x**3 + a**3*d**3*e**3*x**4/4 + 3*a*
*2*b*c**4*e**3*asin(c + d*x)/(4*d) + 3*a**2*b*c**3*e**3*x*asin(c + d*x) + 3*a**2*b*c**3*e**3*sqrt(-c**2 - 2*c*
d*x - d**2*x**2 + 1)/(16*d) + 9*a**2*b*c**2*d*e**3*x**2*asin(c + d*x)/2 + 9*a**2*b*c**2*e**3*x*sqrt(-c**2 - 2*
c*d*x - d**2*x**2 + 1)/16 + 3*a**2*b*c*d**2*e**3*x**3*asin(c + d*x) + 9*a**2*b*c*d*e**3*x**2*sqrt(-c**2 - 2*c*
d*x - d**2*x**2 + 1)/16 + 9*a**2*b*c*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(32*d) + 3*a**2*b*d**3*e**3*x*
*4*asin(c + d*x)/4 + 3*a**2*b*d**2*e**3*x**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/16 + 9*a**2*b*e**3*x*sqrt(-
c**2 - 2*c*d*x - d**2*x**2 + 1)/32 - 9*a**2*b*e**3*asin(c + d*x)/(32*d) + 3*a*b**2*c**4*e**3*asin(c + d*x)**2/
(4*d) + 3*a*b**2*c**3*e**3*x*asin(c + d*x)**2 - 3*a*b**2*c**3*e**3*x/8 + 3*a*b**2*c**3*e**3*sqrt(-c**2 - 2*c*d
*x - d**2*x**2 + 1)*asin(c + d*x)/(8*d) + 9*a*b**2*c**2*d*e**3*x**2*asin(c + d*x)**2/2 - 9*a*b**2*c**2*d*e**3*
x**2/16 + 9*a*b**2*c**2*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/8 + 3*a*b**2*c*d**2*e**3*x*
*3*asin(c + d*x)**2 - 3*a*b**2*c*d**2*e**3*x**3/8 + 9*a*b**2*c*d*e**3*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 +
1)*asin(c + d*x)/8 - 9*a*b**2*c*e**3*x/16 + 9*a*b**2*c*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x
)/(16*d) + 3*a*b**2*d**3*e**3*x**4*asin(c + d*x)**2/4 - 3*a*b**2*d**3*e**3*x**4/32 + 3*a*b**2*d**2*e**3*x**3*s
qrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/8 - 9*a*b**2*d*e**3*x**2/32 + 9*a*b**2*e**3*x*sqrt(-c**2 -
2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/16 - 9*a*b**2*e**3*asin(c + d*x)**2/(32*d) + b**3*c**4*e**3*asin(c + d*
x)**3/(4*d) - 3*b**3*c**4*e**3*asin(c + d*x)/(32*d) + b**3*c**3*e**3*x*asin(c + d*x)**3 - 3*b**3*c**3*e**3*x*a
sin(c + d*x)/8 + 3*b**3*c**3*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/(16*d) - 3*b**3*c**3*
e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(128*d) + 3*b**3*c**2*d*e**3*x**2*asin(c + d*x)**3/2 - 9*b**3*c**2*
d*e**3*x**2*asin(c + d*x)/16 + 9*b**3*c**2*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/16 -
9*b**3*c**2*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/128 - 9*b**3*c**2*e**3*asin(c + d*x)/(32*d) + b**3*c*
d**2*e**3*x**3*asin(c + d*x)**3 - 3*b**3*c*d**2*e**3*x**3*asin(c + d*x)/8 + 9*b**3*c*d*e**3*x**2*sqrt(-c**2 -
2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/16 - 9*b**3*c*d*e**3*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/128
- 9*b**3*c*e**3*x*asin(c + d*x)/16 + 9*b**3*c*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/(32*
d) - 45*b**3*c*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(256*d) + b**3*d**3*e**3*x**4*asin(c + d*x)**3/4 - 3
*b**3*d**3*e**3*x**4*asin(c + d*x)/32 + 3*b**3*d**2*e**3*x**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d
*x)**2/16 - 3*b**3*d**2*e**3*x**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/128 - 9*b**3*d*e**3*x**2*asin(c + d*x)
/32 + 9*b**3*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/32 - 45*b**3*e**3*x*sqrt(-c**2 - 2*
c*d*x - d**2*x**2 + 1)/256 - 3*b**3*e**3*asin(c + d*x)**3/(32*d) + 45*b**3*e**3*asin(c + d*x)/(256*d), Ne(d, 0
)), (c**3*e**3*x*(a + b*asin(c))**3, True))

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Giac [B]  time = 1.32353, size = 864, normalized size = 3.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((d*x + c)^2 - 1)^2*b^3*arcsin(d*x + c)^3*e^3/d - 3/16*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*b^3*arcsin(d*x +
c)^2*e^3/d + 3/4*((d*x + c)^2 - 1)^2*a*b^2*arcsin(d*x + c)^2*e^3/d + 1/2*((d*x + c)^2 - 1)*b^3*arcsin(d*x + c
)^3*e^3/d - 3/8*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*a*b^2*arcsin(d*x + c)*e^3/d + 15/32*sqrt(-(d*x + c)^2 + 1)*
(d*x + c)*b^3*arcsin(d*x + c)^2*e^3/d + 3/4*((d*x + c)^2 - 1)^2*a^2*b*arcsin(d*x + c)*e^3/d - 3/32*((d*x + c)^
2 - 1)^2*b^3*arcsin(d*x + c)*e^3/d + 3/2*((d*x + c)^2 - 1)*a*b^2*arcsin(d*x + c)^2*e^3/d + 5/32*b^3*arcsin(d*x
+ c)^3*e^3/d - 3/16*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*a^2*b*e^3/d + 3/128*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)
*b^3*e^3/d + 15/16*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a*b^2*arcsin(d*x + c)*e^3/d + 1/4*((d*x + c)^2 - 1)^2*a^3*
e^3/d - 3/32*((d*x + c)^2 - 1)^2*a*b^2*e^3/d + 3/2*((d*x + c)^2 - 1)*a^2*b*arcsin(d*x + c)*e^3/d - 15/32*((d*x
+ c)^2 - 1)*b^3*arcsin(d*x + c)*e^3/d + 15/32*a*b^2*arcsin(d*x + c)^2*e^3/d + 15/32*sqrt(-(d*x + c)^2 + 1)*(d
*x + c)*a^2*b*e^3/d - 51/256*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^3*e^3/d + 1/2*((d*x + c)^2 - 1)*a^3*e^3/d - 15
/32*((d*x + c)^2 - 1)*a*b^2*e^3/d + 15/32*a^2*b*arcsin(d*x + c)*e^3/d - 51/256*b^3*arcsin(d*x + c)*e^3/d - 51/
256*a*b^2*e^3/d