3.206 \(\int (c e+d e x)^3 (a+b \sin ^{-1}(c+d x))^4 \, dx\)

Optimal. Leaf size=357 \[ -\frac{3 b^3 e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}-\frac{45 b^3 e^3 (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{64 d}-\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}+\frac{45 b^2 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{128 d}-\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}+\frac{b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}+\frac{3 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{8 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^4}{32 d}+\frac{3 b^4 e^3 (c+d x)^4}{128 d}+\frac{45 b^4 e^3 (c+d x)^2}{128 d} \]

[Out]

(45*b^4*e^3*(c + d*x)^2)/(128*d) + (3*b^4*e^3*(c + d*x)^4)/(128*d) - (45*b^3*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^
2]*(a + b*ArcSin[c + d*x]))/(64*d) - (3*b^3*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(32
*d) + (45*b^2*e^3*(a + b*ArcSin[c + d*x])^2)/(128*d) - (9*b^2*e^3*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2)/(16*d
) - (3*b^2*e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^2)/(16*d) + (3*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b
*ArcSin[c + d*x])^3)/(8*d) + (b*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/(4*d) - (3*e^
3*(a + b*ArcSin[c + d*x])^4)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^4)/(4*d)

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Rubi [A]  time = 0.640142, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {4805, 12, 4627, 4707, 4641, 30} \[ -\frac{3 b^3 e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}-\frac{45 b^3 e^3 (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{64 d}-\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}+\frac{45 b^2 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{128 d}-\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}+\frac{b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}+\frac{3 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{8 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^4}{32 d}+\frac{3 b^4 e^3 (c+d x)^4}{128 d}+\frac{45 b^4 e^3 (c+d x)^2}{128 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^4,x]

[Out]

(45*b^4*e^3*(c + d*x)^2)/(128*d) + (3*b^4*e^3*(c + d*x)^4)/(128*d) - (45*b^3*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^
2]*(a + b*ArcSin[c + d*x]))/(64*d) - (3*b^3*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(32
*d) + (45*b^2*e^3*(a + b*ArcSin[c + d*x])^2)/(128*d) - (9*b^2*e^3*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2)/(16*d
) - (3*b^2*e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^2)/(16*d) + (3*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b
*ArcSin[c + d*x])^3)/(8*d) + (b*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/(4*d) - (3*e^
3*(a + b*ArcSin[c + d*x])^4)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^4)/(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 30

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

Rubi steps

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

Mathematica [A]  time = 0.474308, size = 287, normalized size = 0.8 \[ \frac{e^3 \left (-3 b^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )-\frac{45}{2} b^3 (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )-6 b^2 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2-18 b^2 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2+\frac{45}{4} b^2 \left (a+b \sin ^{-1}(c+d x)\right )^2+8 b (c+d x)^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3+12 b (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3+8 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^4-3 \left (a+b \sin ^{-1}(c+d x)\right )^4+\frac{3}{4} b^4 (c+d x)^4+\frac{45}{4} b^4 (c+d x)^2\right )}{32 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^4,x]

[Out]

(e^3*((45*b^4*(c + d*x)^2)/4 + (3*b^4*(c + d*x)^4)/4 - (45*b^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c
+ d*x]))/2 - 3*b^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) + (45*b^2*(a + b*ArcSin[c + d*x]
)^2)/4 - 18*b^2*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2 - 6*b^2*(c + d*x)^4*(a + b*ArcSin[c + d*x])^2 + 12*b*(c
+ d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3 + 8*b*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c
+ d*x])^3 - 3*(a + b*ArcSin[c + d*x])^4 + 8*(c + d*x)^4*(a + b*ArcSin[c + d*x])^4))/(32*d)

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Maple [A]  time = 0.048, size = 654, normalized size = 1.8 \begin{align*} \text{result too large to display} \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))^4,x)

[Out]

1/d*(1/4*e^3*(d*x+c)^4*a^4+e^3*b^4*(1/4*(d*x+c)^4*arcsin(d*x+c)^4-1/8*arcsin(d*x+c)^3*(-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))-3/16*arcsin(d*x+c)^2*(d*x+c)^4+3/64*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))+27/128*arcsin(d*x+c)^2+3/128*(d*
x+c)^4+45/128*(d*x+c)^2-9/16*arcsin(d*x+c)^2*((d*x+c)^2-1)-9/16*arcsin(d*x+c)*((d*x+c)*(1-(d*x+c)^2)^(1/2)+arc
sin(d*x+c))+9/32*arcsin(d*x+c)^4)+4*e^3*a*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)+6*e^3*a^2*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)+4*e^3*a^3*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))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.21609, size = 2390, normalized size = 6.69 \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))^4,x, algorithm="fricas")

[Out]

1/128*((32*a^4 - 24*a^2*b^2 + 3*b^4)*d^4*e^3*x^4 + 4*(32*a^4 - 24*a^2*b^2 + 3*b^4)*c*d^3*e^3*x^3 - 3*(24*a^2*b
^2 - 15*b^4 - 2*(32*a^4 - 24*a^2*b^2 + 3*b^4)*c^2)*d^2*e^3*x^2 + 2*(2*(32*a^4 - 24*a^2*b^2 + 3*b^4)*c^3 - 9*(8
*a^2*b^2 - 5*b^4)*c)*d*e^3*x + 4*(8*b^4*d^4*e^3*x^4 + 32*b^4*c*d^3*e^3*x^3 + 48*b^4*c^2*d^2*e^3*x^2 + 32*b^4*c
^3*d*e^3*x + (8*b^4*c^4 - 3*b^4)*e^3)*arcsin(d*x + c)^4 + 16*(8*a*b^3*d^4*e^3*x^4 + 32*a*b^3*c*d^3*e^3*x^3 + 4
8*a*b^3*c^2*d^2*e^3*x^2 + 32*a*b^3*c^3*d*e^3*x + (8*a*b^3*c^4 - 3*a*b^3)*e^3)*arcsin(d*x + c)^3 + 3*(8*(8*a^2*
b^2 - b^4)*d^4*e^3*x^4 + 32*(8*a^2*b^2 - b^4)*c*d^3*e^3*x^3 - 24*(b^4 - 2*(8*a^2*b^2 - b^4)*c^2)*d^2*e^3*x^2 -
16*(3*b^4*c - 2*(8*a^2*b^2 - b^4)*c^3)*d*e^3*x - (24*b^4*c^2 - 8*(8*a^2*b^2 - b^4)*c^4 + 24*a^2*b^2 - 15*b^4)
*e^3)*arcsin(d*x + c)^2 + 2*(8*(8*a^3*b - 3*a*b^3)*d^4*e^3*x^4 + 32*(8*a^3*b - 3*a*b^3)*c*d^3*e^3*x^3 - 24*(3*
a*b^3 - 2*(8*a^3*b - 3*a*b^3)*c^2)*d^2*e^3*x^2 - 16*(9*a*b^3*c - 2*(8*a^3*b - 3*a*b^3)*c^3)*d*e^3*x - (72*a*b^
3*c^2 - 8*(8*a^3*b - 3*a*b^3)*c^4 + 24*a^3*b - 45*a*b^3)*e^3)*arcsin(d*x + c) + 2*(2*(8*a^3*b - 3*a*b^3)*d^3*e
^3*x^3 + 6*(8*a^3*b - 3*a*b^3)*c*d^2*e^3*x^2 + 3*(8*a^3*b - 15*a*b^3 + 2*(8*a^3*b - 3*a*b^3)*c^2)*d*e^3*x + (2
*(8*a^3*b - 3*a*b^3)*c^3 + 3*(8*a^3*b - 15*a*b^3)*c)*e^3 + 8*(2*b^4*d^3*e^3*x^3 + 6*b^4*c*d^2*e^3*x^2 + 3*(2*b
^4*c^2 + b^4)*d*e^3*x + (2*b^4*c^3 + 3*b^4*c)*e^3)*arcsin(d*x + c)^3 + 24*(2*a*b^3*d^3*e^3*x^3 + 6*a*b^3*c*d^2
*e^3*x^2 + 3*(2*a*b^3*c^2 + a*b^3)*d*e^3*x + (2*a*b^3*c^3 + 3*a*b^3*c)*e^3)*arcsin(d*x + c)^2 + 3*(2*(8*a^2*b^
2 - b^4)*d^3*e^3*x^3 + 6*(8*a^2*b^2 - b^4)*c*d^2*e^3*x^2 + 3*(8*a^2*b^2 - 5*b^4 + 2*(8*a^2*b^2 - b^4)*c^2)*d*e
^3*x + (2*(8*a^2*b^2 - b^4)*c^3 + 3*(8*a^2*b^2 - 5*b^4)*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 = 31.0822, size = 2876, normalized size = 8.06 \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))**4,x)

[Out]

Piecewise((a**4*c**3*e**3*x + 3*a**4*c**2*d*e**3*x**2/2 + a**4*c*d**2*e**3*x**3 + a**4*d**3*e**3*x**4/4 + a**3
*b*c**4*e**3*asin(c + d*x)/d + 4*a**3*b*c**3*e**3*x*asin(c + d*x) + a**3*b*c**3*e**3*sqrt(-c**2 - 2*c*d*x - d*
*2*x**2 + 1)/(4*d) + 6*a**3*b*c**2*d*e**3*x**2*asin(c + d*x) + 3*a**3*b*c**2*e**3*x*sqrt(-c**2 - 2*c*d*x - d**
2*x**2 + 1)/4 + 4*a**3*b*c*d**2*e**3*x**3*asin(c + d*x) + 3*a**3*b*c*d*e**3*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x
**2 + 1)/4 + 3*a**3*b*c*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(8*d) + a**3*b*d**3*e**3*x**4*asin(c + d*x)
+ a**3*b*d**2*e**3*x**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/4 + 3*a**3*b*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2
*x**2 + 1)/8 - 3*a**3*b*e**3*asin(c + d*x)/(8*d) + 3*a**2*b**2*c**4*e**3*asin(c + d*x)**2/(2*d) + 6*a**2*b**2*
c**3*e**3*x*asin(c + d*x)**2 - 3*a**2*b**2*c**3*e**3*x/4 + 3*a**2*b**2*c**3*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x
**2 + 1)*asin(c + d*x)/(4*d) + 9*a**2*b**2*c**2*d*e**3*x**2*asin(c + d*x)**2 - 9*a**2*b**2*c**2*d*e**3*x**2/8
+ 9*a**2*b**2*c**2*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/4 + 6*a**2*b**2*c*d**2*e**3*x**3
*asin(c + d*x)**2 - 3*a**2*b**2*c*d**2*e**3*x**3/4 + 9*a**2*b**2*c*d*e**3*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**
2 + 1)*asin(c + d*x)/4 - 9*a**2*b**2*c*e**3*x/8 + 9*a**2*b**2*c*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asi
n(c + d*x)/(8*d) + 3*a**2*b**2*d**3*e**3*x**4*asin(c + d*x)**2/2 - 3*a**2*b**2*d**3*e**3*x**4/16 + 3*a**2*b**2
*d**2*e**3*x**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/4 - 9*a**2*b**2*d*e**3*x**2/16 + 9*a**2*b*
*2*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/8 - 9*a**2*b**2*e**3*asin(c + d*x)**2/(16*d) + a
*b**3*c**4*e**3*asin(c + d*x)**3/d - 3*a*b**3*c**4*e**3*asin(c + d*x)/(8*d) + 4*a*b**3*c**3*e**3*x*asin(c + d*
x)**3 - 3*a*b**3*c**3*e**3*x*asin(c + d*x)/2 + 3*a*b**3*c**3*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c
+ d*x)**2/(4*d) - 3*a*b**3*c**3*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(32*d) + 6*a*b**3*c**2*d*e**3*x**2
*asin(c + d*x)**3 - 9*a*b**3*c**2*d*e**3*x**2*asin(c + d*x)/4 + 9*a*b**3*c**2*e**3*x*sqrt(-c**2 - 2*c*d*x - d*
*2*x**2 + 1)*asin(c + d*x)**2/4 - 9*a*b**3*c**2*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/32 - 9*a*b**3*c**
2*e**3*asin(c + d*x)/(8*d) + 4*a*b**3*c*d**2*e**3*x**3*asin(c + d*x)**3 - 3*a*b**3*c*d**2*e**3*x**3*asin(c + d
*x)/2 + 9*a*b**3*c*d*e**3*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/4 - 9*a*b**3*c*d*e**3*x*
*2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/32 - 9*a*b**3*c*e**3*x*asin(c + d*x)/4 + 9*a*b**3*c*e**3*sqrt(-c**2 -
2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/(8*d) - 45*a*b**3*c*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(64*
d) + a*b**3*d**3*e**3*x**4*asin(c + d*x)**3 - 3*a*b**3*d**3*e**3*x**4*asin(c + d*x)/8 + 3*a*b**3*d**2*e**3*x**
3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/4 - 3*a*b**3*d**2*e**3*x**3*sqrt(-c**2 - 2*c*d*x - d*
*2*x**2 + 1)/32 - 9*a*b**3*d*e**3*x**2*asin(c + d*x)/8 + 9*a*b**3*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)
*asin(c + d*x)**2/8 - 45*a*b**3*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/64 - 3*a*b**3*e**3*asin(c + d*x)*
*3/(8*d) + 45*a*b**3*e**3*asin(c + d*x)/(64*d) + b**4*c**4*e**3*asin(c + d*x)**4/(4*d) - 3*b**4*c**4*e**3*asin
(c + d*x)**2/(16*d) + b**4*c**3*e**3*x*asin(c + d*x)**4 - 3*b**4*c**3*e**3*x*asin(c + d*x)**2/4 + 3*b**4*c**3*
e**3*x/32 + b**4*c**3*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**3/(4*d) - 3*b**4*c**3*e**3*sqr
t(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(32*d) + 3*b**4*c**2*d*e**3*x**2*asin(c + d*x)**4/2 - 9*b**4*
c**2*d*e**3*x**2*asin(c + d*x)**2/8 + 9*b**4*c**2*d*e**3*x**2/64 + 3*b**4*c**2*e**3*x*sqrt(-c**2 - 2*c*d*x - d
**2*x**2 + 1)*asin(c + d*x)**3/4 - 9*b**4*c**2*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/32 -
9*b**4*c**2*e**3*asin(c + d*x)**2/(16*d) + b**4*c*d**2*e**3*x**3*asin(c + d*x)**4 - 3*b**4*c*d**2*e**3*x**3*a
sin(c + d*x)**2/4 + 3*b**4*c*d**2*e**3*x**3/32 + 3*b**4*c*d*e**3*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*as
in(c + d*x)**3/4 - 9*b**4*c*d*e**3*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/32 - 9*b**4*c*e**3
*x*asin(c + d*x)**2/8 + 45*b**4*c*e**3*x/64 + 3*b**4*c*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x
)**3/(8*d) - 45*b**4*c*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(64*d) + b**4*d**3*e**3*x**4*a
sin(c + d*x)**4/4 - 3*b**4*d**3*e**3*x**4*asin(c + d*x)**2/16 + 3*b**4*d**3*e**3*x**4/128 + b**4*d**2*e**3*x**
3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**3/4 - 3*b**4*d**2*e**3*x**3*sqrt(-c**2 - 2*c*d*x - d**2
*x**2 + 1)*asin(c + d*x)/32 - 9*b**4*d*e**3*x**2*asin(c + d*x)**2/16 + 45*b**4*d*e**3*x**2/128 + 3*b**4*e**3*x
*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**3/8 - 45*b**4*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 +
1)*asin(c + d*x)/64 - 3*b**4*e**3*asin(c + d*x)**4/(32*d) + 45*b**4*e**3*asin(c + d*x)**2/(128*d), Ne(d, 0)),
(c**3*e**3*x*(a + b*asin(c))**4, True))

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Giac [B]  time = 1.42435, size = 1353, normalized size = 3.79 \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))^4,x, algorithm="giac")

[Out]

1/4*((d*x + c)^2 - 1)^2*b^4*arcsin(d*x + c)^4*e^3/d - 1/4*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*b^4*arcsin(d*x +
c)^3*e^3/d + ((d*x + c)^2 - 1)^2*a*b^3*arcsin(d*x + c)^3*e^3/d + 1/2*((d*x + c)^2 - 1)*b^4*arcsin(d*x + c)^4*e
^3/d - 3/4*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*a*b^3*arcsin(d*x + c)^2*e^3/d + 5/8*sqrt(-(d*x + c)^2 + 1)*(d*x
+ c)*b^4*arcsin(d*x + c)^3*e^3/d + 3/2*((d*x + c)^2 - 1)^2*a^2*b^2*arcsin(d*x + c)^2*e^3/d - 3/16*((d*x + c)^2
- 1)^2*b^4*arcsin(d*x + c)^2*e^3/d + 2*((d*x + c)^2 - 1)*a*b^3*arcsin(d*x + c)^3*e^3/d + 5/32*b^4*arcsin(d*x
+ c)^4*e^3/d - 3/4*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*a^2*b^2*arcsin(d*x + c)*e^3/d + 3/32*(-(d*x + c)^2 + 1)^
(3/2)*(d*x + c)*b^4*arcsin(d*x + c)*e^3/d + 15/8*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a*b^3*arcsin(d*x + c)^2*e^3/
d + ((d*x + c)^2 - 1)^2*a^3*b*arcsin(d*x + c)*e^3/d - 3/8*((d*x + c)^2 - 1)^2*a*b^3*arcsin(d*x + c)*e^3/d + 3*
((d*x + c)^2 - 1)*a^2*b^2*arcsin(d*x + c)^2*e^3/d - 15/16*((d*x + c)^2 - 1)*b^4*arcsin(d*x + c)^2*e^3/d + 5/8*
a*b^3*arcsin(d*x + c)^3*e^3/d - 1/4*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*a^3*b*e^3/d + 3/32*(-(d*x + c)^2 + 1)^(
3/2)*(d*x + c)*a*b^3*e^3/d + 15/8*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a^2*b^2*arcsin(d*x + c)*e^3/d - 51/64*sqrt(
-(d*x + c)^2 + 1)*(d*x + c)*b^4*arcsin(d*x + c)*e^3/d + 1/4*((d*x + c)^2 - 1)^2*a^4*e^3/d - 3/16*((d*x + c)^2
- 1)^2*a^2*b^2*e^3/d + 3/128*((d*x + c)^2 - 1)^2*b^4*e^3/d + 2*((d*x + c)^2 - 1)*a^3*b*arcsin(d*x + c)*e^3/d -
15/8*((d*x + c)^2 - 1)*a*b^3*arcsin(d*x + c)*e^3/d + 15/16*a^2*b^2*arcsin(d*x + c)^2*e^3/d - 51/128*b^4*arcsi
n(d*x + c)^2*e^3/d + 5/8*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a^3*b*e^3/d - 51/64*sqrt(-(d*x + c)^2 + 1)*(d*x + c)
*a*b^3*e^3/d + 1/2*((d*x + c)^2 - 1)*a^4*e^3/d - 15/16*((d*x + c)^2 - 1)*a^2*b^2*e^3/d + 51/128*((d*x + c)^2 -
1)*b^4*e^3/d + 5/8*a^3*b*arcsin(d*x + c)*e^3/d - 51/64*a*b^3*arcsin(d*x + c)*e^3/d - 51/128*a^2*b^2*e^3/d + 1
95/1024*b^4*e^3/d