3.3.6 \(\int (c e+d e x)^3 (a+b \text {ArcSin}(c+d x))^4 \, dx\) [206]

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

[Out]

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

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Rubi [A]
time = 0.44, antiderivative size = 357, normalized size of antiderivative = 1.00, 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 = {4889, 12, 4723, 4795, 4737, 30} \begin {gather*} -\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))}{32 d}-\frac {45 b^3 e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))}{64 d}-\frac {3 b^2 e^3 (c+d x)^4 (a+b \text {ArcSin}(c+d x))^2}{16 d}+\frac {45 b^2 e^3 (a+b \text {ArcSin}(c+d x))^2}{128 d}-\frac {9 b^2 e^3 (c+d x)^2 (a+b \text {ArcSin}(c+d x))^2}{16 d}+\frac {b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^3}{4 d}+\frac {3 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^3}{8 d}+\frac {e^3 (c+d x)^4 (a+b \text {ArcSin}(c+d x))^4}{4 d}-\frac {3 e^3 (a+b \text {ArcSin}(c+d x))^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} \end {gather*}

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 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*} \int (c e+d e x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^4 \, dx &=\frac {\text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}(\int x \, dx,x,c+d x)}{64 d}+\frac {\left (9 b^4 e^3\right ) \text {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*}

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Mathematica [A]
time = 0.33, size = 287, normalized size = 0.80 \begin {gather*} \frac {e^3 \left (\frac {45}{4} b^4 (c+d x)^2+\frac {3}{4} b^4 (c+d x)^4-\frac {45}{2} b^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))-3 b^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))+\frac {45}{4} b^2 (a+b \text {ArcSin}(c+d x))^2-18 b^2 (c+d x)^2 (a+b \text {ArcSin}(c+d x))^2-6 b^2 (c+d x)^4 (a+b \text {ArcSin}(c+d x))^2+12 b (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^3+8 b (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^3-3 (a+b \text {ArcSin}(c+d x))^4+8 (c+d x)^4 (a+b \text {ArcSin}(c+d x))^4\right )}{32 d} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(656\) vs. \(2(327)=654\).
time = 0.12, size = 657, normalized size = 1.84 Too large to display

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,method=_RETURNVERBOSE)

[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*(d*x+c)^4*arcsin(d*x+c)^2+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+3/512*(2*
(d*x+c)^2+3)^2-9/16*((d*x+c)^2-1)*arcsin(d*x+c)^2-9/16*arcsin(d*x+c)*((d*x+c)*(1-(d*x+c)^2)^(1/2)+arcsin(d*x+c
))+9/32*(d*x+c)^2+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*((d*x+c)^2-1)*arcsin(d*x+c)-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*(d*x+c)^4*arcsin(d*x+c)^2-1/16*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)^2-1/128*(2*(d
*x+c)^2+3)^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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

1/4*a^4*d^3*x^4*e^3 + a^4*c*d^2*x^3*e^3 + 3/2*a^4*c^2*d*x^2*e^3 + 3*(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*c^2*d*
e^3 + 2/3*(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^3*b*c*d^2*e^3 + 1/24*(24*x^4*arcsin(d*x + c) + (6*sqrt(-d^2*x^2 -
2*c*d*x - c^2 + 1)*x^3/d^2 - 14*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c*x^2/d^3 + 105*c^4*arcsin(-(d^2*x + c*d)/s
qrt(c^2*d^2 - (c^2 - 1)*d^2))/d^5 + 35*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c^2*x/d^4 - 90*(c^2 - 1)*c^2*arcsin(
-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^5 - 105*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c^3/d^5 - 9*sqrt(-d
^2*x^2 - 2*c*d*x - c^2 + 1)*(c^2 - 1)*x/d^4 + 9*(c^2 - 1)^2*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2
))/d^5 + 55*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(c^2 - 1)*c/d^5)*d)*a^3*b*d^3*e^3 + a^4*c^3*x*e^3 + 4*((d*x + c
)*arcsin(d*x + c) + sqrt(-(d*x + c)^2 + 1))*a^3*b*c^3*e^3/d + 1/4*(b^4*d^3*x^4*e^3 + 4*b^4*c*d^2*x^3*e^3 + 6*b
^4*c^2*d*x^2*e^3 + 4*b^4*c^3*x*e^3)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^4 + integrate(((b^4
*d^4*x^4*e^3 + 4*b^4*c*d^3*x^3*e^3 + 6*b^4*c^2*d^2*x^2*e^3 + 4*b^4*c^3*d*x*e^3)*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 + 4*(a*b^3*d^5*x^5*e^3 + 5*a*b^3*c*d^4*x^4*e^3
+ a*b^3*c^5*e^3 - a*b^3*c^3*e^3 + (10*a*b^3*c^2*e^3 - a*b^3*e^3)*d^3*x^3 + (10*a*b^3*c^3*e^3 - 3*a*b^3*c*e^3)
*d^2*x^2 + (5*a*b^3*c^4*e^3 - 3*a*b^3*c^2*e^3)*d*x)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 +
6*(a^2*b^2*d^5*x^5*e^3 + 5*a^2*b^2*c*d^4*x^4*e^3 + a^2*b^2*c^5*e^3 - a^2*b^2*c^3*e^3 + (10*a^2*b^2*c^2*e^3 -
a^2*b^2*e^3)*d^3*x^3 + (10*a^2*b^2*c^3*e^3 - 3*a^2*b^2*c*e^3)*d^2*x^2 + (5*a^2*b^2*c^4*e^3 - 3*a^2*b^2*c^2*e^3
)*d*x)*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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1032 vs. \(2 (316) = 632\).
time = 1.23, size = 1032, normalized size = 2.89 \begin {gather*} \frac {4 \, {\left (8 \, b^{4} d^{4} x^{4} + 32 \, b^{4} c d^{3} x^{3} + 48 \, b^{4} c^{2} d^{2} x^{2} + 32 \, b^{4} c^{3} d x + 8 \, b^{4} c^{4} - 3 \, b^{4}\right )} \arcsin \left (d x + c\right )^{4} e^{3} + 16 \, {\left (8 \, a b^{3} d^{4} x^{4} + 32 \, a b^{3} c d^{3} x^{3} + 48 \, a b^{3} c^{2} d^{2} x^{2} + 32 \, a b^{3} c^{3} d x + 8 \, a b^{3} c^{4} - 3 \, a b^{3}\right )} \arcsin \left (d x + c\right )^{3} e^{3} + 3 \, {\left (8 \, {\left (8 \, a^{2} b^{2} - b^{4}\right )} d^{4} x^{4} + 32 \, {\left (8 \, a^{2} b^{2} - b^{4}\right )} c d^{3} x^{3} - 24 \, b^{4} c^{2} + 8 \, {\left (8 \, a^{2} b^{2} - b^{4}\right )} c^{4} - 24 \, {\left (b^{4} - 2 \, {\left (8 \, a^{2} b^{2} - b^{4}\right )} c^{2}\right )} d^{2} x^{2} - 24 \, a^{2} b^{2} + 15 \, b^{4} - 16 \, {\left (3 \, b^{4} c - 2 \, {\left (8 \, a^{2} b^{2} - b^{4}\right )} c^{3}\right )} d x\right )} \arcsin \left (d x + c\right )^{2} e^{3} + 2 \, {\left (8 \, {\left (8 \, a^{3} b - 3 \, a b^{3}\right )} d^{4} x^{4} + 32 \, {\left (8 \, a^{3} b - 3 \, a b^{3}\right )} c d^{3} x^{3} - 72 \, a b^{3} c^{2} + 8 \, {\left (8 \, a^{3} b - 3 \, a b^{3}\right )} c^{4} - 24 \, {\left (3 \, a b^{3} - 2 \, {\left (8 \, a^{3} b - 3 \, a b^{3}\right )} c^{2}\right )} d^{2} x^{2} - 24 \, a^{3} b + 45 \, a b^{3} - 16 \, {\left (9 \, a b^{3} c - 2 \, {\left (8 \, a^{3} b - 3 \, a b^{3}\right )} c^{3}\right )} d x\right )} \arcsin \left (d x + c\right ) e^{3} + {\left ({\left (32 \, a^{4} - 24 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{4} x^{4} + 4 \, {\left (32 \, a^{4} - 24 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d^{3} x^{3} - 3 \, {\left (24 \, a^{2} b^{2} - 15 \, b^{4} - 2 \, {\left (32 \, a^{4} - 24 \, a^{2} b^{2} + 3 \, b^{4}\right )} c^{2}\right )} d^{2} x^{2} + 2 \, {\left (2 \, {\left (32 \, a^{4} - 24 \, a^{2} b^{2} + 3 \, b^{4}\right )} c^{3} - 9 \, {\left (8 \, a^{2} b^{2} - 5 \, b^{4}\right )} c\right )} d x\right )} e^{3} + 2 \, {\left (8 \, {\left (2 \, b^{4} d^{3} x^{3} + 6 \, b^{4} c d^{2} x^{2} + 2 \, b^{4} c^{3} + 3 \, b^{4} c + 3 \, {\left (2 \, b^{4} c^{2} + b^{4}\right )} d x\right )} \arcsin \left (d x + c\right )^{3} e^{3} + 24 \, {\left (2 \, a b^{3} d^{3} x^{3} + 6 \, a b^{3} c d^{2} x^{2} + 2 \, a b^{3} c^{3} + 3 \, a b^{3} c + 3 \, {\left (2 \, a b^{3} c^{2} + a b^{3}\right )} d x\right )} \arcsin \left (d x + c\right )^{2} e^{3} + 3 \, {\left (2 \, {\left (8 \, a^{2} b^{2} - b^{4}\right )} d^{3} x^{3} + 6 \, {\left (8 \, a^{2} b^{2} - b^{4}\right )} c d^{2} x^{2} + 2 \, {\left (8 \, a^{2} b^{2} - b^{4}\right )} c^{3} + 3 \, {\left (8 \, a^{2} b^{2} - 5 \, b^{4} + 2 \, {\left (8 \, a^{2} b^{2} - b^{4}\right )} c^{2}\right )} d x + 3 \, {\left (8 \, a^{2} b^{2} - 5 \, b^{4}\right )} c\right )} \arcsin \left (d x + c\right ) e^{3} + {\left (2 \, {\left (8 \, a^{3} b - 3 \, a b^{3}\right )} d^{3} x^{3} + 6 \, {\left (8 \, a^{3} b - 3 \, a b^{3}\right )} c d^{2} x^{2} + 2 \, {\left (8 \, a^{3} b - 3 \, a b^{3}\right )} c^{3} + 3 \, {\left (8 \, a^{3} b - 15 \, a b^{3} + 2 \, {\left (8 \, a^{3} b - 3 \, a b^{3}\right )} c^{2}\right )} d x + 3 \, {\left (8 \, a^{3} b - 15 \, a b^{3}\right )} c\right )} e^{3}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{128 \, d} \end {gather*}

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*(4*(8*b^4*d^4*x^4 + 32*b^4*c*d^3*x^3 + 48*b^4*c^2*d^2*x^2 + 32*b^4*c^3*d*x + 8*b^4*c^4 - 3*b^4)*arcsin(d
*x + c)^4*e^3 + 16*(8*a*b^3*d^4*x^4 + 32*a*b^3*c*d^3*x^3 + 48*a*b^3*c^2*d^2*x^2 + 32*a*b^3*c^3*d*x + 8*a*b^3*c
^4 - 3*a*b^3)*arcsin(d*x + c)^3*e^3 + 3*(8*(8*a^2*b^2 - b^4)*d^4*x^4 + 32*(8*a^2*b^2 - b^4)*c*d^3*x^3 - 24*b^4
*c^2 + 8*(8*a^2*b^2 - b^4)*c^4 - 24*(b^4 - 2*(8*a^2*b^2 - b^4)*c^2)*d^2*x^2 - 24*a^2*b^2 + 15*b^4 - 16*(3*b^4*
c - 2*(8*a^2*b^2 - b^4)*c^3)*d*x)*arcsin(d*x + c)^2*e^3 + 2*(8*(8*a^3*b - 3*a*b^3)*d^4*x^4 + 32*(8*a^3*b - 3*a
*b^3)*c*d^3*x^3 - 72*a*b^3*c^2 + 8*(8*a^3*b - 3*a*b^3)*c^4 - 24*(3*a*b^3 - 2*(8*a^3*b - 3*a*b^3)*c^2)*d^2*x^2
- 24*a^3*b + 45*a*b^3 - 16*(9*a*b^3*c - 2*(8*a^3*b - 3*a*b^3)*c^3)*d*x)*arcsin(d*x + c)*e^3 + ((32*a^4 - 24*a^
2*b^2 + 3*b^4)*d^4*x^4 + 4*(32*a^4 - 24*a^2*b^2 + 3*b^4)*c*d^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*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*x)*e^3 + 2*
(8*(2*b^4*d^3*x^3 + 6*b^4*c*d^2*x^2 + 2*b^4*c^3 + 3*b^4*c + 3*(2*b^4*c^2 + b^4)*d*x)*arcsin(d*x + c)^3*e^3 + 2
4*(2*a*b^3*d^3*x^3 + 6*a*b^3*c*d^2*x^2 + 2*a*b^3*c^3 + 3*a*b^3*c + 3*(2*a*b^3*c^2 + a*b^3)*d*x)*arcsin(d*x + c
)^2*e^3 + 3*(2*(8*a^2*b^2 - b^4)*d^3*x^3 + 6*(8*a^2*b^2 - b^4)*c*d^2*x^2 + 2*(8*a^2*b^2 - b^4)*c^3 + 3*(8*a^2*
b^2 - 5*b^4 + 2*(8*a^2*b^2 - b^4)*c^2)*d*x + 3*(8*a^2*b^2 - 5*b^4)*c)*arcsin(d*x + c)*e^3 + (2*(8*a^3*b - 3*a*
b^3)*d^3*x^3 + 6*(8*a^3*b - 3*a*b^3)*c*d^2*x^2 + 2*(8*a^3*b - 3*a*b^3)*c^3 + 3*(8*a^3*b - 15*a*b^3 + 2*(8*a^3*
b - 3*a*b^3)*c^2)*d*x + 3*(8*a^3*b - 15*a*b^3)*c)*e^3)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2876 vs. \(2 (325) = 650\).
time = 1.54, size = 2876, normalized size = 8.06 \begin {gather*} \text {Too large to display} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 1016 vs. \(2 (327) = 654\).
time = 0.48, size = 1016, normalized size = 2.85 \begin {gather*} \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{4} e^{3} \arcsin \left (d x + c\right )^{4}}{4 \, d} - \frac {{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} b^{4} e^{3} \arcsin \left (d x + c\right )^{3}}{4 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a b^{3} e^{3} \arcsin \left (d x + c\right )^{3}}{d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{4} e^{3} \arcsin \left (d x + c\right )^{4}}{2 \, d} - \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} a b^{3} e^{3} \arcsin \left (d x + c\right )^{2}}{4 \, d} + \frac {5 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{4} e^{3} \arcsin \left (d x + c\right )^{3}}{8 \, d} + \frac {{\left (d x + c\right )}^{4} a^{4} e^{3}}{4 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a^{2} b^{2} e^{3} \arcsin \left (d x + c\right )^{2}}{2 \, d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{4} e^{3} \arcsin \left (d x + c\right )^{2}}{16 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{3} e^{3} \arcsin \left (d x + c\right )^{3}}{d} + \frac {5 \, b^{4} e^{3} \arcsin \left (d x + c\right )^{4}}{32 \, d} - \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} a^{2} b^{2} e^{3} \arcsin \left (d x + c\right )}{4 \, d} + \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} b^{4} e^{3} \arcsin \left (d x + c\right )}{32 \, d} + \frac {15 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b^{3} e^{3} \arcsin \left (d x + c\right )^{2}}{8 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a^{3} b e^{3} \arcsin \left (d x + c\right )}{d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a b^{3} e^{3} \arcsin \left (d x + c\right )}{8 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} b^{2} e^{3} \arcsin \left (d x + c\right )^{2}}{d} - \frac {15 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} b^{4} e^{3} \arcsin \left (d x + c\right )^{2}}{16 \, d} + \frac {5 \, a b^{3} e^{3} \arcsin \left (d x + c\right )^{3}}{8 \, d} - \frac {{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} a^{3} b e^{3}}{4 \, d} + \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} a b^{3} e^{3}}{32 \, d} + \frac {15 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a^{2} b^{2} e^{3} \arcsin \left (d x + c\right )}{8 \, d} - \frac {51 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{4} e^{3} \arcsin \left (d x + c\right )}{64 \, d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a^{2} b^{2} e^{3}}{16 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{4} e^{3}}{128 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a^{3} b e^{3} \arcsin \left (d x + c\right )}{d} - \frac {15 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{3} e^{3} \arcsin \left (d x + c\right )}{8 \, d} + \frac {15 \, a^{2} b^{2} e^{3} \arcsin \left (d x + c\right )^{2}}{16 \, d} - \frac {51 \, b^{4} e^{3} \arcsin \left (d x + c\right )^{2}}{128 \, d} + \frac {5 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a^{3} b e^{3}}{8 \, d} - \frac {51 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b^{3} e^{3}}{64 \, d} - \frac {15 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} b^{2} e^{3}}{16 \, d} + \frac {51 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} b^{4} e^{3}}{128 \, d} + \frac {5 \, a^{3} b e^{3} \arcsin \left (d x + c\right )}{8 \, d} - \frac {51 \, a b^{3} e^{3} \arcsin \left (d x + c\right )}{64 \, d} - \frac {51 \, a^{2} b^{2} e^{3}}{128 \, d} + \frac {195 \, b^{4} e^{3}}{1024 \, d} \end {gather*}

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*e^3*arcsin(d*x + c)^4/d - 1/4*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*b^4*e^3*arcsin(d*
x + c)^3/d + ((d*x + c)^2 - 1)^2*a*b^3*e^3*arcsin(d*x + c)^3/d + 1/2*((d*x + c)^2 - 1)*b^4*e^3*arcsin(d*x + c)
^4/d - 3/4*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*a*b^3*e^3*arcsin(d*x + c)^2/d + 5/8*sqrt(-(d*x + c)^2 + 1)*(d*x
+ c)*b^4*e^3*arcsin(d*x + c)^3/d + 1/4*(d*x + c)^4*a^4*e^3/d + 3/2*((d*x + c)^2 - 1)^2*a^2*b^2*e^3*arcsin(d*x
+ c)^2/d - 3/16*((d*x + c)^2 - 1)^2*b^4*e^3*arcsin(d*x + c)^2/d + 2*((d*x + c)^2 - 1)*a*b^3*e^3*arcsin(d*x + c
)^3/d + 5/32*b^4*e^3*arcsin(d*x + c)^4/d - 3/4*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*a^2*b^2*e^3*arcsin(d*x + c)/
d + 3/32*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*b^4*e^3*arcsin(d*x + c)/d + 15/8*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*
a*b^3*e^3*arcsin(d*x + c)^2/d + ((d*x + c)^2 - 1)^2*a^3*b*e^3*arcsin(d*x + c)/d - 3/8*((d*x + c)^2 - 1)^2*a*b^
3*e^3*arcsin(d*x + c)/d + 3*((d*x + c)^2 - 1)*a^2*b^2*e^3*arcsin(d*x + c)^2/d - 15/16*((d*x + c)^2 - 1)*b^4*e^
3*arcsin(d*x + c)^2/d + 5/8*a*b^3*e^3*arcsin(d*x + c)^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*e^3*arc
sin(d*x + c)/d - 51/64*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^4*e^3*arcsin(d*x + c)/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*e^3*arcsin(d*x + c)/d - 15/8*((
d*x + c)^2 - 1)*a*b^3*e^3*arcsin(d*x + c)/d + 15/16*a^2*b^2*e^3*arcsin(d*x + c)^2/d - 51/128*b^4*e^3*arcsin(d*
x + c)^2/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 - 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*e^3*arcsin(d*x +
c)/d - 51/64*a*b^3*e^3*arcsin(d*x + c)/d - 51/128*a^2*b^2*e^3/d + 195/1024*b^4*e^3/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^3*(a + b*asin(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^3*(a + b*asin(c + d*x))^4, x)

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