3.207 \(\int (c e+d e x)^2 (a+b \sin ^{-1}(c+d x))^4 \, dx\)

Optimal. Leaf size=289 \[ -\frac {160 b^3 e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{27 d}-\frac {4 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{9 d}-\frac {8 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {8 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {4 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^4}{3 d}+\frac {8 b^4 e^2 (c+d x)^3}{81 d}+\frac {160}{27} b^4 e^2 x \]

[Out]

160/27*b^4*e^2*x+8/81*b^4*e^2*(d*x+c)^3/d-8/3*b^2*e^2*(d*x+c)*(a+b*arcsin(d*x+c))^2/d-4/9*b^2*e^2*(d*x+c)^3*(a
+b*arcsin(d*x+c))^2/d+1/3*e^2*(d*x+c)^3*(a+b*arcsin(d*x+c))^4/d-160/27*b^3*e^2*(a+b*arcsin(d*x+c))*(1-(d*x+c)^
2)^(1/2)/d-8/27*b^3*e^2*(d*x+c)^2*(a+b*arcsin(d*x+c))*(1-(d*x+c)^2)^(1/2)/d+8/9*b*e^2*(a+b*arcsin(d*x+c))^3*(1
-(d*x+c)^2)^(1/2)/d+4/9*b*e^2*(d*x+c)^2*(a+b*arcsin(d*x+c))^3*(1-(d*x+c)^2)^(1/2)/d

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Rubi [A]  time = 0.48, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4805, 12, 4627, 4707, 4677, 4619, 8, 30} \[ -\frac {160 b^3 e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{27 d}-\frac {4 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{9 d}-\frac {8 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {8 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {4 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^4}{3 d}+\frac {8 b^4 e^2 (c+d x)^3}{81 d}+\frac {160}{27} b^4 e^2 x \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^2*(a + b*ArcSin[c + d*x])^4,x]

[Out]

(160*b^4*e^2*x)/27 + (8*b^4*e^2*(c + d*x)^3)/(81*d) - (160*b^3*e^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x
]))/(27*d) - (8*b^3*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(27*d) - (8*b^2*e^2*(c + d*
x)*(a + b*ArcSin[c + d*x])^2)/(3*d) - (4*b^2*e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^2)/(9*d) + (8*b*e^2*Sqrt[
1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/(9*d) + (4*b*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c
+ d*x])^3)/(9*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^4)/(3*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 30

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

Rule 4619

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

Rule 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 4677

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

Rule 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 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]

Rubi steps

\begin {align*} \int (c e+d e x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int e^2 x^2 \left (a+b \sin ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \operatorname {Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^4}{3 d}-\frac {\left (4 b e^2\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (a+b \sin ^{-1}(x)\right )^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d}\\ &=\frac {4 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^4}{3 d}-\frac {\left (8 b e^2\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d}-\frac {\left (4 b^2 e^2\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{3 d}\\ &=-\frac {4 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{9 d}+\frac {8 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {4 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^4}{3 d}-\frac {\left (8 b^2 e^2\right ) \operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{3 d}+\frac {\left (8 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d}\\ &=-\frac {8 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}-\frac {4 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{9 d}+\frac {8 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {4 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^4}{3 d}+\frac {\left (16 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{27 d}+\frac {\left (16 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d}+\frac {\left (8 b^4 e^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,c+d x\right )}{27 d}\\ &=\frac {8 b^4 e^2 (c+d x)^3}{81 d}-\frac {160 b^3 e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}-\frac {4 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{9 d}+\frac {8 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {4 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^4}{3 d}+\frac {\left (16 b^4 e^2\right ) \operatorname {Subst}(\int 1 \, dx,x,c+d x)}{27 d}+\frac {\left (16 b^4 e^2\right ) \operatorname {Subst}(\int 1 \, dx,x,c+d x)}{3 d}\\ &=\frac {160}{27} b^4 e^2 x+\frac {8 b^4 e^2 (c+d x)^3}{81 d}-\frac {160 b^3 e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}-\frac {4 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{9 d}+\frac {8 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {4 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^4}{3 d}\\ \end {align*}

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Mathematica [A]  time = 0.78, size = 235, normalized size = 0.81 \[ \frac {e^2 \left (\frac {1}{3} (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^4-\frac {4}{9} b \left (\frac {2}{3} b^2 \sqrt {1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )-\frac {40}{3} b^2 \left (b d x-\sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )\right )+b (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2-\sqrt {1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^3+6 b (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2-2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3-\frac {2}{9} b^3 (c+d x)^3\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^2*(a + b*ArcSin[c + d*x])^4,x]

[Out]

(e^2*(((c + d*x)^3*(a + b*ArcSin[c + d*x])^4)/3 - (4*b*((-2*b^3*(c + d*x)^3)/9 + (2*b^2*(c + d*x)^2*Sqrt[1 - (
c + d*x)^2]*(a + b*ArcSin[c + d*x]))/3 + 6*b*(c + d*x)*(a + b*ArcSin[c + d*x])^2 + b*(c + d*x)^3*(a + b*ArcSin
[c + d*x])^2 - 2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3 - (c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*Ar
cSin[c + d*x])^3 - (40*b^2*(b*d*x - Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])))/3))/9))/d

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fricas [B]  time = 0.56, size = 784, normalized size = 2.71 \[ \frac {{\left (27 \, a^{4} - 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (27 \, a^{4} - 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (72 \, a^{2} b^{2} - 160 \, b^{4} - {\left (27 \, a^{4} - 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} c^{2}\right )} d e^{2} x + 27 \, {\left (b^{4} d^{3} e^{2} x^{3} + 3 \, b^{4} c d^{2} e^{2} x^{2} + 3 \, b^{4} c^{2} d e^{2} x + b^{4} c^{3} e^{2}\right )} \arcsin \left (d x + c\right )^{4} + 108 \, {\left (a b^{3} d^{3} e^{2} x^{3} + 3 \, a b^{3} c d^{2} e^{2} x^{2} + 3 \, a b^{3} c^{2} d e^{2} x + a b^{3} c^{3} e^{2}\right )} \arcsin \left (d x + c\right )^{3} + 18 \, {\left ({\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (4 \, b^{4} - {\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} c^{2}\right )} d e^{2} x - {\left (12 \, b^{4} c - {\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} c^{3}\right )} e^{2}\right )} \arcsin \left (d x + c\right )^{2} + 36 \, {\left ({\left (3 \, a^{3} b - 2 \, a b^{3}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (3 \, a^{3} b - 2 \, a b^{3}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (4 \, a b^{3} - {\left (3 \, a^{3} b - 2 \, a b^{3}\right )} c^{2}\right )} d e^{2} x - {\left (12 \, a b^{3} c - {\left (3 \, a^{3} b - 2 \, a b^{3}\right )} c^{3}\right )} e^{2}\right )} \arcsin \left (d x + c\right ) + 12 \, {\left ({\left (3 \, a^{3} b - 2 \, a b^{3}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (3 \, a^{3} b - 2 \, a b^{3}\right )} c d e^{2} x + 3 \, {\left (b^{4} d^{2} e^{2} x^{2} + 2 \, b^{4} c d e^{2} x + {\left (b^{4} c^{2} + 2 \, b^{4}\right )} e^{2}\right )} \arcsin \left (d x + c\right )^{3} + {\left (6 \, a^{3} b - 40 \, a b^{3} + {\left (3 \, a^{3} b - 2 \, a b^{3}\right )} c^{2}\right )} e^{2} + 9 \, {\left (a b^{3} d^{2} e^{2} x^{2} + 2 \, a b^{3} c d e^{2} x + {\left (a b^{3} c^{2} + 2 \, a b^{3}\right )} e^{2}\right )} \arcsin \left (d x + c\right )^{2} + {\left ({\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} c d e^{2} x + {\left (18 \, a^{2} b^{2} - 40 \, b^{4} + {\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} c^{2}\right )} e^{2}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{81 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2*(a+b*arcsin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/81*((27*a^4 - 36*a^2*b^2 + 8*b^4)*d^3*e^2*x^3 + 3*(27*a^4 - 36*a^2*b^2 + 8*b^4)*c*d^2*e^2*x^2 - 3*(72*a^2*b^
2 - 160*b^4 - (27*a^4 - 36*a^2*b^2 + 8*b^4)*c^2)*d*e^2*x + 27*(b^4*d^3*e^2*x^3 + 3*b^4*c*d^2*e^2*x^2 + 3*b^4*c
^2*d*e^2*x + b^4*c^3*e^2)*arcsin(d*x + c)^4 + 108*(a*b^3*d^3*e^2*x^3 + 3*a*b^3*c*d^2*e^2*x^2 + 3*a*b^3*c^2*d*e
^2*x + a*b^3*c^3*e^2)*arcsin(d*x + c)^3 + 18*((9*a^2*b^2 - 2*b^4)*d^3*e^2*x^3 + 3*(9*a^2*b^2 - 2*b^4)*c*d^2*e^
2*x^2 - 3*(4*b^4 - (9*a^2*b^2 - 2*b^4)*c^2)*d*e^2*x - (12*b^4*c - (9*a^2*b^2 - 2*b^4)*c^3)*e^2)*arcsin(d*x + c
)^2 + 36*((3*a^3*b - 2*a*b^3)*d^3*e^2*x^3 + 3*(3*a^3*b - 2*a*b^3)*c*d^2*e^2*x^2 - 3*(4*a*b^3 - (3*a^3*b - 2*a*
b^3)*c^2)*d*e^2*x - (12*a*b^3*c - (3*a^3*b - 2*a*b^3)*c^3)*e^2)*arcsin(d*x + c) + 12*((3*a^3*b - 2*a*b^3)*d^2*
e^2*x^2 + 2*(3*a^3*b - 2*a*b^3)*c*d*e^2*x + 3*(b^4*d^2*e^2*x^2 + 2*b^4*c*d*e^2*x + (b^4*c^2 + 2*b^4)*e^2)*arcs
in(d*x + c)^3 + (6*a^3*b - 40*a*b^3 + (3*a^3*b - 2*a*b^3)*c^2)*e^2 + 9*(a*b^3*d^2*e^2*x^2 + 2*a*b^3*c*d*e^2*x
+ (a*b^3*c^2 + 2*a*b^3)*e^2)*arcsin(d*x + c)^2 + ((9*a^2*b^2 - 2*b^4)*d^2*e^2*x^2 + 2*(9*a^2*b^2 - 2*b^4)*c*d*
e^2*x + (18*a^2*b^2 - 40*b^4 + (9*a^2*b^2 - 2*b^4)*c^2)*e^2)*arcsin(d*x + c))*sqrt(-d^2*x^2 - 2*c*d*x - c^2 +
1))/d

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giac [B]  time = 0.88, size = 780, normalized size = 2.70 \[ \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{4} \arcsin \left (d x + c\right )^{4} e^{2}}{3 \, d} + \frac {4 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a b^{3} \arcsin \left (d x + c\right )^{3} e^{2}}{3 \, d} + \frac {{\left (d x + c\right )} b^{4} \arcsin \left (d x + c\right )^{4} e^{2}}{3 \, d} - \frac {4 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b^{4} \arcsin \left (d x + c\right )^{3} e^{2}}{9 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a^{2} b^{2} \arcsin \left (d x + c\right )^{2} e^{2}}{d} - \frac {4 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{4} \arcsin \left (d x + c\right )^{2} e^{2}}{9 \, d} + \frac {4 \, {\left (d x + c\right )} a b^{3} \arcsin \left (d x + c\right )^{3} e^{2}}{3 \, d} - \frac {4 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a b^{3} \arcsin \left (d x + c\right )^{2} e^{2}}{3 \, d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{4} \arcsin \left (d x + c\right )^{3} e^{2}}{3 \, d} + \frac {{\left (d x + c\right )}^{3} a^{4} e^{2}}{3 \, d} + \frac {4 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a^{3} b \arcsin \left (d x + c\right ) e^{2}}{3 \, d} - \frac {8 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a b^{3} \arcsin \left (d x + c\right ) e^{2}}{9 \, d} + \frac {2 \, {\left (d x + c\right )} a^{2} b^{2} \arcsin \left (d x + c\right )^{2} e^{2}}{d} - \frac {28 \, {\left (d x + c\right )} b^{4} \arcsin \left (d x + c\right )^{2} e^{2}}{9 \, d} - \frac {4 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a^{2} b^{2} \arcsin \left (d x + c\right ) e^{2}}{3 \, d} + \frac {8 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b^{4} \arcsin \left (d x + c\right ) e^{2}}{27 \, d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{3} \arcsin \left (d x + c\right )^{2} e^{2}}{d} - \frac {4 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a^{2} b^{2} e^{2}}{9 \, d} + \frac {8 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{4} e^{2}}{81 \, d} + \frac {4 \, {\left (d x + c\right )} a^{3} b \arcsin \left (d x + c\right ) e^{2}}{3 \, d} - \frac {56 \, {\left (d x + c\right )} a b^{3} \arcsin \left (d x + c\right ) e^{2}}{9 \, d} - \frac {4 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a^{3} b e^{2}}{9 \, d} + \frac {8 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a b^{3} e^{2}}{27 \, d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b^{2} \arcsin \left (d x + c\right ) e^{2}}{d} - \frac {56 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{4} \arcsin \left (d x + c\right ) e^{2}}{9 \, d} - \frac {28 \, {\left (d x + c\right )} a^{2} b^{2} e^{2}}{9 \, d} + \frac {488 \, {\left (d x + c\right )} b^{4} e^{2}}{81 \, d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{3} b e^{2}}{3 \, d} - \frac {56 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{3} e^{2}}{9 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2*(a+b*arcsin(d*x+c))^4,x, algorithm="giac")

[Out]

1/3*((d*x + c)^2 - 1)*(d*x + c)*b^4*arcsin(d*x + c)^4*e^2/d + 4/3*((d*x + c)^2 - 1)*(d*x + c)*a*b^3*arcsin(d*x
+ c)^3*e^2/d + 1/3*(d*x + c)*b^4*arcsin(d*x + c)^4*e^2/d - 4/9*(-(d*x + c)^2 + 1)^(3/2)*b^4*arcsin(d*x + c)^3
*e^2/d + 2*((d*x + c)^2 - 1)*(d*x + c)*a^2*b^2*arcsin(d*x + c)^2*e^2/d - 4/9*((d*x + c)^2 - 1)*(d*x + c)*b^4*a
rcsin(d*x + c)^2*e^2/d + 4/3*(d*x + c)*a*b^3*arcsin(d*x + c)^3*e^2/d - 4/3*(-(d*x + c)^2 + 1)^(3/2)*a*b^3*arcs
in(d*x + c)^2*e^2/d + 4/3*sqrt(-(d*x + c)^2 + 1)*b^4*arcsin(d*x + c)^3*e^2/d + 1/3*(d*x + c)^3*a^4*e^2/d + 4/3
*((d*x + c)^2 - 1)*(d*x + c)*a^3*b*arcsin(d*x + c)*e^2/d - 8/9*((d*x + c)^2 - 1)*(d*x + c)*a*b^3*arcsin(d*x +
c)*e^2/d + 2*(d*x + c)*a^2*b^2*arcsin(d*x + c)^2*e^2/d - 28/9*(d*x + c)*b^4*arcsin(d*x + c)^2*e^2/d - 4/3*(-(d
*x + c)^2 + 1)^(3/2)*a^2*b^2*arcsin(d*x + c)*e^2/d + 8/27*(-(d*x + c)^2 + 1)^(3/2)*b^4*arcsin(d*x + c)*e^2/d +
4*sqrt(-(d*x + c)^2 + 1)*a*b^3*arcsin(d*x + c)^2*e^2/d - 4/9*((d*x + c)^2 - 1)*(d*x + c)*a^2*b^2*e^2/d + 8/81
*((d*x + c)^2 - 1)*(d*x + c)*b^4*e^2/d + 4/3*(d*x + c)*a^3*b*arcsin(d*x + c)*e^2/d - 56/9*(d*x + c)*a*b^3*arcs
in(d*x + c)*e^2/d - 4/9*(-(d*x + c)^2 + 1)^(3/2)*a^3*b*e^2/d + 8/27*(-(d*x + c)^2 + 1)^(3/2)*a*b^3*e^2/d + 4*s
qrt(-(d*x + c)^2 + 1)*a^2*b^2*arcsin(d*x + c)*e^2/d - 56/9*sqrt(-(d*x + c)^2 + 1)*b^4*arcsin(d*x + c)*e^2/d -
28/9*(d*x + c)*a^2*b^2*e^2/d + 488/81*(d*x + c)*b^4*e^2/d + 4/3*sqrt(-(d*x + c)^2 + 1)*a^3*b*e^2/d - 56/9*sqrt
(-(d*x + c)^2 + 1)*a*b^3*e^2/d

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maple [A]  time = 0.05, size = 440, normalized size = 1.52 \[ \frac {\frac {e^{2} \left (d x +c \right )^{3} a^{4}}{3}+e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{4}}{3}+\frac {4 \arcsin \left (d x +c \right )^{3} \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {8 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )}{3}+\frac {160 d x}{27}+\frac {160 c}{27}-\frac {16 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )^{3}}{9}-\frac {8 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )+4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{3}}{3}+\frac {\arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \arcsin \left (d x +c \right )}{3}-\frac {2 \arcsin \left (d x +c \right ) \left (d x +c \right )^{3}}{9}-\frac {2 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{27}\right )+6 e^{2} a^{2} b^{2} \left (\frac {\arcsin \left (d x +c \right )^{2} \left (d x +c \right )^{3}}{3}+\frac {2 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {2 \left (d x +c \right )^{3}}{27}-\frac {4 d x}{9}-\frac {4 c}{9}\right )+4 e^{2} a^{3} b \left (\frac {\arcsin \left (d x +c \right ) \left (d x +c \right )^{3}}{3}+\frac {\left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (d x +c \right )^{2}}}{9}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^2*(a+b*arcsin(d*x+c))^4,x)

[Out]

1/d*(1/3*e^2*(d*x+c)^3*a^4+e^2*b^4*(1/3*(d*x+c)^3*arcsin(d*x+c)^4+4/9*arcsin(d*x+c)^3*((d*x+c)^2+2)*(1-(d*x+c)
^2)^(1/2)-8/3*arcsin(d*x+c)^2*(d*x+c)+160/27*d*x+160/27*c-16/3*arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2)-4/9*arcsin(d*
x+c)^2*(d*x+c)^3-8/27*arcsin(d*x+c)*((d*x+c)^2+2)*(1-(d*x+c)^2)^(1/2)+8/81*(d*x+c)^3)+4*e^2*a*b^3*(1/3*(d*x+c)
^3*arcsin(d*x+c)^3+1/3*arcsin(d*x+c)^2*((d*x+c)^2+2)*(1-(d*x+c)^2)^(1/2)-4/3*(1-(d*x+c)^2)^(1/2)-4/3*(d*x+c)*a
rcsin(d*x+c)-2/9*arcsin(d*x+c)*(d*x+c)^3-2/27*((d*x+c)^2+2)*(1-(d*x+c)^2)^(1/2))+6*e^2*a^2*b^2*(1/3*arcsin(d*x
+c)^2*(d*x+c)^3+2/9*arcsin(d*x+c)*((d*x+c)^2+2)*(1-(d*x+c)^2)^(1/2)-2/27*(d*x+c)^3-4/9*d*x-4/9*c)+4*e^2*a^3*b*
(1/3*arcsin(d*x+c)*(d*x+c)^3+1/9*(d*x+c)^2*(1-(d*x+c)^2)^(1/2)+2/9*(1-(d*x+c)^2)^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a^{4} d^{2} e^{2} x^{3} + a^{4} c d e^{2} x^{2} + 2 \, {\left (2 \, x^{2} \arcsin \left (d x + c\right ) + d {\left (\frac {3 \, c^{2} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{3}} + \frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{3}} - \frac {3 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} c}{d^{3}}\right )}\right )} a^{3} b c d e^{2} + \frac {2}{9} \, {\left (6 \, x^{3} \arcsin \left (d x + c\right ) + d {\left (\frac {2 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} x^{2}}{d^{2}} - \frac {15 \, c^{3} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{4}} - \frac {5 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} c x}{d^{3}} + \frac {9 \, {\left (c^{2} - 1\right )} c \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{4}} + \frac {15 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} c^{2}}{d^{4}} - \frac {4 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (c^{2} - 1\right )}}{d^{4}}\right )}\right )} a^{3} b d^{2} e^{2} + a^{4} c^{2} e^{2} x + \frac {4 \, {\left ({\left (d x + c\right )} \arcsin \left (d x + c\right ) + \sqrt {-{\left (d x + c\right )}^{2} + 1}\right )} a^{3} b c^{2} e^{2}}{d} + \frac {1}{3} \, {\left (b^{4} d^{2} e^{2} x^{3} + 3 \, b^{4} c d e^{2} x^{2} + 3 \, b^{4} c^{2} e^{2} x\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{4} + \int \frac {2 \, {\left (2 \, {\left (b^{4} d^{3} e^{2} x^{3} + 3 \, b^{4} c d^{2} e^{2} x^{2} + 3 \, b^{4} c^{2} d e^{2} x\right )} \sqrt {d x + c + 1} \sqrt {-d x - c + 1} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{3} + 6 \, {\left (a b^{3} d^{4} e^{2} x^{4} + 4 \, a b^{3} c d^{3} e^{2} x^{3} + {\left (6 \, a b^{3} c^{2} - a b^{3}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (2 \, a b^{3} c^{3} - a b^{3} c\right )} d e^{2} x + {\left (a b^{3} c^{4} - a b^{3} c^{2}\right )} e^{2}\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{3} + 9 \, {\left (a^{2} b^{2} d^{4} e^{2} x^{4} + 4 \, a^{2} b^{2} c d^{3} e^{2} x^{3} + {\left (6 \, a^{2} b^{2} c^{2} - a^{2} b^{2}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (2 \, a^{2} b^{2} c^{3} - a^{2} b^{2} c\right )} d e^{2} x + {\left (a^{2} b^{2} c^{4} - a^{2} b^{2} c^{2}\right )} e^{2}\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{2}\right )}}{3 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2*(a+b*arcsin(d*x+c))^4,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^2*(a + b*asin(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^2*(a + b*asin(c + d*x))^4, x)

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sympy [A]  time = 8.78, size = 1889, normalized size = 6.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**2*(a+b*asin(d*x+c))**4,x)

[Out]

Piecewise((a**4*c**2*e**2*x + a**4*c*d*e**2*x**2 + a**4*d**2*e**2*x**3/3 + 4*a**3*b*c**3*e**2*asin(c + d*x)/(3
*d) + 4*a**3*b*c**2*e**2*x*asin(c + d*x) + 4*a**3*b*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(9*d) + 4*
a**3*b*c*d*e**2*x**2*asin(c + d*x) + 8*a**3*b*c*e**2*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/9 + 4*a**3*b*d**2
*e**2*x**3*asin(c + d*x)/3 + 4*a**3*b*d*e**2*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/9 + 8*a**3*b*e**2*sqrt
(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(9*d) + 2*a**2*b**2*c**3*e**2*asin(c + d*x)**2/d + 6*a**2*b**2*c**2*e**2*x*a
sin(c + d*x)**2 - 4*a**2*b**2*c**2*e**2*x/3 + 4*a**2*b**2*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin
(c + d*x)/(3*d) + 6*a**2*b**2*c*d*e**2*x**2*asin(c + d*x)**2 - 4*a**2*b**2*c*d*e**2*x**2/3 + 8*a**2*b**2*c*e**
2*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/3 + 2*a**2*b**2*d**2*e**2*x**3*asin(c + d*x)**2 - 4*a*
*2*b**2*d**2*e**2*x**3/9 + 4*a**2*b**2*d*e**2*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/3 - 8*a
**2*b**2*e**2*x/3 + 8*a**2*b**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(3*d) + 4*a*b**3*c**3
*e**2*asin(c + d*x)**3/(3*d) - 8*a*b**3*c**3*e**2*asin(c + d*x)/(9*d) + 4*a*b**3*c**2*e**2*x*asin(c + d*x)**3
- 8*a*b**3*c**2*e**2*x*asin(c + d*x)/3 + 4*a*b**3*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x
)**2/(3*d) - 8*a*b**3*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(27*d) + 4*a*b**3*c*d*e**2*x**2*asin(c +
d*x)**3 - 8*a*b**3*c*d*e**2*x**2*asin(c + d*x)/3 + 8*a*b**3*c*e**2*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*as
in(c + d*x)**2/3 - 16*a*b**3*c*e**2*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/27 - 16*a*b**3*c*e**2*asin(c + d*x
)/(3*d) + 4*a*b**3*d**2*e**2*x**3*asin(c + d*x)**3/3 - 8*a*b**3*d**2*e**2*x**3*asin(c + d*x)/9 + 4*a*b**3*d*e*
*2*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/3 - 8*a*b**3*d*e**2*x**2*sqrt(-c**2 - 2*c*d*x -
d**2*x**2 + 1)/27 - 16*a*b**3*e**2*x*asin(c + d*x)/3 + 8*a*b**3*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*as
in(c + d*x)**2/(3*d) - 160*a*b**3*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(27*d) + b**4*c**3*e**2*asin(c +
d*x)**4/(3*d) - 4*b**4*c**3*e**2*asin(c + d*x)**2/(9*d) + b**4*c**2*e**2*x*asin(c + d*x)**4 - 4*b**4*c**2*e**2
*x*asin(c + d*x)**2/3 + 8*b**4*c**2*e**2*x/27 + 4*b**4*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c
+ d*x)**3/(9*d) - 8*b**4*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(27*d) + b**4*c*d*e**2*
x**2*asin(c + d*x)**4 - 4*b**4*c*d*e**2*x**2*asin(c + d*x)**2/3 + 8*b**4*c*d*e**2*x**2/27 + 8*b**4*c*e**2*x*sq
rt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**3/9 - 16*b**4*c*e**2*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1
)*asin(c + d*x)/27 - 8*b**4*c*e**2*asin(c + d*x)**2/(3*d) + b**4*d**2*e**2*x**3*asin(c + d*x)**4/3 - 4*b**4*d*
*2*e**2*x**3*asin(c + d*x)**2/9 + 8*b**4*d**2*e**2*x**3/81 + 4*b**4*d*e**2*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x*
*2 + 1)*asin(c + d*x)**3/9 - 8*b**4*d*e**2*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/27 - 8*b**
4*e**2*x*asin(c + d*x)**2/3 + 160*b**4*e**2*x/27 + 8*b**4*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c +
d*x)**3/(9*d) - 160*b**4*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(27*d), Ne(d, 0)), (c**2*e**
2*x*(a + b*asin(c))**4, True))

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