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]

45/256*b^3*e^3*arcsin(d*x+c)/d-9/32*b^2*e^3*(d*x+c)^2*(a+b*arcsin(d*x+c))/d-3/32*b^2*e^3*(d*x+c)^4*(a+b*arcsin
(d*x+c))/d-3/32*e^3*(a+b*arcsin(d*x+c))^3/d+1/4*e^3*(d*x+c)^4*(a+b*arcsin(d*x+c))^3/d-45/256*b^3*e^3*(d*x+c)*(
1-(d*x+c)^2)^(1/2)/d-3/128*b^3*e^3*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)/d+9/32*b*e^3*(d*x+c)*(a+b*arcsin(d*x+c))^2*(1
-(d*x+c)^2)^(1/2)/d+3/16*b*e^3*(d*x+c)^3*(a+b*arcsin(d*x+c))^2*(1-(d*x+c)^2)^(1/2)/d

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Rubi [A]  time = 0.40, antiderivative size = 287, normalized size of antiderivative = 1.00, 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 12

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

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 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 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 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)^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*}

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Mathematica [A]  time = 0.60, 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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fricas [B]  time = 0.62, size = 769, normalized size = 2.68 \[ \frac {8 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} d^{4} e^{3} x^{4} + 32 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} c d^{3} e^{3} x^{3} - 24 \, {\left (3 \, a b^{2} - 2 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} c^{2}\right )} d^{2} e^{3} x^{2} - 16 \, {\left (9 \, a b^{2} c - 2 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} c^{3}\right )} d e^{3} x + 8 \, {\left (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 + {\left (8 \, b^{3} c^{4} - 3 \, b^{3}\right )} e^{3}\right )} \arcsin \left (d x + c\right )^{3} + 24 \, {\left (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 + {\left (8 \, a b^{2} c^{4} - 3 \, a b^{2}\right )} e^{3}\right )} \arcsin \left (d x + c\right )^{2} + 3 \, {\left (8 \, {\left (8 \, a^{2} b - b^{3}\right )} d^{4} e^{3} x^{4} + 32 \, {\left (8 \, a^{2} b - b^{3}\right )} c d^{3} e^{3} x^{3} - 24 \, {\left (b^{3} - 2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{2}\right )} d^{2} e^{3} x^{2} - 16 \, {\left (3 \, b^{3} c - 2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{3}\right )} d e^{3} x - {\left (24 \, b^{3} c^{2} - 8 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{4} + 24 \, a^{2} b - 15 \, b^{3}\right )} e^{3}\right )} \arcsin \left (d x + c\right ) + 3 \, {\left (2 \, {\left (8 \, a^{2} b - b^{3}\right )} d^{3} e^{3} x^{3} + 6 \, {\left (8 \, a^{2} b - b^{3}\right )} c d^{2} e^{3} x^{2} + 3 \, {\left (8 \, a^{2} b - 5 \, b^{3} + 2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{2}\right )} d e^{3} x + {\left (2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{3} + 3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} c\right )} e^{3} + 8 \, {\left (2 \, b^{3} d^{3} e^{3} x^{3} + 6 \, b^{3} c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, b^{3} c^{2} + b^{3}\right )} d e^{3} x + {\left (2 \, b^{3} c^{3} + 3 \, b^{3} c\right )} e^{3}\right )} \arcsin \left (d x + c\right )^{2} + 16 \, {\left (2 \, a b^{2} d^{3} e^{3} x^{3} + 6 \, a b^{2} c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, a b^{2} c^{2} + a b^{2}\right )} d e^{3} x + {\left (2 \, a b^{2} c^{3} + 3 \, a b^{2} c\right )} e^{3}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{256 \, d} \]

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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giac [B]  time = 0.73, size = 617, normalized size = 2.15 \[ \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{3} \arcsin \left (d x + c\right )^{3} e^{3}}{4 \, d} - \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right )^{2} e^{3}}{16 \, d} + \frac {{\left (d x + c\right )}^{4} a^{3} e^{3}}{4 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a b^{2} \arcsin \left (d x + c\right )^{2} e^{3}}{4 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{3} \arcsin \left (d x + c\right )^{3} e^{3}}{2 \, d} - \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} a b^{2} \arcsin \left (d x + c\right ) e^{3}}{8 \, d} + \frac {15 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right )^{2} e^{3}}{32 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a^{2} b \arcsin \left (d x + c\right ) e^{3}}{4 \, d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{3} \arcsin \left (d x + c\right ) e^{3}}{32 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{2} \arcsin \left (d x + c\right )^{2} e^{3}}{2 \, d} + \frac {5 \, b^{3} \arcsin \left (d x + c\right )^{3} e^{3}}{32 \, d} - \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} a^{2} b e^{3}}{16 \, d} + \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} b^{3} e^{3}}{128 \, d} + \frac {15 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b^{2} \arcsin \left (d x + c\right ) e^{3}}{16 \, d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a b^{2} e^{3}}{32 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} b \arcsin \left (d x + c\right ) e^{3}}{2 \, d} - \frac {15 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} b^{3} \arcsin \left (d x + c\right ) e^{3}}{32 \, d} + \frac {15 \, a b^{2} \arcsin \left (d x + c\right )^{2} e^{3}}{32 \, d} + \frac {15 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a^{2} b e^{3}}{32 \, d} - \frac {51 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{3} e^{3}}{256 \, d} - \frac {15 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{2} e^{3}}{32 \, d} + \frac {15 \, a^{2} b \arcsin \left (d x + c\right ) e^{3}}{32 \, d} - \frac {51 \, b^{3} \arcsin \left (d x + c\right ) e^{3}}{256 \, d} - \frac {51 \, a b^{2} e^{3}}{256 \, d} \]

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 + 1/4*(d*x + c)^4*a^3*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 + 1
5/32*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 -
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 - 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

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maple [A]  time = 0.07, size = 397, normalized size = 1.38 \[ \frac {\frac {e^{3} \left (d x +c \right )^{4} a^{3}}{4}+e^{3} b^{3} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{3}}{4}-\frac {3 \arcsin \left (d x +c \right )^{2} \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{32}-\frac {3 \left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{32}-\frac {3 \left (d x +c \right ) \left (2 \left (d x +c \right )^{2}+3\right ) \sqrt {1-\left (d x +c \right )^{2}}}{256}-\frac {27 \arcsin \left (d x +c \right )}{256}-\frac {9 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}-1\right )}{32}-\frac {9 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{64}+\frac {3 \arcsin \left (d x +c \right )^{3}}{16}\right )+3 e^{3} a \,b^{2} \left (\frac {\arcsin \left (d x +c \right )^{2} \left (d x +c \right )^{4}}{4}-\frac {\arcsin \left (d x +c \right ) \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{16}+\frac {3 \arcsin \left (d x +c \right )^{2}}{32}-\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}\right )+3 e^{3} a^{2} b \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsin \left (d x +c \right )}{32}\right )}{d} \]

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]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

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]

1/4*a^3*d^3*e^3*x^4 + a^3*c*d^2*e^3*x^3 + 3/2*a^3*c^2*d*e^3*x^2 + 9/4*(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)*arcs
in(-(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^2*b*c^2*
d*e^3 + 1/2*(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^2*b*c*d^2*e^3 + 1/32*(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)
/sqrt(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*arcsi
n(-(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^2*b*d^3*e^3 + a^3*c^3*e^3*x + 3*((d*x +
c)*arcsin(d*x + c) + sqrt(-(d*x + c)^2 + 1))*a^2*b*c^3*e^3/d + 1/4*(b^3*d^3*e^3*x^4 + 4*b^3*c*d^2*e^3*x^3 + 6
*b^3*c^2*d*e^3*x^2 + 4*b^3*c^3*e^3*x)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 + integrate(3/4
*((b^3*d^4*e^3*x^4 + 4*b^3*c*d^3*e^3*x^3 + 6*b^3*c^2*d^2*e^3*x^2 + 4*b^3*c^3*d*e^3*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))^2 + 4*(a*b^2*d^5*e^3*x^5 + 5*a*b^2*c*d^4*e
^3*x^4 + (10*a*b^2*c^2 - a*b^2)*d^3*e^3*x^3 + (10*a*b^2*c^3 - 3*a*b^2*c)*d^2*e^3*x^2 + (5*a*b^2*c^4 - 3*a*b^2*
c^2)*d*e^3*x + (a*b^2*c^5 - a*b^2*c^3)*e^3)*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 \[ \int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 9.22, size = 1828, normalized size = 6.37 \[ \text {result too large to display} \]

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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