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

Optimal. Leaf size=338 \[ -\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{125 d}-\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{75 d}-\frac{16}{25} a b^2 e^4 x+\frac{e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}+\frac{3 b e^4 \sqrt{1-(c+d x)^2} (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{4 b e^4 \sqrt{1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{8 b e^4 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}-\frac{6 b^3 e^4 \left (1-(c+d x)^2\right )^{5/2}}{625 d}+\frac{76 b^3 e^4 \left (1-(c+d x)^2\right )^{3/2}}{1125 d}-\frac{298 b^3 e^4 \sqrt{1-(c+d x)^2}}{375 d}-\frac{16 b^3 e^4 (c+d x) \sin ^{-1}(c+d x)}{25 d} \]

[Out]

(-16*a*b^2*e^4*x)/25 - (298*b^3*e^4*Sqrt[1 - (c + d*x)^2])/(375*d) + (76*b^3*e^4*(1 - (c + d*x)^2)^(3/2))/(112
5*d) - (6*b^3*e^4*(1 - (c + d*x)^2)^(5/2))/(625*d) - (16*b^3*e^4*(c + d*x)*ArcSin[c + d*x])/(25*d) - (8*b^2*e^
4*(c + d*x)^3*(a + b*ArcSin[c + d*x]))/(75*d) - (6*b^2*e^4*(c + d*x)^5*(a + b*ArcSin[c + d*x]))/(125*d) + (8*b
*e^4*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(25*d) + (4*b*e^4*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a +
b*ArcSin[c + d*x])^2)/(25*d) + (3*b*e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(25*d) +
(e^4*(c + d*x)^5*(a + b*ArcSin[c + d*x])^3)/(5*d)

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Rubi [A]  time = 0.493582, antiderivative size = 338, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {4805, 12, 4627, 4707, 4677, 4619, 261, 266, 43} \[ -\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{125 d}-\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{75 d}-\frac{16}{25} a b^2 e^4 x+\frac{e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}+\frac{3 b e^4 \sqrt{1-(c+d x)^2} (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{4 b e^4 \sqrt{1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{8 b e^4 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}-\frac{6 b^3 e^4 \left (1-(c+d x)^2\right )^{5/2}}{625 d}+\frac{76 b^3 e^4 \left (1-(c+d x)^2\right )^{3/2}}{1125 d}-\frac{298 b^3 e^4 \sqrt{1-(c+d x)^2}}{375 d}-\frac{16 b^3 e^4 (c+d x) \sin ^{-1}(c+d x)}{25 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*a*b^2*e^4*x)/25 - (298*b^3*e^4*Sqrt[1 - (c + d*x)^2])/(375*d) + (76*b^3*e^4*(1 - (c + d*x)^2)^(3/2))/(112
5*d) - (6*b^3*e^4*(1 - (c + d*x)^2)^(5/2))/(625*d) - (16*b^3*e^4*(c + d*x)*ArcSin[c + d*x])/(25*d) - (8*b^2*e^
4*(c + d*x)^3*(a + b*ArcSin[c + d*x]))/(75*d) - (6*b^2*e^4*(c + d*x)^5*(a + b*ArcSin[c + d*x]))/(125*d) + (8*b
*e^4*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(25*d) + (4*b*e^4*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a +
b*ArcSin[c + d*x])^2)/(25*d) + (3*b*e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(25*d) +
(e^4*(c + d*x)^5*(a + b*ArcSin[c + d*x])^3)/(5*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 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 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 261

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (c e+d e x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int e^4 x^4 \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 \operatorname{Subst}\left (\int x^4 \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}-\frac{\left (3 b e^4\right ) \operatorname{Subst}\left (\int \frac{x^5 \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{5 d}\\ &=\frac{3 b e^4 (c+d x)^4 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}-\frac{\left (12 b e^4\right ) \operatorname{Subst}\left (\int \frac{x^3 \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{25 d}-\frac{\left (6 b^2 e^4\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}\\ &=-\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{125 d}+\frac{4 b e^4 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{3 b e^4 (c+d x)^4 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}-\frac{\left (8 b e^4\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{25 d}-\frac{\left (8 b^2 e^4\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}+\frac{\left (6 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{125 d}\\ &=-\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{75 d}-\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{125 d}+\frac{8 b e^4 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{4 b e^4 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{3 b e^4 (c+d x)^4 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}-\frac{\left (16 b^2 e^4\right ) \operatorname{Subst}\left (\int \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}+\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x}} \, dx,x,(c+d x)^2\right )}{125 d}+\frac{\left (8 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{75 d}\\ &=-\frac{16}{25} a b^2 e^4 x-\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{75 d}-\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{125 d}+\frac{8 b e^4 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{4 b e^4 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{3 b e^4 (c+d x)^4 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}+\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \left (\frac{1}{\sqrt{1-x}}-2 \sqrt{1-x}+(1-x)^{3/2}\right ) \, dx,x,(c+d x)^2\right )}{125 d}+\frac{\left (4 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x}} \, dx,x,(c+d x)^2\right )}{75 d}-\frac{\left (16 b^3 e^4\right ) \operatorname{Subst}\left (\int \sin ^{-1}(x) \, dx,x,c+d x\right )}{25 d}\\ &=-\frac{16}{25} a b^2 e^4 x-\frac{6 b^3 e^4 \sqrt{1-(c+d x)^2}}{125 d}+\frac{4 b^3 e^4 \left (1-(c+d x)^2\right )^{3/2}}{125 d}-\frac{6 b^3 e^4 \left (1-(c+d x)^2\right )^{5/2}}{625 d}-\frac{16 b^3 e^4 (c+d x) \sin ^{-1}(c+d x)}{25 d}-\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{75 d}-\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{125 d}+\frac{8 b e^4 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{4 b e^4 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{3 b e^4 (c+d x)^4 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}+\frac{\left (4 b^3 e^4\right ) \operatorname{Subst}\left (\int \left (\frac{1}{\sqrt{1-x}}-\sqrt{1-x}\right ) \, dx,x,(c+d x)^2\right )}{75 d}+\frac{\left (16 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{25 d}\\ &=-\frac{16}{25} a b^2 e^4 x-\frac{298 b^3 e^4 \sqrt{1-(c+d x)^2}}{375 d}+\frac{76 b^3 e^4 \left (1-(c+d x)^2\right )^{3/2}}{1125 d}-\frac{6 b^3 e^4 \left (1-(c+d x)^2\right )^{5/2}}{625 d}-\frac{16 b^3 e^4 (c+d x) \sin ^{-1}(c+d x)}{25 d}-\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{75 d}-\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{125 d}+\frac{8 b e^4 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{4 b e^4 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{3 b e^4 (c+d x)^4 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.94015, size = 307, normalized size = 0.91 \[ \frac{e^4 \left ((c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3-\frac{1}{25} b \left (6 b (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )-15 \sqrt{1-(c+d x)^2} (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2+\frac{40}{3} b (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )-20 \sqrt{1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2-40 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2+80 b \left (a d x+b \sqrt{1-(c+d x)^2}+b (c+d x) \sin ^{-1}(c+d x)\right )+\frac{40}{9} b^2 \left (c^2+2 c d x+d^2 x^2+2\right ) \sqrt{1-(c+d x)^2}-\frac{2}{5} b^2 \sqrt{1-(c+d x)^2} \left (-3 \left ((c+d x)^2-1\right )^2+10 \left (1-(c+d x)^2\right )-15\right )\right )\right )}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*((c + d*x)^5*(a + b*ArcSin[c + d*x])^3 - (b*((40*b^2*(2 + c^2 + 2*c*d*x + d^2*x^2)*Sqrt[1 - (c + d*x)^2])
/9 - (2*b^2*Sqrt[1 - (c + d*x)^2]*(-15 + 10*(1 - (c + d*x)^2) - 3*(-1 + (c + d*x)^2)^2))/5 + (40*b*(c + d*x)^3
*(a + b*ArcSin[c + d*x]))/3 + 6*b*(c + d*x)^5*(a + b*ArcSin[c + d*x]) - 40*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin
[c + d*x])^2 - 20*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 - 15*(c + d*x)^4*Sqrt[1 - (c + d
*x)^2]*(a + b*ArcSin[c + d*x])^2 + 80*b*(a*d*x + b*Sqrt[1 - (c + d*x)^2] + b*(c + d*x)*ArcSin[c + d*x])))/25))
/(5*d)

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Maple [A]  time = 0.041, size = 383, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{{e}^{4} \left ( dx+c \right ) ^{5}{a}^{3}}{5}}+{e}^{4}{b}^{3} \left ({\frac{ \left ( dx+c \right ) ^{5} \left ( \arcsin \left ( dx+c \right ) \right ) ^{3}}{5}}+{\frac{ \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( 3\, \left ( dx+c \right ) ^{4}+4\, \left ( dx+c \right ) ^{2}+8 \right ) }{25}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{16}{25}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{ \left ( 16\,dx+16\,c \right ) \arcsin \left ( dx+c \right ) }{25}}-{\frac{6\, \left ( dx+c \right ) ^{5}\arcsin \left ( dx+c \right ) }{125}}-{\frac{6\, \left ( dx+c \right ) ^{4}+8\, \left ( dx+c \right ) ^{2}+16}{625}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{8\, \left ( dx+c \right ) ^{3}\arcsin \left ( dx+c \right ) }{75}}-{\frac{8\, \left ( dx+c \right ) ^{2}+16}{225}\sqrt{1- \left ( dx+c \right ) ^{2}}} \right ) +3\,{e}^{4}a{b}^{2} \left ( 1/5\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) ^{5}+{\frac{2\,\arcsin \left ( dx+c \right ) \left ( 3\, \left ( dx+c \right ) ^{4}+4\, \left ( dx+c \right ) ^{2}+8 \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}}{75}}-{\frac{2\, \left ( dx+c \right ) ^{5}}{125}}-{\frac{8\, \left ( dx+c \right ) ^{3}}{225}}-{\frac{16\,dx}{75}}-{\frac{16\,c}{75}} \right ) +3\,{e}^{4}{a}^{2}b \left ( 1/5\, \left ( dx+c \right ) ^{5}\arcsin \left ( dx+c \right ) +1/25\, \left ( dx+c \right ) ^{4}\sqrt{1- \left ( dx+c \right ) ^{2}}+{\frac{4\, \left ( dx+c \right ) ^{2}\sqrt{1- \left ( dx+c \right ) ^{2}}}{75}}+{\frac{8\,\sqrt{1- \left ( dx+c \right ) ^{2}}}{75}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/5*e^4*(d*x+c)^5*a^3+e^4*b^3*(1/5*(d*x+c)^5*arcsin(d*x+c)^3+1/25*arcsin(d*x+c)^2*(3*(d*x+c)^4+4*(d*x+c)^
2+8)*(1-(d*x+c)^2)^(1/2)-16/25*(1-(d*x+c)^2)^(1/2)-16/25*(d*x+c)*arcsin(d*x+c)-6/125*(d*x+c)^5*arcsin(d*x+c)-2
/625*(3*(d*x+c)^4+4*(d*x+c)^2+8)*(1-(d*x+c)^2)^(1/2)-8/75*(d*x+c)^3*arcsin(d*x+c)-8/225*((d*x+c)^2+2)*(1-(d*x+
c)^2)^(1/2))+3*e^4*a*b^2*(1/5*arcsin(d*x+c)^2*(d*x+c)^5+2/75*arcsin(d*x+c)*(3*(d*x+c)^4+4*(d*x+c)^2+8)*(1-(d*x
+c)^2)^(1/2)-2/125*(d*x+c)^5-8/225*(d*x+c)^3-16/75*d*x-16/75*c)+3*e^4*a^2*b*(1/5*(d*x+c)^5*arcsin(d*x+c)+1/25*
(d*x+c)^4*(1-(d*x+c)^2)^(1/2)+4/75*(d*x+c)^2*(1-(d*x+c)^2)^(1/2)+8/75*(1-(d*x+c)^2)^(1/2)))

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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)^4*(a+b*arcsin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.58772, size = 2141, normalized size = 6.33 \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)^4*(a+b*arcsin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/5625*(45*(25*a^3 - 6*a*b^2)*d^5*e^4*x^5 + 225*(25*a^3 - 6*a*b^2)*c*d^4*e^4*x^4 - 150*(4*a*b^2 - 3*(25*a^3 -
6*a*b^2)*c^2)*d^3*e^4*x^3 - 450*(4*a*b^2*c - (25*a^3 - 6*a*b^2)*c^3)*d^2*e^4*x^2 - 225*(8*a*b^2*c^2 - (25*a^3
- 6*a*b^2)*c^4 + 16*a*b^2)*d*e^4*x + 1125*(b^3*d^5*e^4*x^5 + 5*b^3*c*d^4*e^4*x^4 + 10*b^3*c^2*d^3*e^4*x^3 + 10
*b^3*c^3*d^2*e^4*x^2 + 5*b^3*c^4*d*e^4*x + b^3*c^5*e^4)*arcsin(d*x + c)^3 + 3375*(a*b^2*d^5*e^4*x^5 + 5*a*b^2*
c*d^4*e^4*x^4 + 10*a*b^2*c^2*d^3*e^4*x^3 + 10*a*b^2*c^3*d^2*e^4*x^2 + 5*a*b^2*c^4*d*e^4*x + a*b^2*c^5*e^4)*arc
sin(d*x + c)^2 + 15*(9*(25*a^2*b - 2*b^3)*d^5*e^4*x^5 + 45*(25*a^2*b - 2*b^3)*c*d^4*e^4*x^4 - 10*(4*b^3 - 9*(2
5*a^2*b - 2*b^3)*c^2)*d^3*e^4*x^3 - 30*(4*b^3*c - 3*(25*a^2*b - 2*b^3)*c^3)*d^2*e^4*x^2 - 15*(8*b^3*c^2 - 3*(2
5*a^2*b - 2*b^3)*c^4 + 16*b^3)*d*e^4*x - (40*b^3*c^3 - 9*(25*a^2*b - 2*b^3)*c^5 + 240*b^3*c)*e^4)*arcsin(d*x +
c) + (27*(25*a^2*b - 2*b^3)*d^4*e^4*x^4 + 108*(25*a^2*b - 2*b^3)*c*d^3*e^4*x^3 + 2*(450*a^2*b - 136*b^3 + 81*
(25*a^2*b - 2*b^3)*c^2)*d^2*e^4*x^2 + 4*(27*(25*a^2*b - 2*b^3)*c^3 + 2*(225*a^2*b - 68*b^3)*c)*d*e^4*x + (27*(
25*a^2*b - 2*b^3)*c^4 + 1800*a^2*b - 4144*b^3 + 4*(225*a^2*b - 68*b^3)*c^2)*e^4 + 225*(3*b^3*d^4*e^4*x^4 + 12*
b^3*c*d^3*e^4*x^3 + 2*(9*b^3*c^2 + 2*b^3)*d^2*e^4*x^2 + 4*(3*b^3*c^3 + 2*b^3*c)*d*e^4*x + (3*b^3*c^4 + 4*b^3*c
^2 + 8*b^3)*e^4)*arcsin(d*x + c)^2 + 450*(3*a*b^2*d^4*e^4*x^4 + 12*a*b^2*c*d^3*e^4*x^3 + 2*(9*a*b^2*c^2 + 2*a*
b^2)*d^2*e^4*x^2 + 4*(3*a*b^2*c^3 + 2*a*b^2*c)*d*e^4*x + (3*a*b^2*c^4 + 4*a*b^2*c^2 + 8*a*b^2)*e^4)*arcsin(d*x
+ c))*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1))/d

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Sympy [A]  time = 26.3756, size = 2518, normalized size = 7.45 \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)**4*(a+b*asin(d*x+c))**3,x)

[Out]

Piecewise((a**3*c**4*e**4*x + 2*a**3*c**3*d*e**4*x**2 + 2*a**3*c**2*d**2*e**4*x**3 + a**3*c*d**3*e**4*x**4 + a
**3*d**4*e**4*x**5/5 + 3*a**2*b*c**5*e**4*asin(c + d*x)/(5*d) + 3*a**2*b*c**4*e**4*x*asin(c + d*x) + 3*a**2*b*
c**4*e**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(25*d) + 6*a**2*b*c**3*d*e**4*x**2*asin(c + d*x) + 12*a**2*b*c
**3*e**4*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/25 + 6*a**2*b*c**2*d**2*e**4*x**3*asin(c + d*x) + 18*a**2*b*c
**2*d*e**4*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/25 + 4*a**2*b*c**2*e**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2
+ 1)/(25*d) + 3*a**2*b*c*d**3*e**4*x**4*asin(c + d*x) + 12*a**2*b*c*d**2*e**4*x**3*sqrt(-c**2 - 2*c*d*x - d**
2*x**2 + 1)/25 + 8*a**2*b*c*e**4*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/25 + 3*a**2*b*d**4*e**4*x**5*asin(c +
d*x)/5 + 3*a**2*b*d**3*e**4*x**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/25 + 4*a**2*b*d*e**4*x**2*sqrt(-c**2 -
2*c*d*x - d**2*x**2 + 1)/25 + 8*a**2*b*e**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(25*d) + 3*a*b**2*c**5*e**4
*asin(c + d*x)**2/(5*d) + 3*a*b**2*c**4*e**4*x*asin(c + d*x)**2 - 6*a*b**2*c**4*e**4*x/25 + 6*a*b**2*c**4*e**4
*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(25*d) + 6*a*b**2*c**3*d*e**4*x**2*asin(c + d*x)**2 - 12*
a*b**2*c**3*d*e**4*x**2/25 + 24*a*b**2*c**3*e**4*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/25 + 6*
a*b**2*c**2*d**2*e**4*x**3*asin(c + d*x)**2 - 12*a*b**2*c**2*d**2*e**4*x**3/25 + 36*a*b**2*c**2*d*e**4*x**2*sq
rt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/25 - 8*a*b**2*c**2*e**4*x/25 + 8*a*b**2*c**2*e**4*sqrt(-c**2
- 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(25*d) + 3*a*b**2*c*d**3*e**4*x**4*asin(c + d*x)**2 - 6*a*b**2*c*d**
3*e**4*x**4/25 + 24*a*b**2*c*d**2*e**4*x**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/25 - 8*a*b**2*
c*d*e**4*x**2/25 + 16*a*b**2*c*e**4*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/25 + 3*a*b**2*d**4*e
**4*x**5*asin(c + d*x)**2/5 - 6*a*b**2*d**4*e**4*x**5/125 + 6*a*b**2*d**3*e**4*x**4*sqrt(-c**2 - 2*c*d*x - d**
2*x**2 + 1)*asin(c + d*x)/25 - 8*a*b**2*d**2*e**4*x**3/75 + 8*a*b**2*d*e**4*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x
**2 + 1)*asin(c + d*x)/25 - 16*a*b**2*e**4*x/25 + 16*a*b**2*e**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c
+ d*x)/(25*d) + b**3*c**5*e**4*asin(c + d*x)**3/(5*d) - 6*b**3*c**5*e**4*asin(c + d*x)/(125*d) + b**3*c**4*e**
4*x*asin(c + d*x)**3 - 6*b**3*c**4*e**4*x*asin(c + d*x)/25 + 3*b**3*c**4*e**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2
+ 1)*asin(c + d*x)**2/(25*d) - 6*b**3*c**4*e**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(625*d) + 2*b**3*c**3*d
*e**4*x**2*asin(c + d*x)**3 - 12*b**3*c**3*d*e**4*x**2*asin(c + d*x)/25 + 12*b**3*c**3*e**4*x*sqrt(-c**2 - 2*c
*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/25 - 24*b**3*c**3*e**4*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/625 - 8*
b**3*c**3*e**4*asin(c + d*x)/(75*d) + 2*b**3*c**2*d**2*e**4*x**3*asin(c + d*x)**3 - 12*b**3*c**2*d**2*e**4*x**
3*asin(c + d*x)/25 + 18*b**3*c**2*d*e**4*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/25 - 36*b
**3*c**2*d*e**4*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/625 - 8*b**3*c**2*e**4*x*asin(c + d*x)/25 + 4*b**3*
c**2*e**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/(25*d) - 272*b**3*c**2*e**4*sqrt(-c**2 - 2*c*
d*x - d**2*x**2 + 1)/(5625*d) + b**3*c*d**3*e**4*x**4*asin(c + d*x)**3 - 6*b**3*c*d**3*e**4*x**4*asin(c + d*x)
/25 + 12*b**3*c*d**2*e**4*x**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/25 - 24*b**3*c*d**2*e**4
*x**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/625 - 8*b**3*c*d*e**4*x**2*asin(c + d*x)/25 + 8*b**3*c*e**4*x*sqrt
(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/25 - 544*b**3*c*e**4*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1
)/5625 - 16*b**3*c*e**4*asin(c + d*x)/(25*d) + b**3*d**4*e**4*x**5*asin(c + d*x)**3/5 - 6*b**3*d**4*e**4*x**5*
asin(c + d*x)/125 + 3*b**3*d**3*e**4*x**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/25 - 6*b**3*d
**3*e**4*x**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/625 - 8*b**3*d**2*e**4*x**3*asin(c + d*x)/75 + 4*b**3*d*e*
*4*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/25 - 272*b**3*d*e**4*x**2*sqrt(-c**2 - 2*c*d*x
- d**2*x**2 + 1)/5625 - 16*b**3*e**4*x*asin(c + d*x)/25 + 8*b**3*e**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*as
in(c + d*x)**2/(25*d) - 4144*b**3*e**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(5625*d), Ne(d, 0)), (c**4*e**4*x
*(a + b*asin(c))**3, True))

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Giac [B]  time = 1.35541, size = 1085, normalized size = 3.21 \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)^4*(a+b*arcsin(d*x+c))^3,x, algorithm="giac")

[Out]

1/5*((d*x + c)^2 - 1)^2*(d*x + c)*b^3*arcsin(d*x + c)^3*e^4/d + 1/5*(d*x + c)^5*a^3*e^4/d + 3/5*((d*x + c)^2 -
1)^2*(d*x + c)*a*b^2*arcsin(d*x + c)^2*e^4/d + 2/5*((d*x + c)^2 - 1)*(d*x + c)*b^3*arcsin(d*x + c)^3*e^4/d +
3/25*((d*x + c)^2 - 1)^2*sqrt(-(d*x + c)^2 + 1)*b^3*arcsin(d*x + c)^2*e^4/d + 3/5*((d*x + c)^2 - 1)^2*(d*x + c
)*a^2*b*arcsin(d*x + c)*e^4/d - 6/125*((d*x + c)^2 - 1)^2*(d*x + c)*b^3*arcsin(d*x + c)*e^4/d + 6/5*((d*x + c)
^2 - 1)*(d*x + c)*a*b^2*arcsin(d*x + c)^2*e^4/d + 1/5*(d*x + c)*b^3*arcsin(d*x + c)^3*e^4/d + 6/25*((d*x + c)^
2 - 1)^2*sqrt(-(d*x + c)^2 + 1)*a*b^2*arcsin(d*x + c)*e^4/d - 2/5*(-(d*x + c)^2 + 1)^(3/2)*b^3*arcsin(d*x + c)
^2*e^4/d - 6/125*((d*x + c)^2 - 1)^2*(d*x + c)*a*b^2*e^4/d + 6/5*((d*x + c)^2 - 1)*(d*x + c)*a^2*b*arcsin(d*x
+ c)*e^4/d - 76/375*((d*x + c)^2 - 1)*(d*x + c)*b^3*arcsin(d*x + c)*e^4/d + 3/5*(d*x + c)*a*b^2*arcsin(d*x + c
)^2*e^4/d + 3/25*((d*x + c)^2 - 1)^2*sqrt(-(d*x + c)^2 + 1)*a^2*b*e^4/d - 6/625*((d*x + c)^2 - 1)^2*sqrt(-(d*x
+ c)^2 + 1)*b^3*e^4/d - 4/5*(-(d*x + c)^2 + 1)^(3/2)*a*b^2*arcsin(d*x + c)*e^4/d + 3/5*sqrt(-(d*x + c)^2 + 1)
*b^3*arcsin(d*x + c)^2*e^4/d - 76/375*((d*x + c)^2 - 1)*(d*x + c)*a*b^2*e^4/d + 3/5*(d*x + c)*a^2*b*arcsin(d*x
+ c)*e^4/d - 298/375*(d*x + c)*b^3*arcsin(d*x + c)*e^4/d - 2/5*(-(d*x + c)^2 + 1)^(3/2)*a^2*b*e^4/d + 76/1125
*(-(d*x + c)^2 + 1)^(3/2)*b^3*e^4/d + 6/5*sqrt(-(d*x + c)^2 + 1)*a*b^2*arcsin(d*x + c)*e^4/d - 298/375*(d*x +
c)*a*b^2*e^4/d + 3/5*sqrt(-(d*x + c)^2 + 1)*a^2*b*e^4/d - 298/375*sqrt(-(d*x + c)^2 + 1)*b^3*e^4/d