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

Optimal. Leaf size=198 \[ -\frac{3 b^3 e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}-\frac{3 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d}+\frac{3 b^2 e \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}+\frac{b e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^4}{2 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 d}+\frac{3 b^4 e (c+d x)^2}{4 d} \]

[Out]

(3*b^4*e*(c + d*x)^2)/(4*d) - (3*b^3*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(2*d) + (3*b^2
*e*(a + b*ArcSin[c + d*x])^2)/(4*d) - (3*b^2*e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2)/(2*d) + (b*e*(c + d*x)*S
qrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/d - (e*(a + b*ArcSin[c + d*x])^4)/(4*d) + (e*(c + d*x)^2*(a +
b*ArcSin[c + d*x])^4)/(2*d)

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Rubi [A]  time = 0.305862, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4805, 12, 4627, 4707, 4641, 30} \[ -\frac{3 b^3 e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}-\frac{3 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d}+\frac{3 b^2 e \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}+\frac{b e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^4}{2 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 d}+\frac{3 b^4 e (c+d x)^2}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^4*e*(c + d*x)^2)/(4*d) - (3*b^3*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(2*d) + (3*b^2
*e*(a + b*ArcSin[c + d*x])^2)/(4*d) - (3*b^2*e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2)/(2*d) + (b*e*(c + d*x)*S
qrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/d - (e*(a + b*ArcSin[c + d*x])^4)/(4*d) + (e*(c + d*x)^2*(a +
b*ArcSin[c + d*x])^4)/(2*d)

Rule 4805

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rule 12

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

Rule 4627

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

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 4641

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

Rule 30

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

Rubi steps

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

Mathematica [A]  time = 0.263336, size = 163, normalized size = 0.82 \[ -\frac{e \left (3 b^2 \left (2 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2+2 b \sqrt{1-(c+d x)^2} (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )-\left (a+b \sin ^{-1}(c+d x)\right )^2-b^2 (c+d x)^2\right )-2 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^4+\left (a+b \sin ^{-1}(c+d x)\right )^4-4 b (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.04, size = 412, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ({\frac{e \left ( dx+c \right ) ^{2}{a}^{4}}{2}}+e{b}^{4} \left ({\frac{ \left ( \left ( dx+c \right ) ^{2}-1 \right ) \left ( \arcsin \left ( dx+c \right ) \right ) ^{4}}{2}}+ \left ( \arcsin \left ( dx+c \right ) \right ) ^{3} \left ( \left ( dx+c \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}+\arcsin \left ( dx+c \right ) \right ) -{\frac{3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( \left ( dx+c \right ) ^{2}-1 \right ) }{2}}-{\frac{3\,\arcsin \left ( dx+c \right ) }{2} \left ( \left ( dx+c \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}+\arcsin \left ( dx+c \right ) \right ) }+{\frac{3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}}{4}}+{\frac{3\, \left ( dx+c \right ) ^{2}}{4}}-{\frac{3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{4}}{4}} \right ) +4\,ea{b}^{3} \left ( 1/2\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{3} \left ( \left ( dx+c \right ) ^{2}-1 \right ) +3/4\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( \left ( dx+c \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}+\arcsin \left ( dx+c \right ) \right ) -3/4\,\arcsin \left ( dx+c \right ) \left ( \left ( dx+c \right ) ^{2}-1 \right ) -3/8\, \left ( dx+c \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}-3/8\,\arcsin \left ( dx+c \right ) -1/2\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{3} \right ) +6\,e{a}^{2}{b}^{2} \left ( 1/2\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( \left ( dx+c \right ) ^{2}-1 \right ) +1/2\,\arcsin \left ( dx+c \right ) \left ( \left ( dx+c \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}+\arcsin \left ( dx+c \right ) \right ) -1/4\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}-1/4\, \left ( dx+c \right ) ^{2} \right ) +4\,e{a}^{3}b \left ( 1/2\,\arcsin \left ( dx+c \right ) \left ( dx+c \right ) ^{2}+1/4\, \left ( dx+c \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}-1/4\,\arcsin \left ( dx+c \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/2*e*(d*x+c)^2*a^4+e*b^4*(1/2*((d*x+c)^2-1)*arcsin(d*x+c)^4+arcsin(d*x+c)^3*((d*x+c)*(1-(d*x+c)^2)^(1/2)
+arcsin(d*x+c))-3/2*arcsin(d*x+c)^2*((d*x+c)^2-1)-3/2*arcsin(d*x+c)*((d*x+c)*(1-(d*x+c)^2)^(1/2)+arcsin(d*x+c)
)+3/4*arcsin(d*x+c)^2+3/4*(d*x+c)^2-3/4*arcsin(d*x+c)^4)+4*e*a*b^3*(1/2*arcsin(d*x+c)^3*((d*x+c)^2-1)+3/4*arcs
in(d*x+c)^2*((d*x+c)*(1-(d*x+c)^2)^(1/2)+arcsin(d*x+c))-3/4*arcsin(d*x+c)*((d*x+c)^2-1)-3/8*(d*x+c)*(1-(d*x+c)
^2)^(1/2)-3/8*arcsin(d*x+c)-1/2*arcsin(d*x+c)^3)+6*e*a^2*b^2*(1/2*arcsin(d*x+c)^2*((d*x+c)^2-1)+1/2*arcsin(d*x
+c)*((d*x+c)*(1-(d*x+c)^2)^(1/2)+arcsin(d*x+c))-1/4*arcsin(d*x+c)^2-1/4*(d*x+c)^2)+4*e*a^3*b*(1/2*arcsin(d*x+c
)*(d*x+c)^2+1/4*(d*x+c)*(1-(d*x+c)^2)^(1/2)-1/4*arcsin(d*x+c)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.58579, size = 1045, normalized size = 5.28 \begin{align*} \frac{{\left (2 \, a^{4} - 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{2} e x^{2} + 2 \,{\left (2 \, a^{4} - 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d e x +{\left (2 \, b^{4} d^{2} e x^{2} + 4 \, b^{4} c d e x +{\left (2 \, b^{4} c^{2} - b^{4}\right )} e\right )} \arcsin \left (d x + c\right )^{4} + 4 \,{\left (2 \, a b^{3} d^{2} e x^{2} + 4 \, a b^{3} c d e x +{\left (2 \, a b^{3} c^{2} - a b^{3}\right )} e\right )} \arcsin \left (d x + c\right )^{3} + 3 \,{\left (2 \,{\left (2 \, a^{2} b^{2} - b^{4}\right )} d^{2} e x^{2} + 4 \,{\left (2 \, a^{2} b^{2} - b^{4}\right )} c d e x -{\left (2 \, a^{2} b^{2} - b^{4} - 2 \,{\left (2 \, a^{2} b^{2} - b^{4}\right )} c^{2}\right )} e\right )} \arcsin \left (d x + c\right )^{2} + 2 \,{\left (2 \,{\left (2 \, a^{3} b - 3 \, a b^{3}\right )} d^{2} e x^{2} + 4 \,{\left (2 \, a^{3} b - 3 \, a b^{3}\right )} c d e x -{\left (2 \, a^{3} b - 3 \, a b^{3} - 2 \,{\left (2 \, a^{3} b - 3 \, a b^{3}\right )} c^{2}\right )} e\right )} \arcsin \left (d x + c\right ) + 2 \,{\left ({\left (2 \, a^{3} b - 3 \, a b^{3}\right )} d e x + 2 \,{\left (b^{4} d e x + b^{4} c e\right )} \arcsin \left (d x + c\right )^{3} +{\left (2 \, a^{3} b - 3 \, a b^{3}\right )} c e + 6 \,{\left (a b^{3} d e x + a b^{3} c e\right )} \arcsin \left (d x + c\right )^{2} + 3 \,{\left ({\left (2 \, a^{2} b^{2} - b^{4}\right )} d e x +{\left (2 \, a^{2} b^{2} - b^{4}\right )} c e\right )} \arcsin \left (d x + c\right )\right )} \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 6.88189, size = 1027, normalized size = 5.19 \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)*(a+b*asin(d*x+c))**4,x)

[Out]

Piecewise((a**4*c*e*x + a**4*d*e*x**2/2 + 2*a**3*b*c**2*e*asin(c + d*x)/d + 4*a**3*b*c*e*x*asin(c + d*x) + a**
3*b*c*e*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/d + 2*a**3*b*d*e*x**2*asin(c + d*x) + a**3*b*e*x*sqrt(-c**2 - 2*
c*d*x - d**2*x**2 + 1) - a**3*b*e*asin(c + d*x)/d + 3*a**2*b**2*c**2*e*asin(c + d*x)**2/d + 6*a**2*b**2*c*e*x*
asin(c + d*x)**2 - 3*a**2*b**2*c*e*x + 3*a**2*b**2*c*e*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/d +
3*a**2*b**2*d*e*x**2*asin(c + d*x)**2 - 3*a**2*b**2*d*e*x**2/2 + 3*a**2*b**2*e*x*sqrt(-c**2 - 2*c*d*x - d**2*
x**2 + 1)*asin(c + d*x) - 3*a**2*b**2*e*asin(c + d*x)**2/(2*d) + 2*a*b**3*c**2*e*asin(c + d*x)**3/d - 3*a*b**3
*c**2*e*asin(c + d*x)/d + 4*a*b**3*c*e*x*asin(c + d*x)**3 - 6*a*b**3*c*e*x*asin(c + d*x) + 3*a*b**3*c*e*sqrt(-
c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/d - 3*a*b**3*c*e*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(2*d)
+ 2*a*b**3*d*e*x**2*asin(c + d*x)**3 - 3*a*b**3*d*e*x**2*asin(c + d*x) + 3*a*b**3*e*x*sqrt(-c**2 - 2*c*d*x - d
**2*x**2 + 1)*asin(c + d*x)**2 - 3*a*b**3*e*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/2 - a*b**3*e*asin(c + d*x)
**3/d + 3*a*b**3*e*asin(c + d*x)/(2*d) + b**4*c**2*e*asin(c + d*x)**4/(2*d) - 3*b**4*c**2*e*asin(c + d*x)**2/(
2*d) + b**4*c*e*x*asin(c + d*x)**4 - 3*b**4*c*e*x*asin(c + d*x)**2 + 3*b**4*c*e*x/2 + b**4*c*e*sqrt(-c**2 - 2*
c*d*x - d**2*x**2 + 1)*asin(c + d*x)**3/d - 3*b**4*c*e*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(2*
d) + b**4*d*e*x**2*asin(c + d*x)**4/2 - 3*b**4*d*e*x**2*asin(c + d*x)**2/2 + 3*b**4*d*e*x**2/4 + b**4*e*x*sqrt
(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**3 - 3*b**4*e*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c +
d*x)/2 - b**4*e*asin(c + d*x)**4/(4*d) + 3*b**4*e*asin(c + d*x)**2/(4*d), Ne(d, 0)), (c*e*x*(a + b*asin(c))**
4, True))

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Giac [B]  time = 1.34008, size = 751, normalized size = 3.79 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{4} \arcsin \left (d x + c\right )^{4} e}{2 \, d} + \frac{\sqrt{-{\left (d x + c\right )}^{2} + 1}{\left (d x + c\right )} b^{4} \arcsin \left (d x + c\right )^{3} e}{d} + \frac{2 \,{\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{3} \arcsin \left (d x + c\right )^{3} e}{d} + \frac{b^{4} \arcsin \left (d x + c\right )^{4} e}{4 \, d} + \frac{3 \, \sqrt{-{\left (d x + c\right )}^{2} + 1}{\left (d x + c\right )} a b^{3} \arcsin \left (d x + c\right )^{2} e}{d} + \frac{3 \,{\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} b^{2} \arcsin \left (d x + c\right )^{2} e}{d} - \frac{3 \,{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{4} \arcsin \left (d x + c\right )^{2} e}{2 \, d} + \frac{a b^{3} \arcsin \left (d x + c\right )^{3} e}{d} + \frac{3 \, \sqrt{-{\left (d x + c\right )}^{2} + 1}{\left (d x + c\right )} a^{2} b^{2} \arcsin \left (d x + c\right ) e}{d} - \frac{3 \, \sqrt{-{\left (d x + c\right )}^{2} + 1}{\left (d x + c\right )} b^{4} \arcsin \left (d x + c\right ) e}{2 \, d} + \frac{2 \,{\left ({\left (d x + c\right )}^{2} - 1\right )} a^{3} b \arcsin \left (d x + c\right ) e}{d} - \frac{3 \,{\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{3} \arcsin \left (d x + c\right ) e}{d} + \frac{3 \, a^{2} b^{2} \arcsin \left (d x + c\right )^{2} e}{2 \, d} - \frac{3 \, b^{4} \arcsin \left (d x + c\right )^{2} e}{4 \, d} + \frac{\sqrt{-{\left (d x + c\right )}^{2} + 1}{\left (d x + c\right )} a^{3} b e}{d} - \frac{3 \, \sqrt{-{\left (d x + c\right )}^{2} + 1}{\left (d x + c\right )} a b^{3} e}{2 \, d} + \frac{{\left ({\left (d x + c\right )}^{2} - 1\right )} a^{4} e}{2 \, d} - \frac{3 \,{\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} b^{2} e}{2 \, d} + \frac{3 \,{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{4} e}{4 \, d} + \frac{a^{3} b \arcsin \left (d x + c\right ) e}{d} - \frac{3 \, a b^{3} \arcsin \left (d x + c\right ) e}{2 \, d} - \frac{3 \, a^{2} b^{2} e}{4 \, d} + \frac{3 \, b^{4} e}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((d*x + c)^2 - 1)*b^4*arcsin(d*x + c)^4*e/d + sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^4*arcsin(d*x + c)^3*e/d +
2*((d*x + c)^2 - 1)*a*b^3*arcsin(d*x + c)^3*e/d + 1/4*b^4*arcsin(d*x + c)^4*e/d + 3*sqrt(-(d*x + c)^2 + 1)*(d
*x + c)*a*b^3*arcsin(d*x + c)^2*e/d + 3*((d*x + c)^2 - 1)*a^2*b^2*arcsin(d*x + c)^2*e/d - 3/2*((d*x + c)^2 - 1
)*b^4*arcsin(d*x + c)^2*e/d + a*b^3*arcsin(d*x + c)^3*e/d + 3*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a^2*b^2*arcsin(
d*x + c)*e/d - 3/2*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^4*arcsin(d*x + c)*e/d + 2*((d*x + c)^2 - 1)*a^3*b*arcsin
(d*x + c)*e/d - 3*((d*x + c)^2 - 1)*a*b^3*arcsin(d*x + c)*e/d + 3/2*a^2*b^2*arcsin(d*x + c)^2*e/d - 3/4*b^4*ar
csin(d*x + c)^2*e/d + sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a^3*b*e/d - 3/2*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a*b^3*
e/d + 1/2*((d*x + c)^2 - 1)*a^4*e/d - 3/2*((d*x + c)^2 - 1)*a^2*b^2*e/d + 3/4*((d*x + c)^2 - 1)*b^4*e/d + a^3*
b*arcsin(d*x + c)*e/d - 3/2*a*b^3*arcsin(d*x + c)*e/d - 3/4*a^2*b^2*e/d + 3/8*b^4*e/d