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*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)
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Rubi [A] time = 0.481905, antiderivative size = 289, normalized size of antiderivative = 1.,
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 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 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
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)^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*}
Mathematica [A] time = 0.598003, 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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Maple [A] time = 0.038, size = 440, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ({\frac{{e}^{2} \left ( dx+c \right ) ^{3}{a}^{4}}{3}}+{e}^{2}{b}^{4} \left ({\frac{ \left ( dx+c \right ) ^{3} \left ( \arcsin \left ( dx+c \right ) \right ) ^{4}}{3}}+{\frac{4\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{3} \left ( \left ( dx+c \right ) ^{2}+2 \right ) }{9}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{8\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) }{3}}+{\frac{160\,dx}{27}}+{\frac{160\,c}{27}}-{\frac{16\,\arcsin \left ( dx+c \right ) }{3}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{4\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) ^{3}}{9}}-{\frac{8\,\arcsin \left ( dx+c \right ) \left ( \left ( dx+c \right ) ^{2}+2 \right ) }{27}\sqrt{1- \left ( dx+c \right ) ^{2}}}+{\frac{8\, \left ( dx+c \right ) ^{3}}{81}} \right ) +4\,{e}^{2}a{b}^{3} \left ( 1/3\, \left ( dx+c \right ) ^{3} \left ( \arcsin \left ( dx+c \right ) \right ) ^{3}+1/3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( \left ( dx+c \right ) ^{2}+2 \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}-4/3\,\sqrt{1- \left ( dx+c \right ) ^{2}}-4/3\, \left ( dx+c \right ) \arcsin \left ( dx+c \right ) -2/9\, \left ( dx+c \right ) ^{3}\arcsin \left ( dx+c \right ) -{\frac{ \left ( 2\, \left ( dx+c \right ) ^{2}+4 \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}}{27}} \right ) +6\,{e}^{2}{a}^{2}{b}^{2} \left ( 1/3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) ^{3}+2/9\,\arcsin \left ( dx+c \right ) \left ( \left ( dx+c \right ) ^{2}+2 \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}-{\frac{2\, \left ( dx+c \right ) ^{3}}{27}}-4/9\,dx-4/9\,c \right ) +4\,{e}^{2}{a}^{3}b \left ( 1/3\, \left ( dx+c \right ) ^{3}\arcsin \left ( dx+c \right ) +1/9\, \left ( dx+c \right ) ^{2}\sqrt{1- \left ( dx+c \right ) ^{2}}+2/9\,\sqrt{1- \left ( dx+c \right ) ^{2}} \right ) \right ) } \end{align*}
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*(1-(d*x+c)^2)^(1/2)*arcsin(d*x+c)-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*(d*x+c)^3*arcsin(d*x+c)-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*(d*x+c)^3*arcsin(d*x+c)+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(-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)^2*(a+b*arcsin(d*x+c))^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.86572, size = 1643, normalized size = 5.69 \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)^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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Sympy [A] time = 13.9753, size = 1889, normalized size = 6.54 \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)**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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Giac [B] time = 1.39374, size = 1053, normalized size = 3.64 \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)^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