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

Optimal. Leaf size=198 \[ \frac{1}{3} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{b \left (1-c^2 x^2\right )^{3/2} \left (35 c^4 d^2+84 c^2 d e+45 e^2\right )}{315 c^7}+\frac{b \sqrt{1-c^2 x^2} \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7}+\frac{b e \left (1-c^2 x^2\right )^{5/2} \left (14 c^2 d+15 e\right )}{175 c^7}-\frac{b e^2 \left (1-c^2 x^2\right )^{7/2}}{49 c^7} \]

[Out]

(b*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*Sqrt[1 - c^2*x^2])/(105*c^7) - (b*(35*c^4*d^2 + 84*c^2*d*e + 45*e^2)*(1
- c^2*x^2)^(3/2))/(315*c^7) + (b*e*(14*c^2*d + 15*e)*(1 - c^2*x^2)^(5/2))/(175*c^7) - (b*e^2*(1 - c^2*x^2)^(7/
2))/(49*c^7) + (d^2*x^3*(a + b*ArcSin[c*x]))/3 + (2*d*e*x^5*(a + b*ArcSin[c*x]))/5 + (e^2*x^7*(a + b*ArcSin[c*
x]))/7

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Rubi [A]  time = 0.221466, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {270, 4731, 12, 1251, 771} \[ \frac{1}{3} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{b \left (1-c^2 x^2\right )^{3/2} \left (35 c^4 d^2+84 c^2 d e+45 e^2\right )}{315 c^7}+\frac{b \sqrt{1-c^2 x^2} \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7}+\frac{b e \left (1-c^2 x^2\right )^{5/2} \left (14 c^2 d+15 e\right )}{175 c^7}-\frac{b e^2 \left (1-c^2 x^2\right )^{7/2}}{49 c^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*Sqrt[1 - c^2*x^2])/(105*c^7) - (b*(35*c^4*d^2 + 84*c^2*d*e + 45*e^2)*(1
- c^2*x^2)^(3/2))/(315*c^7) + (b*e*(14*c^2*d + 15*e)*(1 - c^2*x^2)^(5/2))/(175*c^7) - (b*e^2*(1 - c^2*x^2)^(7/
2))/(49*c^7) + (d^2*x^3*(a + b*ArcSin[c*x]))/3 + (2*d*e*x^5*(a + b*ArcSin[c*x]))/5 + (e^2*x^7*(a + b*ArcSin[c*
x]))/7

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] ||
(IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 12

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

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{3} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{105} (b c) \int \frac{x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{3} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{210} (b c) \operatorname{Subst}\left (\int \frac{x \left (35 d^2+42 d e x+15 e^2 x^2\right )}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{210} (b c) \operatorname{Subst}\left (\int \left (\frac{35 c^4 d^2+42 c^2 d e+15 e^2}{c^6 \sqrt{1-c^2 x}}+\frac{\left (-35 c^4 d^2-84 c^2 d e-45 e^2\right ) \sqrt{1-c^2 x}}{c^6}+\frac{3 e \left (14 c^2 d+15 e\right ) \left (1-c^2 x\right )^{3/2}}{c^6}-\frac{15 e^2 \left (1-c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )\\ &=\frac{b \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \sqrt{1-c^2 x^2}}{105 c^7}-\frac{b \left (35 c^4 d^2+84 c^2 d e+45 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{315 c^7}+\frac{b e \left (14 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^{5/2}}{175 c^7}-\frac{b e^2 \left (1-c^2 x^2\right )^{7/2}}{49 c^7}+\frac{1}{3} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.184045, size = 158, normalized size = 0.8 \[ \frac{105 a x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )+\frac{b \sqrt{1-c^2 x^2} \left (c^6 \left (1225 d^2 x^2+882 d e x^4+225 e^2 x^6\right )+2 c^4 \left (1225 d^2+588 d e x^2+135 e^2 x^4\right )+24 c^2 e \left (98 d+15 e x^2\right )+720 e^2\right )}{c^7}+105 b x^3 \sin ^{-1}(c x) \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{11025} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(105*a*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) + (b*Sqrt[1 - c^2*x^2]*(720*e^2 + 24*c^2*e*(98*d + 15*e*x^2) + 2
*c^4*(1225*d^2 + 588*d*e*x^2 + 135*e^2*x^4) + c^6*(1225*d^2*x^2 + 882*d*e*x^4 + 225*e^2*x^6)))/c^7 + 105*b*x^3
*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcSin[c*x])/11025

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Maple [A]  time = 0.004, size = 279, normalized size = 1.4 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{7}{x}^{7}}{7}}+{\frac{2\,{c}^{7}ed{x}^{5}}{5}}+{\frac{{d}^{2}{c}^{7}{x}^{3}}{3}} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{\arcsin \left ( cx \right ){e}^{2}{c}^{7}{x}^{7}}{7}}+{\frac{2\,\arcsin \left ( cx \right ){c}^{7}ed{x}^{5}}{5}}+{\frac{\arcsin \left ( cx \right ){d}^{2}{c}^{7}{x}^{3}}{3}}-{\frac{{e}^{2}}{7} \left ( -{\frac{{c}^{6}{x}^{6}}{7}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{6\,{c}^{4}{x}^{4}}{35}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{8\,{c}^{2}{x}^{2}}{35}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{16}{35}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{2\,{c}^{2}ed}{5} \left ( -{\frac{{c}^{4}{x}^{4}}{5}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{4\,{c}^{2}{x}^{2}}{15}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{8}{15}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{{d}^{2}{c}^{4}}{3} \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(a/c^4*(1/7*e^2*c^7*x^7+2/5*c^7*e*d*x^5+1/3*d^2*c^7*x^3)+b/c^4*(1/7*arcsin(c*x)*e^2*c^7*x^7+2/5*arcsin(c
*x)*c^7*e*d*x^5+1/3*arcsin(c*x)*d^2*c^7*x^3-1/7*e^2*(-1/7*c^6*x^6*(-c^2*x^2+1)^(1/2)-6/35*c^4*x^4*(-c^2*x^2+1)
^(1/2)-8/35*c^2*x^2*(-c^2*x^2+1)^(1/2)-16/35*(-c^2*x^2+1)^(1/2))-2/5*c^2*e*d*(-1/5*c^4*x^4*(-c^2*x^2+1)^(1/2)-
4/15*c^2*x^2*(-c^2*x^2+1)^(1/2)-8/15*(-c^2*x^2+1)^(1/2))-1/3*d^2*c^4*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^
2*x^2+1)^(1/2))))

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Maxima [A]  time = 1.46714, size = 342, normalized size = 1.73 \begin{align*} \frac{1}{7} \, a e^{2} x^{7} + \frac{2}{5} \, a d e x^{5} + \frac{1}{3} \, a d^{2} x^{3} + \frac{1}{9} \,{\left (3 \, x^{3} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} + \frac{2}{75} \,{\left (15 \, x^{5} \arcsin \left (c x\right ) +{\left (\frac{3 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d e + \frac{1}{245} \,{\left (35 \, x^{7} \arcsin \left (c x\right ) +{\left (\frac{5 \, \sqrt{-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*e^2*x^7 + 2/5*a*d*e*x^5 + 1/3*a*d^2*x^3 + 1/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqr
t(-c^2*x^2 + 1)/c^4))*b*d^2 + 2/75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*
x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*d*e + 1/245*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*s
qrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*b*e^2

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Fricas [A]  time = 2.04131, size = 450, normalized size = 2.27 \begin{align*} \frac{1575 \, a c^{7} e^{2} x^{7} + 4410 \, a c^{7} d e x^{5} + 3675 \, a c^{7} d^{2} x^{3} + 105 \,{\left (15 \, b c^{7} e^{2} x^{7} + 42 \, b c^{7} d e x^{5} + 35 \, b c^{7} d^{2} x^{3}\right )} \arcsin \left (c x\right ) +{\left (225 \, b c^{6} e^{2} x^{6} + 2450 \, b c^{4} d^{2} + 2352 \, b c^{2} d e + 18 \,{\left (49 \, b c^{6} d e + 15 \, b c^{4} e^{2}\right )} x^{4} + 720 \, b e^{2} +{\left (1225 \, b c^{6} d^{2} + 1176 \, b c^{4} d e + 360 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt{-c^{2} x^{2} + 1}}{11025 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11025*(1575*a*c^7*e^2*x^7 + 4410*a*c^7*d*e*x^5 + 3675*a*c^7*d^2*x^3 + 105*(15*b*c^7*e^2*x^7 + 42*b*c^7*d*e*x
^5 + 35*b*c^7*d^2*x^3)*arcsin(c*x) + (225*b*c^6*e^2*x^6 + 2450*b*c^4*d^2 + 2352*b*c^2*d*e + 18*(49*b*c^6*d*e +
15*b*c^4*e^2)*x^4 + 720*b*e^2 + (1225*b*c^6*d^2 + 1176*b*c^4*d*e + 360*b*c^2*e^2)*x^2)*sqrt(-c^2*x^2 + 1))/c^
7

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Sympy [A]  time = 8.28768, size = 333, normalized size = 1.68 \begin{align*} \begin{cases} \frac{a d^{2} x^{3}}{3} + \frac{2 a d e x^{5}}{5} + \frac{a e^{2} x^{7}}{7} + \frac{b d^{2} x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{2 b d e x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{b e^{2} x^{7} \operatorname{asin}{\left (c x \right )}}{7} + \frac{b d^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{2 b d e x^{4} \sqrt{- c^{2} x^{2} + 1}}{25 c} + \frac{b e^{2} x^{6} \sqrt{- c^{2} x^{2} + 1}}{49 c} + \frac{2 b d^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + \frac{8 b d e x^{2} \sqrt{- c^{2} x^{2} + 1}}{75 c^{3}} + \frac{6 b e^{2} x^{4} \sqrt{- c^{2} x^{2} + 1}}{245 c^{3}} + \frac{16 b d e \sqrt{- c^{2} x^{2} + 1}}{75 c^{5}} + \frac{8 b e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{245 c^{5}} + \frac{16 b e^{2} \sqrt{- c^{2} x^{2} + 1}}{245 c^{7}} & \text{for}\: c \neq 0 \\a \left (\frac{d^{2} x^{3}}{3} + \frac{2 d e x^{5}}{5} + \frac{e^{2} x^{7}}{7}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x**3/3 + 2*a*d*e*x**5/5 + a*e**2*x**7/7 + b*d**2*x**3*asin(c*x)/3 + 2*b*d*e*x**5*asin(c*x)/5
+ b*e**2*x**7*asin(c*x)/7 + b*d**2*x**2*sqrt(-c**2*x**2 + 1)/(9*c) + 2*b*d*e*x**4*sqrt(-c**2*x**2 + 1)/(25*c)
+ b*e**2*x**6*sqrt(-c**2*x**2 + 1)/(49*c) + 2*b*d**2*sqrt(-c**2*x**2 + 1)/(9*c**3) + 8*b*d*e*x**2*sqrt(-c**2*
x**2 + 1)/(75*c**3) + 6*b*e**2*x**4*sqrt(-c**2*x**2 + 1)/(245*c**3) + 16*b*d*e*sqrt(-c**2*x**2 + 1)/(75*c**5)
+ 8*b*e**2*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5) + 16*b*e**2*sqrt(-c**2*x**2 + 1)/(245*c**7), Ne(c, 0)), (a*(d*
*2*x**3/3 + 2*d*e*x**5/5 + e**2*x**7/7), True))

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Giac [B]  time = 1.26082, size = 576, normalized size = 2.91 \begin{align*} \frac{1}{7} \, a x^{7} e^{2} + \frac{2}{5} \, a d x^{5} e + \frac{1}{3} \, a d^{2} x^{3} + \frac{{\left (c^{2} x^{2} - 1\right )} b d^{2} x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac{b d^{2} x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{2} b d x \arcsin \left (c x\right ) e}{5 \, c^{4}} + \frac{4 \,{\left (c^{2} x^{2} - 1\right )} b d x \arcsin \left (c x\right ) e}{5 \, c^{4}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d^{2}}{9 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b x \arcsin \left (c x\right ) e^{2}}{7 \, c^{6}} + \frac{2 \, b d x \arcsin \left (c x\right ) e}{5 \, c^{4}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d^{2}}{3 \, c^{3}} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b d e}{25 \, c^{5}} + \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{2} b x \arcsin \left (c x\right ) e^{2}}{7 \, c^{6}} - \frac{4 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d e}{15 \, c^{5}} + \frac{3 \,{\left (c^{2} x^{2} - 1\right )} b x \arcsin \left (c x\right ) e^{2}}{7 \, c^{6}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} b e^{2}}{49 \, c^{7}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b d e}{5 \, c^{5}} + \frac{b x \arcsin \left (c x\right ) e^{2}}{7 \, c^{6}} + \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b e^{2}}{35 \, c^{7}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b e^{2}}{7 \, c^{7}} + \frac{\sqrt{-c^{2} x^{2} + 1} b e^{2}}{7 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*x^7*e^2 + 2/5*a*d*x^5*e + 1/3*a*d^2*x^3 + 1/3*(c^2*x^2 - 1)*b*d^2*x*arcsin(c*x)/c^2 + 1/3*b*d^2*x*arcsin
(c*x)/c^2 + 2/5*(c^2*x^2 - 1)^2*b*d*x*arcsin(c*x)*e/c^4 + 4/5*(c^2*x^2 - 1)*b*d*x*arcsin(c*x)*e/c^4 - 1/9*(-c^
2*x^2 + 1)^(3/2)*b*d^2/c^3 + 1/7*(c^2*x^2 - 1)^3*b*x*arcsin(c*x)*e^2/c^6 + 2/5*b*d*x*arcsin(c*x)*e/c^4 + 1/3*s
qrt(-c^2*x^2 + 1)*b*d^2/c^3 + 2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d*e/c^5 + 3/7*(c^2*x^2 - 1)^2*b*x*arcs
in(c*x)*e^2/c^6 - 4/15*(-c^2*x^2 + 1)^(3/2)*b*d*e/c^5 + 3/7*(c^2*x^2 - 1)*b*x*arcsin(c*x)*e^2/c^6 + 1/49*(c^2*
x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*e^2/c^7 + 2/5*sqrt(-c^2*x^2 + 1)*b*d*e/c^5 + 1/7*b*x*arcsin(c*x)*e^2/c^6 + 3/3
5*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e^2/c^7 - 1/7*(-c^2*x^2 + 1)^(3/2)*b*e^2/c^7 + 1/7*sqrt(-c^2*x^2 + 1)*b
*e^2/c^7