3.21 \(\int x^2 (d-c^2 d x^2)^3 (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=207 \[ -\frac {1}{9} c^6 d^3 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {b d^3 \left (1-c^2 x^2\right )^{9/2}}{81 c^3}+\frac {b d^3 \left (1-c^2 x^2\right )^{7/2}}{441 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{5/2}}{525 c^3}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{945 c^3}+\frac {16 b d^3 \sqrt {1-c^2 x^2}}{315 c^3} \]

[Out]

8/945*b*d^3*(-c^2*x^2+1)^(3/2)/c^3+2/525*b*d^3*(-c^2*x^2+1)^(5/2)/c^3+1/441*b*d^3*(-c^2*x^2+1)^(7/2)/c^3-1/81*
b*d^3*(-c^2*x^2+1)^(9/2)/c^3+1/3*d^3*x^3*(a+b*arcsin(c*x))-3/5*c^2*d^3*x^5*(a+b*arcsin(c*x))+3/7*c^4*d^3*x^7*(
a+b*arcsin(c*x))-1/9*c^6*d^3*x^9*(a+b*arcsin(c*x))+16/315*b*d^3*(-c^2*x^2+1)^(1/2)/c^3

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Rubi [A]  time = 0.26, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {270, 4687, 12, 1799, 1620} \[ -\frac {1}{9} c^6 d^3 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {b d^3 \left (1-c^2 x^2\right )^{9/2}}{81 c^3}+\frac {b d^3 \left (1-c^2 x^2\right )^{7/2}}{441 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{5/2}}{525 c^3}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{945 c^3}+\frac {16 b d^3 \sqrt {1-c^2 x^2}}{315 c^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]),x]

[Out]

(16*b*d^3*Sqrt[1 - c^2*x^2])/(315*c^3) + (8*b*d^3*(1 - c^2*x^2)^(3/2))/(945*c^3) + (2*b*d^3*(1 - c^2*x^2)^(5/2
))/(525*c^3) + (b*d^3*(1 - c^2*x^2)^(7/2))/(441*c^3) - (b*d^3*(1 - c^2*x^2)^(9/2))/(81*c^3) + (d^3*x^3*(a + b*
ArcSin[c*x]))/3 - (3*c^2*d^3*x^5*(a + b*ArcSin[c*x]))/5 + (3*c^4*d^3*x^7*(a + b*ArcSin[c*x]))/7 - (c^6*d^3*x^9
*(a + b*ArcSin[c*x]))/9

Rule 12

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

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 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 4687

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = I
ntHide[(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] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d^3 x^3 \left (105-189 c^2 x^2+135 c^4 x^4-35 c^6 x^6\right )}{315 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{315} \left (b c d^3\right ) \int \frac {x^3 \left (105-189 c^2 x^2+135 c^4 x^4-35 c^6 x^6\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{630} \left (b c d^3\right ) \operatorname {Subst}\left (\int \frac {x \left (105-189 c^2 x+135 c^4 x^2-35 c^6 x^3\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{630} \left (b c d^3\right ) \operatorname {Subst}\left (\int \left (\frac {16}{c^2 \sqrt {1-c^2 x}}+\frac {8 \sqrt {1-c^2 x}}{c^2}+\frac {6 \left (1-c^2 x\right )^{3/2}}{c^2}+\frac {5 \left (1-c^2 x\right )^{5/2}}{c^2}-\frac {35 \left (1-c^2 x\right )^{7/2}}{c^2}\right ) \, dx,x,x^2\right )\\ &=\frac {16 b d^3 \sqrt {1-c^2 x^2}}{315 c^3}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{945 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{5/2}}{525 c^3}+\frac {b d^3 \left (1-c^2 x^2\right )^{7/2}}{441 c^3}-\frac {b d^3 \left (1-c^2 x^2\right )^{9/2}}{81 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 135, normalized size = 0.65 \[ \frac {d^3 \left (-315 a c^3 x^3 \left (35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right )+b \sqrt {1-c^2 x^2} \left (-1225 c^8 x^8+4675 c^6 x^6-6297 c^4 x^4+2629 c^2 x^2+5258\right )-315 b c^3 x^3 \left (35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right ) \sin ^{-1}(c x)\right )}{99225 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]),x]

[Out]

(d^3*(-315*a*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(5258 + 2629*c^2*x^
2 - 6297*c^4*x^4 + 4675*c^6*x^6 - 1225*c^8*x^8) - 315*b*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6
)*ArcSin[c*x]))/(99225*c^3)

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fricas [A]  time = 0.50, size = 177, normalized size = 0.86 \[ -\frac {11025 \, a c^{9} d^{3} x^{9} - 42525 \, a c^{7} d^{3} x^{7} + 59535 \, a c^{5} d^{3} x^{5} - 33075 \, a c^{3} d^{3} x^{3} + 315 \, {\left (35 \, b c^{9} d^{3} x^{9} - 135 \, b c^{7} d^{3} x^{7} + 189 \, b c^{5} d^{3} x^{5} - 105 \, b c^{3} d^{3} x^{3}\right )} \arcsin \left (c x\right ) + {\left (1225 \, b c^{8} d^{3} x^{8} - 4675 \, b c^{6} d^{3} x^{6} + 6297 \, b c^{4} d^{3} x^{4} - 2629 \, b c^{2} d^{3} x^{2} - 5258 \, b d^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{99225 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

-1/99225*(11025*a*c^9*d^3*x^9 - 42525*a*c^7*d^3*x^7 + 59535*a*c^5*d^3*x^5 - 33075*a*c^3*d^3*x^3 + 315*(35*b*c^
9*d^3*x^9 - 135*b*c^7*d^3*x^7 + 189*b*c^5*d^3*x^5 - 105*b*c^3*d^3*x^3)*arcsin(c*x) + (1225*b*c^8*d^3*x^8 - 467
5*b*c^6*d^3*x^6 + 6297*b*c^4*d^3*x^4 - 2629*b*c^2*d^3*x^2 - 5258*b*d^3)*sqrt(-c^2*x^2 + 1))/c^3

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giac [A]  time = 0.43, size = 296, normalized size = 1.43 \[ -\frac {1}{9} \, a c^{6} d^{3} x^{9} + \frac {3}{7} \, a c^{4} d^{3} x^{7} - \frac {3}{5} \, a c^{2} d^{3} x^{5} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b d^{3} x \arcsin \left (c x\right )}{9 \, c^{2}} + \frac {1}{3} \, a d^{3} x^{3} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d^{3} x \arcsin \left (c x\right )}{63 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d^{3} x \arcsin \left (c x\right )}{105 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{81 \, c^{3}} - \frac {8 \, {\left (c^{2} x^{2} - 1\right )} b d^{3} x \arcsin \left (c x\right )}{315 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{441 \, c^{3}} + \frac {16 \, b d^{3} x \arcsin \left (c x\right )}{315 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{525 \, c^{3}} + \frac {8 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{3}}{945 \, c^{3}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b d^{3}}{315 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

-1/9*a*c^6*d^3*x^9 + 3/7*a*c^4*d^3*x^7 - 3/5*a*c^2*d^3*x^5 - 1/9*(c^2*x^2 - 1)^4*b*d^3*x*arcsin(c*x)/c^2 + 1/3
*a*d^3*x^3 - 1/63*(c^2*x^2 - 1)^3*b*d^3*x*arcsin(c*x)/c^2 + 2/105*(c^2*x^2 - 1)^2*b*d^3*x*arcsin(c*x)/c^2 - 1/
81*(c^2*x^2 - 1)^4*sqrt(-c^2*x^2 + 1)*b*d^3/c^3 - 8/315*(c^2*x^2 - 1)*b*d^3*x*arcsin(c*x)/c^2 - 1/441*(c^2*x^2
- 1)^3*sqrt(-c^2*x^2 + 1)*b*d^3/c^3 + 16/315*b*d^3*x*arcsin(c*x)/c^2 + 2/525*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 +
1)*b*d^3/c^3 + 8/945*(-c^2*x^2 + 1)^(3/2)*b*d^3/c^3 + 16/315*sqrt(-c^2*x^2 + 1)*b*d^3/c^3

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maple [A]  time = 0.01, size = 194, normalized size = 0.94 \[ \frac {-d^{3} a \left (\frac {1}{9} c^{9} x^{9}-\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {2099 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {5258 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x)),x)

[Out]

1/c^3*(-d^3*a*(1/9*c^9*x^9-3/7*c^7*x^7+3/5*c^5*x^5-1/3*c^3*x^3)-d^3*b*(1/9*arcsin(c*x)*c^9*x^9-3/7*arcsin(c*x)
*c^7*x^7+3/5*arcsin(c*x)*c^5*x^5-1/3*c^3*x^3*arcsin(c*x)+1/81*c^8*x^8*(-c^2*x^2+1)^(1/2)-187/3969*c^6*x^6*(-c^
2*x^2+1)^(1/2)+2099/33075*c^4*x^4*(-c^2*x^2+1)^(1/2)-2629/99225*c^2*x^2*(-c^2*x^2+1)^(1/2)-5258/99225*(-c^2*x^
2+1)^(1/2)))

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maxima [B]  time = 0.46, size = 398, normalized size = 1.92 \[ -\frac {1}{9} \, a c^{6} d^{3} x^{9} + \frac {3}{7} \, a c^{4} d^{3} x^{7} - \frac {1}{2835} \, {\left (315 \, x^{9} \arcsin \left (c x\right ) + {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{6} d^{3} - \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {3}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{4} d^{3} - \frac {1}{25} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{3} \, a d^{3} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

-1/9*a*c^6*d^3*x^9 + 3/7*a*c^4*d^3*x^7 - 1/2835*(315*x^9*arcsin(c*x) + (35*sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*sqr
t(-c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(-c^2*x^2 + 1)*x^4/c^6 + 64*sqrt(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^2*x^2 +
1)/c^10)*c)*b*c^6*d^3 - 3/5*a*c^2*d^3*x^5 + 3/245*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt
(-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*c^4*d^3 - 1/25*(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*c
^2*d^3 + 1/3*a*d^3*x^3 + 1/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b
*d^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*asin(c*x))*(d - c^2*d*x^2)^3,x)

[Out]

int(x^2*(a + b*asin(c*x))*(d - c^2*d*x^2)^3, x)

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sympy [A]  time = 16.15, size = 265, normalized size = 1.28 \[ \begin {cases} - \frac {a c^{6} d^{3} x^{9}}{9} + \frac {3 a c^{4} d^{3} x^{7}}{7} - \frac {3 a c^{2} d^{3} x^{5}}{5} + \frac {a d^{3} x^{3}}{3} - \frac {b c^{6} d^{3} x^{9} \operatorname {asin}{\left (c x \right )}}{9} - \frac {b c^{5} d^{3} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81} + \frac {3 b c^{4} d^{3} x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {187 b c^{3} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{3969} - \frac {3 b c^{2} d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} - \frac {2099 b c d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{33075} + \frac {b d^{3} x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {2629 b d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c} + \frac {5258 b d^{3} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{3}}{3} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(-c**2*d*x**2+d)**3*(a+b*asin(c*x)),x)

[Out]

Piecewise((-a*c**6*d**3*x**9/9 + 3*a*c**4*d**3*x**7/7 - 3*a*c**2*d**3*x**5/5 + a*d**3*x**3/3 - b*c**6*d**3*x**
9*asin(c*x)/9 - b*c**5*d**3*x**8*sqrt(-c**2*x**2 + 1)/81 + 3*b*c**4*d**3*x**7*asin(c*x)/7 + 187*b*c**3*d**3*x*
*6*sqrt(-c**2*x**2 + 1)/3969 - 3*b*c**2*d**3*x**5*asin(c*x)/5 - 2099*b*c*d**3*x**4*sqrt(-c**2*x**2 + 1)/33075
+ b*d**3*x**3*asin(c*x)/3 + 2629*b*d**3*x**2*sqrt(-c**2*x**2 + 1)/(99225*c) + 5258*b*d**3*sqrt(-c**2*x**2 + 1)
/(99225*c**3), Ne(c, 0)), (a*d**3*x**3/3, True))

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