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

Optimal. Leaf size=183 \[ \frac {\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{36 c}+\frac {5 b x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )}{144 c^3}-\frac {b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sin ^{-1}(c x)}{96 c^6 e}+\frac {b x \sqrt {1-c^2 x^2} \left (44 c^4 d^2+44 c^2 d e+15 e^2\right )}{288 c^5} \]

[Out]

-1/96*b*(2*c^2*d+e)*(8*c^4*d^2+8*c^2*d*e+5*e^2)*arcsin(c*x)/c^6/e+1/6*(e*x^2+d)^3*(a+b*arcsin(c*x))/e+1/288*b*
(44*c^4*d^2+44*c^2*d*e+15*e^2)*x*(-c^2*x^2+1)^(1/2)/c^5+5/144*b*(2*c^2*d+e)*x*(e*x^2+d)*(-c^2*x^2+1)^(1/2)/c^3
+1/36*b*x*(e*x^2+d)^2*(-c^2*x^2+1)^(1/2)/c

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Rubi [A]  time = 0.18, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4729, 416, 528, 388, 216} \[ \frac {\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}+\frac {b x \sqrt {1-c^2 x^2} \left (44 c^4 d^2+44 c^2 d e+15 e^2\right )}{288 c^5}-\frac {b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sin ^{-1}(c x)}{96 c^6 e}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{36 c}+\frac {5 b x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )}{144 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(44*c^4*d^2 + 44*c^2*d*e + 15*e^2)*x*Sqrt[1 - c^2*x^2])/(288*c^5) + (5*b*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2]*
(d + e*x^2))/(144*c^3) + (b*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^2)/(36*c) - (b*(2*c^2*d + e)*(8*c^4*d^2 + 8*c^2*d*
e + 5*e^2)*ArcSin[c*x])/(96*c^6*e) + ((d + e*x^2)^3*(a + b*ArcSin[c*x]))/(6*e)

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 388

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

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 4729

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

Rubi steps

\begin {align*} \int x \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}-\frac {(b c) \int \frac {\left (d+e x^2\right )^3}{\sqrt {1-c^2 x^2}} \, dx}{6 e}\\ &=\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{36 c}+\frac {\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}+\frac {b \int \frac {\left (d+e x^2\right ) \left (-d \left (6 c^2 d+e\right )-5 e \left (2 c^2 d+e\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{36 c e}\\ &=\frac {5 b \left (2 c^2 d+e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{144 c^3}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{36 c}+\frac {\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}-\frac {b \int \frac {d \left (24 c^4 d^2+14 c^2 d e+5 e^2\right )+e \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x^2}{\sqrt {1-c^2 x^2}} \, dx}{144 c^3 e}\\ &=\frac {b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \sqrt {1-c^2 x^2}}{288 c^5}+\frac {5 b \left (2 c^2 d+e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{144 c^3}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{36 c}+\frac {\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}-\frac {\left (b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{96 c^5 e}\\ &=\frac {b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \sqrt {1-c^2 x^2}}{288 c^5}+\frac {5 b \left (2 c^2 d+e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{144 c^3}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{36 c}-\frac {b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sin ^{-1}(c x)}{96 c^6 e}+\frac {\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 159, normalized size = 0.87 \[ \frac {c x \left (48 a c^5 x \left (3 d^2+3 d e x^2+e^2 x^4\right )+b \sqrt {1-c^2 x^2} \left (4 c^4 \left (18 d^2+9 d e x^2+2 e^2 x^4\right )+2 c^2 e \left (27 d+5 e x^2\right )+15 e^2\right )\right )+3 b \sin ^{-1}(c x) \left (16 c^6 \left (3 d^2 x^2+3 d e x^4+e^2 x^6\right )-24 c^4 d^2-18 c^2 d e-5 e^2\right )}{288 c^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(48*a*c^5*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) + b*Sqrt[1 - c^2*x^2]*(15*e^2 + 2*c^2*e*(27*d + 5*e*x^2) + 4*c^
4*(18*d^2 + 9*d*e*x^2 + 2*e^2*x^4))) + 3*b*(-24*c^4*d^2 - 18*c^2*d*e - 5*e^2 + 16*c^6*(3*d^2*x^2 + 3*d*e*x^4 +
e^2*x^6))*ArcSin[c*x])/(288*c^6)

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fricas [A]  time = 0.68, size = 183, normalized size = 1.00 \[ \frac {48 \, a c^{6} e^{2} x^{6} + 144 \, a c^{6} d e x^{4} + 144 \, a c^{6} d^{2} x^{2} + 3 \, {\left (16 \, b c^{6} e^{2} x^{6} + 48 \, b c^{6} d e x^{4} + 48 \, b c^{6} d^{2} x^{2} - 24 \, b c^{4} d^{2} - 18 \, b c^{2} d e - 5 \, b e^{2}\right )} \arcsin \left (c x\right ) + {\left (8 \, b c^{5} e^{2} x^{5} + 2 \, {\left (18 \, b c^{5} d e + 5 \, b c^{3} e^{2}\right )} x^{3} + 3 \, {\left (24 \, b c^{5} d^{2} + 18 \, b c^{3} d e + 5 \, b c e^{2}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*(48*a*c^6*e^2*x^6 + 144*a*c^6*d*e*x^4 + 144*a*c^6*d^2*x^2 + 3*(16*b*c^6*e^2*x^6 + 48*b*c^6*d*e*x^4 + 48*
b*c^6*d^2*x^2 - 24*b*c^4*d^2 - 18*b*c^2*d*e - 5*b*e^2)*arcsin(c*x) + (8*b*c^5*e^2*x^5 + 2*(18*b*c^5*d*e + 5*b*
c^3*e^2)*x^3 + 3*(24*b*c^5*d^2 + 18*b*c^3*d*e + 5*b*c*e^2)*x)*sqrt(-c^2*x^2 + 1))/c^6

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giac [B]  time = 0.53, size = 348, normalized size = 1.90 \[ \frac {1}{6} \, a x^{6} e^{2} + \frac {1}{2} \, a d x^{4} e + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} x}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} \arcsin \left (c x\right )}{2 \, c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d x e}{8 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d^{2}}{2 \, c^{2}} + \frac {b d^{2} \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d \arcsin \left (c x\right ) e}{2 \, c^{4}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d x e}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d \arcsin \left (c x\right ) e}{c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b x e^{2}}{36 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b \arcsin \left (c x\right ) e^{2}}{6 \, c^{6}} + \frac {5 \, b d \arcsin \left (c x\right ) e}{16 \, c^{4}} - \frac {13 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b x e^{2}}{144 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b \arcsin \left (c x\right ) e^{2}}{2 \, c^{6}} + \frac {11 \, \sqrt {-c^{2} x^{2} + 1} b x e^{2}}{96 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )} b \arcsin \left (c x\right ) e^{2}}{2 \, c^{6}} + \frac {11 \, b \arcsin \left (c x\right ) e^{2}}{96 \, c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*x^6*e^2 + 1/2*a*d*x^4*e + 1/4*sqrt(-c^2*x^2 + 1)*b*d^2*x/c + 1/2*(c^2*x^2 - 1)*b*d^2*arcsin(c*x)/c^2 - 1
/8*(-c^2*x^2 + 1)^(3/2)*b*d*x*e/c^3 + 1/2*(c^2*x^2 - 1)*a*d^2/c^2 + 1/4*b*d^2*arcsin(c*x)/c^2 + 1/2*(c^2*x^2 -
1)^2*b*d*arcsin(c*x)*e/c^4 + 5/16*sqrt(-c^2*x^2 + 1)*b*d*x*e/c^3 + (c^2*x^2 - 1)*b*d*arcsin(c*x)*e/c^4 + 1/36
*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*x*e^2/c^5 + 1/6*(c^2*x^2 - 1)^3*b*arcsin(c*x)*e^2/c^6 + 5/16*b*d*arcsin(
c*x)*e/c^4 - 13/144*(-c^2*x^2 + 1)^(3/2)*b*x*e^2/c^5 + 1/2*(c^2*x^2 - 1)^2*b*arcsin(c*x)*e^2/c^6 + 11/96*sqrt(
-c^2*x^2 + 1)*b*x*e^2/c^5 + 1/2*(c^2*x^2 - 1)*b*arcsin(c*x)*e^2/c^6 + 11/96*b*arcsin(c*x)*e^2/c^6

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maple [A]  time = 0.00, size = 243, normalized size = 1.33 \[ \frac {\frac {a \left (\frac {1}{6} e^{2} c^{6} x^{6}+\frac {1}{2} c^{6} e d \,x^{4}+\frac {1}{2} x^{2} c^{6} d^{2}\right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{2} c^{6} x^{6}}{6}+\frac {\arcsin \left (c x \right ) c^{6} e d \,x^{4}}{2}+\frac {\arcsin \left (c x \right ) d^{2} c^{6} x^{2}}{2}-\frac {e^{2} \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{6}-\frac {c^{2} e d \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{2}-\frac {d^{2} c^{4} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}\right )}{c^{4}}}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(a/c^4*(1/6*e^2*c^6*x^6+1/2*c^6*e*d*x^4+1/2*x^2*c^6*d^2)+b/c^4*(1/6*arcsin(c*x)*e^2*c^6*x^6+1/2*arcsin(c
*x)*c^6*e*d*x^4+1/2*arcsin(c*x)*d^2*c^6*x^2-1/6*e^2*(-1/6*c^5*x^5*(-c^2*x^2+1)^(1/2)-5/24*c^3*x^3*(-c^2*x^2+1)
^(1/2)-5/16*c*x*(-c^2*x^2+1)^(1/2)+5/16*arcsin(c*x))-1/2*c^2*e*d*(-1/4*c^3*x^3*(-c^2*x^2+1)^(1/2)-3/8*c*x*(-c^
2*x^2+1)^(1/2)+3/8*arcsin(c*x))-1/2*d^2*c^4*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))))

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maxima [A]  time = 0.62, size = 223, normalized size = 1.22 \[ \frac {1}{6} \, a e^{2} x^{6} + \frac {1}{2} \, a d e x^{4} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d e + \frac {1}{288} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*e^2*x^6 + 1/2*a*d*e*x^4 + 1/2*a*d^2*x^2 + 1/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(
c*x)/c^3))*b*d^2 + 1/16*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*ar
csin(c*x)/c^5)*c)*b*d*e + 1/288*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^
3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*b*e^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 4.06, size = 299, normalized size = 1.63 \[ \begin {cases} \frac {a d^{2} x^{2}}{2} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{6}}{6} + \frac {b d^{2} x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b d e x^{4} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e^{2} x^{6} \operatorname {asin}{\left (c x \right )}}{6} + \frac {b d^{2} x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b d e x^{3} \sqrt {- c^{2} x^{2} + 1}}{8 c} + \frac {b e^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{36 c} - \frac {b d^{2} \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {3 b d e x \sqrt {- c^{2} x^{2} + 1}}{16 c^{3}} + \frac {5 b e^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{144 c^{3}} - \frac {3 b d e \operatorname {asin}{\left (c x \right )}}{16 c^{4}} + \frac {5 b e^{2} x \sqrt {- c^{2} x^{2} + 1}}{96 c^{5}} - \frac {5 b e^{2} \operatorname {asin}{\left (c x \right )}}{96 c^{6}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{2} x^{2}}{2} + \frac {d e x^{4}}{2} + \frac {e^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x**2/2 + a*d*e*x**4/2 + a*e**2*x**6/6 + b*d**2*x**2*asin(c*x)/2 + b*d*e*x**4*asin(c*x)/2 + b
*e**2*x**6*asin(c*x)/6 + b*d**2*x*sqrt(-c**2*x**2 + 1)/(4*c) + b*d*e*x**3*sqrt(-c**2*x**2 + 1)/(8*c) + b*e**2*
x**5*sqrt(-c**2*x**2 + 1)/(36*c) - b*d**2*asin(c*x)/(4*c**2) + 3*b*d*e*x*sqrt(-c**2*x**2 + 1)/(16*c**3) + 5*b*
e**2*x**3*sqrt(-c**2*x**2 + 1)/(144*c**3) - 3*b*d*e*asin(c*x)/(16*c**4) + 5*b*e**2*x*sqrt(-c**2*x**2 + 1)/(96*
c**5) - 5*b*e**2*asin(c*x)/(96*c**6), Ne(c, 0)), (a*(d**2*x**2/2 + d*e*x**4/2 + e**2*x**6/6), True))

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