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

Optimal. Leaf size=184 \[ \frac{1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{64} b c^3 d^2 x^7 \sqrt{1-c^2 x^2}-\frac{43 b c d^2 x^5 \sqrt{1-c^2 x^2}}{1152}+\frac{73 b d^2 x^3 \sqrt{1-c^2 x^2}}{4608 c}+\frac{73 b d^2 x \sqrt{1-c^2 x^2}}{3072 c^3}-\frac{73 b d^2 \sin ^{-1}(c x)}{3072 c^4} \]

[Out]

(73*b*d^2*x*Sqrt[1 - c^2*x^2])/(3072*c^3) + (73*b*d^2*x^3*Sqrt[1 - c^2*x^2])/(4608*c) - (43*b*c*d^2*x^5*Sqrt[1
- c^2*x^2])/1152 + (b*c^3*d^2*x^7*Sqrt[1 - c^2*x^2])/64 - (73*b*d^2*ArcSin[c*x])/(3072*c^4) + (d^2*x^4*(a + b
*ArcSin[c*x]))/4 - (c^2*d^2*x^6*(a + b*ArcSin[c*x]))/3 + (c^4*d^2*x^8*(a + b*ArcSin[c*x]))/8

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Rubi [A]  time = 0.169913, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {266, 43, 4687, 12, 1267, 459, 321, 216} \[ \frac{1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{64} b c^3 d^2 x^7 \sqrt{1-c^2 x^2}-\frac{43 b c d^2 x^5 \sqrt{1-c^2 x^2}}{1152}+\frac{73 b d^2 x^3 \sqrt{1-c^2 x^2}}{4608 c}+\frac{73 b d^2 x \sqrt{1-c^2 x^2}}{3072 c^3}-\frac{73 b d^2 \sin ^{-1}(c x)}{3072 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(73*b*d^2*x*Sqrt[1 - c^2*x^2])/(3072*c^3) + (73*b*d^2*x^3*Sqrt[1 - c^2*x^2])/(4608*c) - (43*b*c*d^2*x^5*Sqrt[1
- c^2*x^2])/1152 + (b*c^3*d^2*x^7*Sqrt[1 - c^2*x^2])/64 - (73*b*d^2*ArcSin[c*x])/(3072*c^4) + (d^2*x^4*(a + b
*ArcSin[c*x]))/4 - (c^2*d^2*x^6*(a + b*ArcSin[c*x]))/3 + (c^4*d^2*x^8*(a + b*ArcSin[c*x]))/8

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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]

Rule 12

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

Rule 1267

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rule 459

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

Rule 321

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

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]

Rubi steps

\begin{align*} \int x^3 \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{d^2 x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{24} \left (b c d^2\right ) \int \frac{x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{64} b c^3 d^2 x^7 \sqrt{1-c^2 x^2}+\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (b d^2\right ) \int \frac{x^4 \left (-48 c^2+43 c^4 x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{192 c}\\ &=-\frac{43 b c d^2 x^5 \sqrt{1-c^2 x^2}}{1152}+\frac{1}{64} b c^3 d^2 x^7 \sqrt{1-c^2 x^2}+\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (73 b c d^2\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{1152}\\ &=\frac{73 b d^2 x^3 \sqrt{1-c^2 x^2}}{4608 c}-\frac{43 b c d^2 x^5 \sqrt{1-c^2 x^2}}{1152}+\frac{1}{64} b c^3 d^2 x^7 \sqrt{1-c^2 x^2}+\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (73 b d^2\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{1536 c}\\ &=\frac{73 b d^2 x \sqrt{1-c^2 x^2}}{3072 c^3}+\frac{73 b d^2 x^3 \sqrt{1-c^2 x^2}}{4608 c}-\frac{43 b c d^2 x^5 \sqrt{1-c^2 x^2}}{1152}+\frac{1}{64} b c^3 d^2 x^7 \sqrt{1-c^2 x^2}+\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (73 b d^2\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{3072 c^3}\\ &=\frac{73 b d^2 x \sqrt{1-c^2 x^2}}{3072 c^3}+\frac{73 b d^2 x^3 \sqrt{1-c^2 x^2}}{4608 c}-\frac{43 b c d^2 x^5 \sqrt{1-c^2 x^2}}{1152}+\frac{1}{64} b c^3 d^2 x^7 \sqrt{1-c^2 x^2}-\frac{73 b d^2 \sin ^{-1}(c x)}{3072 c^4}+\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0998687, size = 115, normalized size = 0.62 \[ \frac{d^2 \left (384 a c^4 x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )+b c x \sqrt{1-c^2 x^2} \left (144 c^6 x^6-344 c^4 x^4+146 c^2 x^2+219\right )+3 b \left (384 c^8 x^8-1024 c^6 x^6+768 c^4 x^4-73\right ) \sin ^{-1}(c x)\right )}{9216 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(384*a*c^4*x^4*(6 - 8*c^2*x^2 + 3*c^4*x^4) + b*c*x*Sqrt[1 - c^2*x^2]*(219 + 146*c^2*x^2 - 344*c^4*x^4 + 1
44*c^6*x^6) + 3*b*(-73 + 768*c^4*x^4 - 1024*c^6*x^6 + 384*c^8*x^8)*ArcSin[c*x]))/(9216*c^4)

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Maple [A]  time = 0.007, size = 160, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{4}} \left ({d}^{2}a \left ({\frac{{c}^{8}{x}^{8}}{8}}-{\frac{{c}^{6}{x}^{6}}{3}}+{\frac{{c}^{4}{x}^{4}}{4}} \right ) +{d}^{2}b \left ({\frac{\arcsin \left ( cx \right ){c}^{8}{x}^{8}}{8}}-{\frac{\arcsin \left ( cx \right ){c}^{6}{x}^{6}}{3}}+{\frac{{c}^{4}{x}^{4}\arcsin \left ( cx \right ) }{4}}+{\frac{{c}^{7}{x}^{7}}{64}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{43\,{c}^{5}{x}^{5}}{1152}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{73\,{c}^{3}{x}^{3}}{4608}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{73\,cx}{3072}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{73\,\arcsin \left ( cx \right ) }{3072}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(d^2*a*(1/8*c^8*x^8-1/3*c^6*x^6+1/4*c^4*x^4)+d^2*b*(1/8*arcsin(c*x)*c^8*x^8-1/3*arcsin(c*x)*c^6*x^6+1/4*
c^4*x^4*arcsin(c*x)+1/64*c^7*x^7*(-c^2*x^2+1)^(1/2)-43/1152*c^5*x^5*(-c^2*x^2+1)^(1/2)+73/4608*c^3*x^3*(-c^2*x
^2+1)^(1/2)+73/3072*c*x*(-c^2*x^2+1)^(1/2)-73/3072*arcsin(c*x)))

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Maxima [B]  time = 1.58398, size = 451, normalized size = 2.45 \begin{align*} \frac{1}{8} \, a c^{4} d^{2} x^{8} - \frac{1}{3} \, a c^{2} d^{2} x^{6} + \frac{1}{3072} \,{\left (384 \, x^{8} \arcsin \left (c x\right ) +{\left (\frac{48 \, \sqrt{-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac{56 \, \sqrt{-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac{70 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac{105 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{8}} - \frac{105 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{8}}\right )} c\right )} b c^{4} d^{2} + \frac{1}{4} \, a d^{2} x^{4} - \frac{1}{144} \,{\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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b c^{2} d^{2} + \frac{1}{32} \,{\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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b d^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*c^4*d^2*x^8 - 1/3*a*c^2*d^2*x^6 + 1/3072*(384*x^8*arcsin(c*x) + (48*sqrt(-c^2*x^2 + 1)*x^7/c^2 + 56*sqrt
(-c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(-c^2*x^2 + 1)*x^3/c^6 + 105*sqrt(-c^2*x^2 + 1)*x/c^8 - 105*arcsin(c^2*x/sqrt(
c^2))/(sqrt(c^2)*c^8))*c)*b*c^4*d^2 + 1/4*a*d^2*x^4 - 1/144*(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^2*x/sqrt(c^2))/(sqrt(c^2)*c^6))*
c)*b*c^2*d^2 + 1/32*(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*arcsin
(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*d^2

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Fricas [A]  time = 2.53678, size = 351, normalized size = 1.91 \begin{align*} \frac{1152 \, a c^{8} d^{2} x^{8} - 3072 \, a c^{6} d^{2} x^{6} + 2304 \, a c^{4} d^{2} x^{4} + 3 \,{\left (384 \, b c^{8} d^{2} x^{8} - 1024 \, b c^{6} d^{2} x^{6} + 768 \, b c^{4} d^{2} x^{4} - 73 \, b d^{2}\right )} \arcsin \left (c x\right ) +{\left (144 \, b c^{7} d^{2} x^{7} - 344 \, b c^{5} d^{2} x^{5} + 146 \, b c^{3} d^{2} x^{3} + 219 \, b c d^{2} x\right )} \sqrt{-c^{2} x^{2} + 1}}{9216 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9216*(1152*a*c^8*d^2*x^8 - 3072*a*c^6*d^2*x^6 + 2304*a*c^4*d^2*x^4 + 3*(384*b*c^8*d^2*x^8 - 1024*b*c^6*d^2*x
^6 + 768*b*c^4*d^2*x^4 - 73*b*d^2)*arcsin(c*x) + (144*b*c^7*d^2*x^7 - 344*b*c^5*d^2*x^5 + 146*b*c^3*d^2*x^3 +
219*b*c*d^2*x)*sqrt(-c^2*x^2 + 1))/c^4

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Sympy [A]  time = 20.7344, size = 218, normalized size = 1.18 \begin{align*} \begin{cases} \frac{a c^{4} d^{2} x^{8}}{8} - \frac{a c^{2} d^{2} x^{6}}{3} + \frac{a d^{2} x^{4}}{4} + \frac{b c^{4} d^{2} x^{8} \operatorname{asin}{\left (c x \right )}}{8} + \frac{b c^{3} d^{2} x^{7} \sqrt{- c^{2} x^{2} + 1}}{64} - \frac{b c^{2} d^{2} x^{6} \operatorname{asin}{\left (c x \right )}}{3} - \frac{43 b c d^{2} x^{5} \sqrt{- c^{2} x^{2} + 1}}{1152} + \frac{b d^{2} x^{4} \operatorname{asin}{\left (c x \right )}}{4} + \frac{73 b d^{2} x^{3} \sqrt{- c^{2} x^{2} + 1}}{4608 c} + \frac{73 b d^{2} x \sqrt{- c^{2} x^{2} + 1}}{3072 c^{3}} - \frac{73 b d^{2} \operatorname{asin}{\left (c x \right )}}{3072 c^{4}} & \text{for}\: c \neq 0 \\\frac{a d^{2} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c**4*d**2*x**8/8 - a*c**2*d**2*x**6/3 + a*d**2*x**4/4 + b*c**4*d**2*x**8*asin(c*x)/8 + b*c**3*d**
2*x**7*sqrt(-c**2*x**2 + 1)/64 - b*c**2*d**2*x**6*asin(c*x)/3 - 43*b*c*d**2*x**5*sqrt(-c**2*x**2 + 1)/1152 + b
*d**2*x**4*asin(c*x)/4 + 73*b*d**2*x**3*sqrt(-c**2*x**2 + 1)/(4608*c) + 73*b*d**2*x*sqrt(-c**2*x**2 + 1)/(3072
*c**3) - 73*b*d**2*asin(c*x)/(3072*c**4), Ne(c, 0)), (a*d**2*x**4/4, True))

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Giac [A]  time = 1.27356, size = 286, normalized size = 1.55 \begin{align*} \frac{{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} b d^{2} x}{64 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{4} b d^{2} \arcsin \left (c x\right )}{8 \, c^{4}} + \frac{11 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b d^{2} x}{1152 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{4} a d^{2}}{8 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b d^{2} \arcsin \left (c x\right )}{6 \, c^{4}} + \frac{55 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d^{2} x}{4608 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} a d^{2}}{6 \, c^{4}} + \frac{55 \, \sqrt{-c^{2} x^{2} + 1} b d^{2} x}{3072 \, c^{3}} + \frac{55 \, b d^{2} \arcsin \left (c x\right )}{3072 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*d^2*x/c^3 + 1/8*(c^2*x^2 - 1)^4*b*d^2*arcsin(c*x)/c^4 + 11/1152*(c^2
*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^2*x/c^3 + 1/8*(c^2*x^2 - 1)^4*a*d^2/c^4 + 1/6*(c^2*x^2 - 1)^3*b*d^2*arcsin(
c*x)/c^4 + 55/4608*(-c^2*x^2 + 1)^(3/2)*b*d^2*x/c^3 + 1/6*(c^2*x^2 - 1)^3*a*d^2/c^4 + 55/3072*sqrt(-c^2*x^2 +
1)*b*d^2*x/c^3 + 55/3072*b*d^2*arcsin(c*x)/c^4