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

Optimal. Leaf size=209 \[ \frac{b d^2 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{72 c}+\frac{5 b d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}-\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}+\frac{5 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{96 c^2}+\frac{5}{288} b^2 c^2 d^2 x^4+\frac{b^2 d^2 \left (1-c^2 x^2\right )^3}{108 c^2}-\frac{25}{288} b^2 d^2 x^2 \]

[Out]

(-25*b^2*d^2*x^2)/288 + (5*b^2*c^2*d^2*x^4)/288 + (b^2*d^2*(1 - c^2*x^2)^3)/(108*c^2) + (5*b*d^2*x*Sqrt[1 - c^
2*x^2]*(a + b*ArcSin[c*x]))/(48*c) + (5*b*d^2*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(72*c) + (b*d^2*x*(1
- c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(18*c) + (5*d^2*(a + b*ArcSin[c*x])^2)/(96*c^2) - (d^2*(1 - c^2*x^2)^3*(
a + b*ArcSin[c*x])^2)/(6*c^2)

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Rubi [A]  time = 0.197679, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {4677, 4649, 4647, 4641, 30, 14, 261} \[ \frac{b d^2 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{72 c}+\frac{5 b d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}-\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}+\frac{5 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{96 c^2}+\frac{5}{288} b^2 c^2 d^2 x^4+\frac{b^2 d^2 \left (1-c^2 x^2\right )^3}{108 c^2}-\frac{25}{288} b^2 d^2 x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-25*b^2*d^2*x^2)/288 + (5*b^2*c^2*d^2*x^4)/288 + (b^2*d^2*(1 - c^2*x^2)^3)/(108*c^2) + (5*b*d^2*x*Sqrt[1 - c^
2*x^2]*(a + b*ArcSin[c*x]))/(48*c) + (5*b*d^2*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(72*c) + (b*d^2*x*(1
- c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(18*c) + (5*d^2*(a + b*ArcSin[c*x])^2)/(96*c^2) - (d^2*(1 - c^2*x^2)^3*(
a + b*ArcSin[c*x])^2)/(6*c^2)

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 4649

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

Rule 4647

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

Rule 4641

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 261

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

Rubi steps

\begin{align*} \int x \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=-\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}+\frac{\left (b d^2\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 c}\\ &=\frac{b d^2 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}-\frac{1}{18} \left (b^2 d^2\right ) \int x \left (1-c^2 x^2\right )^2 \, dx+\frac{\left (5 b d^2\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{18 c}\\ &=\frac{b^2 d^2 \left (1-c^2 x^2\right )^3}{108 c^2}+\frac{5 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{72 c}+\frac{b d^2 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}-\frac{1}{72} \left (5 b^2 d^2\right ) \int x \left (1-c^2 x^2\right ) \, dx+\frac{\left (5 b d^2\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{24 c}\\ &=\frac{b^2 d^2 \left (1-c^2 x^2\right )^3}{108 c^2}+\frac{5 b d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{5 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{72 c}+\frac{b d^2 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}-\frac{1}{72} \left (5 b^2 d^2\right ) \int \left (x-c^2 x^3\right ) \, dx-\frac{1}{48} \left (5 b^2 d^2\right ) \int x \, dx+\frac{\left (5 b d^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{48 c}\\ &=-\frac{25}{288} b^2 d^2 x^2+\frac{5}{288} b^2 c^2 d^2 x^4+\frac{b^2 d^2 \left (1-c^2 x^2\right )^3}{108 c^2}+\frac{5 b d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{5 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{72 c}+\frac{b d^2 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{96 c^2}-\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}\\ \end{align*}

Mathematica [A]  time = 0.285541, size = 209, normalized size = 1. \[ \frac{d^2 \left (c x \left (144 a^2 c x \left (c^4 x^4-3 c^2 x^2+3\right )+6 a b \sqrt{1-c^2 x^2} \left (8 c^4 x^4-26 c^2 x^2+33\right )+b^2 c x \left (-8 c^4 x^4+39 c^2 x^2-99\right )\right )+6 b \sin ^{-1}(c x) \left (3 a \left (16 c^6 x^6-48 c^4 x^4+48 c^2 x^2-11\right )+b c x \sqrt{1-c^2 x^2} \left (8 c^4 x^4-26 c^2 x^2+33\right )\right )+9 b^2 \left (16 c^6 x^6-48 c^4 x^4+48 c^2 x^2-11\right ) \sin ^{-1}(c x)^2\right )}{864 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(c*x*(b^2*c*x*(-99 + 39*c^2*x^2 - 8*c^4*x^4) + 144*a^2*c*x*(3 - 3*c^2*x^2 + c^4*x^4) + 6*a*b*Sqrt[1 - c^2
*x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4)) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 3*a*(-11 +
48*c^2*x^2 - 48*c^4*x^4 + 16*c^6*x^6))*ArcSin[c*x] + 9*b^2*(-11 + 48*c^2*x^2 - 48*c^4*x^4 + 16*c^6*x^6)*ArcSi
n[c*x]^2))/(864*c^2)

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Maple [A]  time = 0.038, size = 283, normalized size = 1.4 \begin{align*}{\frac{1}{{c}^{2}} \left ({d}^{2}{a}^{2} \left ({\frac{{c}^{6}{x}^{6}}{6}}-{\frac{{c}^{4}{x}^{4}}{2}}+{\frac{{c}^{2}{x}^{2}}{2}} \right ) +{d}^{2}{b}^{2} \left ({\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}-1 \right ) ^{3}}{6}}+{\frac{\arcsin \left ( cx \right ) }{144} \left ( 8\,{c}^{5}{x}^{5}\sqrt{-{c}^{2}{x}^{2}+1}-26\,{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}+33\,cx\sqrt{-{c}^{2}{x}^{2}+1}+15\,\arcsin \left ( cx \right ) \right ) }-{\frac{5\, \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{96}}-{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) ^{3}}{108}}+{\frac{5\, \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}{288}}-{\frac{5\,{c}^{2}{x}^{2}}{96}}+{\frac{5}{96}} \right ) +2\,{d}^{2}ab \left ( 1/6\,\arcsin \left ( cx \right ){c}^{6}{x}^{6}-1/2\,{c}^{4}{x}^{4}\arcsin \left ( cx \right ) +1/2\,{c}^{2}{x}^{2}\arcsin \left ( cx \right ) +1/36\,{c}^{5}{x}^{5}\sqrt{-{c}^{2}{x}^{2}+1}-{\frac{13\,{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}}{144}}+{\frac{11\,cx\sqrt{-{c}^{2}{x}^{2}+1}}{96}}-{\frac{11\,\arcsin \left ( cx \right ) }{96}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(d^2*a^2*(1/6*c^6*x^6-1/2*c^4*x^4+1/2*c^2*x^2)+d^2*b^2*(1/6*arcsin(c*x)^2*(c^2*x^2-1)^3+1/144*arcsin(c*x
)*(8*c^5*x^5*(-c^2*x^2+1)^(1/2)-26*c^3*x^3*(-c^2*x^2+1)^(1/2)+33*c*x*(-c^2*x^2+1)^(1/2)+15*arcsin(c*x))-5/96*a
rcsin(c*x)^2-1/108*(c^2*x^2-1)^3+5/288*(c^2*x^2-1)^2-5/96*c^2*x^2+5/96)+2*d^2*a*b*(1/6*arcsin(c*x)*c^6*x^6-1/2
*c^4*x^4*arcsin(c*x)+1/2*c^2*x^2*arcsin(c*x)+1/36*c^5*x^5*(-c^2*x^2+1)^(1/2)-13/144*c^3*x^3*(-c^2*x^2+1)^(1/2)
+11/96*c*x*(-c^2*x^2+1)^(1/2)-11/96*arcsin(c*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \, a^{2} c^{4} d^{2} x^{6} - \frac{1}{2} \, a^{2} c^{2} 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 )} a b c^{4} d^{2} - \frac{1}{8} \,{\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 )} a b c^{2} d^{2} + \frac{1}{2} \, a^{2} d^{2} x^{2} + \frac{1}{2} \,{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} a b d^{2} + \frac{1}{6} \,{\left (b^{2} c^{4} d^{2} x^{6} - 3 \, b^{2} c^{2} d^{2} x^{4} + 3 \, b^{2} d^{2} x^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + \int \frac{{\left (b^{2} c^{5} d^{2} x^{6} - 3 \, b^{2} c^{3} d^{2} x^{4} + 3 \, b^{2} c d^{2} x^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{3 \,{\left (c^{2} x^{2} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a^2*c^4*d^2*x^6 - 1/2*a^2*c^2*d^2*x^4 + 1/144*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqr
t(-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)*a*b*c^4
*d^2 - 1/8*(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/sq
rt(c^2))/(sqrt(c^2)*c^4))*c)*a*b*c^2*d^2 + 1/2*a^2*d^2*x^2 + 1/2*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/
c^2 - arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*a*b*d^2 + 1/6*(b^2*c^4*d^2*x^6 - 3*b^2*c^2*d^2*x^4 + 3*b^2*d^2
*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + integrate(1/3*(b^2*c^5*d^2*x^6 - 3*b^2*c^3*d^2*x^4 + 3*b^
2*c*d^2*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*x^2 - 1), x)

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Fricas [A]  time = 1.80314, size = 609, normalized size = 2.91 \begin{align*} \frac{8 \,{\left (18 \, a^{2} - b^{2}\right )} c^{6} d^{2} x^{6} - 3 \,{\left (144 \, a^{2} - 13 \, b^{2}\right )} c^{4} d^{2} x^{4} + 9 \,{\left (48 \, a^{2} - 11 \, b^{2}\right )} c^{2} d^{2} x^{2} + 9 \,{\left (16 \, b^{2} c^{6} d^{2} x^{6} - 48 \, b^{2} c^{4} d^{2} x^{4} + 48 \, b^{2} c^{2} d^{2} x^{2} - 11 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 18 \,{\left (16 \, a b c^{6} d^{2} x^{6} - 48 \, a b c^{4} d^{2} x^{4} + 48 \, a b c^{2} d^{2} x^{2} - 11 \, a b d^{2}\right )} \arcsin \left (c x\right ) + 6 \,{\left (8 \, a b c^{5} d^{2} x^{5} - 26 \, a b c^{3} d^{2} x^{3} + 33 \, a b c d^{2} x +{\left (8 \, b^{2} c^{5} d^{2} x^{5} - 26 \, b^{2} c^{3} d^{2} x^{3} + 33 \, b^{2} c d^{2} x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{864 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/864*(8*(18*a^2 - b^2)*c^6*d^2*x^6 - 3*(144*a^2 - 13*b^2)*c^4*d^2*x^4 + 9*(48*a^2 - 11*b^2)*c^2*d^2*x^2 + 9*(
16*b^2*c^6*d^2*x^6 - 48*b^2*c^4*d^2*x^4 + 48*b^2*c^2*d^2*x^2 - 11*b^2*d^2)*arcsin(c*x)^2 + 18*(16*a*b*c^6*d^2*
x^6 - 48*a*b*c^4*d^2*x^4 + 48*a*b*c^2*d^2*x^2 - 11*a*b*d^2)*arcsin(c*x) + 6*(8*a*b*c^5*d^2*x^5 - 26*a*b*c^3*d^
2*x^3 + 33*a*b*c*d^2*x + (8*b^2*c^5*d^2*x^5 - 26*b^2*c^3*d^2*x^3 + 33*b^2*c*d^2*x)*arcsin(c*x))*sqrt(-c^2*x^2
+ 1))/c^2

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Sympy [A]  time = 13.7133, size = 430, normalized size = 2.06 \begin{align*} \begin{cases} \frac{a^{2} c^{4} d^{2} x^{6}}{6} - \frac{a^{2} c^{2} d^{2} x^{4}}{2} + \frac{a^{2} d^{2} x^{2}}{2} + \frac{a b c^{4} d^{2} x^{6} \operatorname{asin}{\left (c x \right )}}{3} + \frac{a b c^{3} d^{2} x^{5} \sqrt{- c^{2} x^{2} + 1}}{18} - a b c^{2} d^{2} x^{4} \operatorname{asin}{\left (c x \right )} - \frac{13 a b c d^{2} x^{3} \sqrt{- c^{2} x^{2} + 1}}{72} + a b d^{2} x^{2} \operatorname{asin}{\left (c x \right )} + \frac{11 a b d^{2} x \sqrt{- c^{2} x^{2} + 1}}{48 c} - \frac{11 a b d^{2} \operatorname{asin}{\left (c x \right )}}{48 c^{2}} + \frac{b^{2} c^{4} d^{2} x^{6} \operatorname{asin}^{2}{\left (c x \right )}}{6} - \frac{b^{2} c^{4} d^{2} x^{6}}{108} + \frac{b^{2} c^{3} d^{2} x^{5} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{18} - \frac{b^{2} c^{2} d^{2} x^{4} \operatorname{asin}^{2}{\left (c x \right )}}{2} + \frac{13 b^{2} c^{2} d^{2} x^{4}}{288} - \frac{13 b^{2} c d^{2} x^{3} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{72} + \frac{b^{2} d^{2} x^{2} \operatorname{asin}^{2}{\left (c x \right )}}{2} - \frac{11 b^{2} d^{2} x^{2}}{96} + \frac{11 b^{2} d^{2} x \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{48 c} - \frac{11 b^{2} d^{2} \operatorname{asin}^{2}{\left (c x \right )}}{96 c^{2}} & \text{for}\: c \neq 0 \\\frac{a^{2} d^{2} x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**4*d**2*x**6/6 - a**2*c**2*d**2*x**4/2 + a**2*d**2*x**2/2 + a*b*c**4*d**2*x**6*asin(c*x)/3 +
a*b*c**3*d**2*x**5*sqrt(-c**2*x**2 + 1)/18 - a*b*c**2*d**2*x**4*asin(c*x) - 13*a*b*c*d**2*x**3*sqrt(-c**2*x**
2 + 1)/72 + a*b*d**2*x**2*asin(c*x) + 11*a*b*d**2*x*sqrt(-c**2*x**2 + 1)/(48*c) - 11*a*b*d**2*asin(c*x)/(48*c*
*2) + b**2*c**4*d**2*x**6*asin(c*x)**2/6 - b**2*c**4*d**2*x**6/108 + b**2*c**3*d**2*x**5*sqrt(-c**2*x**2 + 1)*
asin(c*x)/18 - b**2*c**2*d**2*x**4*asin(c*x)**2/2 + 13*b**2*c**2*d**2*x**4/288 - 13*b**2*c*d**2*x**3*sqrt(-c**
2*x**2 + 1)*asin(c*x)/72 + b**2*d**2*x**2*asin(c*x)**2/2 - 11*b**2*d**2*x**2/96 + 11*b**2*d**2*x*sqrt(-c**2*x*
*2 + 1)*asin(c*x)/(48*c) - 11*b**2*d**2*asin(c*x)**2/(96*c**2), Ne(c, 0)), (a**2*d**2*x**2/2, True))

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Giac [A]  time = 1.48457, size = 482, normalized size = 2.31 \begin{align*} \frac{{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right )}{18 \, c} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2} \arcsin \left (c x\right )^{2}}{6 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} a b d^{2} x}{18 \, c} + \frac{5 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} d^{2} x \arcsin \left (c x\right )}{72 \, c} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} a b d^{2} \arcsin \left (c x\right )}{3 \, c^{2}} + \frac{5 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b d^{2} x}{72 \, c} + \frac{5 \, \sqrt{-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right )}{48 \, c} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} a^{2} d^{2}}{6 \, c^{2}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2}}{108 \, c^{2}} + \frac{5 \, \sqrt{-c^{2} x^{2} + 1} a b d^{2} x}{48 \, c} + \frac{5 \,{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2}}{288 \, c^{2}} + \frac{5 \, b^{2} d^{2} \arcsin \left (c x\right )^{2}}{96 \, c^{2}} - \frac{5 \,{\left (c^{2} x^{2} - 1\right )} b^{2} d^{2}}{96 \, c^{2}} + \frac{5 \, a b d^{2} \arcsin \left (c x\right )}{48 \, c^{2}} - \frac{245 \, b^{2} d^{2}}{6912 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^2*x*arcsin(c*x)/c + 1/6*(c^2*x^2 - 1)^3*b^2*d^2*arcsin(c*x)^2/c^
2 + 1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d^2*x/c + 5/72*(-c^2*x^2 + 1)^(3/2)*b^2*d^2*x*arcsin(c*x)/c +
1/3*(c^2*x^2 - 1)^3*a*b*d^2*arcsin(c*x)/c^2 + 5/72*(-c^2*x^2 + 1)^(3/2)*a*b*d^2*x/c + 5/48*sqrt(-c^2*x^2 + 1)*
b^2*d^2*x*arcsin(c*x)/c + 1/6*(c^2*x^2 - 1)^3*a^2*d^2/c^2 - 1/108*(c^2*x^2 - 1)^3*b^2*d^2/c^2 + 5/48*sqrt(-c^2
*x^2 + 1)*a*b*d^2*x/c + 5/288*(c^2*x^2 - 1)^2*b^2*d^2/c^2 + 5/96*b^2*d^2*arcsin(c*x)^2/c^2 - 5/96*(c^2*x^2 - 1
)*b^2*d^2/c^2 + 5/48*a*b*d^2*arcsin(c*x)/c^2 - 245/6912*b^2*d^2/c^2