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

Optimal. Leaf size=219 \[ \frac{1}{5} d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4}{15} d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac{16 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 c}+\frac{8}{15} d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2}{125} b^2 c^4 d^2 x^5+\frac{76}{675} b^2 c^2 d^2 x^3-\frac{298}{225} b^2 d^2 x \]

[Out]

(-298*b^2*d^2*x)/225 + (76*b^2*c^2*d^2*x^3)/675 - (2*b^2*c^4*d^2*x^5)/125 + (16*b*d^2*Sqrt[1 - c^2*x^2]*(a + b
*ArcSin[c*x]))/(15*c) + (8*b*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(45*c) + (2*b*d^2*(1 - c^2*x^2)^(5/2
)*(a + b*ArcSin[c*x]))/(25*c) + (8*d^2*x*(a + b*ArcSin[c*x])^2)/15 + (4*d^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x]
)^2)/15 + (d^2*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/5

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Rubi [A]  time = 0.255607, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4649, 4619, 4677, 8, 194} \[ \frac{1}{5} d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4}{15} d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac{16 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 c}+\frac{8}{15} d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2}{125} b^2 c^4 d^2 x^5+\frac{76}{675} b^2 c^2 d^2 x^3-\frac{298}{225} b^2 d^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-298*b^2*d^2*x)/225 + (76*b^2*c^2*d^2*x^3)/675 - (2*b^2*c^4*d^2*x^5)/125 + (16*b*d^2*Sqrt[1 - c^2*x^2]*(a + b
*ArcSin[c*x]))/(15*c) + (8*b*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(45*c) + (2*b*d^2*(1 - c^2*x^2)^(5/2
)*(a + b*ArcSin[c*x]))/(25*c) + (8*d^2*x*(a + b*ArcSin[c*x])^2)/15 + (4*d^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x]
)^2)/15 + (d^2*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/5

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 4619

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

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 194

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

Rubi steps

\begin{align*} \int \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{1}{5} d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} (4 d) \int \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{5} \left (2 b c d^2\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{4}{15} d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{15} \left (8 d^2\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{25} \left (2 b^2 d^2\right ) \int \left (1-c^2 x^2\right )^2 \, dx-\frac{1}{15} \left (8 b c d^2\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac{8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{8}{15} d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4}{15} d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{25} \left (2 b^2 d^2\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx-\frac{1}{45} \left (8 b^2 d^2\right ) \int \left (1-c^2 x^2\right ) \, dx-\frac{1}{15} \left (16 b c d^2\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{58}{225} b^2 d^2 x+\frac{76}{675} b^2 c^2 d^2 x^3-\frac{2}{125} b^2 c^4 d^2 x^5+\frac{16 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 c}+\frac{8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{8}{15} d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4}{15} d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{15} \left (16 b^2 d^2\right ) \int 1 \, dx\\ &=-\frac{298}{225} b^2 d^2 x+\frac{76}{675} b^2 c^2 d^2 x^3-\frac{2}{125} b^2 c^4 d^2 x^5+\frac{16 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 c}+\frac{8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{8}{15} d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4}{15} d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.257989, size = 193, normalized size = 0.88 \[ \frac{d^2 \left (225 a^2 c x \left (3 c^4 x^4-10 c^2 x^2+15\right )+30 a b \sqrt{1-c^2 x^2} \left (9 c^4 x^4-38 c^2 x^2+149\right )+30 b \sin ^{-1}(c x) \left (15 a c x \left (3 c^4 x^4-10 c^2 x^2+15\right )+b \sqrt{1-c^2 x^2} \left (9 c^4 x^4-38 c^2 x^2+149\right )\right )-2 b^2 c x \left (27 c^4 x^4-190 c^2 x^2+2235\right )+225 b^2 c x \left (3 c^4 x^4-10 c^2 x^2+15\right ) \sin ^{-1}(c x)^2\right )}{3375 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(225*a^2*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 30*a*b*Sqrt[1 - c^2*x^2]*(149 - 38*c^2*x^2 + 9*c^4*x^4) - 2*
b^2*c*x*(2235 - 190*c^2*x^2 + 27*c^4*x^4) + 30*b*(15*a*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 - c^2*x^2]
*(149 - 38*c^2*x^2 + 9*c^4*x^4))*ArcSin[c*x] + 225*b^2*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]^2))/(3375
*c)

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Maple [A]  time = 0.037, size = 275, normalized size = 1.3 \begin{align*}{\frac{1}{c} \left ({d}^{2}{a}^{2} \left ({\frac{{c}^{5}{x}^{5}}{5}}-{\frac{2\,{c}^{3}{x}^{3}}{3}}+cx \right ) +{d}^{2}{b}^{2} \left ({\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ( 3\,{c}^{4}{x}^{4}-10\,{c}^{2}{x}^{2}+15 \right ) cx}{15}}-{\frac{16\,cx}{15}}+{\frac{16\,\arcsin \left ( cx \right ) }{15}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{2\,\arcsin \left ( cx \right ) \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}{25}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{ \left ( 6\,{c}^{4}{x}^{4}-20\,{c}^{2}{x}^{2}+30 \right ) cx}{375}}-{\frac{8\, \left ({c}^{2}{x}^{2}-1 \right ) \arcsin \left ( cx \right ) }{45}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{ \left ( 8\,{c}^{2}{x}^{2}-24 \right ) cx}{135}} \right ) +2\,{d}^{2}ab \left ( 1/5\,\arcsin \left ( cx \right ){c}^{5}{x}^{5}-2/3\,{c}^{3}{x}^{3}\arcsin \left ( cx \right ) +cx\arcsin \left ( cx \right ) +1/25\,{c}^{4}{x}^{4}\sqrt{-{c}^{2}{x}^{2}+1}-{\frac{38\,{c}^{2}{x}^{2}\sqrt{-{c}^{2}{x}^{2}+1}}{225}}+{\frac{149\,\sqrt{-{c}^{2}{x}^{2}+1}}{225}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(d^2*a^2*(1/5*c^5*x^5-2/3*c^3*x^3+c*x)+d^2*b^2*(1/15*arcsin(c*x)^2*(3*c^4*x^4-10*c^2*x^2+15)*c*x-16/15*c*x
+16/15*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+2/25*arcsin(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/2)-2/375*(3*c^4*x^4-10*c^
2*x^2+15)*c*x-8/45*arcsin(c*x)*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+8/135*(c^2*x^2-3)*c*x)+2*d^2*a*b*(1/5*arcsin(c*x
)*c^5*x^5-2/3*c^3*x^3*arcsin(c*x)+c*x*arcsin(c*x)+1/25*c^4*x^4*(-c^2*x^2+1)^(1/2)-38/225*c^2*x^2*(-c^2*x^2+1)^
(1/2)+149/225*(-c^2*x^2+1)^(1/2)))

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Maxima [B]  time = 1.75827, size = 628, normalized size = 2.87 \begin{align*} \frac{1}{5} \, b^{2} c^{4} d^{2} x^{5} \arcsin \left (c x\right )^{2} + \frac{1}{5} \, a^{2} c^{4} d^{2} x^{5} - \frac{2}{3} \, b^{2} c^{2} d^{2} x^{3} \arcsin \left (c x\right )^{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 )} a b c^{4} d^{2} + \frac{2}{1125} \,{\left (15 \,{\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 \arcsin \left (c x\right ) - \frac{9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{4} d^{2} - \frac{2}{3} \, a^{2} c^{2} d^{2} x^{3} - \frac{4}{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 )} a b c^{2} d^{2} - \frac{4}{27} \,{\left (3 \, c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac{c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} c^{2} d^{2} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} - 2 \, b^{2} d^{2}{\left (x - \frac{\sqrt{-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac{2 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*c^4*d^2*x^5*arcsin(c*x)^2 + 1/5*a^2*c^4*d^2*x^5 - 2/3*b^2*c^2*d^2*x^3*arcsin(c*x)^2 + 2/75*(15*x^5*arc
sin(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)*a*b*c^4
*d^2 + 2/1125*(15*(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*a
rcsin(c*x) - (9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4)*b^2*c^4*d^2 - 2/3*a^2*c^2*d^2*x^3 - 4/9*(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))*a*b*c^2*d^2 - 4/27*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c
^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arcsin(c*x) - (c^2*x^3 + 6*x)/c^2)*b^2*c^2*d^2 + b^2*d^2*x*arcsin(c*x)^2 - 2*b^
2*d^2*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^2*d^2*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d^2/c

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Fricas [A]  time = 1.87606, size = 560, normalized size = 2.56 \begin{align*} \frac{27 \,{\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{5} d^{2} x^{5} - 10 \,{\left (225 \, a^{2} - 38 \, b^{2}\right )} c^{3} d^{2} x^{3} + 15 \,{\left (225 \, a^{2} - 298 \, b^{2}\right )} c d^{2} x + 225 \,{\left (3 \, b^{2} c^{5} d^{2} x^{5} - 10 \, b^{2} c^{3} d^{2} x^{3} + 15 \, b^{2} c d^{2} x\right )} \arcsin \left (c x\right )^{2} + 450 \,{\left (3 \, a b c^{5} d^{2} x^{5} - 10 \, a b c^{3} d^{2} x^{3} + 15 \, a b c d^{2} x\right )} \arcsin \left (c x\right ) + 30 \,{\left (9 \, a b c^{4} d^{2} x^{4} - 38 \, a b c^{2} d^{2} x^{2} + 149 \, a b d^{2} +{\left (9 \, b^{2} c^{4} d^{2} x^{4} - 38 \, b^{2} c^{2} d^{2} x^{2} + 149 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{3375 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3375*(27*(25*a^2 - 2*b^2)*c^5*d^2*x^5 - 10*(225*a^2 - 38*b^2)*c^3*d^2*x^3 + 15*(225*a^2 - 298*b^2)*c*d^2*x +
225*(3*b^2*c^5*d^2*x^5 - 10*b^2*c^3*d^2*x^3 + 15*b^2*c*d^2*x)*arcsin(c*x)^2 + 450*(3*a*b*c^5*d^2*x^5 - 10*a*b
*c^3*d^2*x^3 + 15*a*b*c*d^2*x)*arcsin(c*x) + 30*(9*a*b*c^4*d^2*x^4 - 38*a*b*c^2*d^2*x^2 + 149*a*b*d^2 + (9*b^2
*c^4*d^2*x^4 - 38*b^2*c^2*d^2*x^2 + 149*b^2*d^2)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c

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Sympy [A]  time = 7.15306, size = 389, normalized size = 1.78 \begin{align*} \begin{cases} \frac{a^{2} c^{4} d^{2} x^{5}}{5} - \frac{2 a^{2} c^{2} d^{2} x^{3}}{3} + a^{2} d^{2} x + \frac{2 a b c^{4} d^{2} x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{2 a b c^{3} d^{2} x^{4} \sqrt{- c^{2} x^{2} + 1}}{25} - \frac{4 a b c^{2} d^{2} x^{3} \operatorname{asin}{\left (c x \right )}}{3} - \frac{76 a b c d^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{225} + 2 a b d^{2} x \operatorname{asin}{\left (c x \right )} + \frac{298 a b d^{2} \sqrt{- c^{2} x^{2} + 1}}{225 c} + \frac{b^{2} c^{4} d^{2} x^{5} \operatorname{asin}^{2}{\left (c x \right )}}{5} - \frac{2 b^{2} c^{4} d^{2} x^{5}}{125} + \frac{2 b^{2} c^{3} d^{2} x^{4} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{25} - \frac{2 b^{2} c^{2} d^{2} x^{3} \operatorname{asin}^{2}{\left (c x \right )}}{3} + \frac{76 b^{2} c^{2} d^{2} x^{3}}{675} - \frac{76 b^{2} c d^{2} x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{225} + b^{2} d^{2} x \operatorname{asin}^{2}{\left (c x \right )} - \frac{298 b^{2} d^{2} x}{225} + \frac{298 b^{2} d^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{225 c} & \text{for}\: c \neq 0 \\a^{2} d^{2} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**4*d**2*x**5/5 - 2*a**2*c**2*d**2*x**3/3 + a**2*d**2*x + 2*a*b*c**4*d**2*x**5*asin(c*x)/5 +
2*a*b*c**3*d**2*x**4*sqrt(-c**2*x**2 + 1)/25 - 4*a*b*c**2*d**2*x**3*asin(c*x)/3 - 76*a*b*c*d**2*x**2*sqrt(-c**
2*x**2 + 1)/225 + 2*a*b*d**2*x*asin(c*x) + 298*a*b*d**2*sqrt(-c**2*x**2 + 1)/(225*c) + b**2*c**4*d**2*x**5*asi
n(c*x)**2/5 - 2*b**2*c**4*d**2*x**5/125 + 2*b**2*c**3*d**2*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/25 - 2*b**2*c**
2*d**2*x**3*asin(c*x)**2/3 + 76*b**2*c**2*d**2*x**3/675 - 76*b**2*c*d**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/2
25 + b**2*d**2*x*asin(c*x)**2 - 298*b**2*d**2*x/225 + 298*b**2*d**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(225*c), Ne
(c, 0)), (a**2*d**2*x, True))

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Giac [A]  time = 1.44649, size = 505, normalized size = 2.31 \begin{align*} \frac{1}{5} \, a^{2} c^{4} d^{2} x^{5} - \frac{2}{3} \, a^{2} c^{2} d^{2} x^{3} + \frac{1}{5} \,{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2} x \arcsin \left (c x\right )^{2} + \frac{2}{5} \,{\left (c^{2} x^{2} - 1\right )}^{2} a b d^{2} x \arcsin \left (c x\right ) - \frac{4}{15} \,{\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} x \arcsin \left (c x\right )^{2} - \frac{2}{125} \,{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2} x - \frac{8}{15} \,{\left (c^{2} x^{2} - 1\right )} a b d^{2} x \arcsin \left (c x\right ) + \frac{8}{15} \, b^{2} d^{2} x \arcsin \left (c x\right )^{2} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{25 \, c} + \frac{272}{3375} \,{\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} x + \frac{16}{15} \, a b d^{2} x \arcsin \left (c x\right ) + \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} a b d^{2}}{25 \, c} + \frac{8 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} d^{2} \arcsin \left (c x\right )}{45 \, c} + a^{2} d^{2} x - \frac{4144}{3375} \, b^{2} d^{2} x + \frac{8 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b d^{2}}{45 \, c} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{15 \, c} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1} a b d^{2}}{15 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^2*c^4*d^2*x^5 - 2/3*a^2*c^2*d^2*x^3 + 1/5*(c^2*x^2 - 1)^2*b^2*d^2*x*arcsin(c*x)^2 + 2/5*(c^2*x^2 - 1)^2*
a*b*d^2*x*arcsin(c*x) - 4/15*(c^2*x^2 - 1)*b^2*d^2*x*arcsin(c*x)^2 - 2/125*(c^2*x^2 - 1)^2*b^2*d^2*x - 8/15*(c
^2*x^2 - 1)*a*b*d^2*x*arcsin(c*x) + 8/15*b^2*d^2*x*arcsin(c*x)^2 + 2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2
*d^2*arcsin(c*x)/c + 272/3375*(c^2*x^2 - 1)*b^2*d^2*x + 16/15*a*b*d^2*x*arcsin(c*x) + 2/25*(c^2*x^2 - 1)^2*sqr
t(-c^2*x^2 + 1)*a*b*d^2/c + 8/45*(-c^2*x^2 + 1)^(3/2)*b^2*d^2*arcsin(c*x)/c + a^2*d^2*x - 4144/3375*b^2*d^2*x
+ 8/45*(-c^2*x^2 + 1)^(3/2)*a*b*d^2/c + 16/15*sqrt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)/c + 16/15*sqrt(-c^2*x^2 +
1)*a*b*d^2/c