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

Optimal. Leaf size=268 \[ \frac{b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{192 c}+\frac{35 b d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{768 c}+\frac{35 b d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{512 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac{35 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{1024 c^2}+\frac{35 b^2 c^2 d^3 x^4}{3072}+\frac{b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac{7 b^2 d^3 \left (1-c^2 x^2\right )^3}{1152 c^2}-\frac{175 b^2 d^3 x^2}{3072} \]

[Out]

(-175*b^2*d^3*x^2)/3072 + (35*b^2*c^2*d^3*x^4)/3072 + (7*b^2*d^3*(1 - c^2*x^2)^3)/(1152*c^2) + (b^2*d^3*(1 - c
^2*x^2)^4)/(256*c^2) + (35*b*d^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(512*c) + (35*b*d^3*x*(1 - c^2*x^2)^
(3/2)*(a + b*ArcSin[c*x]))/(768*c) + (7*b*d^3*x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(192*c) + (b*d^3*x*(1
- c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(32*c) + (35*d^3*(a + b*ArcSin[c*x])^2)/(1024*c^2) - (d^3*(1 - c^2*x^2)
^4*(a + b*ArcSin[c*x])^2)/(8*c^2)

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Rubi [A]  time = 0.246654, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{192 c}+\frac{35 b d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{768 c}+\frac{35 b d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{512 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac{35 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{1024 c^2}+\frac{35 b^2 c^2 d^3 x^4}{3072}+\frac{b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac{7 b^2 d^3 \left (1-c^2 x^2\right )^3}{1152 c^2}-\frac{175 b^2 d^3 x^2}{3072} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-175*b^2*d^3*x^2)/3072 + (35*b^2*c^2*d^3*x^4)/3072 + (7*b^2*d^3*(1 - c^2*x^2)^3)/(1152*c^2) + (b^2*d^3*(1 - c
^2*x^2)^4)/(256*c^2) + (35*b*d^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(512*c) + (35*b*d^3*x*(1 - c^2*x^2)^
(3/2)*(a + b*ArcSin[c*x]))/(768*c) + (7*b*d^3*x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(192*c) + (b*d^3*x*(1
- c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(32*c) + (35*d^3*(a + b*ArcSin[c*x])^2)/(1024*c^2) - (d^3*(1 - c^2*x^2)
^4*(a + b*ArcSin[c*x])^2)/(8*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 )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac{\left (b d^3\right ) \int \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 c}\\ &=\frac{b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac{1}{32} \left (b^2 d^3\right ) \int x \left (1-c^2 x^2\right )^3 \, dx+\frac{\left (7 b d^3\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{32 c}\\ &=\frac{b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{192 c}+\frac{b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac{1}{192} \left (7 b^2 d^3\right ) \int x \left (1-c^2 x^2\right )^2 \, dx+\frac{\left (35 b d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{192 c}\\ &=\frac{7 b^2 d^3 \left (1-c^2 x^2\right )^3}{1152 c^2}+\frac{b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac{35 b d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{768 c}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{192 c}+\frac{b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac{1}{768} \left (35 b^2 d^3\right ) \int x \left (1-c^2 x^2\right ) \, dx+\frac{\left (35 b d^3\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{256 c}\\ &=\frac{7 b^2 d^3 \left (1-c^2 x^2\right )^3}{1152 c^2}+\frac{b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac{35 b d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{512 c}+\frac{35 b d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{768 c}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{192 c}+\frac{b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac{1}{768} \left (35 b^2 d^3\right ) \int \left (x-c^2 x^3\right ) \, dx-\frac{1}{512} \left (35 b^2 d^3\right ) \int x \, dx+\frac{\left (35 b d^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{512 c}\\ &=-\frac{175 b^2 d^3 x^2}{3072}+\frac{35 b^2 c^2 d^3 x^4}{3072}+\frac{7 b^2 d^3 \left (1-c^2 x^2\right )^3}{1152 c^2}+\frac{b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac{35 b d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{512 c}+\frac{35 b d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{768 c}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{192 c}+\frac{b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}+\frac{35 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{1024 c^2}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}\\ \end{align*}

Mathematica [A]  time = 0.336713, size = 257, normalized size = 0.96 \[ -\frac{d^3 \left (c x \left (1152 a^2 c x \left (c^6 x^6-4 c^4 x^4+6 c^2 x^2-4\right )+6 a b \sqrt{1-c^2 x^2} \left (48 c^6 x^6-200 c^4 x^4+326 c^2 x^2-279\right )+b^2 c x \left (-36 c^6 x^6+200 c^4 x^4-489 c^2 x^2+837\right )\right )+6 b \sin ^{-1}(c x) \left (3 a \left (128 c^8 x^8-512 c^6 x^6+768 c^4 x^4-512 c^2 x^2+93\right )+b c x \sqrt{1-c^2 x^2} \left (48 c^6 x^6-200 c^4 x^4+326 c^2 x^2-279\right )\right )+9 b^2 \left (128 c^8 x^8-512 c^6 x^6+768 c^4 x^4-512 c^2 x^2+93\right ) \sin ^{-1}(c x)^2\right )}{9216 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^3*(c*x*(b^2*c*x*(837 - 489*c^2*x^2 + 200*c^4*x^4 - 36*c^6*x^6) + 1152*a^2*c*x*(-4 + 6*c^2*x^2 - 4*c^4*x^4
+ c^6*x^6) + 6*a*b*Sqrt[1 - c^2*x^2]*(-279 + 326*c^2*x^2 - 200*c^4*x^4 + 48*c^6*x^6)) + 6*b*(b*c*x*Sqrt[1 - c^
2*x^2]*(-279 + 326*c^2*x^2 - 200*c^4*x^4 + 48*c^6*x^6) + 3*a*(93 - 512*c^2*x^2 + 768*c^4*x^4 - 512*c^6*x^6 + 1
28*c^8*x^8))*ArcSin[c*x] + 9*b^2*(93 - 512*c^2*x^2 + 768*c^4*x^4 - 512*c^6*x^6 + 128*c^8*x^8)*ArcSin[c*x]^2))/
(9216*c^2)

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Maple [A]  time = 0.041, size = 358, normalized size = 1.3 \begin{align*}{\frac{1}{{c}^{2}} \left ( -{d}^{3}{a}^{2} \left ({\frac{{c}^{8}{x}^{8}}{8}}-{\frac{{c}^{6}{x}^{6}}{2}}+{\frac{3\,{c}^{4}{x}^{4}}{4}}-{\frac{{c}^{2}{x}^{2}}{2}} \right ) -{d}^{3}{b}^{2} \left ({\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}-1 \right ) ^{4}}{8}}-{\frac{\arcsin \left ( cx \right ) }{1536} \left ( -48\,{c}^{7}{x}^{7}\sqrt{-{c}^{2}{x}^{2}+1}+200\,{c}^{5}{x}^{5}\sqrt{-{c}^{2}{x}^{2}+1}-326\,{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}+279\,cx\sqrt{-{c}^{2}{x}^{2}+1}+105\,\arcsin \left ( cx \right ) \right ) }+{\frac{35\, \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{1024}}-{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) ^{4}}{256}}+{\frac{7\, \left ({c}^{2}{x}^{2}-1 \right ) ^{3}}{1152}}-{\frac{35\, \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}{3072}}+{\frac{35\,{c}^{2}{x}^{2}}{1024}}-{\frac{35}{1024}} \right ) -2\,{d}^{3}ab \left ( 1/8\,\arcsin \left ( cx \right ){c}^{8}{x}^{8}-1/2\,\arcsin \left ( cx \right ){c}^{6}{x}^{6}+3/4\,{c}^{4}{x}^{4}\arcsin \left ( cx \right ) -1/2\,{c}^{2}{x}^{2}\arcsin \left ( cx \right ) +{\frac{{c}^{7}{x}^{7}\sqrt{-{c}^{2}{x}^{2}+1}}{64}}-{\frac{25\,{c}^{5}{x}^{5}\sqrt{-{c}^{2}{x}^{2}+1}}{384}}+{\frac{163\,{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}}{1536}}-{\frac{93\,cx\sqrt{-{c}^{2}{x}^{2}+1}}{1024}}+{\frac{93\,\arcsin \left ( cx \right ) }{1024}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(-d^3*a^2*(1/8*c^8*x^8-1/2*c^6*x^6+3/4*c^4*x^4-1/2*c^2*x^2)-d^3*b^2*(1/8*arcsin(c*x)^2*(c^2*x^2-1)^4-1/1
536*arcsin(c*x)*(-48*c^7*x^7*(-c^2*x^2+1)^(1/2)+200*c^5*x^5*(-c^2*x^2+1)^(1/2)-326*c^3*x^3*(-c^2*x^2+1)^(1/2)+
279*c*x*(-c^2*x^2+1)^(1/2)+105*arcsin(c*x))+35/1024*arcsin(c*x)^2-1/256*(c^2*x^2-1)^4+7/1152*(c^2*x^2-1)^3-35/
3072*(c^2*x^2-1)^2+35/1024*c^2*x^2-35/1024)-2*d^3*a*b*(1/8*arcsin(c*x)*c^8*x^8-1/2*arcsin(c*x)*c^6*x^6+3/4*c^4
*x^4*arcsin(c*x)-1/2*c^2*x^2*arcsin(c*x)+1/64*c^7*x^7*(-c^2*x^2+1)^(1/2)-25/384*c^5*x^5*(-c^2*x^2+1)^(1/2)+163
/1536*c^3*x^3*(-c^2*x^2+1)^(1/2)-93/1024*c*x*(-c^2*x^2+1)^(1/2)+93/1024*arcsin(c*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{8} \, a^{2} c^{6} d^{3} x^{8} + \frac{1}{2} \, a^{2} c^{4} d^{3} x^{6} - \frac{1}{1536} \,{\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 )} a b c^{6} d^{3} - \frac{3}{4} \, a^{2} c^{2} d^{3} x^{4} + \frac{1}{48} \,{\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^{3} - \frac{3}{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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} a b c^{2} d^{3} + \frac{1}{2} \, a^{2} d^{3} 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^{3} - \frac{1}{8} \,{\left (b^{2} c^{6} d^{3} x^{8} - 4 \, b^{2} c^{4} d^{3} x^{6} + 6 \, b^{2} c^{2} d^{3} x^{4} - 4 \, b^{2} d^{3} x^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} - \int \frac{{\left (b^{2} c^{7} d^{3} x^{8} - 4 \, b^{2} c^{5} d^{3} x^{6} + 6 \, b^{2} c^{3} d^{3} x^{4} - 4 \, b^{2} c d^{3} x^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{4 \,{\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)^3*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

-1/8*a^2*c^6*d^3*x^8 + 1/2*a^2*c^4*d^3*x^6 - 1/1536*(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)*a*b*c^6*d^3 - 3/4*a^2*c^2*d^3*x^4 + 1/48*(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)*a*b*c^4*d^3 - 3/16*(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*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*a*b*c^2*d^3 + 1/2*a^2*d^3*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^3 - 1/8*(b^2*c^6*d^3*x^8 - 4*b^2*
c^4*d^3*x^6 + 6*b^2*c^2*d^3*x^4 - 4*b^2*d^3*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 - integrate(1/4*
(b^2*c^7*d^3*x^8 - 4*b^2*c^5*d^3*x^6 + 6*b^2*c^3*d^3*x^4 - 4*b^2*c*d^3*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arcta
n2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*x^2 - 1), x)

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Fricas [A]  time = 2.02357, size = 799, normalized size = 2.98 \begin{align*} -\frac{36 \,{\left (32 \, a^{2} - b^{2}\right )} c^{8} d^{3} x^{8} - 8 \,{\left (576 \, a^{2} - 25 \, b^{2}\right )} c^{6} d^{3} x^{6} + 3 \,{\left (2304 \, a^{2} - 163 \, b^{2}\right )} c^{4} d^{3} x^{4} - 9 \,{\left (512 \, a^{2} - 93 \, b^{2}\right )} c^{2} d^{3} x^{2} + 9 \,{\left (128 \, b^{2} c^{8} d^{3} x^{8} - 512 \, b^{2} c^{6} d^{3} x^{6} + 768 \, b^{2} c^{4} d^{3} x^{4} - 512 \, b^{2} c^{2} d^{3} x^{2} + 93 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )^{2} + 18 \,{\left (128 \, a b c^{8} d^{3} x^{8} - 512 \, a b c^{6} d^{3} x^{6} + 768 \, a b c^{4} d^{3} x^{4} - 512 \, a b c^{2} d^{3} x^{2} + 93 \, a b d^{3}\right )} \arcsin \left (c x\right ) + 6 \,{\left (48 \, a b c^{7} d^{3} x^{7} - 200 \, a b c^{5} d^{3} x^{5} + 326 \, a b c^{3} d^{3} x^{3} - 279 \, a b c d^{3} x +{\left (48 \, b^{2} c^{7} d^{3} x^{7} - 200 \, b^{2} c^{5} d^{3} x^{5} + 326 \, b^{2} c^{3} d^{3} x^{3} - 279 \, b^{2} c d^{3} x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{9216 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9216*(36*(32*a^2 - b^2)*c^8*d^3*x^8 - 8*(576*a^2 - 25*b^2)*c^6*d^3*x^6 + 3*(2304*a^2 - 163*b^2)*c^4*d^3*x^4
- 9*(512*a^2 - 93*b^2)*c^2*d^3*x^2 + 9*(128*b^2*c^8*d^3*x^8 - 512*b^2*c^6*d^3*x^6 + 768*b^2*c^4*d^3*x^4 - 512
*b^2*c^2*d^3*x^2 + 93*b^2*d^3)*arcsin(c*x)^2 + 18*(128*a*b*c^8*d^3*x^8 - 512*a*b*c^6*d^3*x^6 + 768*a*b*c^4*d^3
*x^4 - 512*a*b*c^2*d^3*x^2 + 93*a*b*d^3)*arcsin(c*x) + 6*(48*a*b*c^7*d^3*x^7 - 200*a*b*c^5*d^3*x^5 + 326*a*b*c
^3*d^3*x^3 - 279*a*b*c*d^3*x + (48*b^2*c^7*d^3*x^7 - 200*b^2*c^5*d^3*x^5 + 326*b^2*c^3*d^3*x^3 - 279*b^2*c*d^3
*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^2

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Sympy [A]  time = 38.3857, size = 573, normalized size = 2.14 \begin{align*} \begin{cases} - \frac{a^{2} c^{6} d^{3} x^{8}}{8} + \frac{a^{2} c^{4} d^{3} x^{6}}{2} - \frac{3 a^{2} c^{2} d^{3} x^{4}}{4} + \frac{a^{2} d^{3} x^{2}}{2} - \frac{a b c^{6} d^{3} x^{8} \operatorname{asin}{\left (c x \right )}}{4} - \frac{a b c^{5} d^{3} x^{7} \sqrt{- c^{2} x^{2} + 1}}{32} + a b c^{4} d^{3} x^{6} \operatorname{asin}{\left (c x \right )} + \frac{25 a b c^{3} d^{3} x^{5} \sqrt{- c^{2} x^{2} + 1}}{192} - \frac{3 a b c^{2} d^{3} x^{4} \operatorname{asin}{\left (c x \right )}}{2} - \frac{163 a b c d^{3} x^{3} \sqrt{- c^{2} x^{2} + 1}}{768} + a b d^{3} x^{2} \operatorname{asin}{\left (c x \right )} + \frac{93 a b d^{3} x \sqrt{- c^{2} x^{2} + 1}}{512 c} - \frac{93 a b d^{3} \operatorname{asin}{\left (c x \right )}}{512 c^{2}} - \frac{b^{2} c^{6} d^{3} x^{8} \operatorname{asin}^{2}{\left (c x \right )}}{8} + \frac{b^{2} c^{6} d^{3} x^{8}}{256} - \frac{b^{2} c^{5} d^{3} x^{7} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{32} + \frac{b^{2} c^{4} d^{3} x^{6} \operatorname{asin}^{2}{\left (c x \right )}}{2} - \frac{25 b^{2} c^{4} d^{3} x^{6}}{1152} + \frac{25 b^{2} c^{3} d^{3} x^{5} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{192} - \frac{3 b^{2} c^{2} d^{3} x^{4} \operatorname{asin}^{2}{\left (c x \right )}}{4} + \frac{163 b^{2} c^{2} d^{3} x^{4}}{3072} - \frac{163 b^{2} c d^{3} x^{3} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{768} + \frac{b^{2} d^{3} x^{2} \operatorname{asin}^{2}{\left (c x \right )}}{2} - \frac{93 b^{2} d^{3} x^{2}}{1024} + \frac{93 b^{2} d^{3} x \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{512 c} - \frac{93 b^{2} d^{3} \operatorname{asin}^{2}{\left (c x \right )}}{1024 c^{2}} & \text{for}\: c \neq 0 \\\frac{a^{2} d^{3} 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)**3*(a+b*asin(c*x))**2,x)

[Out]

Piecewise((-a**2*c**6*d**3*x**8/8 + a**2*c**4*d**3*x**6/2 - 3*a**2*c**2*d**3*x**4/4 + a**2*d**3*x**2/2 - a*b*c
**6*d**3*x**8*asin(c*x)/4 - a*b*c**5*d**3*x**7*sqrt(-c**2*x**2 + 1)/32 + a*b*c**4*d**3*x**6*asin(c*x) + 25*a*b
*c**3*d**3*x**5*sqrt(-c**2*x**2 + 1)/192 - 3*a*b*c**2*d**3*x**4*asin(c*x)/2 - 163*a*b*c*d**3*x**3*sqrt(-c**2*x
**2 + 1)/768 + a*b*d**3*x**2*asin(c*x) + 93*a*b*d**3*x*sqrt(-c**2*x**2 + 1)/(512*c) - 93*a*b*d**3*asin(c*x)/(5
12*c**2) - b**2*c**6*d**3*x**8*asin(c*x)**2/8 + b**2*c**6*d**3*x**8/256 - b**2*c**5*d**3*x**7*sqrt(-c**2*x**2
+ 1)*asin(c*x)/32 + b**2*c**4*d**3*x**6*asin(c*x)**2/2 - 25*b**2*c**4*d**3*x**6/1152 + 25*b**2*c**3*d**3*x**5*
sqrt(-c**2*x**2 + 1)*asin(c*x)/192 - 3*b**2*c**2*d**3*x**4*asin(c*x)**2/4 + 163*b**2*c**2*d**3*x**4/3072 - 163
*b**2*c*d**3*x**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/768 + b**2*d**3*x**2*asin(c*x)**2/2 - 93*b**2*d**3*x**2/1024
+ 93*b**2*d**3*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(512*c) - 93*b**2*d**3*asin(c*x)**2/(1024*c**2), Ne(c, 0)), (a
**2*d**3*x**2/2, True))

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Giac [A]  time = 1.47056, size = 610, normalized size = 2.28 \begin{align*} -\frac{{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{32 \, c} - \frac{{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3} \arcsin \left (c x\right )^{2}}{8 \, c^{2}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} a b d^{3} x}{32 \, c} + \frac{7 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{192 \, c} - \frac{{\left (c^{2} x^{2} - 1\right )}^{4} a b d^{3} \arcsin \left (c x\right )}{4 \, c^{2}} + \frac{7 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} a b d^{3} x}{192 \, c} + \frac{35 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} d^{3} x \arcsin \left (c x\right )}{768 \, c} - \frac{{\left (c^{2} x^{2} - 1\right )}^{4} a^{2} d^{3}}{8 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3}}{256 \, c^{2}} + \frac{35 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b d^{3} x}{768 \, c} + \frac{35 \, \sqrt{-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{512 \, c} - \frac{7 \,{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3}}{1152 \, c^{2}} + \frac{35 \, \sqrt{-c^{2} x^{2} + 1} a b d^{3} x}{512 \, c} + \frac{35 \,{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3}}{3072 \, c^{2}} + \frac{35 \, b^{2} d^{3} \arcsin \left (c x\right )^{2}}{1024 \, c^{2}} - \frac{35 \,{\left (c^{2} x^{2} - 1\right )} b^{2} d^{3}}{1024 \, c^{2}} + \frac{35 \, a b d^{3} \arcsin \left (c x\right )}{512 \, c^{2}} - \frac{7175 \, b^{2} d^{3}}{294912 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x)/c - 1/8*(c^2*x^2 - 1)^4*b^2*d^3*arcsin(c*x)^2/c
^2 - 1/32*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a*b*d^3*x/c + 7/192*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*
arcsin(c*x)/c - 1/4*(c^2*x^2 - 1)^4*a*b*d^3*arcsin(c*x)/c^2 + 7/192*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d^3
*x/c + 35/768*(-c^2*x^2 + 1)^(3/2)*b^2*d^3*x*arcsin(c*x)/c - 1/8*(c^2*x^2 - 1)^4*a^2*d^3/c^2 + 1/256*(c^2*x^2
- 1)^4*b^2*d^3/c^2 + 35/768*(-c^2*x^2 + 1)^(3/2)*a*b*d^3*x/c + 35/512*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x)
/c - 7/1152*(c^2*x^2 - 1)^3*b^2*d^3/c^2 + 35/512*sqrt(-c^2*x^2 + 1)*a*b*d^3*x/c + 35/3072*(c^2*x^2 - 1)^2*b^2*
d^3/c^2 + 35/1024*b^2*d^3*arcsin(c*x)^2/c^2 - 35/1024*(c^2*x^2 - 1)*b^2*d^3/c^2 + 35/512*a*b*d^3*arcsin(c*x)/c
^2 - 7175/294912*b^2*d^3/c^2