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

Optimal. Leaf size=302 \[ -\frac{1}{32} b c d^2 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{25}{576} b c d^2 x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{12} d^2 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{73 b d^2 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2304 c}+\frac{73 b d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{1536 c^3}-\frac{73 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3072 c^4}+\frac{1}{24} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{256} b^2 c^4 d^2 x^8+\frac{43 b^2 c^2 d^2 x^6}{3456}-\frac{73 b^2 d^2 x^2}{3072 c^2}-\frac{73 b^2 d^2 x^4}{9216} \]

[Out]

(-73*b^2*d^2*x^2)/(3072*c^2) - (73*b^2*d^2*x^4)/9216 + (43*b^2*c^2*d^2*x^6)/3456 - (b^2*c^4*d^2*x^8)/256 + (73
*b*d^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(1536*c^3) + (73*b*d^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x
]))/(2304*c) - (25*b*c*d^2*x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/576 - (b*c*d^2*x^5*(1 - c^2*x^2)^(3/2)*(
a + b*ArcSin[c*x]))/32 - (73*d^2*(a + b*ArcSin[c*x])^2)/(3072*c^4) + (d^2*x^4*(a + b*ArcSin[c*x])^2)/24 + (d^2
*x^4*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/12 + (d^2*x^4*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/8

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Rubi [A]  time = 1.00818, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {4699, 4627, 4707, 4641, 30, 4697, 14} \[ -\frac{1}{32} b c d^2 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{25}{576} b c d^2 x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{12} d^2 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{73 b d^2 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2304 c}+\frac{73 b d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{1536 c^3}-\frac{73 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3072 c^4}+\frac{1}{24} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{256} b^2 c^4 d^2 x^8+\frac{43 b^2 c^2 d^2 x^6}{3456}-\frac{73 b^2 d^2 x^2}{3072 c^2}-\frac{73 b^2 d^2 x^4}{9216} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-73*b^2*d^2*x^2)/(3072*c^2) - (73*b^2*d^2*x^4)/9216 + (43*b^2*c^2*d^2*x^6)/3456 - (b^2*c^4*d^2*x^8)/256 + (73
*b*d^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(1536*c^3) + (73*b*d^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x
]))/(2304*c) - (25*b*c*d^2*x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/576 - (b*c*d^2*x^5*(1 - c^2*x^2)^(3/2)*(
a + b*ArcSin[c*x]))/32 - (73*d^2*(a + b*ArcSin[c*x])^2)/(3072*c^4) + (d^2*x^4*(a + b*ArcSin[c*x])^2)/24 + (d^2
*x^4*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/12 + (d^2*x^4*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/8

Rule 4699

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

Rule 4627

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

Rule 4707

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

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 4697

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

Rubi steps

\begin{align*} \int x^3 \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d \int x^3 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{4} \left (b c d^2\right ) \int x^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac{1}{32} b c d^2 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{12} d^2 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 \int x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{32} \left (3 b c d^2\right ) \int x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac{1}{6} \left (b c d^2\right ) \int x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac{1}{32} \left (b^2 c^2 d^2\right ) \int x^5 \left (1-c^2 x^2\right ) \, dx\\ &=-\frac{25}{576} b c d^2 x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{32} b c d^2 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{24} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{12} d^2 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{64} \left (b c d^2\right ) \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{36} \left (b c d^2\right ) \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{12} \left (b c d^2\right ) \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx+\frac{1}{64} \left (b^2 c^2 d^2\right ) \int x^5 \, dx+\frac{1}{36} \left (b^2 c^2 d^2\right ) \int x^5 \, dx+\frac{1}{32} \left (b^2 c^2 d^2\right ) \int \left (x^5-c^2 x^7\right ) \, dx\\ &=\frac{43 b^2 c^2 d^2 x^6}{3456}-\frac{1}{256} b^2 c^4 d^2 x^8+\frac{73 b d^2 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2304 c}-\frac{25}{576} b c d^2 x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{32} b c d^2 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{24} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{12} d^2 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{256} \left (b^2 d^2\right ) \int x^3 \, dx-\frac{1}{144} \left (b^2 d^2\right ) \int x^3 \, dx-\frac{1}{48} \left (b^2 d^2\right ) \int x^3 \, dx-\frac{\left (3 b d^2\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{256 c}-\frac{\left (b d^2\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{48 c}-\frac{\left (b d^2\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{16 c}\\ &=-\frac{73 b^2 d^2 x^4}{9216}+\frac{43 b^2 c^2 d^2 x^6}{3456}-\frac{1}{256} b^2 c^4 d^2 x^8+\frac{73 b d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{1536 c^3}+\frac{73 b d^2 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2304 c}-\frac{25}{576} b c d^2 x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{32} b c d^2 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{24} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{12} d^2 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (3 b d^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{512 c^3}-\frac{\left (b d^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{96 c^3}-\frac{\left (b d^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{32 c^3}-\frac{\left (3 b^2 d^2\right ) \int x \, dx}{512 c^2}-\frac{\left (b^2 d^2\right ) \int x \, dx}{96 c^2}-\frac{\left (b^2 d^2\right ) \int x \, dx}{32 c^2}\\ &=-\frac{73 b^2 d^2 x^2}{3072 c^2}-\frac{73 b^2 d^2 x^4}{9216}+\frac{43 b^2 c^2 d^2 x^6}{3456}-\frac{1}{256} b^2 c^4 d^2 x^8+\frac{73 b d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{1536 c^3}+\frac{73 b d^2 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2304 c}-\frac{25}{576} b c d^2 x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{32} b c d^2 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{73 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3072 c^4}+\frac{1}{24} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{12} d^2 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.238716, size = 239, normalized size = 0.79 \[ \frac{d^2 \left (c x \left (1152 a^2 c^3 x^3 \left (3 c^4 x^4-8 c^2 x^2+6\right )+6 a b \sqrt{1-c^2 x^2} \left (144 c^6 x^6-344 c^4 x^4+146 c^2 x^2+219\right )-b^2 c x \left (108 c^6 x^6-344 c^4 x^4+219 c^2 x^2+657\right )\right )+6 b \sin ^{-1}(c x) \left (3 a \left (384 c^8 x^8-1024 c^6 x^6+768 c^4 x^4-73\right )+b c x \sqrt{1-c^2 x^2} \left (144 c^6 x^6-344 c^4 x^4+146 c^2 x^2+219\right )\right )+9 b^2 \left (384 c^8 x^8-1024 c^6 x^6+768 c^4 x^4-73\right ) \sin ^{-1}(c x)^2\right )}{27648 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(c*x*(1152*a^2*c^3*x^3*(6 - 8*c^2*x^2 + 3*c^4*x^4) - b^2*c*x*(657 + 219*c^2*x^2 - 344*c^4*x^4 + 108*c^6*x
^6) + 6*a*b*Sqrt[1 - c^2*x^2]*(219 + 146*c^2*x^2 - 344*c^4*x^4 + 144*c^6*x^6)) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*
(219 + 146*c^2*x^2 - 344*c^4*x^4 + 144*c^6*x^6) + 3*a*(-73 + 768*c^4*x^4 - 1024*c^6*x^6 + 384*c^8*x^8))*ArcSin
[c*x] + 9*b^2*(-73 + 768*c^4*x^4 - 1024*c^6*x^6 + 384*c^8*x^8)*ArcSin[c*x]^2))/(27648*c^4)

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Maple [A]  time = 0.139, size = 424, normalized size = 1.4 \begin{align*}{\frac{1}{{c}^{4}} \left ({d}^{2}{a}^{2} \left ({\frac{{c}^{8}{x}^{8}}{8}}-{\frac{{c}^{6}{x}^{6}}{3}}+{\frac{{c}^{4}{x}^{4}}{4}} \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{55\, \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{3072}}-{\frac{11\, \left ({c}^{2}{x}^{2}-1 \right ) ^{3}}{3456}}+{\frac{55\, \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}{9216}}-{\frac{55\,{c}^{2}{x}^{2}}{3072}}+{\frac{55}{3072}}+{\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{ \left ({c}^{2}{x}^{2}-1 \right ) ^{4}}{256}} \right ) +2\,{d}^{2}ab \left ( 1/8\,\arcsin \left ( cx \right ){c}^{8}{x}^{8}-1/3\,\arcsin \left ( cx \right ){c}^{6}{x}^{6}+1/4\,{c}^{4}{x}^{4}\arcsin \left ( cx \right ) +{\frac{{c}^{7}{x}^{7}\sqrt{-{c}^{2}{x}^{2}+1}}{64}}-{\frac{43\,{c}^{5}{x}^{5}\sqrt{-{c}^{2}{x}^{2}+1}}{1152}}+{\frac{73\,{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}}{4608}}+{\frac{73\,cx\sqrt{-{c}^{2}{x}^{2}+1}}{3072}}-{\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))^2,x)

[Out]

1/c^4*(d^2*a^2*(1/8*c^8*x^8-1/3*c^6*x^6+1/4*c^4*x^4)+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))-55/307
2*arcsin(c*x)^2-11/3456*(c^2*x^2-1)^3+55/9216*(c^2*x^2-1)^2-55/3072*c^2*x^2+55/3072+1/8*arcsin(c*x)^2*(c^2*x^2
-1)^4-1/1536*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))-1/256*(c^2*x^2-1)^4)+2*d^2*a*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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{8} \, a^{2} c^{4} d^{2} x^{8} - \frac{1}{3} \, a^{2} c^{2} d^{2} 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^{4} d^{2} + \frac{1}{4} \, a^{2} d^{2} x^{4} - \frac{1}{72} \,{\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^{2} 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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} a b d^{2} + \frac{1}{24} \,{\left (3 \, b^{2} c^{4} d^{2} x^{8} - 8 \, b^{2} c^{2} d^{2} x^{6} + 6 \, b^{2} d^{2} x^{4}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + \int \frac{{\left (3 \, b^{2} c^{5} d^{2} x^{8} - 8 \, b^{2} c^{3} d^{2} x^{6} + 6 \, b^{2} c d^{2} x^{4}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{12 \,{\left (c^{2} x^{2} - 1\right )}}\,{d x} \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))^2,x, algorithm="maxima")

[Out]

1/8*a^2*c^4*d^2*x^8 - 1/3*a^2*c^2*d^2*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/s
qrt(c^2))/(sqrt(c^2)*c^8))*c)*a*b*c^4*d^2 + 1/4*a^2*d^2*x^4 - 1/72*(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^2*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*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*a*b*d^2 + 1/24*(3*b^2*c^4*d^2*x^8 - 8*b^2*c^2*d^2*x^6 + 6*b^2*d
^2*x^4)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + integrate(1/12*(3*b^2*c^5*d^2*x^8 - 8*b^2*c^3*d^2*x^6 +
6*b^2*c*d^2*x^4)*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 = 1.88243, size = 729, normalized size = 2.41 \begin{align*} \frac{108 \,{\left (32 \, a^{2} - b^{2}\right )} c^{8} d^{2} x^{8} - 8 \,{\left (1152 \, a^{2} - 43 \, b^{2}\right )} c^{6} d^{2} x^{6} + 3 \,{\left (2304 \, a^{2} - 73 \, b^{2}\right )} c^{4} d^{2} x^{4} - 657 \, b^{2} c^{2} d^{2} x^{2} + 9 \,{\left (384 \, b^{2} c^{8} d^{2} x^{8} - 1024 \, b^{2} c^{6} d^{2} x^{6} + 768 \, b^{2} c^{4} d^{2} x^{4} - 73 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 18 \,{\left (384 \, a b c^{8} d^{2} x^{8} - 1024 \, a b c^{6} d^{2} x^{6} + 768 \, a b c^{4} d^{2} x^{4} - 73 \, a b d^{2}\right )} \arcsin \left (c x\right ) + 6 \,{\left (144 \, a b c^{7} d^{2} x^{7} - 344 \, a b c^{5} d^{2} x^{5} + 146 \, a b c^{3} d^{2} x^{3} + 219 \, a b c d^{2} x +{\left (144 \, b^{2} c^{7} d^{2} x^{7} - 344 \, b^{2} c^{5} d^{2} x^{5} + 146 \, b^{2} c^{3} d^{2} x^{3} + 219 \, b^{2} c d^{2} x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{27648 \, 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))^2,x, algorithm="fricas")

[Out]

1/27648*(108*(32*a^2 - b^2)*c^8*d^2*x^8 - 8*(1152*a^2 - 43*b^2)*c^6*d^2*x^6 + 3*(2304*a^2 - 73*b^2)*c^4*d^2*x^
4 - 657*b^2*c^2*d^2*x^2 + 9*(384*b^2*c^8*d^2*x^8 - 1024*b^2*c^6*d^2*x^6 + 768*b^2*c^4*d^2*x^4 - 73*b^2*d^2)*ar
csin(c*x)^2 + 18*(384*a*b*c^8*d^2*x^8 - 1024*a*b*c^6*d^2*x^6 + 768*a*b*c^4*d^2*x^4 - 73*a*b*d^2)*arcsin(c*x) +
6*(144*a*b*c^7*d^2*x^7 - 344*a*b*c^5*d^2*x^5 + 146*a*b*c^3*d^2*x^3 + 219*a*b*c*d^2*x + (144*b^2*c^7*d^2*x^7 -
344*b^2*c^5*d^2*x^5 + 146*b^2*c^3*d^2*x^3 + 219*b^2*c*d^2*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^4

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Sympy [A]  time = 38.1413, size = 515, normalized size = 1.71 \begin{align*} \begin{cases} \frac{a^{2} c^{4} d^{2} x^{8}}{8} - \frac{a^{2} c^{2} d^{2} x^{6}}{3} + \frac{a^{2} d^{2} x^{4}}{4} + \frac{a b c^{4} d^{2} x^{8} \operatorname{asin}{\left (c x \right )}}{4} + \frac{a b c^{3} d^{2} x^{7} \sqrt{- c^{2} x^{2} + 1}}{32} - \frac{2 a b c^{2} d^{2} x^{6} \operatorname{asin}{\left (c x \right )}}{3} - \frac{43 a b c d^{2} x^{5} \sqrt{- c^{2} x^{2} + 1}}{576} + \frac{a b d^{2} x^{4} \operatorname{asin}{\left (c x \right )}}{2} + \frac{73 a b d^{2} x^{3} \sqrt{- c^{2} x^{2} + 1}}{2304 c} + \frac{73 a b d^{2} x \sqrt{- c^{2} x^{2} + 1}}{1536 c^{3}} - \frac{73 a b d^{2} \operatorname{asin}{\left (c x \right )}}{1536 c^{4}} + \frac{b^{2} c^{4} d^{2} x^{8} \operatorname{asin}^{2}{\left (c x \right )}}{8} - \frac{b^{2} c^{4} d^{2} x^{8}}{256} + \frac{b^{2} c^{3} d^{2} x^{7} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{32} - \frac{b^{2} c^{2} d^{2} x^{6} \operatorname{asin}^{2}{\left (c x \right )}}{3} + \frac{43 b^{2} c^{2} d^{2} x^{6}}{3456} - \frac{43 b^{2} c d^{2} x^{5} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{576} + \frac{b^{2} d^{2} x^{4} \operatorname{asin}^{2}{\left (c x \right )}}{4} - \frac{73 b^{2} d^{2} x^{4}}{9216} + \frac{73 b^{2} d^{2} x^{3} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{2304 c} - \frac{73 b^{2} d^{2} x^{2}}{3072 c^{2}} + \frac{73 b^{2} d^{2} x \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{1536 c^{3}} - \frac{73 b^{2} d^{2} \operatorname{asin}^{2}{\left (c x \right )}}{3072 c^{4}} & \text{for}\: c \neq 0 \\\frac{a^{2} 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))**2,x)

[Out]

Piecewise((a**2*c**4*d**2*x**8/8 - a**2*c**2*d**2*x**6/3 + a**2*d**2*x**4/4 + a*b*c**4*d**2*x**8*asin(c*x)/4 +
a*b*c**3*d**2*x**7*sqrt(-c**2*x**2 + 1)/32 - 2*a*b*c**2*d**2*x**6*asin(c*x)/3 - 43*a*b*c*d**2*x**5*sqrt(-c**2
*x**2 + 1)/576 + a*b*d**2*x**4*asin(c*x)/2 + 73*a*b*d**2*x**3*sqrt(-c**2*x**2 + 1)/(2304*c) + 73*a*b*d**2*x*sq
rt(-c**2*x**2 + 1)/(1536*c**3) - 73*a*b*d**2*asin(c*x)/(1536*c**4) + b**2*c**4*d**2*x**8*asin(c*x)**2/8 - b**2
*c**4*d**2*x**8/256 + b**2*c**3*d**2*x**7*sqrt(-c**2*x**2 + 1)*asin(c*x)/32 - b**2*c**2*d**2*x**6*asin(c*x)**2
/3 + 43*b**2*c**2*d**2*x**6/3456 - 43*b**2*c*d**2*x**5*sqrt(-c**2*x**2 + 1)*asin(c*x)/576 + b**2*d**2*x**4*asi
n(c*x)**2/4 - 73*b**2*d**2*x**4/9216 + 73*b**2*d**2*x**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(2304*c) - 73*b**2*d**
2*x**2/(3072*c**2) + 73*b**2*d**2*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(1536*c**3) - 73*b**2*d**2*asin(c*x)**2/(30
72*c**4), Ne(c, 0)), (a**2*d**2*x**4/4, True))

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Giac [A]  time = 1.57187, size = 711, normalized size = 2.35 \begin{align*} \frac{{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right )}{32 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{2} \arcsin \left (c x\right )^{2}}{8 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} a b d^{2} x}{32 \, c^{3}} + \frac{11 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right )}{576 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{4} a b d^{2} \arcsin \left (c x\right )}{4 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2} \arcsin \left (c x\right )^{2}}{6 \, c^{4}} + \frac{11 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} a b d^{2} x}{576 \, c^{3}} + \frac{55 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} d^{2} x \arcsin \left (c x\right )}{2304 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{4} a^{2} d^{2}}{8 \, c^{4}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{2}}{256 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} a b d^{2} \arcsin \left (c x\right )}{3 \, c^{4}} + \frac{55 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b d^{2} x}{2304 \, c^{3}} + \frac{55 \, \sqrt{-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right )}{1536 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} a^{2} d^{2}}{6 \, c^{4}} - \frac{11 \,{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2}}{3456 \, c^{4}} + \frac{55 \, \sqrt{-c^{2} x^{2} + 1} a b d^{2} x}{1536 \, c^{3}} + \frac{55 \,{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2}}{9216 \, c^{4}} + \frac{55 \, b^{2} d^{2} \arcsin \left (c x\right )^{2}}{3072 \, c^{4}} - \frac{55 \,{\left (c^{2} x^{2} - 1\right )} b^{2} d^{2}}{3072 \, c^{4}} + \frac{55 \, a b d^{2} \arcsin \left (c x\right )}{1536 \, c^{4}} - \frac{9835 \, b^{2} d^{2}}{884736 \, 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))^2,x, algorithm="giac")

[Out]

1/32*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*d^2*x*arcsin(c*x)/c^3 + 1/8*(c^2*x^2 - 1)^4*b^2*d^2*arcsin(c*x)^2/
c^4 + 1/32*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a*b*d^2*x/c^3 + 11/576*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^
2*x*arcsin(c*x)/c^3 + 1/4*(c^2*x^2 - 1)^4*a*b*d^2*arcsin(c*x)/c^4 + 1/6*(c^2*x^2 - 1)^3*b^2*d^2*arcsin(c*x)^2/
c^4 + 11/576*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d^2*x/c^3 + 55/2304*(-c^2*x^2 + 1)^(3/2)*b^2*d^2*x*arcsin(
c*x)/c^3 + 1/8*(c^2*x^2 - 1)^4*a^2*d^2/c^4 - 1/256*(c^2*x^2 - 1)^4*b^2*d^2/c^4 + 1/3*(c^2*x^2 - 1)^3*a*b*d^2*a
rcsin(c*x)/c^4 + 55/2304*(-c^2*x^2 + 1)^(3/2)*a*b*d^2*x/c^3 + 55/1536*sqrt(-c^2*x^2 + 1)*b^2*d^2*x*arcsin(c*x)
/c^3 + 1/6*(c^2*x^2 - 1)^3*a^2*d^2/c^4 - 11/3456*(c^2*x^2 - 1)^3*b^2*d^2/c^4 + 55/1536*sqrt(-c^2*x^2 + 1)*a*b*
d^2*x/c^3 + 55/9216*(c^2*x^2 - 1)^2*b^2*d^2/c^4 + 55/3072*b^2*d^2*arcsin(c*x)^2/c^4 - 55/3072*(c^2*x^2 - 1)*b^
2*d^2/c^4 + 55/1536*a*b*d^2*arcsin(c*x)/c^4 - 9835/884736*b^2*d^2/c^4