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

Optimal. Leaf size=310 \[ \frac{1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4}{35} d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{16 b d^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{315 c}-\frac{2 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^3}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{175 c^3}+\frac{8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 c^3}+\frac{32 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^3}+\frac{8}{105} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2}{343} b^2 c^4 d^2 x^7+\frac{136 b^2 c^2 d^2 x^5}{6125}-\frac{1636 b^2 d^2 x}{11025 c^2}-\frac{818 b^2 d^2 x^3}{33075} \]

[Out]

(-1636*b^2*d^2*x)/(11025*c^2) - (818*b^2*d^2*x^3)/33075 + (136*b^2*c^2*d^2*x^5)/6125 - (2*b^2*c^4*d^2*x^7)/343
+ (32*b*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(315*c^3) + (16*b*d^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[
c*x]))/(315*c) + (8*b*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(105*c^3) + (2*b*d^2*(1 - c^2*x^2)^(5/2)*(a
+ b*ArcSin[c*x]))/(175*c^3) - (2*b*d^2*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(49*c^3) + (8*d^2*x^3*(a + b*
ArcSin[c*x])^2)/105 + (4*d^2*x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/35 + (d^2*x^3*(1 - c^2*x^2)^2*(a + b*Arc
Sin[c*x])^2)/7

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Rubi [A]  time = 0.570934, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {4699, 4627, 4707, 4677, 8, 30, 266, 43, 4689, 12, 373} \[ \frac{1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4}{35} d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{16 b d^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{315 c}-\frac{2 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^3}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{175 c^3}+\frac{8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 c^3}+\frac{32 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^3}+\frac{8}{105} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2}{343} b^2 c^4 d^2 x^7+\frac{136 b^2 c^2 d^2 x^5}{6125}-\frac{1636 b^2 d^2 x}{11025 c^2}-\frac{818 b^2 d^2 x^3}{33075} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1636*b^2*d^2*x)/(11025*c^2) - (818*b^2*d^2*x^3)/33075 + (136*b^2*c^2*d^2*x^5)/6125 - (2*b^2*c^4*d^2*x^7)/343
+ (32*b*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(315*c^3) + (16*b*d^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[
c*x]))/(315*c) + (8*b*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(105*c^3) + (2*b*d^2*(1 - c^2*x^2)^(5/2)*(a
+ b*ArcSin[c*x]))/(175*c^3) - (2*b*d^2*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(49*c^3) + (8*d^2*x^3*(a + b*
ArcSin[c*x])^2)/105 + (4*d^2*x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/35 + (d^2*x^3*(1 - c^2*x^2)^2*(a + b*Arc
Sin[c*x])^2)/7

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 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 30

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

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 4689

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(1 - c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcSin[c*x]), u, x] - Dist[b*c*d^p, Int[SimplifyIntegrand[u/Sqrt[1 - c
^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2
, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d, 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 373

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

Rubi steps

\begin{align*} \int x^2 \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} (4 d) \int x^2 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{7} \left (2 b c d^2\right ) \int x^3 \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 )}{35 c^3}-\frac{2 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^3}+\frac{4}{35} d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{35} \left (8 d^2\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{35} \left (8 b c d^2\right ) \int x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac{1}{7} \left (2 b^2 c^2 d^2\right ) \int \frac{\left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2}{35 c^4} \, dx\\ &=\frac{8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 c^3}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{175 c^3}-\frac{2 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^3}+\frac{8}{105} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4}{35} d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (2 b^2 d^2\right ) \int \left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2 \, dx}{245 c^2}-\frac{1}{105} \left (16 b c d^2\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx+\frac{1}{35} \left (8 b^2 c^2 d^2\right ) \int \frac{-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx\\ &=\frac{16 b d^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{315 c}+\frac{8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 c^3}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{175 c^3}-\frac{2 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^3}+\frac{8}{105} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4}{35} d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{315} \left (16 b^2 d^2\right ) \int x^2 \, dx+\frac{\left (2 b^2 d^2\right ) \int \left (-2-c^2 x^2+8 c^4 x^4-5 c^6 x^6\right ) \, dx}{245 c^2}+\frac{\left (8 b^2 d^2\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{525 c^2}-\frac{\left (32 b d^2\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{315 c}\\ &=-\frac{172 b^2 d^2 x}{3675 c^2}-\frac{818 b^2 d^2 x^3}{33075}+\frac{136 b^2 c^2 d^2 x^5}{6125}-\frac{2}{343} b^2 c^4 d^2 x^7+\frac{32 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^3}+\frac{16 b d^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{315 c}+\frac{8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 c^3}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{175 c^3}-\frac{2 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^3}+\frac{8}{105} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4}{35} d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (32 b^2 d^2\right ) \int 1 \, dx}{315 c^2}\\ &=-\frac{1636 b^2 d^2 x}{11025 c^2}-\frac{818 b^2 d^2 x^3}{33075}+\frac{136 b^2 c^2 d^2 x^5}{6125}-\frac{2}{343} b^2 c^4 d^2 x^7+\frac{32 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^3}+\frac{16 b d^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{315 c}+\frac{8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 c^3}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{175 c^3}-\frac{2 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^3}+\frac{8}{105} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4}{35} d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.21692, size = 229, normalized size = 0.74 \[ \frac{d^2 \left (11025 a^2 c^3 x^3 \left (15 c^4 x^4-42 c^2 x^2+35\right )+210 a b \sqrt{1-c^2 x^2} \left (225 c^6 x^6-612 c^4 x^4+409 c^2 x^2+818\right )+210 b \sin ^{-1}(c x) \left (105 a c^3 x^3 \left (15 c^4 x^4-42 c^2 x^2+35\right )+b \sqrt{1-c^2 x^2} \left (225 c^6 x^6-612 c^4 x^4+409 c^2 x^2+818\right )\right )-2 b^2 c x \left (3375 c^6 x^6-12852 c^4 x^4+14315 c^2 x^2+85890\right )+11025 b^2 c^3 x^3 \left (15 c^4 x^4-42 c^2 x^2+35\right ) \sin ^{-1}(c x)^2\right )}{1157625 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(11025*a^2*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4) + 210*a*b*Sqrt[1 - c^2*x^2]*(818 + 409*c^2*x^2 - 612*c^
4*x^4 + 225*c^6*x^6) - 2*b^2*c*x*(85890 + 14315*c^2*x^2 - 12852*c^4*x^4 + 3375*c^6*x^6) + 210*b*(105*a*c^3*x^3
*(35 - 42*c^2*x^2 + 15*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(818 + 409*c^2*x^2 - 612*c^4*x^4 + 225*c^6*x^6))*ArcSin[
c*x] + 11025*b^2*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4)*ArcSin[c*x]^2))/(1157625*c^3)

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Maple [A]  time = 0.044, size = 400, normalized size = 1.3 \begin{align*}{\frac{1}{{c}^{3}} \left ({d}^{2}{a}^{2} \left ({\frac{{c}^{7}{x}^{7}}{7}}-{\frac{2\,{c}^{5}{x}^{5}}{5}}+{\frac{{c}^{3}{x}^{3}}{3}} \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}{105}}+{\frac{16\,\arcsin \left ( cx \right ) }{105}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{2\,\arcsin \left ( cx \right ) \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}{175}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{ \left ( 6\,{c}^{4}{x}^{4}-20\,{c}^{2}{x}^{2}+30 \right ) cx}{2625}}-{\frac{8\, \left ({c}^{2}{x}^{2}-1 \right ) \arcsin \left ( cx \right ) }{315}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{ \left ( 8\,{c}^{2}{x}^{2}-24 \right ) cx}{945}}+{\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ( 5\,{c}^{6}{x}^{6}-21\,{c}^{4}{x}^{4}+35\,{c}^{2}{x}^{2}-35 \right ) cx}{35}}+{\frac{2\,\arcsin \left ( cx \right ) \left ({c}^{2}{x}^{2}-1 \right ) ^{3}}{49}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{ \left ( 10\,{c}^{6}{x}^{6}-42\,{c}^{4}{x}^{4}+70\,{c}^{2}{x}^{2}-70 \right ) cx}{1715}} \right ) +2\,{d}^{2}ab \left ( 1/7\,\arcsin \left ( cx \right ){c}^{7}{x}^{7}-2/5\,\arcsin \left ( cx \right ){c}^{5}{x}^{5}+1/3\,{c}^{3}{x}^{3}\arcsin \left ( cx \right ) +1/49\,{c}^{6}{x}^{6}\sqrt{-{c}^{2}{x}^{2}+1}-{\frac{68\,{c}^{4}{x}^{4}\sqrt{-{c}^{2}{x}^{2}+1}}{1225}}+{\frac{409\,{c}^{2}{x}^{2}\sqrt{-{c}^{2}{x}^{2}+1}}{11025}}+{\frac{818\,\sqrt{-{c}^{2}{x}^{2}+1}}{11025}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(d^2*a^2*(1/7*c^7*x^7-2/5*c^5*x^5+1/3*c^3*x^3)+d^2*b^2*(1/15*arcsin(c*x)^2*(3*c^4*x^4-10*c^2*x^2+15)*c*x
-16/105*c*x+16/105*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+2/175*arcsin(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/2)-2/2625*(3
*c^4*x^4-10*c^2*x^2+15)*c*x-8/315*arcsin(c*x)*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+8/945*(c^2*x^2-3)*c*x+1/35*arcsin
(c*x)^2*(5*c^6*x^6-21*c^4*x^4+35*c^2*x^2-35)*c*x+2/49*arcsin(c*x)*(c^2*x^2-1)^3*(-c^2*x^2+1)^(1/2)-2/1715*(5*c
^6*x^6-21*c^4*x^4+35*c^2*x^2-35)*c*x)+2*d^2*a*b*(1/7*arcsin(c*x)*c^7*x^7-2/5*arcsin(c*x)*c^5*x^5+1/3*c^3*x^3*a
rcsin(c*x)+1/49*c^6*x^6*(-c^2*x^2+1)^(1/2)-68/1225*c^4*x^4*(-c^2*x^2+1)^(1/2)+409/11025*c^2*x^2*(-c^2*x^2+1)^(
1/2)+818/11025*(-c^2*x^2+1)^(1/2)))

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Maxima [B]  time = 1.86786, size = 856, normalized size = 2.76 \begin{align*} \frac{1}{7} \, b^{2} c^{4} d^{2} x^{7} \arcsin \left (c x\right )^{2} + \frac{1}{7} \, a^{2} c^{4} d^{2} x^{7} - \frac{2}{5} \, b^{2} c^{2} d^{2} x^{5} \arcsin \left (c x\right )^{2} - \frac{2}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac{2}{245} \,{\left (35 \, x^{7} \arcsin \left (c x\right ) +{\left (\frac{5 \, \sqrt{-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{4} d^{2} + \frac{2}{25725} \,{\left (105 \,{\left (\frac{5 \, \sqrt{-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1}}{c^{8}}\right )} c \arcsin \left (c x\right ) - \frac{75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} c^{4} d^{2} + \frac{1}{3} \, b^{2} d^{2} x^{3} \arcsin \left (c x\right )^{2} - \frac{4}{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^{2} d^{2} - \frac{4}{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^{2} d^{2} + \frac{1}{3} \, a^{2} d^{2} x^{3} + \frac{2}{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 d^{2} + \frac{2}{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} d^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*c^4*d^2*x^7*arcsin(c*x)^2 + 1/7*a^2*c^4*d^2*x^7 - 2/5*b^2*c^2*d^2*x^5*arcsin(c*x)^2 - 2/5*a^2*c^2*d^2*
x^5 + 2/245*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x
^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*a*b*c^4*d^2 + 2/25725*(105*(5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*s
qrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c*arcsin(c*x) - (75*c^6*
x^7 + 126*c^4*x^5 + 280*c^2*x^3 + 1680*x)/c^6)*b^2*c^4*d^2 + 1/3*b^2*d^2*x^3*arcsin(c*x)^2 - 4/75*(15*x^5*arcs
in(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^2*
d^2 - 4/1125*(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*ar
csin(c*x) - (9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4)*b^2*c^2*d^2 + 1/3*a^2*d^2*x^3 + 2/9*(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*d^2 + 2/27*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sq
rt(-c^2*x^2 + 1)/c^4)*arcsin(c*x) - (c^2*x^3 + 6*x)/c^2)*b^2*d^2

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Fricas [A]  time = 1.90088, size = 697, normalized size = 2.25 \begin{align*} \frac{3375 \,{\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{7} d^{2} x^{7} - 378 \,{\left (1225 \, a^{2} - 68 \, b^{2}\right )} c^{5} d^{2} x^{5} + 35 \,{\left (11025 \, a^{2} - 818 \, b^{2}\right )} c^{3} d^{2} x^{3} - 171780 \, b^{2} c d^{2} x + 11025 \,{\left (15 \, b^{2} c^{7} d^{2} x^{7} - 42 \, b^{2} c^{5} d^{2} x^{5} + 35 \, b^{2} c^{3} d^{2} x^{3}\right )} \arcsin \left (c x\right )^{2} + 22050 \,{\left (15 \, a b c^{7} d^{2} x^{7} - 42 \, a b c^{5} d^{2} x^{5} + 35 \, a b c^{3} d^{2} x^{3}\right )} \arcsin \left (c x\right ) + 210 \,{\left (225 \, a b c^{6} d^{2} x^{6} - 612 \, a b c^{4} d^{2} x^{4} + 409 \, a b c^{2} d^{2} x^{2} + 818 \, a b d^{2} +{\left (225 \, b^{2} c^{6} d^{2} x^{6} - 612 \, b^{2} c^{4} d^{2} x^{4} + 409 \, b^{2} c^{2} d^{2} x^{2} + 818 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{1157625 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1157625*(3375*(49*a^2 - 2*b^2)*c^7*d^2*x^7 - 378*(1225*a^2 - 68*b^2)*c^5*d^2*x^5 + 35*(11025*a^2 - 818*b^2)*
c^3*d^2*x^3 - 171780*b^2*c*d^2*x + 11025*(15*b^2*c^7*d^2*x^7 - 42*b^2*c^5*d^2*x^5 + 35*b^2*c^3*d^2*x^3)*arcsin
(c*x)^2 + 22050*(15*a*b*c^7*d^2*x^7 - 42*a*b*c^5*d^2*x^5 + 35*a*b*c^3*d^2*x^3)*arcsin(c*x) + 210*(225*a*b*c^6*
d^2*x^6 - 612*a*b*c^4*d^2*x^4 + 409*a*b*c^2*d^2*x^2 + 818*a*b*d^2 + (225*b^2*c^6*d^2*x^6 - 612*b^2*c^4*d^2*x^4
+ 409*b^2*c^2*d^2*x^2 + 818*b^2*d^2)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^3

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Sympy [A]  time = 19.8862, size = 483, normalized size = 1.56 \begin{align*} \begin{cases} \frac{a^{2} c^{4} d^{2} x^{7}}{7} - \frac{2 a^{2} c^{2} d^{2} x^{5}}{5} + \frac{a^{2} d^{2} x^{3}}{3} + \frac{2 a b c^{4} d^{2} x^{7} \operatorname{asin}{\left (c x \right )}}{7} + \frac{2 a b c^{3} d^{2} x^{6} \sqrt{- c^{2} x^{2} + 1}}{49} - \frac{4 a b c^{2} d^{2} x^{5} \operatorname{asin}{\left (c x \right )}}{5} - \frac{136 a b c d^{2} x^{4} \sqrt{- c^{2} x^{2} + 1}}{1225} + \frac{2 a b d^{2} x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{818 a b d^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{11025 c} + \frac{1636 a b d^{2} \sqrt{- c^{2} x^{2} + 1}}{11025 c^{3}} + \frac{b^{2} c^{4} d^{2} x^{7} \operatorname{asin}^{2}{\left (c x \right )}}{7} - \frac{2 b^{2} c^{4} d^{2} x^{7}}{343} + \frac{2 b^{2} c^{3} d^{2} x^{6} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{49} - \frac{2 b^{2} c^{2} d^{2} x^{5} \operatorname{asin}^{2}{\left (c x \right )}}{5} + \frac{136 b^{2} c^{2} d^{2} x^{5}}{6125} - \frac{136 b^{2} c d^{2} x^{4} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{1225} + \frac{b^{2} d^{2} x^{3} \operatorname{asin}^{2}{\left (c x \right )}}{3} - \frac{818 b^{2} d^{2} x^{3}}{33075} + \frac{818 b^{2} d^{2} x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{11025 c} - \frac{1636 b^{2} d^{2} x}{11025 c^{2}} + \frac{1636 b^{2} d^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{11025 c^{3}} & \text{for}\: c \neq 0 \\\frac{a^{2} d^{2} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**4*d**2*x**7/7 - 2*a**2*c**2*d**2*x**5/5 + a**2*d**2*x**3/3 + 2*a*b*c**4*d**2*x**7*asin(c*x)
/7 + 2*a*b*c**3*d**2*x**6*sqrt(-c**2*x**2 + 1)/49 - 4*a*b*c**2*d**2*x**5*asin(c*x)/5 - 136*a*b*c*d**2*x**4*sqr
t(-c**2*x**2 + 1)/1225 + 2*a*b*d**2*x**3*asin(c*x)/3 + 818*a*b*d**2*x**2*sqrt(-c**2*x**2 + 1)/(11025*c) + 1636
*a*b*d**2*sqrt(-c**2*x**2 + 1)/(11025*c**3) + b**2*c**4*d**2*x**7*asin(c*x)**2/7 - 2*b**2*c**4*d**2*x**7/343 +
2*b**2*c**3*d**2*x**6*sqrt(-c**2*x**2 + 1)*asin(c*x)/49 - 2*b**2*c**2*d**2*x**5*asin(c*x)**2/5 + 136*b**2*c**
2*d**2*x**5/6125 - 136*b**2*c*d**2*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/1225 + b**2*d**2*x**3*asin(c*x)**2/3 -
818*b**2*d**2*x**3/33075 + 818*b**2*d**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(11025*c) - 1636*b**2*d**2*x/(110
25*c**2) + 1636*b**2*d**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(11025*c**3), Ne(c, 0)), (a**2*d**2*x**3/3, True))

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Giac [B]  time = 1.46981, size = 747, normalized size = 2.41 \begin{align*} \frac{1}{7} \, a^{2} c^{4} d^{2} x^{7} - \frac{2}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{7 \, c^{2}} + \frac{1}{3} \, a^{2} d^{2} x^{3} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{3} a b d^{2} x \arcsin \left (c x\right )}{7 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{35 \, c^{2}} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2} x}{343 \, c^{2}} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{2} a b d^{2} x \arcsin \left (c x\right )}{35 \, c^{2}} - \frac{4 \,{\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{105 \, c^{2}} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{49 \, c^{3}} + \frac{202 \,{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2} x}{42875 \, c^{2}} - \frac{8 \,{\left (c^{2} x^{2} - 1\right )} a b d^{2} x \arcsin \left (c x\right )}{105 \, c^{2}} + \frac{8 \, b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{105 \, c^{2}} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} a b d^{2}}{49 \, c^{3}} + \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 )}{175 \, c^{3}} + \frac{2528 \,{\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} x}{1157625 \, c^{2}} + \frac{16 \, a b d^{2} x \arcsin \left (c x\right )}{105 \, c^{2}} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} a b d^{2}}{175 \, c^{3}} + \frac{8 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} d^{2} \arcsin \left (c x\right )}{315 \, c^{3}} - \frac{181456 \, b^{2} d^{2} x}{1157625 \, c^{2}} + \frac{8 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b d^{2}}{315 \, c^{3}} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{105 \, c^{3}} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1} a b d^{2}}{105 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a^2*c^4*d^2*x^7 - 2/5*a^2*c^2*d^2*x^5 + 1/7*(c^2*x^2 - 1)^3*b^2*d^2*x*arcsin(c*x)^2/c^2 + 1/3*a^2*d^2*x^3
+ 2/7*(c^2*x^2 - 1)^3*a*b*d^2*x*arcsin(c*x)/c^2 + 1/35*(c^2*x^2 - 1)^2*b^2*d^2*x*arcsin(c*x)^2/c^2 - 2/343*(c^
2*x^2 - 1)^3*b^2*d^2*x/c^2 + 2/35*(c^2*x^2 - 1)^2*a*b*d^2*x*arcsin(c*x)/c^2 - 4/105*(c^2*x^2 - 1)*b^2*d^2*x*ar
csin(c*x)^2/c^2 + 2/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)/c^3 + 202/42875*(c^2*x^2 - 1)^2*
b^2*d^2*x/c^2 - 8/105*(c^2*x^2 - 1)*a*b*d^2*x*arcsin(c*x)/c^2 + 8/105*b^2*d^2*x*arcsin(c*x)^2/c^2 + 2/49*(c^2*
x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a*b*d^2/c^3 + 2/175*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)/c^3 +
2528/1157625*(c^2*x^2 - 1)*b^2*d^2*x/c^2 + 16/105*a*b*d^2*x*arcsin(c*x)/c^2 + 2/175*(c^2*x^2 - 1)^2*sqrt(-c^2
*x^2 + 1)*a*b*d^2/c^3 + 8/315*(-c^2*x^2 + 1)^(3/2)*b^2*d^2*arcsin(c*x)/c^3 - 181456/1157625*b^2*d^2*x/c^2 + 8/
315*(-c^2*x^2 + 1)^(3/2)*a*b*d^2/c^3 + 16/105*sqrt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)/c^3 + 16/105*sqrt(-c^2*x^
2 + 1)*a*b*d^2/c^3