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

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

[Out]

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

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Rubi [A]  time = 0.57, antiderivative size = 310, normalized size of antiderivative = 1.00, 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 8

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

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

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 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 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]

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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.21, 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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fricas [A]  time = 0.78, size = 296, normalized size = 0.95 \[ \frac {3375 \, {\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{7} d^{2} x^{7} - 378 \, {\left (1225 \, a^{2} - 68 \, b^{2}\right )} c^{5} d^{2} x^{5} + 35 \, {\left (11025 \, a^{2} - 818 \, b^{2}\right )} c^{3} d^{2} x^{3} - 171780 \, b^{2} c d^{2} x + 11025 \, {\left (15 \, b^{2} c^{7} d^{2} x^{7} - 42 \, b^{2} c^{5} d^{2} x^{5} + 35 \, b^{2} c^{3} d^{2} x^{3}\right )} \arcsin \left (c x\right )^{2} + 22050 \, {\left (15 \, a b c^{7} d^{2} x^{7} - 42 \, a b c^{5} d^{2} x^{5} + 35 \, a b c^{3} d^{2} x^{3}\right )} \arcsin \left (c x\right ) + 210 \, {\left (225 \, a b c^{6} d^{2} x^{6} - 612 \, a b c^{4} d^{2} x^{4} + 409 \, a b c^{2} d^{2} x^{2} + 818 \, a b d^{2} + {\left (225 \, b^{2} c^{6} d^{2} x^{6} - 612 \, b^{2} c^{4} d^{2} x^{4} + 409 \, b^{2} c^{2} d^{2} x^{2} + 818 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{1157625 \, c^{3}} \]

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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giac [B]  time = 0.50, size = 553, normalized size = 1.78 \[ \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}} \]

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

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maple [A]  time = 0.05, size = 400, normalized size = 1.29 \[ \frac {d^{2} a^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {16 c x}{105}+\frac {16 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{105}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{2625}-\frac {8 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{945}+\frac {\arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}\right )+2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {2 \arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {68 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {409 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{11025}+\frac {818 \sqrt {-c^{2} x^{2}+1}}{11025}\right )}{c^{3}} \]

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 = 0.76, size = 634, normalized size = 2.05 \[ \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} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 11.93, size = 483, normalized size = 1.56 \[ \begin {cases} \frac {a^{2} c^{4} d^{2} x^{7}}{7} - \frac {2 a^{2} c^{2} d^{2} x^{5}}{5} + \frac {a^{2} d^{2} x^{3}}{3} + \frac {2 a b c^{4} d^{2} x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {2 a b c^{3} d^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49} - \frac {4 a b c^{2} d^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} - \frac {136 a b c d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{1225} + \frac {2 a b d^{2} x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {818 a b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{11025 c} + \frac {1636 a b d^{2} \sqrt {- c^{2} x^{2} + 1}}{11025 c^{3}} + \frac {b^{2} c^{4} d^{2} x^{7} \operatorname {asin}^{2}{\left (c x \right )}}{7} - \frac {2 b^{2} c^{4} d^{2} x^{7}}{343} + \frac {2 b^{2} c^{3} d^{2} x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{49} - \frac {2 b^{2} c^{2} d^{2} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} + \frac {136 b^{2} c^{2} d^{2} x^{5}}{6125} - \frac {136 b^{2} c d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{1225} + \frac {b^{2} d^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {818 b^{2} d^{2} x^{3}}{33075} + \frac {818 b^{2} d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{11025 c} - \frac {1636 b^{2} d^{2} x}{11025 c^{2}} + \frac {1636 b^{2} d^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{11025 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{2} x^{3}}{3} & \text {otherwise} \end {cases} \]

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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