3.2.58 \(\int x^2 (d-c^2 d x^2) (a+b \text {ArcSin}(c x))^2 \, dx\) [158]

Optimal. Leaf size=211 \[ -\frac {52 b^2 d x}{225 c^2}-\frac {26}{675} b^2 d x^3+\frac {2}{125} b^2 c^2 d x^5+\frac {8 b d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{45 c^3}+\frac {4 b d x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{45 c}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{15 c^3}-\frac {2 b d \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{25 c^3}+\frac {2}{15} d x^3 (a+b \text {ArcSin}(c x))^2+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2 \]

[Out]

-52/225*b^2*d*x/c^2-26/675*b^2*d*x^3+2/125*b^2*c^2*d*x^5+2/15*b*d*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/c^3-2/2
5*b*d*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))/c^3+2/15*d*x^3*(a+b*arcsin(c*x))^2+1/5*d*x^3*(-c^2*x^2+1)*(a+b*arcs
in(c*x))^2+8/45*b*d*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+4/45*b*d*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)
/c

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Rubi [A]
time = 0.24, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4787, 4723, 4795, 4767, 8, 30, 272, 45, 4779, 12} \begin {gather*} \frac {4 b d x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{45 c}+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2-\frac {2 b d \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{25 c^3}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{15 c^3}+\frac {8 b d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{45 c^3}+\frac {2}{15} d x^3 (a+b \text {ArcSin}(c x))^2+\frac {2}{125} b^2 c^2 d x^5-\frac {52 b^2 d x}{225 c^2}-\frac {26}{675} b^2 d x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-52*b^2*d*x)/(225*c^2) - (26*b^2*d*x^3)/675 + (2*b^2*c^2*d*x^5)/125 + (8*b*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[
c*x]))/(45*c^3) + (4*b*d*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(45*c) + (2*b*d*(1 - c^2*x^2)^(3/2)*(a + b
*ArcSin[c*x]))/(15*c^3) - (2*b*d*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(25*c^3) + (2*d*x^3*(a + b*ArcSin[c*
x])^2)/15 + (d*x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/5

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 45

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 272

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 4723

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 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 4779

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

Rule 4787

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/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(
1 - c^2*x^2)^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]

Rule 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps

\begin {align*} \int x^2 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} (2 d) \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{5} (2 b c d) \int x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac {2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 c^3}-\frac {2 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac {2}{15} d x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{15} (4 b c d) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{5} \left (2 b^2 c^2 d\right ) \int \frac {-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx\\ &=\frac {4 b d x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 c^3}-\frac {2 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac {2}{15} d x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{45} \left (4 b^2 d\right ) \int x^2 \, dx+\frac {\left (2 b^2 d\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{75 c^2}-\frac {(8 b d) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{45 c}\\ &=-\frac {4 b^2 d x}{75 c^2}-\frac {26}{675} b^2 d x^3+\frac {2}{125} b^2 c^2 d x^5+\frac {8 b d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3}+\frac {4 b d x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 c^3}-\frac {2 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac {2}{15} d x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (8 b^2 d\right ) \int 1 \, dx}{45 c^2}\\ &=-\frac {52 b^2 d x}{225 c^2}-\frac {26}{675} b^2 d x^3+\frac {2}{125} b^2 c^2 d x^5+\frac {8 b d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3}+\frac {4 b d x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 c^3}-\frac {2 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac {2}{15} d x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 179, normalized size = 0.85 \begin {gather*} -\frac {d \left (225 a^2 c^3 x^3 \left (-5+3 c^2 x^2\right )+30 a b \sqrt {1-c^2 x^2} \left (-26-13 c^2 x^2+9 c^4 x^4\right )+b^2 \left (780 c x+130 c^3 x^3-54 c^5 x^5\right )+30 b \left (15 a c^3 x^3 \left (-5+3 c^2 x^2\right )+b \sqrt {1-c^2 x^2} \left (-26-13 c^2 x^2+9 c^4 x^4\right )\right ) \text {ArcSin}(c x)+225 b^2 c^3 x^3 \left (-5+3 c^2 x^2\right ) \text {ArcSin}(c x)^2\right )}{3375 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3375*(d*(225*a^2*c^3*x^3*(-5 + 3*c^2*x^2) + 30*a*b*Sqrt[1 - c^2*x^2]*(-26 - 13*c^2*x^2 + 9*c^4*x^4) + b^2*(
780*c*x + 130*c^3*x^3 - 54*c^5*x^5) + 30*b*(15*a*c^3*x^3*(-5 + 3*c^2*x^2) + b*Sqrt[1 - c^2*x^2]*(-26 - 13*c^2*
x^2 + 9*c^4*x^4))*ArcSin[c*x] + 225*b^2*c^3*x^3*(-5 + 3*c^2*x^2)*ArcSin[c*x]^2))/c^3

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Maple [A]
time = 0.13, size = 280, normalized size = 1.33

method	result	size
derivativedivides	\(\frac {-d \,a^{2} \left (\frac {1}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d \,b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{15}-\frac {4 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{135}+\frac {\arcsin \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}\right )-2 d a b \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {13 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}-\frac {26 \sqrt {-c^{2} x^{2}+1}}{225}\right )}{c^{3}}\)	\(280\)
default	\(\frac {-d \,a^{2} \left (\frac {1}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d \,b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{15}-\frac {4 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{135}+\frac {\arcsin \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}\right )-2 d a b \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {13 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}-\frac {26 \sqrt {-c^{2} x^{2}+1}}{225}\right )}{c^{3}}\)	\(280\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(-d*a^2*(1/5*c^5*x^5-1/3*c^3*x^3)-d*b^2*(1/3*arcsin(c*x)^2*(c^2*x^2-3)*c*x+4/15*c*x-4/15*arcsin(c*x)*(-c
^2*x^2+1)^(1/2)+2/45*arcsin(c*x)*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-2/135*(c^2*x^2-3)*c*x+1/15*arcsin(c*x)^2*(3*c^
4*x^4-10*c^2*x^2+15)*c*x+2/25*arcsin(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/2)-2/375*(3*c^4*x^4-10*c^2*x^2+15)*c*x
)-2*d*a*b*(1/5*arcsin(c*x)*c^5*x^5-1/3*c^3*x^3*arcsin(c*x)+1/25*c^4*x^4*(-c^2*x^2+1)^(1/2)-13/225*c^2*x^2*(-c^
2*x^2+1)^(1/2)-26/225*(-c^2*x^2+1)^(1/2)))

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Maxima [A]
time = 0.50, size = 354, normalized size = 1.68 \begin {gather*} -\frac {1}{5} \, b^{2} c^{2} d x^{5} \arcsin \left (c x\right )^{2} - \frac {1}{5} \, a^{2} c^{2} d x^{5} + \frac {1}{3} \, b^{2} d x^{3} \arcsin \left (c x\right )^{2} - \frac {2}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{2} d - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{2} d + \frac {1}{3} \, a^{2} d 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 + \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*b^2*c^2*d*x^5*arcsin(c*x)^2 - 1/5*a^2*c^2*d*x^5 + 1/3*b^2*d*x^3*arcsin(c*x)^2 - 2/75*(15*x^5*arcsin(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 - 2/11
25*(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*arcsin(c*x)
- (9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4)*b^2*c^2*d + 1/3*a^2*d*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 + 2/27*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)
/c^4)*arcsin(c*x) - (c^2*x^3 + 6*x)/c^2)*b^2*d

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Fricas [A]
time = 3.06, size = 194, normalized size = 0.92 \begin {gather*} -\frac {27 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{5} d x^{5} - 5 \, {\left (225 \, a^{2} - 26 \, b^{2}\right )} c^{3} d x^{3} + 780 \, b^{2} c d x + 225 \, {\left (3 \, b^{2} c^{5} d x^{5} - 5 \, b^{2} c^{3} d x^{3}\right )} \arcsin \left (c x\right )^{2} + 450 \, {\left (3 \, a b c^{5} d x^{5} - 5 \, a b c^{3} d x^{3}\right )} \arcsin \left (c x\right ) + 30 \, {\left (9 \, a b c^{4} d x^{4} - 13 \, a b c^{2} d x^{2} - 26 \, a b d + {\left (9 \, b^{2} c^{4} d x^{4} - 13 \, b^{2} c^{2} d x^{2} - 26 \, b^{2} d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{3375 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3375*(27*(25*a^2 - 2*b^2)*c^5*d*x^5 - 5*(225*a^2 - 26*b^2)*c^3*d*x^3 + 780*b^2*c*d*x + 225*(3*b^2*c^5*d*x^5
- 5*b^2*c^3*d*x^3)*arcsin(c*x)^2 + 450*(3*a*b*c^5*d*x^5 - 5*a*b*c^3*d*x^3)*arcsin(c*x) + 30*(9*a*b*c^4*d*x^4
- 13*a*b*c^2*d*x^2 - 26*a*b*d + (9*b^2*c^4*d*x^4 - 13*b^2*c^2*d*x^2 - 26*b^2*d)*arcsin(c*x))*sqrt(-c^2*x^2 + 1
))/c^3

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Sympy [A]
time = 0.60, size = 313, normalized size = 1.48 \begin {gather*} \begin {cases} - \frac {a^{2} c^{2} d x^{5}}{5} + \frac {a^{2} d x^{3}}{3} - \frac {2 a b c^{2} d x^{5} \operatorname {asin}{\left (c x \right )}}{5} - \frac {2 a b c d x^{4} \sqrt {- c^{2} x^{2} + 1}}{25} + \frac {2 a b d x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {26 a b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{225 c} + \frac {52 a b d \sqrt {- c^{2} x^{2} + 1}}{225 c^{3}} - \frac {b^{2} c^{2} d x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} + \frac {2 b^{2} c^{2} d x^{5}}{125} - \frac {2 b^{2} c d x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{25} + \frac {b^{2} d x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {26 b^{2} d x^{3}}{675} + \frac {26 b^{2} d x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{225 c} - \frac {52 b^{2} d x}{225 c^{2}} + \frac {52 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{3}}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*c**2*d*x**5/5 + a**2*d*x**3/3 - 2*a*b*c**2*d*x**5*asin(c*x)/5 - 2*a*b*c*d*x**4*sqrt(-c**2*x**
2 + 1)/25 + 2*a*b*d*x**3*asin(c*x)/3 + 26*a*b*d*x**2*sqrt(-c**2*x**2 + 1)/(225*c) + 52*a*b*d*sqrt(-c**2*x**2 +
1)/(225*c**3) - b**2*c**2*d*x**5*asin(c*x)**2/5 + 2*b**2*c**2*d*x**5/125 - 2*b**2*c*d*x**4*sqrt(-c**2*x**2 +
1)*asin(c*x)/25 + b**2*d*x**3*asin(c*x)**2/3 - 26*b**2*d*x**3/675 + 26*b**2*d*x**2*sqrt(-c**2*x**2 + 1)*asin(c
*x)/(225*c) - 52*b**2*d*x/(225*c**2) + 52*b**2*d*sqrt(-c**2*x**2 + 1)*asin(c*x)/(225*c**3), Ne(c, 0)), (a**2*d
*x**3/3, True))

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Giac [A]
time = 0.44, size = 356, normalized size = 1.69 \begin {gather*} -\frac {1}{5} \, a^{2} c^{2} d x^{5} + \frac {1}{3} \, a^{2} d x^{3} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d x \arcsin \left (c x\right )^{2}}{5 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b d x \arcsin \left (c x\right )}{5 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d x \arcsin \left (c x\right )^{2}}{15 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d x}{125 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b d x \arcsin \left (c x\right )}{15 \, c^{2}} + \frac {2 \, b^{2} d x \arcsin \left (c x\right )^{2}}{15 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{25 \, c^{3}} - \frac {22 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d x}{3375 \, c^{2}} + \frac {4 \, a b d x \arcsin \left (c x\right )}{15 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d}{25 \, c^{3}} + \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d \arcsin \left (c x\right )}{45 \, c^{3}} - \frac {856 \, b^{2} d x}{3375 \, c^{2}} + \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d}{45 \, c^{3}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{15 \, c^{3}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b d}{15 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*a^2*c^2*d*x^5 + 1/3*a^2*d*x^3 - 1/5*(c^2*x^2 - 1)^2*b^2*d*x*arcsin(c*x)^2/c^2 - 2/5*(c^2*x^2 - 1)^2*a*b*d
*x*arcsin(c*x)/c^2 - 1/15*(c^2*x^2 - 1)*b^2*d*x*arcsin(c*x)^2/c^2 + 2/125*(c^2*x^2 - 1)^2*b^2*d*x/c^2 - 2/15*(
c^2*x^2 - 1)*a*b*d*x*arcsin(c*x)/c^2 + 2/15*b^2*d*x*arcsin(c*x)^2/c^2 - 2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1
)*b^2*d*arcsin(c*x)/c^3 - 22/3375*(c^2*x^2 - 1)*b^2*d*x/c^2 + 4/15*a*b*d*x*arcsin(c*x)/c^2 - 2/25*(c^2*x^2 - 1
)^2*sqrt(-c^2*x^2 + 1)*a*b*d/c^3 + 2/45*(-c^2*x^2 + 1)^(3/2)*b^2*d*arcsin(c*x)/c^3 - 856/3375*b^2*d*x/c^2 + 2/
45*(-c^2*x^2 + 1)^(3/2)*a*b*d/c^3 + 4/15*sqrt(-c^2*x^2 + 1)*b^2*d*arcsin(c*x)/c^3 + 4/15*sqrt(-c^2*x^2 + 1)*a*
b*d/c^3

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

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),x)

[Out]

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

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