3.68 \(\int x^4 (d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=340 \[ \frac{1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{16} d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{64 c^2}-\frac{3 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^4}+\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{256 b c^5 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}-\frac{b c d x^6 \sqrt{d-c^2 d x^2}}{32 \sqrt{1-c^2 x^2}}+\frac{b d x^4 \sqrt{d-c^2 d x^2}}{256 c \sqrt{1-c^2 x^2}}+\frac{3 b d x^2 \sqrt{d-c^2 d x^2}}{256 c^3 \sqrt{1-c^2 x^2}} \]

[Out]

(3*b*d*x^2*Sqrt[d - c^2*d*x^2])/(256*c^3*Sqrt[1 - c^2*x^2]) + (b*d*x^4*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^
2*x^2]) - (b*c*d*x^6*Sqrt[d - c^2*d*x^2])/(32*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[
1 - c^2*x^2]) - (3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c^4) - (d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*
ArcSin[c*x]))/(64*c^2) + (d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/16 + (x^5*(d - c^2*d*x^2)^(3/2)*(a +
b*ArcSin[c*x]))/8 + (3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(256*b*c^5*Sqrt[1 - c^2*x^2])

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Rubi [A]  time = 0.405304, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4699, 4697, 4707, 4641, 30, 14} \[ \frac{1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{16} d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{64 c^2}-\frac{3 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^4}+\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{256 b c^5 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}-\frac{b c d x^6 \sqrt{d-c^2 d x^2}}{32 \sqrt{1-c^2 x^2}}+\frac{b d x^4 \sqrt{d-c^2 d x^2}}{256 c \sqrt{1-c^2 x^2}}+\frac{3 b d x^2 \sqrt{d-c^2 d x^2}}{256 c^3 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]

[Out]

(3*b*d*x^2*Sqrt[d - c^2*d*x^2])/(256*c^3*Sqrt[1 - c^2*x^2]) + (b*d*x^4*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^
2*x^2]) - (b*c*d*x^6*Sqrt[d - c^2*d*x^2])/(32*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[
1 - c^2*x^2]) - (3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c^4) - (d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*
ArcSin[c*x]))/(64*c^2) + (d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/16 + (x^5*(d - c^2*d*x^2)^(3/2)*(a +
b*ArcSin[c*x]))/8 + (3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(256*b*c^5*Sqrt[1 - c^2*x^2])

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 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 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 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^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} (3 d) \int x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int x^5 \left (1-c^2 x^2\right ) \, dx}{8 \sqrt{1-c^2 x^2}}\\ &=\frac{1}{16} d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{16 \sqrt{1-c^2 x^2}}-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int x^5 \, dx}{16 \sqrt{1-c^2 x^2}}-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \left (x^5-c^2 x^7\right ) \, dx}{8 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d x^6 \sqrt{d-c^2 d x^2}}{32 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}-\frac{d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{64 c^2}+\frac{1}{16} d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (3 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{64 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (b d \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{64 c \sqrt{1-c^2 x^2}}\\ &=\frac{b d x^4 \sqrt{d-c^2 d x^2}}{256 c \sqrt{1-c^2 x^2}}-\frac{b c d x^6 \sqrt{d-c^2 d x^2}}{32 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}-\frac{3 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^4}-\frac{d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{64 c^2}+\frac{1}{16} d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (3 d \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{128 c^4 \sqrt{1-c^2 x^2}}+\frac{\left (3 b d \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{128 c^3 \sqrt{1-c^2 x^2}}\\ &=\frac{3 b d x^2 \sqrt{d-c^2 d x^2}}{256 c^3 \sqrt{1-c^2 x^2}}+\frac{b d x^4 \sqrt{d-c^2 d x^2}}{256 c \sqrt{1-c^2 x^2}}-\frac{b c d x^6 \sqrt{d-c^2 d x^2}}{32 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}-\frac{3 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^4}-\frac{d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{64 c^2}+\frac{1}{16} d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{256 b c^5 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.192092, size = 193, normalized size = 0.57 \[ \frac{d \sqrt{d-c^2 d x^2} \left (3 a^2-2 a b c x \sqrt{1-c^2 x^2} \left (16 c^6 x^6-24 c^4 x^4+2 c^2 x^2+3\right )-2 b \sin ^{-1}(c x) \left (b c x \sqrt{1-c^2 x^2} \left (16 c^6 x^6-24 c^4 x^4+2 c^2 x^2+3\right )-3 a\right )+b^2 c^2 x^2 \left (4 c^6 x^6-8 c^4 x^4+c^2 x^2+3\right )+3 b^2 \sin ^{-1}(c x)^2\right )}{256 b c^5 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]

[Out]

(d*Sqrt[d - c^2*d*x^2]*(3*a^2 + b^2*c^2*x^2*(3 + c^2*x^2 - 8*c^4*x^4 + 4*c^6*x^6) - 2*a*b*c*x*Sqrt[1 - c^2*x^2
]*(3 + 2*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6) - 2*b*(-3*a + b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2 - 24*c^4*x^4
+ 16*c^6*x^6))*ArcSin[c*x] + 3*b^2*ArcSin[c*x]^2))/(256*b*c^5*Sqrt[1 - c^2*x^2])

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Maple [B]  time = 0.339, size = 600, normalized size = 1.8 \begin{align*} -{\frac{a{x}^{3}}{8\,{c}^{2}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}-{\frac{ax}{16\,{c}^{4}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{ax}{64\,{c}^{4}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,adx}{128\,{c}^{4}}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{3\,a{d}^{2}}{128\,{c}^{4}}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}-{\frac{3\,b \left ( \arcsin \left ( cx \right ) \right ) ^{2}d}{256\,{c}^{5} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{bd{c}^{3}{x}^{8}}{64\,{c}^{2}{x}^{2}-64}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{bdc{x}^{6}}{32\,{c}^{2}{x}^{2}-32}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{bd{x}^{4}}{256\,c \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{3\,bd{x}^{2}}{256\,{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{bd{c}^{4}\arcsin \left ( cx \right ){x}^{9}}{8\,{c}^{2}{x}^{2}-8}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{5\,b{c}^{2}d\arcsin \left ( cx \right ){x}^{7}}{16\,{c}^{2}{x}^{2}-16}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{13\,bd\arcsin \left ( cx \right ){x}^{5}}{64\,{c}^{2}{x}^{2}-64}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bd\arcsin \left ( cx \right ){x}^{3}}{128\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{3\,bd\arcsin \left ( cx \right ) x}{128\,{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{15\,bd}{8192\,{c}^{5} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x)

[Out]

-1/8*a*x^3*(-c^2*d*x^2+d)^(5/2)/c^2/d-1/16*a/c^4*x*(-c^2*d*x^2+d)^(5/2)/d+1/64*a/c^4*x*(-c^2*d*x^2+d)^(3/2)+3/
128*a/c^4*d*x*(-c^2*d*x^2+d)^(1/2)+3/128*a/c^4*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-
3/256*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/(c^2*x^2-1)*arcsin(c*x)^2*d-1/64*b*(-d*(c^2*x^2-1))^(1/2
)*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^8+1/32*b*(-d*(c^2*x^2-1))^(1/2)*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^
6-1/256*b*(-d*(c^2*x^2-1))^(1/2)*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4-3/256*b*(-d*(c^2*x^2-1))^(1/2)*d/c^3/(
c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-1/8*b*(-d*(c^2*x^2-1))^(1/2)*d*c^4/(c^2*x^2-1)*arcsin(c*x)*x^9+5/16*b*(-d*(c
^2*x^2-1))^(1/2)*d*c^2/(c^2*x^2-1)*arcsin(c*x)*x^7-13/64*b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*arcsin(c*x)*x^
5-1/128*b*(-d*(c^2*x^2-1))^(1/2)*d/c^2/(c^2*x^2-1)*arcsin(c*x)*x^3+3/128*b*(-d*(c^2*x^2-1))^(1/2)*d/c^4/(c^2*x
^2-1)*arcsin(c*x)*x+15/8192*b*(-d*(c^2*x^2-1))^(1/2)*d/c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (a c^{2} d x^{6} - a d x^{4} +{\left (b c^{2} d x^{6} - b d x^{4}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral(-(a*c^2*d*x^6 - a*d*x^4 + (b*c^2*d*x^6 - b*d*x^4)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )} x^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)*x^4, x)