\(\int (c e+d e x)^3 (a+b \arcsin (c+d x))^{3/2} \, dx\) [245]

Optimal result

Integrand size = 25, antiderivative size = 380 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{3/2} \, dx=\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{3/2}}{4 d}+\frac {3 b^{3/2} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {3 b^{3/2} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{64 d}+\frac {3 b^{3/2} e^3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{64 d}-\frac {3 b^{3/2} e^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{512 d} \]

[Out]

-3/32*e^3*(a+b*arcsin(d*x+c))^(3/2)/d+1/4*e^3*(d*x+c)^4*(a+b*arcsin(d*x+c))^(3/2)/d+3/1024*b^(3/2)*e^3*cos(4*a
/b)*FresnelS(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d-3/1024*b^(3/2)*e^3*Fresn
elC(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(4*a/b)*2^(1/2)*Pi^(1/2)/d-3/64*b^(3/2)*e^3*cos(2
*a/b)*FresnelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/d+3/64*b^(3/2)*e^3*FresnelC(2*(a+b*arcsi
n(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/d+9/64*b*e^3*(d*x+c)*(1-(d*x+c)^2)^(1/2)*(a+b*arcsin(d*x
+c))^(1/2)/d+3/32*b*e^3*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {4889, 12, 4725, 4795, 4737, 4731, 4491, 3387, 3386, 3432, 3385, 3433} \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{3/2} \, dx=\frac {3 \sqrt {\pi } b^{3/2} e^3 \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{64 d}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^3 \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{512 d}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^3 \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {3 \sqrt {\pi } b^{3/2} e^3 \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{64 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{3/2}}{4 d}+\frac {3 b e^3 \sqrt {1-(c+d x)^2} (c+d x)^3 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {9 b e^3 \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}}{64 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{3/2}}{32 d} \]

[In]

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

[Out]

(9*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(64*d) + (3*b*e^3*(c + d*x)^3*Sqrt[1 - (
c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(32*d) - (3*e^3*(a + b*ArcSin[c + d*x])^(3/2))/(32*d) + (e^3*(c + d*x
)^4*(a + b*ArcSin[c + d*x])^(3/2))/(4*d) + (3*b^(3/2)*e^3*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[
a + b*ArcSin[c + d*x]])/Sqrt[b]])/(512*d) - (3*b^(3/2)*e^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin
[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(64*d) + (3*b^(3/2)*e^3*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sq
rt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(64*d) - (3*b^(3/2)*e^3*Sqrt[Pi/2]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c +
d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(512*d)

Rule 12

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

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[f*(x^2/d)],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[f*(x^2/d)], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4725

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

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sin[-
a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -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]

Rule 4889

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^3 x^3 (a+b \arcsin (x))^{3/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int x^3 (a+b \arcsin (x))^{3/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{3/2}}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {x^4 \sqrt {a+b \arcsin (x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d} \\ & = \frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{3/2}}{4 d}-\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {a+b \arcsin (x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{32 d}-\frac {\left (3 b^2 e^3\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{64 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{3/2}}{4 d}+\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{64 d}-\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {\sqrt {a+b \arcsin (x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{64 d}-\frac {\left (9 b^2 e^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{128 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{3/2}}{4 d}+\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \left (-\frac {\sin \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 \sqrt {x}}+\frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{64 d}+\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{128 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{3/2}}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{512 d}+\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{256 d}+\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{128 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{3/2}}{4 d}+\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{256 d}-\frac {\left (3 b e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{256 d}+\frac {\left (3 b e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{512 d}+\frac {\left (3 b e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{256 d}-\frac {\left (3 b e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{512 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{3/2}}{4 d}-\frac {\left (3 b e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{128 d}-\frac {\left (9 b e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{256 d}+\frac {\left (3 b e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{256 d}+\frac {\left (3 b e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{128 d}+\frac {\left (9 b e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{256 d}-\frac {\left (3 b e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{256 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{3/2}}{4 d}+\frac {3 b^{3/2} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {3 b^{3/2} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{256 d}+\frac {3 b^{3/2} e^3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{256 d}-\frac {3 b^{3/2} e^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{512 d}-\frac {\left (9 b e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{128 d}+\frac {\left (9 b e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{128 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{3/2}}{4 d}+\frac {3 b^{3/2} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {3 b^{3/2} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{64 d}+\frac {3 b^{3/2} e^3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{64 d}-\frac {3 b^{3/2} e^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{512 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.66 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{3/2} \, dx=-\frac {b^2 e^3 e^{-\frac {4 i a}{b}} \left (-8 \sqrt {2} e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {5}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-8 \sqrt {2} e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {5}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )+\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {5}{2},-\frac {4 i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {5}{2},\frac {4 i (a+b \arcsin (c+d x))}{b}\right )\right )}{512 d \sqrt {a+b \arcsin (c+d x)}} \]

[In]

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

[Out]

-1/512*(b^2*e^3*(-8*Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((-2*I)*(a + b*A
rcSin[c + d*x]))/b] - 8*Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((2*I)*(a + b*A
rcSin[c + d*x]))/b] + Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((-4*I)*(a + b*ArcSin[c + d*x]))/b] +
E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]))/(d*E^(((4*
I)*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.59

method	result	size
default	\(-\frac {e^{3} \left (3 \sqrt {a +b \arcsin \left (d x +c \right )}\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \cos \left (\frac {4 a}{b}\right ) \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, b^{2}+3 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, b^{2}-48 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{2}-48 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{2}+128 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}-32 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) b^{2}-12 \arcsin \left (d x +c \right ) \sin \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) b^{2}+256 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a b +96 \arcsin \left (d x +c \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}-64 \arcsin \left (d x +c \right ) \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) a b -12 \sin \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) a b +128 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2}+96 \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a b -32 \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) a^{2}\right )}{1024 d \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(604\)

[In]

int((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/1024*e^3/d*(3*(a+b*arcsin(d*x+c))^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/
b)*cos(4*a/b)*Pi^(1/2)*2^(1/2)*(-1/b)^(1/2)*b^2+3*(a+b*arcsin(d*x+c))^(1/2)*sin(4*a/b)*FresnelC(2*2^(1/2)/Pi^(
1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*Pi^(1/2)*2^(1/2)*(-1/b)^(1/2)*b^2-48*(-1/b)^(1/2)*Pi^(1/2)*(a+b
*arcsin(d*x+c))^(1/2)*cos(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2-48*
(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsi
n(d*x+c))^(1/2)/b)*b^2+128*arcsin(d*x+c)^2*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^2-32*arcsin(d*x+c)^2*cos(-4*(
a+b*arcsin(d*x+c))/b+4*a/b)*b^2-12*arcsin(d*x+c)*sin(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*b^2+256*arcsin(d*x+c)*cos
(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b+96*arcsin(d*x+c)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^2-64*arcsin(d*x+c)
*cos(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*a*b-12*sin(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*a*b+128*cos(-2*(a+b*arcsin(d*x
+c))/b+2*a/b)*a^2+96*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b-32*cos(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*a^2)/(a+b*
arcsin(d*x+c))^(1/2)

Fricas [F(-2)]

Exception generated. \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{3/2} \, dx=e^{3} \left (\int a c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 3 a c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int 3 a c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 3 b c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 3 b c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \]

[In]

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

[Out]

e**3*(Integral(a*c**3*sqrt(a + b*asin(c + d*x)), x) + Integral(a*d**3*x**3*sqrt(a + b*asin(c + d*x)), x) + Int
egral(b*c**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x), x) + Integral(3*a*c*d**2*x**2*sqrt(a + b*asin(c + d*x)),
x) + Integral(3*a*c**2*d*x*sqrt(a + b*asin(c + d*x)), x) + Integral(b*d**3*x**3*sqrt(a + b*asin(c + d*x))*asi
n(c + d*x), x) + Integral(3*b*c*d**2*x**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x), x) + Integral(3*b*c**2*d*x*
sqrt(a + b*asin(c + d*x))*asin(c + d*x), x))

Maxima [F]

\[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{3/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.29 (sec) , antiderivative size = 2237, normalized size of antiderivative = 5.89 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/16*I*sqrt(pi)*a^2*b^2*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c
) + a)*sqrt(b)/abs(b))*e^(4*I*a/b)/((sqrt(2)*b^(5/2) + I*sqrt(2)*b^(7/2)/abs(b))*d) + 1/8*I*sqrt(pi)*a^2*b^2*e
^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^
(-4*I*a/b)/((sqrt(2)*b^(5/2) - I*sqrt(2)*b^(7/2)/abs(b))*d) + 1/64*sqrt(pi)*a*b^3*e^3*erf(-sqrt(2)*sqrt(b*arcs
in(d*x + c) + a)/sqrt(b) + I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-4*I*a/b)/((sqrt(2)*b^(5/2
) - I*sqrt(2)*b^(7/2)/abs(b))*d) - 1/8*I*sqrt(pi)*a^2*b^(3/2)*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqr
t(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(4*I*a/b)/((sqrt(2)*b^2 + I*sqrt(2)*b^3/abs(b))
*d) + 1/64*sqrt(pi)*a*b^(5/2)*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d
*x + c) + a)*sqrt(b)/abs(b))*e^(4*I*a/b)/((sqrt(2)*b^2 + I*sqrt(2)*b^3/abs(b))*d) + 1/8*I*sqrt(pi)*a^2*b^(3/2)
*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^
2 + I*b^3/abs(b))*d) - 1/16*sqrt(pi)*a*b^(5/2)*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(
d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^2 + I*b^3/abs(b))*d) - 1/8*I*sqrt(pi)*a^2*b^(3/2)*e^3*erf(-sqrt(
b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^2 - I*b^3/abs(
b))*d) - 1/16*sqrt(pi)*a*b^(5/2)*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*
sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^2 - I*b^3/abs(b))*d) - 1/16*I*sqrt(pi)*a^2*b^(3/2)*e^3*erf(-sqrt(2)*sqrt(b*ar
csin(d*x + c) + a)/sqrt(b) + I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-4*I*a/b)/((sqrt(2)*b^2
- I*sqrt(2)*b^3/abs(b))*d) + 1/16*I*sqrt(pi)*a^2*b*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sq
rt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(4*I*a/b)/((sqrt(2)*b^(3/2) + I*sqrt(2)*b^(5/2)/abs(b))*d)
- 1/64*sqrt(pi)*a*b^2*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c)
+ a)*sqrt(b)/abs(b))*e^(4*I*a/b)/((sqrt(2)*b^(3/2) + I*sqrt(2)*b^(5/2)/abs(b))*d) + 1/16*sqrt(pi)*a*b^2*e^3*e
rf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^(3/2)
+ I*b^(5/2)/abs(b))*d) + 1/8*I*sqrt(pi)*a^2*b*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d
*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^(3/2) - I*b^(5/2)/abs(b))*d) + 1/16*sqrt(pi)*a*b^2*e^3*erf(-sqrt
(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^(3/2) - I*b^(
5/2)/abs(b))*d) - 1/16*I*sqrt(pi)*a^2*b*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(2)*sqrt(
b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-4*I*a/b)/((sqrt(2)*b^(3/2) - I*sqrt(2)*b^(5/2)/abs(b))*d) - 1/64*sq
rt(pi)*a*b^2*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt
(b)/abs(b))*e^(-4*I*a/b)/((sqrt(2)*b^(3/2) - I*sqrt(2)*b^(5/2)/abs(b))*d) + 3/1024*I*sqrt(pi)*b^3*e^3*erf(-sqr
t(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-4*I*a/b)/
((sqrt(2)*b^(3/2) - I*sqrt(2)*b^(5/2)/abs(b))*d) - 3/1024*I*sqrt(pi)*b^(5/2)*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*
x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(4*I*a/b)/((sqrt(2)*b + I*sqrt(2
)*b^2/abs(b))*d) - 1/8*I*sqrt(pi)*a^2*sqrt(b)*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d
*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*d) + 3/128*I*sqrt(pi)*b^(5/2)*e^3*erf(-sqrt(b*arc
sin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*d) -
3/128*I*sqrt(pi)*b^(5/2)*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)
/abs(b))*e^(-2*I*a/b)/((b - I*b^2/abs(b))*d) + 1/64*sqrt(b*arcsin(d*x + c) + a)*b*e^3*arcsin(d*x + c)*e^(4*I*a
rcsin(d*x + c))/d - 1/16*sqrt(b*arcsin(d*x + c) + a)*b*e^3*arcsin(d*x + c)*e^(2*I*arcsin(d*x + c))/d - 1/16*sq
rt(b*arcsin(d*x + c) + a)*b*e^3*arcsin(d*x + c)*e^(-2*I*arcsin(d*x + c))/d + 1/64*sqrt(b*arcsin(d*x + c) + a)*
b*e^3*arcsin(d*x + c)*e^(-4*I*arcsin(d*x + c))/d + 1/64*sqrt(b*arcsin(d*x + c) + a)*a*e^3*e^(4*I*arcsin(d*x +
c))/d + 3/512*I*sqrt(b*arcsin(d*x + c) + a)*b*e^3*e^(4*I*arcsin(d*x + c))/d - 1/16*sqrt(b*arcsin(d*x + c) + a)
*a*e^3*e^(2*I*arcsin(d*x + c))/d - 3/64*I*sqrt(b*arcsin(d*x + c) + a)*b*e^3*e^(2*I*arcsin(d*x + c))/d - 1/16*s
qrt(b*arcsin(d*x + c) + a)*a*e^3*e^(-2*I*arcsin(d*x + c))/d + 3/64*I*sqrt(b*arcsin(d*x + c) + a)*b*e^3*e^(-2*I
*arcsin(d*x + c))/d + 1/64*sqrt(b*arcsin(d*x + c) + a)*a*e^3*e^(-4*I*arcsin(d*x + c))/d - 3/512*I*sqrt(b*arcsi
n(d*x + c) + a)*b*e^3*e^(-4*I*arcsin(d*x + c))/d

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{3/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2} \,d x \]

[In]

int((c*e + d*e*x)^3*(a + b*asin(c + d*x))^(3/2),x)

[Out]

int((c*e + d*e*x)^3*(a + b*asin(c + d*x))^(3/2), x)