3.626 \(\int \frac{1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx\)

Optimal. Leaf size=162 \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 e \left (b^2-4 a c\right )^{3/2}}-\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e}+\frac{\log (d+e x)}{a^2 e}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

[Out]

(b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d +
e*x)^4)) + (b*(b^2 - 6*a*c)*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(
2*a^2*(b^2 - 4*a*c)^(3/2)*e) + Log[d + e*x]/(a^2*e) - Log[a + b*(d + e*x)^2 + c*
(d + e*x)^4]/(4*a^2*e)

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Rubi [A]  time = 0.577985, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 e \left (b^2-4 a c\right )^{3/2}}-\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e}+\frac{\log (d+e x)}{a^2 e}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]

[Out]

(b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d +
e*x)^4)) + (b*(b^2 - 6*a*c)*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(
2*a^2*(b^2 - 4*a*c)^(3/2)*e) + Log[d + e*x]/(a^2*e) - Log[a + b*(d + e*x)^2 + c*
(d + e*x)^4]/(4*a^2*e)

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Rubi in Sympy [A]  time = 66.2603, size = 146, normalized size = 0.9 \[ \frac{- 2 a c + b^{2} + b c \left (d + e x\right )^{2}}{2 a e \left (- 4 a c + b^{2}\right ) \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )} + \frac{b \left (- 6 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{2} e \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{\log{\left (\left (d + e x\right )^{2} \right )}}{2 a^{2} e} - \frac{\log{\left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4} \right )}}{4 a^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)

[Out]

(-2*a*c + b**2 + b*c*(d + e*x)**2)/(2*a*e*(-4*a*c + b**2)*(a + b*(d + e*x)**2 +
c*(d + e*x)**4)) + b*(-6*a*c + b**2)*atanh((b + 2*c*(d + e*x)**2)/sqrt(-4*a*c +
b**2))/(2*a**2*e*(-4*a*c + b**2)**(3/2)) + log((d + e*x)**2)/(2*a**2*e) - log(a
+ b*(d + e*x)**2 + c*(d + e*x)**4)/(4*a**2*e)

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Mathematica [A]  time = 0.733271, size = 235, normalized size = 1.45 \[ \frac{\frac{2 a \left (-2 a c+b^2+b c (d+e x)^2\right )}{\left (b^2-4 a c\right ) \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}-\frac{\left (b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\left (-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+4 \log (d+e x)}{4 a^2 e} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]

[Out]

((2*a*(b^2 - 2*a*c + b*c*(d + e*x)^2))/((b^2 - 4*a*c)*(a + (d + e*x)^2*(b + c*(d
+ e*x)^2))) + 4*Log[d + e*x] - ((b^3 - 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*
Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(
3/2) + ((b^3 - 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b
+ Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(3/2))/(4*a^2*e)

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Maple [C]  time = 0.046, size = 693, normalized size = 4.3 \[ -{\frac{bce{x}^{2}}{2\,a \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{bcdx}{a \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{bc{d}^{2}}{2\,a \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) e \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{c}{ \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) e \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{2}}{2\,a \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) e \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{1}{2\,{a}^{2}e}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ( c{e}^{3} \left ( 4\,ac-{b}^{2} \right ){{\it \_R}}^{3}+3\,cd{e}^{2} \left ( 4\,ac-{b}^{2} \right ){{\it \_R}}^{2}+e \left ( 12\,a{c}^{2}{d}^{2}-3\,{b}^{2}c{d}^{2}+5\,abc-{b}^{3} \right ){\it \_R}+4\,a{c}^{2}{d}^{3}-{b}^{2}c{d}^{3}+5\,abcd-{b}^{3}d \right ) \ln \left ( x-{\it \_R} \right ) }{ \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,{d}^{2}ec{\it \_R}+2\,c{d}^{3}+be{\it \_R}+bd \right ) }}}+{\frac{\ln \left ( ex+d \right ) }{{a}^{2}e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x)

[Out]

-1/2/a/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*
d*e*x+b*d^2+a)*b*e*c/(4*a*c-b^2)*x^2-1/a/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^
2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)*b*c*d/(4*a*c-b^2)*x-1/2/a/(c*e^
4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+
a)/e/(4*a*c-b^2)*b*c*d^2+1/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+
b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)/e/(4*a*c-b^2)*c-1/2/a/(c*e^4*x^4+4*c*d*e^3*x^
3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)/e/(4*a*c-b^2)*b
^2-1/2/a^2/e*sum((c*e^3*(4*a*c-b^2)*_R^3+3*c*d*e^2*(4*a*c-b^2)*_R^2+e*(12*a*c^2*
d^2-3*b^2*c*d^2+5*a*b*c-b^3)*_R+4*a*c^2*d^3-b^2*c*d^3+5*a*b*c*d-b^3*d)/(4*a*c-b^
2)/(2*_R^3*c*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)*ln(x-_R),_R=Roo
tOf(c*e^4*_Z^4+4*c*d*e^3*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+c*
d^4+b*d^2+a))+ln(e*x+d)/a^2/e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b c e^{2} x^{2} + 2 \, b c d e x + b c d^{2} + b^{2} - 2 \, a c}{2 \,{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} e^{5} x^{4} + 4 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d e^{4} x^{3} +{\left (a b^{3} - 4 \, a^{2} b c + 6 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{2}\right )} e^{3} x^{2} + 2 \,{\left (2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{3} +{\left (a b^{3} - 4 \, a^{2} b c\right )} d\right )} e^{2} x +{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{4} + a^{2} b^{2} - 4 \, a^{3} c +{\left (a b^{3} - 4 \, a^{2} b c\right )} d^{2}\right )} e\right )}} - \frac{\int \frac{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{3} x^{3} + 3 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e^{2} x^{2} +{\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} +{\left (b^{3} - 5 \, a b c + 3 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2}\right )} e x +{\left (b^{3} - 5 \, a b c\right )} d}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{4} x^{4} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e^{3} x^{3} +{\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} +{\left (b^{3} - 4 \, a b c + 6 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2}\right )} e^{2} x^{2} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} d^{2} + 2 \,{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} +{\left (b^{3} - 4 \, a b c\right )} d\right )} e x}\,{d x}}{a^{2}} + \frac{\log \left (e x + d\right )}{a^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)^2*(e*x + d)),x, algorithm="maxima")

[Out]

1/2*(b*c*e^2*x^2 + 2*b*c*d*e*x + b*c*d^2 + b^2 - 2*a*c)/((a*b^2*c - 4*a^2*c^2)*e
^5*x^4 + 4*(a*b^2*c - 4*a^2*c^2)*d*e^4*x^3 + (a*b^3 - 4*a^2*b*c + 6*(a*b^2*c - 4
*a^2*c^2)*d^2)*e^3*x^2 + 2*(2*(a*b^2*c - 4*a^2*c^2)*d^3 + (a*b^3 - 4*a^2*b*c)*d)
*e^2*x + ((a*b^2*c - 4*a^2*c^2)*d^4 + a^2*b^2 - 4*a^3*c + (a*b^3 - 4*a^2*b*c)*d^
2)*e) - integrate(((b^2*c - 4*a*c^2)*e^3*x^3 + 3*(b^2*c - 4*a*c^2)*d*e^2*x^2 + (
b^2*c - 4*a*c^2)*d^3 + (b^3 - 5*a*b*c + 3*(b^2*c - 4*a*c^2)*d^2)*e*x + (b^3 - 5*
a*b*c)*d)/((b^2*c - 4*a*c^2)*e^4*x^4 + 4*(b^2*c - 4*a*c^2)*d*e^3*x^3 + (b^2*c -
4*a*c^2)*d^4 + (b^3 - 4*a*b*c + 6*(b^2*c - 4*a*c^2)*d^2)*e^2*x^2 + a*b^2 - 4*a^2
*c + (b^3 - 4*a*b*c)*d^2 + 2*(2*(b^2*c - 4*a*c^2)*d^3 + (b^3 - 4*a*b*c)*d)*e*x),
x)/a^2 + log(e*x + d)/(a^2*e)

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Fricas [A]  time = 0.489128, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)^2*(e*x + d)),x, algorithm="fricas")

[Out]

[1/4*(((b^3*c - 6*a*b*c^2)*e^4*x^4 + 4*(b^3*c - 6*a*b*c^2)*d*e^3*x^3 + (b^3*c -
6*a*b*c^2)*d^4 + (b^4 - 6*a*b^2*c + 6*(b^3*c - 6*a*b*c^2)*d^2)*e^2*x^2 + a*b^3 -
6*a^2*b*c + (b^4 - 6*a*b^2*c)*d^2 + 2*(2*(b^3*c - 6*a*b*c^2)*d^3 + (b^4 - 6*a*b
^2*c)*d)*e*x)*log((2*(b^2*c - 4*a*c^2)*e^2*x^2 + 4*(b^2*c - 4*a*c^2)*d*e*x + b^3
- 4*a*b*c + 2*(b^2*c - 4*a*c^2)*d^2 + (2*c^2*e^4*x^4 + 8*c^2*d*e^3*x^3 + 2*c^2*
d^4 + 2*(6*c^2*d^2 + b*c)*e^2*x^2 + 2*b*c*d^2 + 4*(2*c^2*d^3 + b*c*d)*e*x + b^2
- 2*a*c)*sqrt(b^2 - 4*a*c))/(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e
^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a)) + (2*a*b*c*e^2*x^2 + 4*a*b*c*d*e*x
+ 2*a*b*c*d^2 + 2*a*b^2 - 4*a^2*c - ((b^2*c - 4*a*c^2)*e^4*x^4 + 4*(b^2*c - 4*a*
c^2)*d*e^3*x^3 + (b^2*c - 4*a*c^2)*d^4 + (b^3 - 4*a*b*c + 6*(b^2*c - 4*a*c^2)*d^
2)*e^2*x^2 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*d^2 + 2*(2*(b^2*c - 4*a*c^2)*d^3
+ (b^3 - 4*a*b*c)*d)*e*x)*log(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*
e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a) + 4*((b^2*c - 4*a*c^2)*e^4*x^4 + 4*
(b^2*c - 4*a*c^2)*d*e^3*x^3 + (b^2*c - 4*a*c^2)*d^4 + (b^3 - 4*a*b*c + 6*(b^2*c
- 4*a*c^2)*d^2)*e^2*x^2 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*d^2 + 2*(2*(b^2*c -
4*a*c^2)*d^3 + (b^3 - 4*a*b*c)*d)*e*x)*log(e*x + d))*sqrt(b^2 - 4*a*c))/(((a^2*b
^2*c - 4*a^3*c^2)*e^5*x^4 + 4*(a^2*b^2*c - 4*a^3*c^2)*d*e^4*x^3 + (a^2*b^3 - 4*a
^3*b*c + 6*(a^2*b^2*c - 4*a^3*c^2)*d^2)*e^3*x^2 + 2*(2*(a^2*b^2*c - 4*a^3*c^2)*d
^3 + (a^2*b^3 - 4*a^3*b*c)*d)*e^2*x + (a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^
2)*d^4 + (a^2*b^3 - 4*a^3*b*c)*d^2)*e)*sqrt(b^2 - 4*a*c)), -1/4*(2*((b^3*c - 6*a
*b*c^2)*e^4*x^4 + 4*(b^3*c - 6*a*b*c^2)*d*e^3*x^3 + (b^3*c - 6*a*b*c^2)*d^4 + (b
^4 - 6*a*b^2*c + 6*(b^3*c - 6*a*b*c^2)*d^2)*e^2*x^2 + a*b^3 - 6*a^2*b*c + (b^4 -
6*a*b^2*c)*d^2 + 2*(2*(b^3*c - 6*a*b*c^2)*d^3 + (b^4 - 6*a*b^2*c)*d)*e*x)*arcta
n(-(2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (
2*a*b*c*e^2*x^2 + 4*a*b*c*d*e*x + 2*a*b*c*d^2 + 2*a*b^2 - 4*a^2*c - ((b^2*c - 4*
a*c^2)*e^4*x^4 + 4*(b^2*c - 4*a*c^2)*d*e^3*x^3 + (b^2*c - 4*a*c^2)*d^4 + (b^3 -
4*a*b*c + 6*(b^2*c - 4*a*c^2)*d^2)*e^2*x^2 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*d
^2 + 2*(2*(b^2*c - 4*a*c^2)*d^3 + (b^3 - 4*a*b*c)*d)*e*x)*log(c*e^4*x^4 + 4*c*d*
e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a) + 4
*((b^2*c - 4*a*c^2)*e^4*x^4 + 4*(b^2*c - 4*a*c^2)*d*e^3*x^3 + (b^2*c - 4*a*c^2)*
d^4 + (b^3 - 4*a*b*c + 6*(b^2*c - 4*a*c^2)*d^2)*e^2*x^2 + a*b^2 - 4*a^2*c + (b^3
- 4*a*b*c)*d^2 + 2*(2*(b^2*c - 4*a*c^2)*d^3 + (b^3 - 4*a*b*c)*d)*e*x)*log(e*x +
d))*sqrt(-b^2 + 4*a*c))/(((a^2*b^2*c - 4*a^3*c^2)*e^5*x^4 + 4*(a^2*b^2*c - 4*a^
3*c^2)*d*e^4*x^3 + (a^2*b^3 - 4*a^3*b*c + 6*(a^2*b^2*c - 4*a^3*c^2)*d^2)*e^3*x^2
+ 2*(2*(a^2*b^2*c - 4*a^3*c^2)*d^3 + (a^2*b^3 - 4*a^3*b*c)*d)*e^2*x + (a^3*b^2
- 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*d^4 + (a^2*b^3 - 4*a^3*b*c)*d^2)*e)*sqrt(-b^
2 + 4*a*c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left ({\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a\right )}^{2}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)^2*(e*x + d)),x, algorithm="giac")

[Out]

integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)^2*(e*x + d)), x)