3.1782 \(\int \frac{A+B x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=388 \[ -\frac{e^3 (a+b x) (B d-A e)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}-\frac{e^3 (a+b x) \log (a+b x) (a B e-5 A b e+4 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac{e^3 (a+b x) \log (d+e x) (a B e-5 A b e+4 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac{e^2 (a B e-4 A b e+3 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{e (a B e-3 A b e+2 b B d)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{a B e-2 A b e+b B d}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{A b-a B}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

[Out]

-((e^2*(3*b*B*d - 4*A*b*e + a*B*e))/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)) - (A*b - a*B)/(4*(b*d - a*e)^2*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
b*B*d - 2*A*b*e + a*B*e)/(3*(b*d - a*e)^3*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) + (e*(2*b*B*d - 3*A*b*e + a*B*e))/(2*(b*d - a*e)^4*(a + b*x)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2]) - (e^3*(B*d - A*e)*(a + b*x))/((b*d - a*e)^5*(d + e*x)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (e^3*(4*b*B*d - 5*A*b*e + a*B*e)*(a + b*x)*Log[a + b*x]
)/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^3*(4*b*B*d - 5*A*b*e + a*B*
e)*(a + b*x)*Log[d + e*x])/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 1.00862, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{e^3 (a+b x) (B d-A e)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}-\frac{e^3 (a+b x) \log (a+b x) (a B e-5 A b e+4 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac{e^3 (a+b x) \log (d+e x) (a B e-5 A b e+4 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac{e^2 (a B e-4 A b e+3 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{e (a B e-3 A b e+2 b B d)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{a B e-2 A b e+b B d}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{A b-a B}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

-((e^2*(3*b*B*d - 4*A*b*e + a*B*e))/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)) - (A*b - a*B)/(4*(b*d - a*e)^2*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
b*B*d - 2*A*b*e + a*B*e)/(3*(b*d - a*e)^3*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) + (e*(2*b*B*d - 3*A*b*e + a*B*e))/(2*(b*d - a*e)^4*(a + b*x)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2]) - (e^3*(B*d - A*e)*(a + b*x))/((b*d - a*e)^5*(d + e*x)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (e^3*(4*b*B*d - 5*A*b*e + a*B*e)*(a + b*x)*Log[a + b*x]
)/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^3*(4*b*B*d - 5*A*b*e + a*B*
e)*(a + b*x)*Log[d + e*x])/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 96.8589, size = 394, normalized size = 1.02 \[ \frac{e^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - B a e - 4 B b d\right ) \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{6}} - \frac{e^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - B a e - 4 B b d\right ) \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{6}} - \frac{e^{2} \left (5 A b e - B a e - 4 B b d\right )}{\left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{e \left (2 a + 2 b x\right ) \left (5 A b e - B a e - 4 B b d\right )}{4 \left (a e - b d\right )^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 A b e - B a e - 4 B b d}{3 \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{\left (2 a + 2 b x\right ) \left (5 A b e - B a e - 4 B b d\right )}{8 e \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right )}{2 e \left (d + e x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

e**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(5*A*b*e - B*a*e - 4*B*b*d)*log(a + b*x)/(
(a + b*x)*(a*e - b*d)**6) - e**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(5*A*b*e - B*a
*e - 4*B*b*d)*log(d + e*x)/((a + b*x)*(a*e - b*d)**6) - e**2*(5*A*b*e - B*a*e -
4*B*b*d)/((a*e - b*d)**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - e*(2*a + 2*b*x)*(5*
A*b*e - B*a*e - 4*B*b*d)/(4*(a*e - b*d)**4*(a**2 + 2*a*b*x + b**2*x**2)**(3/2))
- (5*A*b*e - B*a*e - 4*B*b*d)/(3*(a*e - b*d)**3*(a**2 + 2*a*b*x + b**2*x**2)**(3
/2)) - (2*a + 2*b*x)*(5*A*b*e - B*a*e - 4*B*b*d)/(8*e*(a*e - b*d)**2*(a**2 + 2*a
*b*x + b**2*x**2)**(5/2)) - (2*a + 2*b*x)*(A*e - B*d)/(2*e*(d + e*x)*(a*e - b*d)
*(a**2 + 2*a*b*x + b**2*x**2)**(5/2))

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Mathematica [A]  time = 0.419687, size = 250, normalized size = 0.64 \[ \frac{\frac{12 e^3 (a+b x)^3 (b d-a e) (A e-B d)}{d+e x}-12 e^3 (a+b x)^3 \log (a+b x) (a B e-5 A b e+4 b B d)+12 e^3 (a+b x)^3 \log (d+e x) (a B e-5 A b e+4 b B d)+12 e^2 (a+b x)^2 (b d-a e) (-a B e+4 A b e-3 b B d)+\frac{3 (a B-A b) (b d-a e)^4}{a+b x}-6 e (a+b x) (b d-a e)^2 (-a B e+3 A b e-2 b B d)-4 (b d-a e)^3 (a B e-2 A b e+b B d)}{12 \left ((a+b x)^2\right )^{3/2} (b d-a e)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-4*(b*d - a*e)^3*(b*B*d - 2*A*b*e + a*B*e) + (3*(-(A*b) + a*B)*(b*d - a*e)^4)/(
a + b*x) - 6*e*(b*d - a*e)^2*(-2*b*B*d + 3*A*b*e - a*B*e)*(a + b*x) + 12*e^2*(b*
d - a*e)*(-3*b*B*d + 4*A*b*e - a*B*e)*(a + b*x)^2 + (12*e^3*(b*d - a*e)*(-(B*d)
+ A*e)*(a + b*x)^3)/(d + e*x) - 12*e^3*(4*b*B*d - 5*A*b*e + a*B*e)*(a + b*x)^3*L
og[a + b*x] + 12*e^3*(4*b*B*d - 5*A*b*e + a*B*e)*(a + b*x)^3*Log[d + e*x])/(12*(
b*d - a*e)^6*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.044, size = 1652, normalized size = 4.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/12*(-3*A*b^5*d^5-12*A*a^5*e^5+37*B*a^5*d*e^4-B*a*b^4*d^5+25*B*a^5*e^5*x-4*B*b^
5*d^5*x-240*A*ln(e*x+d)*x^2*a^3*b^2*e^5-48*B*ln(b*x+a)*x^2*a^4*b*e^5+48*B*ln(e*x
+d)*x^2*a^4*b*e^5+60*A*ln(b*x+a)*x*a^4*b*e^5-60*A*ln(e*x+d)*x*a^4*b*e^5+60*A*ln(
b*x+a)*a^4*b*d*e^4-60*A*ln(e*x+d)*a^4*b*d*e^4-48*B*ln(b*x+a)*a^4*b*d^2*e^3+48*B*
ln(e*x+d)*a^4*b*d^2*e^3+36*B*x^4*a*b^4*d*e^4+120*A*x^2*a*b^4*d^2*e^3-150*B*x^3*a
*b^4*d^2*e^3+180*A*a^2*b^3*d^2*e^3*x-40*A*a*b^4*d^3*e^2*x+104*B*a^4*b*d*e^4*x-20
*B*a^3*b^2*d^2*e^3*x-136*B*a^2*b^3*d^3*e^2*x+150*A*x^2*a^2*b^3*d*e^4+178*B*x^2*a
^3*b^2*d*e^4-144*B*x^2*a^2*b^3*d^2*e^3-94*B*x^2*a*b^4*d^3*e^2-20*A*a^3*b^2*d*e^4
*x+132*B*x^3*a^2*b^3*d*e^4+180*A*x^3*a*b^4*d*e^4+31*B*a*b^4*d^4*e*x-12*B*ln(b*x+
a)*x^5*a*b^4*e^5-48*B*ln(b*x+a)*x^5*b^5*d*e^4+12*B*ln(e*x+d)*x^5*a*b^4*e^5+48*B*
ln(e*x+d)*x^5*b^5*d*e^4+240*A*ln(b*x+a)*x^4*a*b^4*e^5+60*A*ln(b*x+a)*x^4*b^5*d*e
^4-240*A*ln(e*x+d)*x^4*a*b^4*e^5-60*A*ln(e*x+d)*x^4*b^5*d*e^4-48*B*ln(b*x+a)*x^4
*a^2*b^3*e^5-48*B*ln(b*x+a)*x^4*b^5*d^2*e^3+48*B*ln(e*x+d)*x^4*a^2*b^3*e^5+48*B*
ln(e*x+d)*x^4*b^5*d^2*e^3+360*A*ln(b*x+a)*x^3*a^2*b^3*e^5-360*A*ln(e*x+d)*x^3*a^
2*b^3*e^5-72*B*ln(b*x+a)*x^3*a^3*b^2*e^5+72*B*ln(e*x+d)*x^3*a^3*b^2*e^5-96*B*ln(
b*x+a)*x*a^4*b*d*e^4+96*B*ln(e*x+d)*x*a^4*b*d*e^4+192*B*ln(e*x+d)*x*a^3*b^2*d^2*
e^3-336*B*ln(b*x+a)*x^3*a^2*b^3*d*e^4-192*B*ln(b*x+a)*x^3*a*b^4*d^2*e^3+336*B*ln
(e*x+d)*x^3*a^2*b^3*d*e^4+192*B*ln(e*x+d)*x^3*a*b^4*d^2*e^3+360*A*ln(b*x+a)*x^2*
a^2*b^3*d*e^4-360*A*ln(e*x+d)*x^2*a^2*b^3*d*e^4-264*B*ln(b*x+a)*x^2*a^3*b^2*d*e^
4-288*B*ln(b*x+a)*x^2*a^2*b^3*d^2*e^3+264*B*ln(e*x+d)*x^2*a^3*b^2*d*e^4+288*B*ln
(e*x+d)*x^2*a^2*b^3*d^2*e^3+240*A*ln(b*x+a)*x*a^3*b^2*d*e^4-240*A*ln(e*x+d)*x*a^
3*b^2*d*e^4-204*B*ln(b*x+a)*x^4*a*b^4*d*e^4+204*B*ln(e*x+d)*x^4*a*b^4*d*e^4+240*
A*ln(b*x+a)*x^3*a*b^4*d*e^4-240*A*ln(e*x+d)*x^3*a*b^4*d*e^4-192*B*ln(b*x+a)*x*a^
3*b^2*d^2*e^3-24*B*x^3*b^5*d^3*e^2-260*A*x^2*a^3*b^2*e^5+8*B*x^2*b^5*d^4*e-10*A*
x^2*b^5*d^3*e^2-60*A*x^4*a*b^4*e^5-210*A*x^3*a^2*b^3*e^5+60*A*x^4*b^5*d*e^4+12*B
*x^4*a^2*b^3*e^5-48*B*x^4*b^5*d^2*e^3-125*A*a^4*b*e^5*x+5*A*b^5*d^4*e*x-65*A*a^4
*b*d*e^4+120*A*a^3*b^2*d^2*e^3-60*A*a^2*b^3*d^3*e^2+20*A*a*b^4*d^4*e-8*B*a^4*b*d
^2*e^3+8*B*a^2*b^3*d^4*e-36*B*a^3*b^2*d^3*e^2+240*A*ln(b*x+a)*x^2*a^3*b^2*e^5+30
*A*x^3*b^5*d^2*e^3+52*B*x^2*a^4*b*e^5+42*B*x^3*a^3*b^2*e^5+60*A*ln(b*x+a)*x^5*b^
5*e^5-60*A*ln(e*x+d)*x^5*b^5*e^5-12*B*ln(b*x+a)*x*a^5*e^5+12*B*ln(e*x+d)*x*a^5*e
^5-12*B*ln(b*x+a)*a^5*d*e^4+12*B*ln(e*x+d)*a^5*d*e^4)*(b*x+a)/(e*x+d)/(a*e-b*d)^
6/((b*x+a)^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.319206, size = 2327, normalized size = 6. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

-1/12*(12*A*a^5*e^5 + (B*a*b^4 + 3*A*b^5)*d^5 - 4*(2*B*a^2*b^3 + 5*A*a*b^4)*d^4*
e + 12*(3*B*a^3*b^2 + 5*A*a^2*b^3)*d^3*e^2 + 8*(B*a^4*b - 15*A*a^3*b^2)*d^2*e^3
- (37*B*a^5 - 65*A*a^4*b)*d*e^4 + 12*(4*B*b^5*d^2*e^3 - (3*B*a*b^4 + 5*A*b^5)*d*
e^4 - (B*a^2*b^3 - 5*A*a*b^4)*e^5)*x^4 + 6*(4*B*b^5*d^3*e^2 + 5*(5*B*a*b^4 - A*b
^5)*d^2*e^3 - 2*(11*B*a^2*b^3 + 15*A*a*b^4)*d*e^4 - 7*(B*a^3*b^2 - 5*A*a^2*b^3)*
e^5)*x^3 - 2*(4*B*b^5*d^4*e - (47*B*a*b^4 + 5*A*b^5)*d^3*e^2 - 12*(6*B*a^2*b^3 -
5*A*a*b^4)*d^2*e^3 + (89*B*a^3*b^2 + 75*A*a^2*b^3)*d*e^4 + 26*(B*a^4*b - 5*A*a^
3*b^2)*e^5)*x^2 + (4*B*b^5*d^5 - (31*B*a*b^4 + 5*A*b^5)*d^4*e + 8*(17*B*a^2*b^3
+ 5*A*a*b^4)*d^3*e^2 + 20*(B*a^3*b^2 - 9*A*a^2*b^3)*d^2*e^3 - 4*(26*B*a^4*b - 5*
A*a^3*b^2)*d*e^4 - 25*(B*a^5 - 5*A*a^4*b)*e^5)*x + 12*(4*B*a^4*b*d^2*e^3 + (B*a^
5 - 5*A*a^4*b)*d*e^4 + (4*B*b^5*d*e^4 + (B*a*b^4 - 5*A*b^5)*e^5)*x^5 + (4*B*b^5*
d^2*e^3 + (17*B*a*b^4 - 5*A*b^5)*d*e^4 + 4*(B*a^2*b^3 - 5*A*a*b^4)*e^5)*x^4 + 2*
(8*B*a*b^4*d^2*e^3 + 2*(7*B*a^2*b^3 - 5*A*a*b^4)*d*e^4 + 3*(B*a^3*b^2 - 5*A*a^2*
b^3)*e^5)*x^3 + 2*(12*B*a^2*b^3*d^2*e^3 + (11*B*a^3*b^2 - 15*A*a^2*b^3)*d*e^4 +
2*(B*a^4*b - 5*A*a^3*b^2)*e^5)*x^2 + (16*B*a^3*b^2*d^2*e^3 + 4*(2*B*a^4*b - 5*A*
a^3*b^2)*d*e^4 + (B*a^5 - 5*A*a^4*b)*e^5)*x)*log(b*x + a) - 12*(4*B*a^4*b*d^2*e^
3 + (B*a^5 - 5*A*a^4*b)*d*e^4 + (4*B*b^5*d*e^4 + (B*a*b^4 - 5*A*b^5)*e^5)*x^5 +
(4*B*b^5*d^2*e^3 + (17*B*a*b^4 - 5*A*b^5)*d*e^4 + 4*(B*a^2*b^3 - 5*A*a*b^4)*e^5)
*x^4 + 2*(8*B*a*b^4*d^2*e^3 + 2*(7*B*a^2*b^3 - 5*A*a*b^4)*d*e^4 + 3*(B*a^3*b^2 -
5*A*a^2*b^3)*e^5)*x^3 + 2*(12*B*a^2*b^3*d^2*e^3 + (11*B*a^3*b^2 - 15*A*a^2*b^3)
*d*e^4 + 2*(B*a^4*b - 5*A*a^3*b^2)*e^5)*x^2 + (16*B*a^3*b^2*d^2*e^3 + 4*(2*B*a^4
*b - 5*A*a^3*b^2)*d*e^4 + (B*a^5 - 5*A*a^4*b)*e^5)*x)*log(e*x + d))/(a^4*b^6*d^7
- 6*a^5*b^5*d^6*e + 15*a^6*b^4*d^5*e^2 - 20*a^7*b^3*d^4*e^3 + 15*a^8*b^2*d^3*e^
4 - 6*a^9*b*d^2*e^5 + a^10*d*e^6 + (b^10*d^6*e - 6*a*b^9*d^5*e^2 + 15*a^2*b^8*d^
4*e^3 - 20*a^3*b^7*d^3*e^4 + 15*a^4*b^6*d^2*e^5 - 6*a^5*b^5*d*e^6 + a^6*b^4*e^7)
*x^5 + (b^10*d^7 - 2*a*b^9*d^6*e - 9*a^2*b^8*d^5*e^2 + 40*a^3*b^7*d^4*e^3 - 65*a
^4*b^6*d^3*e^4 + 54*a^5*b^5*d^2*e^5 - 23*a^6*b^4*d*e^6 + 4*a^7*b^3*e^7)*x^4 + 2*
(2*a*b^9*d^7 - 9*a^2*b^8*d^6*e + 12*a^3*b^7*d^5*e^2 + 5*a^4*b^6*d^4*e^3 - 30*a^5
*b^5*d^3*e^4 + 33*a^6*b^4*d^2*e^5 - 16*a^7*b^3*d*e^6 + 3*a^8*b^2*e^7)*x^3 + 2*(3
*a^2*b^8*d^7 - 16*a^3*b^7*d^6*e + 33*a^4*b^6*d^5*e^2 - 30*a^5*b^5*d^4*e^3 + 5*a^
6*b^4*d^3*e^4 + 12*a^7*b^3*d^2*e^5 - 9*a^8*b^2*d*e^6 + 2*a^9*b*e^7)*x^2 + (4*a^3
*b^7*d^7 - 23*a^4*b^6*d^6*e + 54*a^5*b^5*d^5*e^2 - 65*a^6*b^4*d^4*e^3 + 40*a^7*b
^3*d^3*e^4 - 9*a^8*b^2*d^2*e^5 - 2*a^9*b*d*e^6 + a^10*e^7)*x)

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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((B*x+A)/(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^2),x, algorithm="giac")

[Out]

integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^2), x)