[next] [prev] [prev-tail] [tail] [up]

3.301 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=205 \[ \frac{3 \left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 \sqrt{d} e^4 \sqrt{c d-b e}}-\frac{3 \sqrt{c} (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^4}+\frac{3 \sqrt{b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac{\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2} \]

[Out]

(3*(4*c*d - b*e + 2*c*e*x)*Sqrt[b*x + c*x^2])/(4*e^3*(d + e*x)) - (b*x + c*x^2)^
(3/2)/(2*e*(d + e*x)^2) - (3*Sqrt[c]*(2*c*d - b*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x
+ c*x^2]])/e^4 + (3*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcTanh[(b*d + (2*c*d - b*
e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*Sqrt[d]*e^4*Sqrt[c*d -
b*e])

_______________________________________________________________________________________

Rubi [A]  time = 0.54224, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{3 \left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 \sqrt{d} e^4 \sqrt{c d-b e}}-\frac{3 \sqrt{c} (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^4}+\frac{3 \sqrt{b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac{\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*(4*c*d - b*e + 2*c*e*x)*Sqrt[b*x + c*x^2])/(4*e^3*(d + e*x)) - (b*x + c*x^2)^
(3/2)/(2*e*(d + e*x)^2) - (3*Sqrt[c]*(2*c*d - b*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x
+ c*x^2]])/e^4 + (3*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcTanh[(b*d + (2*c*d - b*
e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*Sqrt[d]*e^4*Sqrt[c*d -
b*e])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 69.9398, size = 189, normalized size = 0.92 \[ \frac{3 \sqrt{c} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{e^{4}} - \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )^{2}} - \frac{3 \sqrt{b x + c x^{2}} \left (b e - 4 c d - 2 c e x\right )}{4 e^{3} \left (d + e x\right )} + \frac{3 \left (b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{8 \sqrt{d} e^{4} \sqrt{b e - c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/(e*x+d)**3,x)

[Out]

3*sqrt(c)*(b*e - 2*c*d)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/e**4 - (b*x + c*x**2
)**(3/2)/(2*e*(d + e*x)**2) - 3*sqrt(b*x + c*x**2)*(b*e - 4*c*d - 2*c*e*x)/(4*e*
*3*(d + e*x)) + 3*(b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)*atan((-b*d + x*(b*e - 2*
c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)))/(8*sqrt(d)*e**4*sqrt(b*e -
 c*d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.690214, size = 212, normalized size = 1.03 \[ \frac{(x (b+c x))^{3/2} \left (\frac{3 \left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{d} (b+c x)^{3/2} \sqrt{b e-c d}}+\frac{e \sqrt{x} \left (2 c \left (6 d^2+9 d e x+2 e^2 x^2\right )-b e (3 d+5 e x)\right )}{(b+c x) (d+e x)^2}-\frac{12 \sqrt{c} (2 c d-b e) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{(b+c x)^{3/2}}\right )}{4 e^4 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((x*(b + c*x))^(3/2)*((e*Sqrt[x]*(-(b*e*(3*d + 5*e*x)) + 2*c*(6*d^2 + 9*d*e*x +
2*e^2*x^2)))/((b + c*x)*(d + e*x)^2) + (3*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcT
an[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*Sqrt[-(c*d) +
 b*e]*(b + c*x)^(3/2)) - (12*Sqrt[c]*(2*c*d - b*e)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[
b + c*x]])/(b + c*x)^(3/2)))/(4*e^4*x^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.015, size = 3466, normalized size = 16.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/(e*x+d)^3,x)

[Out]

-51/16/e^2*d/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(
b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)*b^2+15/4/e^3*d^2/(b*e-c*d)^
2*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*
(b*e-c*d)/e^2)^(1/2))*c^(5/2)*b-1/4*e/d^2/(b*e-c*d)^2/(d/e+x)*(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)*b-3/2/e^5*d^4/(b*e-c*d)^2/(-d*(b*e-c*d)/
e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/
2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^4-3/4/e
^2*d/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^3
+3/32/e/c^(1/2)/d/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^
2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*b^3+3/2/e^5*c^3*d^3/(b*e-c*d)/(-
d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c
*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x
))+9/8/e/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x
*b*c^2-9/4/e^3*c^(3/2)*d/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(
d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*b-3/8/d/(b*e-c*d)^2*(c*(d
/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*b^2*c-27/8/e^2*d/(b*e-c*d
)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2*b-1/2/e*c/d/(b
*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)-3/2/e^3*c^2*d/
(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)-3/2/e^4*d^3/
(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*
(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(7/2)+3/2/e^3*d^2/(b*e-c*d)^2*(c*(d/e+x)^2+(b*
e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^3+1/4*e/d^2/(b*e-c*d)^2*(c*(d/e+x)^2
+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b^2-1/2/d/(b*e-c*d)^2*c^2*(c*(d/e+
x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x-3/4/d/(b*e-c*d)^2*(c*(d/e+x)
^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c*b-3/32/d/(b*e-c*d)^2*ln((1/2*(
b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e
^2)^(1/2))/c^(1/2)*b^4+1/2/d/(b*e-c*d)^2/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e
+x)-d*(b*e-c*d)/e^2)^(5/2)*c+15/8/e^2*c/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/
e+x)-d*(b*e-c*d)/e^2)^(1/2)*b+9/16/e^2*c^(1/2)/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+c
*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*b^2
+3/2/e^4*c^(5/2)*d^2/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+
x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))+3/4/e^2*c^2/(b*e-c*d)*(c*(d/e
+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x-3/16/e/d/(b*e-c*d)*(c*(d/e+
x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^2+39/16/e/(b*e-c*d)^2*(c*(d/
e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^2*c+33/32/e/(b*e-c*d)^2*ln
((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e
-c*d)/e^2)^(1/2))*c^(1/2)*b^3+1/2/e/d/(b*e-c*d)/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*c*
d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)-3/8/e/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*l
n((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x
)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^4-39/8/e^3*d^2/(b*e
-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*
(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/
2))/(d/e+x))*b^2*c^2+1/4*e/d^2/(b*e-c*d)^2*c*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-
d*(b*e-c*d)/e^2)^(3/2)*x*b+3/2/e^3*c*d/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2
*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(
b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2-3/e^4*c^2*d^2/(b*e-c*d
)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b
*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d
/e+x))*b-3/8/e*c/d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)
^(1/2)*x*b+9/2/e^4*d^3/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e
^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(
d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^3*b+9/4/e^2*d/(b*e-c*d)^2/(-d*(b*e-c*d
)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(
1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^3*c-9
/16/d/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^3+
1/2/e/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^2

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.31127, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^3,x, algorithm="fricas")

[Out]

[-1/8*(12*(2*c*d^3 - b*d^2*e + (2*c*d*e^2 - b*e^3)*x^2 + 2*(2*c*d^2*e - b*d*e^2)
*x)*sqrt(c*d^2 - b*d*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2
*(4*c*e^3*x^2 + 12*c*d^2*e - 3*b*d*e^2 + (18*c*d*e^2 - 5*b*e^3)*x)*sqrt(c*d^2 -
b*d*e)*sqrt(c*x^2 + b*x) - 3*(8*c^2*d^4 - 8*b*c*d^3*e + b^2*d^2*e^2 + (8*c^2*d^2
*e^2 - 8*b*c*d*e^3 + b^2*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + b^2*d*e^3)*
x)*log((2*(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) + sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d
- b*e)*x))/(e*x + d)))/((e^6*x^2 + 2*d*e^5*x + d^2*e^4)*sqrt(c*d^2 - b*d*e)), -1
/4*(6*(2*c*d^3 - b*d^2*e + (2*c*d*e^2 - b*e^3)*x^2 + 2*(2*c*d^2*e - b*d*e^2)*x)*
sqrt(-c*d^2 + b*d*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - (4*c
*e^3*x^2 + 12*c*d^2*e - 3*b*d*e^2 + (18*c*d*e^2 - 5*b*e^3)*x)*sqrt(-c*d^2 + b*d*
e)*sqrt(c*x^2 + b*x) + 3*(8*c^2*d^4 - 8*b*c*d^3*e + b^2*d^2*e^2 + (8*c^2*d^2*e^2
 - 8*b*c*d*e^3 + b^2*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + b^2*d*e^3)*x)*a
rctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)))/((e^6*x^2 + 2*d*
e^5*x + d^2*e^4)*sqrt(-c*d^2 + b*d*e)), -1/8*(24*(2*c*d^3 - b*d^2*e + (2*c*d*e^2
 - b*e^3)*x^2 + 2*(2*c*d^2*e - b*d*e^2)*x)*sqrt(c*d^2 - b*d*e)*sqrt(-c)*arctan(s
qrt(c*x^2 + b*x)/(sqrt(-c)*x)) - 2*(4*c*e^3*x^2 + 12*c*d^2*e - 3*b*d*e^2 + (18*c
*d*e^2 - 5*b*e^3)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) - 3*(8*c^2*d^4 - 8*b*
c*d^3*e + b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + b^2*e^4)*x^2 + 2*(8*c^2*d
^3*e - 8*b*c*d^2*e^2 + b^2*d*e^3)*x)*log((2*(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) +
sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)))/((e^6*x^2 + 2*d*e^5*x +
 d^2*e^4)*sqrt(c*d^2 - b*d*e)), -1/4*(12*(2*c*d^3 - b*d^2*e + (2*c*d*e^2 - b*e^3
)*x^2 + 2*(2*c*d^2*e - b*d*e^2)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(-c)*arctan(sqrt(c*x
^2 + b*x)/(sqrt(-c)*x)) - (4*c*e^3*x^2 + 12*c*d^2*e - 3*b*d*e^2 + (18*c*d*e^2 -
5*b*e^3)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x) + 3*(8*c^2*d^4 - 8*b*c*d^3*e
+ b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + b^2*e^4)*x^2 + 2*(8*c^2*d^3*e - 8
*b*c*d^2*e^2 + b^2*d*e^3)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*
d - b*e)*x)))/((e^6*x^2 + 2*d*e^5*x + d^2*e^4)*sqrt(-c*d^2 + b*d*e))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{\left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(3/2)/(e*x+d)**3,x)

[Out]

Integral((x*(b + c*x))**(3/2)/(d + e*x)**3, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.620214, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^3,x, algorithm="giac")

[Out]

sage0*x

[next] [prev] [prev-tail] [front] [up]