3.3.67 \(\int \frac {x^3 \sqrt {c+d x^3}}{4 c+d x^3} \, dx\) [267]
Optimal. Leaf size=66 \[ \frac {x^4 \sqrt {c+d x^3} F_1\left (\frac {4}{3};1,-\frac {1}{2};\frac {7}{3};-\frac {d x^3}{4 c},-\frac {d x^3}{c}\right )}{16 c \sqrt {1+\frac {d x^3}{c}}} \]
[Out]
1/16*x^4*AppellF1(4/3,-1/2,1,7/3,-d*x^3/c,-1/4*d*x^3/c)*(d*x^3+c)^(1/2)/c/(1+d*x^3/c)^(1/2)
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Rubi [A]
time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {525, 524}
\begin {gather*} \frac {x^4 \sqrt {c+d x^3} F_1\left (\frac {4}{3};1,-\frac {1}{2};\frac {7}{3};-\frac {d x^3}{4 c},-\frac {d x^3}{c}\right )}{16 c \sqrt {\frac {d x^3}{c}+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x^3*Sqrt[c + d*x^3])/(4*c + d*x^3),x]
[Out]
(x^4*Sqrt[c + d*x^3]*AppellF1[4/3, 1, -1/2, 7/3, -1/4*(d*x^3)/c, -((d*x^3)/c)])/(16*c*Sqrt[1 + (d*x^3)/c])
Rule 524
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])
Rule 525
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && !(IntegerQ[
p] || GtQ[a, 0])
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {c+d x^3}}{4 c+d x^3} \, dx &=\frac {\sqrt {c+d x^3} \int \frac {x^3 \sqrt {1+\frac {d x^3}{c}}}{4 c+d x^3} \, dx}{\sqrt {1+\frac {d x^3}{c}}}\\ &=\frac {x^4 \sqrt {c+d x^3} F_1\left (\frac {4}{3};1,-\frac {1}{2};\frac {7}{3};-\frac {d x^3}{4 c},-\frac {d x^3}{c}\right )}{16 c \sqrt {1+\frac {d x^3}{c}}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(236\) vs. \(2(66)=132\).
time = 3.88, size = 236, normalized size = 3.58 \begin {gather*} \frac {x \left (-17 x^3 \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {4}{3};\frac {1}{2},1;\frac {7}{3};-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )+32 \left (\frac {c}{d}+x^3+\frac {64 c^3 F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )}{d \left (4 c+d x^3\right ) \left (-16 c F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )+3 d x^3 \left (F_1\left (\frac {4}{3};\frac {1}{2},2;\frac {7}{3};-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )+2 F_1\left (\frac {4}{3};\frac {3}{2},1;\frac {7}{3};-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )\right )\right )}\right )\right )}{80 \sqrt {c+d x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(x^3*Sqrt[c + d*x^3])/(4*c + d*x^3),x]
[Out]
(x*(-17*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + 32*(c/d + x^3 + (64
*c^3*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(d*(4*c + d*x^3)*(-16*c*AppellF1[1/3, 1/2, 1, 4
/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + 2*Appe
llF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c]))))))/(80*Sqrt[c + d*x^3])
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
6.
time = 0.40, size = 1003, normalized size = 15.20 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*(d*x^3+c)^(1/2)/(d*x^3+4*c),x,method=_RETURNVERBOSE)
[Out]
1/d*(2/5*x*(d*x^3+c)^(1/2)-2/5*I*c*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2
)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^
2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(
d*x^3+c)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*
d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)))-
4*c/d*(-2/3*I*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-
c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*
(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*Ellipt
icF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*
3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))+1/3*I/d^3*2^(1/2)*sum(
1/_alpha^2*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*
(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-
c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/
2)*(-c*d^2)^(2/3)+2*_alpha^2*d^2-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-
c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),1/6/d*(2*I*(-c*d^2)^(1/3)*3^(1/2)
*_alpha^2*d-I*(-c*d^2)^(2/3)*3^(1/2)*_alpha+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha-3*c*d)/c,(I*3^(1/2)/d*(-c*d^
2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d+4*c)))
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="maxima")
[Out]
integrate(sqrt(d*x^3 + c)*x^3/(d*x^3 + 4*c), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3821 vs.
\(2 (52) = 104\).
time = 18.90, size = 3821, normalized size = 57.89 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="fricas")
[Out]
-1/60*(20*sqrt(3)*(16/27)^(1/6)*d^2*(-c^5/d^8)^(1/6)*arctan(1/24*(96*sqrt(3)*2^(2/3)*(c^5*d^11*x^17 - 271*c^6*
d^10*x^14 + 112*c^7*d^9*x^11 + 1216*c^8*d^8*x^8 + 1088*c^9*d^7*x^5 + 256*c^10*d^6*x^2)*(-c^5/d^8)^(2/3) + 1728
*sqrt(3)*2^(1/3)*(c^7*d^8*x^16 - 39*c^8*d^7*x^13 - 72*c^9*d^6*x^10 - 32*c^10*d^5*x^7)*(-c^5/d^8)^(1/3) - 3*sqr
t(1/3)*(9*sqrt(3)*(16/27)^(5/6)*(d^13*x^18 + 1098*c*d^12*x^15 - 24720*c^2*d^11*x^12 - 56704*c^3*d^10*x^9 - 449
28*c^4*d^9*x^6 - 15360*c^5*d^8*x^3 - 2048*c^6*d^7)*(-c^5/d^8)^(5/6) + 96*sqrt(3)*sqrt(1/3)*(c^2*d^10*x^17 + 73
7*c^3*d^9*x^14 + 2704*c^4*d^8*x^11 + 3376*c^5*d^7*x^8 + 1664*c^6*d^6*x^5 + 256*c^7*d^5*x^2)*sqrt(-c^5/d^8) + 5
76*sqrt(3)*(16/27)^(1/6)*(c^4*d^7*x^16 + 229*c^5*d^6*x^13 + 492*c^6*d^5*x^10 + 328*c^7*d^4*x^7 + 64*c^8*d^3*x^
4)*(-c^5/d^8)^(1/6) - 16*sqrt(d*x^3 + c)*(864*sqrt(3)*2^(2/3)*(c^2*d^10*x^13 + 2*c^3*d^9*x^10 + c^4*d^8*x^7)*(
-c^5/d^8)^(2/3) - sqrt(3)*2^(1/3)*(5*c^3*d^8*x^15 - 3272*c^4*d^7*x^12 - 12544*c^5*d^6*x^9 - 14656*c^6*d^5*x^6
- 6656*c^7*d^4*x^3 - 1024*c^8*d^3)*(-c^5/d^8)^(1/3) - 6*sqrt(3)*(17*c^5*d^5*x^14 - 1456*c^6*d^4*x^11 - 2544*c^
7*d^3*x^8 - 1408*c^8*d^2*x^5 - 256*c^9*d*x^2)))*sqrt((24*c^8*d^2*x^8 - 168*c^9*d*x^5 - 192*c^10*x^2 + 24*2^(2/
3)*(c^5*d^7*x^7 + 5*c^6*d^6*x^4 + 4*c^7*d^5*x)*(-c^5/d^8)^(2/3) + 2*2^(1/3)*(c^6*d^5*x^9 + 60*c^7*d^4*x^6 - 32
*c^9*d^2)*(-c^5/d^8)^(1/3) + 3*(72*sqrt(1/3)*c^6*d^5*x^5*sqrt(-c^5/d^8) - 9*(16/27)^(5/6)*(c^4*d^8*x^6 - 16*c^
5*d^7*x^3 - 8*c^6*d^6)*(-c^5/d^8)^(5/6) + 4*(16/27)^(1/6)*(c^7*d^3*x^7 + 2*c^8*d^2*x^4 - 8*c^9*d*x)*(-c^5/d^8)
^(1/6))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) - 8*sqrt(3)*(c^8*d^6*x^18 - 1416*c^
9*d^5*x^15 + 14352*c^10*d^4*x^12 + 44480*c^11*d^3*x^9 + 49920*c^12*d^2*x^6 + 24576*c^13*d*x^3 + 4096*c^14) + 3
6*sqrt(d*x^3 + c)*(3*sqrt(3)*(16/27)^(5/6)*(c^4*d^12*x^16 + 686*c^5*d^11*x^13 + 7072*c^6*d^10*x^10 + 11008*c^7
*d^9*x^7 + 5888*c^8*d^8*x^4 + 1024*c^9*d^7*x)*(-c^5/d^8)^(5/6) + 32*sqrt(3)*sqrt(1/3)*(c^6*d^9*x^15 + 157*c^7*
d^8*x^12 + 348*c^8*d^7*x^9 + 256*c^9*d^6*x^6 + 64*c^10*d^5*x^3)*sqrt(-c^5/d^8) + 8*sqrt(3)*(16/27)^(1/6)*(31*c
^8*d^6*x^14 + 1744*c^9*d^5*x^11 + 2976*c^10*d^4*x^8 + 1600*c^11*d^3*x^5 + 256*c^12*d^2*x^2)*(-c^5/d^8)^(1/6)))
/(c^8*d^6*x^18 + 2184*c^9*d^5*x^15 + 57696*c^10*d^4*x^12 + 125696*c^11*d^3*x^9 + 100608*c^12*d^2*x^6 + 33792*c
^13*d*x^3 + 4096*c^14)) - 20*sqrt(3)*(16/27)^(1/6)*d^2*(-c^5/d^8)^(1/6)*arctan(1/24*(96*sqrt(3)*2^(2/3)*(c^5*d
^11*x^17 - 271*c^6*d^10*x^14 + 112*c^7*d^9*x^11 + 1216*c^8*d^8*x^8 + 1088*c^9*d^7*x^5 + 256*c^10*d^6*x^2)*(-c^
5/d^8)^(2/3) + 1728*sqrt(3)*2^(1/3)*(c^7*d^8*x^16 - 39*c^8*d^7*x^13 - 72*c^9*d^6*x^10 - 32*c^10*d^5*x^7)*(-c^5
/d^8)^(1/3) + 3*sqrt(1/3)*(9*sqrt(3)*(16/27)^(5/6)*(d^13*x^18 + 1098*c*d^12*x^15 - 24720*c^2*d^11*x^12 - 56704
*c^3*d^10*x^9 - 44928*c^4*d^9*x^6 - 15360*c^5*d^8*x^3 - 2048*c^6*d^7)*(-c^5/d^8)^(5/6) + 96*sqrt(3)*sqrt(1/3)*
(c^2*d^10*x^17 + 737*c^3*d^9*x^14 + 2704*c^4*d^8*x^11 + 3376*c^5*d^7*x^8 + 1664*c^6*d^6*x^5 + 256*c^7*d^5*x^2)
*sqrt(-c^5/d^8) + 576*sqrt(3)*(16/27)^(1/6)*(c^4*d^7*x^16 + 229*c^5*d^6*x^13 + 492*c^6*d^5*x^10 + 328*c^7*d^4*
x^7 + 64*c^8*d^3*x^4)*(-c^5/d^8)^(1/6) + 16*sqrt(d*x^3 + c)*(864*sqrt(3)*2^(2/3)*(c^2*d^10*x^13 + 2*c^3*d^9*x^
10 + c^4*d^8*x^7)*(-c^5/d^8)^(2/3) - sqrt(3)*2^(1/3)*(5*c^3*d^8*x^15 - 3272*c^4*d^7*x^12 - 12544*c^5*d^6*x^9 -
14656*c^6*d^5*x^6 - 6656*c^7*d^4*x^3 - 1024*c^8*d^3)*(-c^5/d^8)^(1/3) - 6*sqrt(3)*(17*c^5*d^5*x^14 - 1456*c^6
*d^4*x^11 - 2544*c^7*d^3*x^8 - 1408*c^8*d^2*x^5 - 256*c^9*d*x^2)))*sqrt((24*c^8*d^2*x^8 - 168*c^9*d*x^5 - 192*
c^10*x^2 + 24*2^(2/3)*(c^5*d^7*x^7 + 5*c^6*d^6*x^4 + 4*c^7*d^5*x)*(-c^5/d^8)^(2/3) + 2*2^(1/3)*(c^6*d^5*x^9 +
60*c^7*d^4*x^6 - 32*c^9*d^2)*(-c^5/d^8)^(1/3) - 3*(72*sqrt(1/3)*c^6*d^5*x^5*sqrt(-c^5/d^8) - 9*(16/27)^(5/6)*(
c^4*d^8*x^6 - 16*c^5*d^7*x^3 - 8*c^6*d^6)*(-c^5/d^8)^(5/6) + 4*(16/27)^(1/6)*(c^7*d^3*x^7 + 2*c^8*d^2*x^4 - 8*
c^9*d*x)*(-c^5/d^8)^(1/6))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) - 8*sqrt(3)*(c^8
*d^6*x^18 - 1416*c^9*d^5*x^15 + 14352*c^10*d^4*x^12 + 44480*c^11*d^3*x^9 + 49920*c^12*d^2*x^6 + 24576*c^13*d*x
^3 + 4096*c^14) - 36*sqrt(d*x^3 + c)*(3*sqrt(3)*(16/27)^(5/6)*(c^4*d^12*x^16 + 686*c^5*d^11*x^13 + 7072*c^6*d^
10*x^10 + 11008*c^7*d^9*x^7 + 5888*c^8*d^8*x^4 + 1024*c^9*d^7*x)*(-c^5/d^8)^(5/6) + 32*sqrt(3)*sqrt(1/3)*(c^6*
d^9*x^15 + 157*c^7*d^8*x^12 + 348*c^8*d^7*x^9 + 256*c^9*d^6*x^6 + 64*c^10*d^5*x^3)*sqrt(-c^5/d^8) + 8*sqrt(3)*
(16/27)^(1/6)*(31*c^8*d^6*x^14 + 1744*c^9*d^5*x^11 + 2976*c^10*d^4*x^8 + 1600*c^11*d^3*x^5 + 256*c^12*d^2*x^2)
*(-c^5/d^8)^(1/6)))/(c^8*d^6*x^18 + 2184*c^9*d^5*x^15 + 57696*c^10*d^4*x^12 + 125696*c^11*d^3*x^9 + 100608*c^1
2*d^2*x^6 + 33792*c^13*d*x^3 + 4096*c^14)) - 5*(16/27)^(1/6)*d^2*(-c^5/d^8)^(1/6)*log(16/3*(24*c^8*d^2*x^8 - 1
68*c^9*d*x^5 - 192*c^10*x^2 + 24*2^(2/3)*(c^5*d^7*x^7 + 5*c^6*d^6*x^4 + 4*c^7*d^5*x)*(-c^5/d^8)^(2/3) + 2*2^(1
/3)*(c^6*d^5*x^9 + 60*c^7*d^4*x^6 - 32*c^9*d^2)*(-c^5/d^8)^(1/3) + 3*(72*sqrt(1/3)*c^6*d^5*x^5*sqrt(-c^5/d^8)
- 9*(16/27)^(5/6)*(c^4*d^8*x^6 - 16*c^5*d^7*x^3...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \sqrt {c + d x^{3}}}{4 c + d x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3*(d*x**3+c)**(1/2)/(d*x**3+4*c),x)
[Out]
Integral(x**3*sqrt(c + d*x**3)/(4*c + d*x**3), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="giac")
[Out]
integrate(sqrt(d*x^3 + c)*x^3/(d*x^3 + 4*c), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^3\,\sqrt {d\,x^3+c}}{d\,x^3+4\,c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^3*(c + d*x^3)^(1/2))/(4*c + d*x^3),x)
[Out]
int((x^3*(c + d*x^3)^(1/2))/(4*c + d*x^3), x)
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