\(\int \frac {(d+e x)^6}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [1964]

Optimal result

Integrand size = 37, antiderivative size = 294 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {35 e^{3/2} \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}} \]

[Out]

-2/3*(e*x+d)^5/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+35/8*e^(3/2)*(-a*e^2+c*d^2)^2*arctanh(1/2*(2*c*d*e*
x+a*e^2+c*d^2)/c^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(9/2)/d^(9/2)-14/3*e*(e*x+d)
^3/c^2/d^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/4*e^2*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)/c^4/d^4+35/6*e^2*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {682, 684, 654, 635, 212} \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {35 e^{3/2} \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}}+\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c^3 d^3}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 (d+e x)^5}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[In]

Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^5)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (14*e*(d + e*x)^3)/(3*c^2*d^2*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*e^2*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(
4*c^4*d^4) + (35*e^2*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(6*c^3*d^3) + (35*e^(3/2)*(c*d^2 -
a*e^2)^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2])])/(8*c^(9/2)*d^(9/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 682

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] - Dist[e^2*((m + p)/(c*(p + 1))), Int[(d + e*x)^(m - 2)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &
& LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 684

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(7 e) \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d} \\ & = -\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (35 e^2\right ) \int \frac {(d+e x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 c^2 d^2} \\ & = -\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {\left (35 e^2 \left (c d^2-a e^2\right )\right ) \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 c^3 d^3} \\ & = -\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {\left (35 e^2 \left (c d^2-a e^2\right )^2\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 c^4 d^4} \\ & = -\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {\left (35 e^2 \left (c d^2-a e^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^4 d^4} \\ & = -\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {35 e^{3/2} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {c} \sqrt {d} \left (105 a^3 e^6-35 a^2 c d e^4 (5 d-4 e x)+7 a c^2 d^2 e^2 \left (8 d^2-34 d e x+3 e^2 x^2\right )+c^3 d^3 \left (8 d^3+80 d^2 e x-39 d e^2 x^2-6 e^3 x^3\right )\right )}{(a e+c d x)^2}+\frac {105 e^{3/2} \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{12 c^{9/2} d^{9/2}} \]

[In]

Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[c]*Sqrt[d]*(105*a^3*e^6 - 35*a^2*c*d*e^4*(5*d - 4*e*x) + 7*a*c^2*d^2*e
^2*(8*d^2 - 34*d*e*x + 3*e^2*x^2) + c^3*d^3*(8*d^3 + 80*d^2*e*x - 39*d*e^2*x^2 - 6*e^3*x^3)))/(a*e + c*d*x)^2)
+ (105*e^(3/2)*(c*d^2 - a*e^2)^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(Sqrt[
a*e + c*d*x]*Sqrt[d + e*x])))/(12*c^(9/2)*d^(9/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(7419\) vs. \(2(260)=520\).

Time = 3.83 (sec) , antiderivative size = 7420, normalized size of antiderivative = 25.24

method	result	size
default	\(\text {Expression too large to display}\)	\(7420\)

[In]

int((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [A] (verification not implemented)

none

Time = 1.80 (sec) , antiderivative size = 833, normalized size of antiderivative = 2.83 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\left [\frac {105 \, {\left (a^{2} c^{2} d^{4} e^{3} - 2 \, a^{3} c d^{2} e^{5} + a^{4} e^{7} + {\left (c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + 2 \, {\left (a c^{3} d^{5} e^{2} - 2 \, a^{2} c^{2} d^{3} e^{4} + a^{3} c d e^{6}\right )} x\right )} \sqrt {\frac {e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \, {\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {e}{c d}}\right ) + 4 \, {\left (6 \, c^{3} d^{3} e^{3} x^{3} - 8 \, c^{3} d^{6} - 56 \, a c^{2} d^{4} e^{2} + 175 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \, {\left (13 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (40 \, c^{3} d^{5} e - 119 \, a c^{2} d^{3} e^{3} + 70 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}}, -\frac {105 \, {\left (a^{2} c^{2} d^{4} e^{3} - 2 \, a^{3} c d^{2} e^{5} + a^{4} e^{7} + {\left (c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + 2 \, {\left (a c^{3} d^{5} e^{2} - 2 \, a^{2} c^{2} d^{3} e^{4} + a^{3} c d e^{6}\right )} x\right )} \sqrt {-\frac {e}{c d}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-\frac {e}{c d}}}{2 \, {\left (c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x\right )}}\right ) - 2 \, {\left (6 \, c^{3} d^{3} e^{3} x^{3} - 8 \, c^{3} d^{6} - 56 \, a c^{2} d^{4} e^{2} + 175 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \, {\left (13 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (40 \, c^{3} d^{5} e - 119 \, a c^{2} d^{3} e^{3} + 70 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{24 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}}\right ] \]

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(105*(a^2*c^2*d^4*e^3 - 2*a^3*c*d^2*e^5 + a^4*e^7 + (c^4*d^6*e - 2*a*c^3*d^4*e^3 + a^2*c^2*d^2*e^5)*x^2
+ 2*(a*c^3*d^5*e^2 - 2*a^2*c^2*d^3*e^4 + a^3*c*d*e^6)*x)*sqrt(e/(c*d))*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c
*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3)*x + 4*(2*c^2*d^2*e*x + c^2*d^3 + a*c*d*e^2)*sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d))) + 4*(6*c^3*d^3*e^3*x^3 - 8*c^3*d^6 - 56*a*c^2*d^4*e^2 + 175*a^2*c*d^2*
e^4 - 105*a^3*e^6 + 3*(13*c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 - 2*(40*c^3*d^5*e - 119*a*c^2*d^3*e^3 + 70*a^2*c*
d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*x^2 + 2*a*c^5*d^5*e*x + a^2*c^4*d^4*e^2), -1/2
4*(105*(a^2*c^2*d^4*e^3 - 2*a^3*c*d^2*e^5 + a^4*e^7 + (c^4*d^6*e - 2*a*c^3*d^4*e^3 + a^2*c^2*d^2*e^5)*x^2 + 2*
(a*c^3*d^5*e^2 - 2*a^2*c^2*d^3*e^4 + a^3*c*d*e^6)*x)*sqrt(-e/(c*d))*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-e/(c*d))/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) - 2*(6*
c^3*d^3*e^3*x^3 - 8*c^3*d^6 - 56*a*c^2*d^4*e^2 + 175*a^2*c*d^2*e^4 - 105*a^3*e^6 + 3*(13*c^3*d^4*e^2 - 7*a*c^2
*d^2*e^4)*x^2 - 2*(40*c^3*d^5*e - 119*a*c^2*d^3*e^3 + 70*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e
^2)*x))/(c^6*d^6*x^2 + 2*a*c^5*d^5*e*x + a^2*c^4*d^4*e^2)]

Sympy [F]

\[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{6}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Integral((d + e*x)**6/((d + e*x)*(a*e + c*d*x))**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?`
for more de

Giac [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%{[%%%{8,[6,6,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[4,
4]%%%}+%%%{

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^6}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \]

[In]

int((d + e*x)^6/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

int((d + e*x)^6/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2), x)