Integrand size = 18, antiderivative size = 169 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=-\frac {7 (9 A b-5 a B)}{20 a^3 b x^{5/2}}+\frac {7 (9 A b-5 a B)}{12 a^4 x^{3/2}}-\frac {7 b (9 A b-5 a B)}{4 a^5 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}+\frac {9 A b-5 a B}{4 a^2 b x^{5/2} (a+b x)}-\frac {7 b^{3/2} (9 A b-5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{11/2}} \]
-7/20*(9*A*b-5*B*a)/a^3/b/x^(5/2)+7/12*(9*A*b-5*B*a)/a^4/x^(3/2)+1/2*(A*b- B*a)/a/b/x^(5/2)/(b*x+a)^2+1/4*(9*A*b-5*B*a)/a^2/b/x^(5/2)/(b*x+a)-7/4*b^( 3/2)*(9*A*b-5*B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(11/2)-7/4*b*(9*A*b-5 *B*a)/a^5/x^(1/2)
Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=\frac {-945 A b^4 x^4+525 a b^3 x^3 (-3 A+B x)-8 a^4 (3 A+5 B x)+8 a^3 b x (9 A+35 B x)+7 a^2 b^2 x^2 (-72 A+125 B x)}{60 a^5 x^{5/2} (a+b x)^2}+\frac {7 b^{3/2} (-9 A b+5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{11/2}} \]
(-945*A*b^4*x^4 + 525*a*b^3*x^3*(-3*A + B*x) - 8*a^4*(3*A + 5*B*x) + 8*a^3 *b*x*(9*A + 35*B*x) + 7*a^2*b^2*x^2*(-72*A + 125*B*x))/(60*a^5*x^(5/2)*(a + b*x)^2) + (7*b^(3/2)*(-9*A*b + 5*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]) /(4*a^(11/2))
Time = 0.23 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.90, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {87, 52, 61, 61, 61, 73, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(9 A b-5 a B) \int \frac {1}{x^{7/2} (a+b x)^2}dx}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(9 A b-5 a B) \left (\frac {7 \int \frac {1}{x^{7/2} (a+b x)}dx}{2 a}+\frac {1}{a x^{5/2} (a+b x)}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(9 A b-5 a B) \left (\frac {7 \left (-\frac {b \int \frac {1}{x^{5/2} (a+b x)}dx}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a}+\frac {1}{a x^{5/2} (a+b x)}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(9 A b-5 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \int \frac {1}{x^{3/2} (a+b x)}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a}+\frac {1}{a x^{5/2} (a+b x)}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(9 A b-5 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {1}{\sqrt {x} (a+b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a}+\frac {1}{a x^{5/2} (a+b x)}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(9 A b-5 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {2 b \int \frac {1}{a+b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a}+\frac {1}{a x^{5/2} (a+b x)}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(9 A b-5 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a}+\frac {1}{a x^{5/2} (a+b x)}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}\) |
(A*b - a*B)/(2*a*b*x^(5/2)*(a + b*x)^2) + ((9*A*b - 5*a*B)*(1/(a*x^(5/2)*( a + b*x)) + (7*(-2/(5*a*x^(5/2)) - (b*(-2/(3*a*x^(3/2)) - (b*(-2/(a*Sqrt[x ]) - (2*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(3/2)))/a))/a))/(2*a) ))/(4*a*b)
3.4.70.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Time = 0.50 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.72
| method | result | size |
| derivativedivides | \(-\frac {2 b^{2} \left (\frac {\left (\frac {15}{8} b^{2} A -\frac {11}{8} a b B \right ) x^{\frac {3}{2}}+\frac {a \left (17 A b -13 B a \right ) \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {7 \left (9 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{5}}-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 \left (-3 A b +B a \right )}{3 a^{4} x^{\frac {3}{2}}}-\frac {6 b \left (2 A b -B a \right )}{a^{5} \sqrt {x}}\) | \(121\) |
| default | \(-\frac {2 b^{2} \left (\frac {\left (\frac {15}{8} b^{2} A -\frac {11}{8} a b B \right ) x^{\frac {3}{2}}+\frac {a \left (17 A b -13 B a \right ) \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {7 \left (9 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{5}}-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 \left (-3 A b +B a \right )}{3 a^{4} x^{\frac {3}{2}}}-\frac {6 b \left (2 A b -B a \right )}{a^{5} \sqrt {x}}\) | \(121\) |
| risch | \(-\frac {2 \left (90 A \,b^{2} x^{2}-45 B a b \,x^{2}-15 a A b x +5 a^{2} B x +3 a^{2} A \right )}{15 a^{5} x^{\frac {5}{2}}}-\frac {b^{2} \left (\frac {2 \left (\frac {15}{8} b^{2} A -\frac {11}{8} a b B \right ) x^{\frac {3}{2}}+\frac {a \left (17 A b -13 B a \right ) \sqrt {x}}{4}}{\left (b x +a \right )^{2}}+\frac {7 \left (9 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{a^{5}}\) | \(124\) |
-2/a^5*b^2*(((15/8*b^2*A-11/8*a*b*B)*x^(3/2)+1/8*a*(17*A*b-13*B*a)*x^(1/2) )/(b*x+a)^2+7/8*(9*A*b-5*B*a)/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2)))-2 /5*A/a^3/x^(5/2)-2/3*(-3*A*b+B*a)/a^4/x^(3/2)-6*b*(2*A*b-B*a)/a^5/x^(1/2)
Time = 0.23 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.59 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=\left [-\frac {105 \, {\left ({\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{5} + 2 \, {\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{4} + {\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (24 \, A a^{4} - 105 \, {\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} - 175 \, {\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 56 \, {\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 8 \, {\left (5 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt {x}}{120 \, {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}}, -\frac {105 \, {\left ({\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{5} + 2 \, {\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{4} + {\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (24 \, A a^{4} - 105 \, {\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} - 175 \, {\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 56 \, {\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 8 \, {\left (5 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt {x}}{60 \, {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}}\right ] \]
[-1/120*(105*((5*B*a*b^3 - 9*A*b^4)*x^5 + 2*(5*B*a^2*b^2 - 9*A*a*b^3)*x^4 + (5*B*a^3*b - 9*A*a^2*b^2)*x^3)*sqrt(-b/a)*log((b*x - 2*a*sqrt(x)*sqrt(-b /a) - a)/(b*x + a)) + 2*(24*A*a^4 - 105*(5*B*a*b^3 - 9*A*b^4)*x^4 - 175*(5 *B*a^2*b^2 - 9*A*a*b^3)*x^3 - 56*(5*B*a^3*b - 9*A*a^2*b^2)*x^2 + 8*(5*B*a^ 4 - 9*A*a^3*b)*x)*sqrt(x))/(a^5*b^2*x^5 + 2*a^6*b*x^4 + a^7*x^3), -1/60*(1 05*((5*B*a*b^3 - 9*A*b^4)*x^5 + 2*(5*B*a^2*b^2 - 9*A*a*b^3)*x^4 + (5*B*a^3 *b - 9*A*a^2*b^2)*x^3)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) + (24*A*a ^4 - 105*(5*B*a*b^3 - 9*A*b^4)*x^4 - 175*(5*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 5 6*(5*B*a^3*b - 9*A*a^2*b^2)*x^2 + 8*(5*B*a^4 - 9*A*a^3*b)*x)*sqrt(x))/(a^5 *b^2*x^5 + 2*a^6*b*x^4 + a^7*x^3)]
Leaf count of result is larger than twice the leaf count of optimal. 1882 vs. \(2 (163) = 326\).
Time = 109.15 (sec) , antiderivative size = 1882, normalized size of antiderivative = 11.14 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=\text {Too large to display} \]
Piecewise((zoo*(-2*A/(11*x**(11/2)) - 2*B/(9*x**(9/2))), Eq(a, 0) & Eq(b, 0)), ((-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2)))/a**3, Eq(b, 0)), ((-2*A/(11*x **(11/2)) - 2*B/(9*x**(9/2)))/b**3, Eq(a, 0)), (-48*A*a**4*sqrt(-a/b)/(120 *a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2 *x**(9/2)*sqrt(-a/b)) + 144*A*a**3*b*x*sqrt(-a/b)/(120*a**7*x**(5/2)*sqrt( -a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b) ) - 945*A*a**2*b**2*x**(5/2)*log(sqrt(x) - sqrt(-a/b))/(120*a**7*x**(5/2)* sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt( -a/b)) + 945*A*a**2*b**2*x**(5/2)*log(sqrt(x) + sqrt(-a/b))/(120*a**7*x**( 5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)* sqrt(-a/b)) - 1008*A*a**2*b**2*x**2*sqrt(-a/b)/(120*a**7*x**(5/2)*sqrt(-a/ b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) - 1890*A*a*b**3*x**(7/2)*log(sqrt(x) - sqrt(-a/b))/(120*a**7*x**(5/2)*sqrt( -a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b) ) + 1890*A*a*b**3*x**(7/2)*log(sqrt(x) + sqrt(-a/b))/(120*a**7*x**(5/2)*sq rt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a /b)) - 3150*A*a*b**3*x**3*sqrt(-a/b)/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a **6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) - 945*A*b** 4*x**(9/2)*log(sqrt(x) - sqrt(-a/b))/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a **6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) + 945*A*...
Time = 0.31 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=-\frac {24 \, A a^{4} - 105 \, {\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} - 175 \, {\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 56 \, {\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 8 \, {\left (5 \, B a^{4} - 9 \, A a^{3} b\right )} x}{60 \, {\left (a^{5} b^{2} x^{\frac {9}{2}} + 2 \, a^{6} b x^{\frac {7}{2}} + a^{7} x^{\frac {5}{2}}\right )}} + \frac {7 \, {\left (5 \, B a b^{2} - 9 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{5}} \]
-1/60*(24*A*a^4 - 105*(5*B*a*b^3 - 9*A*b^4)*x^4 - 175*(5*B*a^2*b^2 - 9*A*a *b^3)*x^3 - 56*(5*B*a^3*b - 9*A*a^2*b^2)*x^2 + 8*(5*B*a^4 - 9*A*a^3*b)*x)/ (a^5*b^2*x^(9/2) + 2*a^6*b*x^(7/2) + a^7*x^(5/2)) + 7/4*(5*B*a*b^2 - 9*A*b ^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5)
Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.80 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=\frac {7 \, {\left (5 \, B a b^{2} - 9 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{5}} + \frac {11 \, B a b^{3} x^{\frac {3}{2}} - 15 \, A b^{4} x^{\frac {3}{2}} + 13 \, B a^{2} b^{2} \sqrt {x} - 17 \, A a b^{3} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{5}} + \frac {2 \, {\left (45 \, B a b x^{2} - 90 \, A b^{2} x^{2} - 5 \, B a^{2} x + 15 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{5} x^{\frac {5}{2}}} \]
7/4*(5*B*a*b^2 - 9*A*b^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5) + 1/ 4*(11*B*a*b^3*x^(3/2) - 15*A*b^4*x^(3/2) + 13*B*a^2*b^2*sqrt(x) - 17*A*a*b ^3*sqrt(x))/((b*x + a)^2*a^5) + 2/15*(45*B*a*b*x^2 - 90*A*b^2*x^2 - 5*B*a^ 2*x + 15*A*a*b*x - 3*A*a^2)/(a^5*x^(5/2))
Time = 0.72 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.80 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=-\frac {\frac {2\,A}{5\,a}-\frac {2\,x\,\left (9\,A\,b-5\,B\,a\right )}{15\,a^2}+\frac {35\,b^2\,x^3\,\left (9\,A\,b-5\,B\,a\right )}{12\,a^4}+\frac {7\,b^3\,x^4\,\left (9\,A\,b-5\,B\,a\right )}{4\,a^5}+\frac {14\,b\,x^2\,\left (9\,A\,b-5\,B\,a\right )}{15\,a^3}}{a^2\,x^{5/2}+b^2\,x^{9/2}+2\,a\,b\,x^{7/2}}-\frac {7\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (9\,A\,b-5\,B\,a\right )}{4\,a^{11/2}} \]