Integrand size = 29, antiderivative size = 190 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {(A b-a B) \sqrt {x}}{5 a b (a+b x)^5}+\frac {(9 A b+a B) \sqrt {x}}{40 a^2 b (a+b x)^4}+\frac {7 (9 A b+a B) \sqrt {x}}{240 a^3 b (a+b x)^3}+\frac {7 (9 A b+a B) \sqrt {x}}{192 a^4 b (a+b x)^2}+\frac {7 (9 A b+a B) \sqrt {x}}{128 a^5 b (a+b x)}+\frac {7 (9 A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{11/2} b^{3/2}} \]
7/128*(9*A*b+B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(11/2)/b^(3/2)+1/5*(A* b-B*a)*x^(1/2)/a/b/(b*x+a)^5+1/40*(9*A*b+B*a)*x^(1/2)/a^2/b/(b*x+a)^4+7/24 0*(9*A*b+B*a)*x^(1/2)/a^3/b/(b*x+a)^3+7/192*(9*A*b+B*a)*x^(1/2)/a^4/b/(b*x +a)^2+7/128*(9*A*b+B*a)*x^(1/2)/a^5/b/(b*x+a)
Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\sqrt {x} \left (-105 a^5 B+945 A b^5 x^4+105 a b^4 x^3 (42 A+B x)+14 a^2 b^3 x^2 (576 A+35 B x)+5 a^4 b (579 A+158 B x)+2 a^3 b^2 x (3555 A+448 B x)\right )}{1920 a^5 b (a+b x)^5}+\frac {7 (9 A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{11/2} b^{3/2}} \]
(Sqrt[x]*(-105*a^5*B + 945*A*b^5*x^4 + 105*a*b^4*x^3*(42*A + B*x) + 14*a^2 *b^3*x^2*(576*A + 35*B*x) + 5*a^4*b*(579*A + 158*B*x) + 2*a^3*b^2*x*(3555* A + 448*B*x)))/(1920*a^5*b*(a + b*x)^5) + (7*(9*A*b + a*B)*ArcTan[(Sqrt[b] *Sqrt[x])/Sqrt[a]])/(128*a^(11/2)*b^(3/2))
Time = 0.26 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.92, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1184, 27, 87, 52, 52, 52, 52, 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.
| \(\displaystyle \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {A+B x}{b^6 \sqrt {x} (a+b x)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {A+B x}{\sqrt {x} (a+b x)^6}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(a B+9 A b) \int \frac {1}{\sqrt {x} (a+b x)^5}dx}{10 a b}+\frac {\sqrt {x} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {7 \int \frac {1}{\sqrt {x} (a+b x)^4}dx}{8 a}+\frac {\sqrt {x}}{4 a (a+b x)^4}\right )}{10 a b}+\frac {\sqrt {x} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {7 \left (\frac {5 \int \frac {1}{\sqrt {x} (a+b x)^3}dx}{6 a}+\frac {\sqrt {x}}{3 a (a+b x)^3}\right )}{8 a}+\frac {\sqrt {x}}{4 a (a+b x)^4}\right )}{10 a b}+\frac {\sqrt {x} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{\sqrt {x} (a+b x)^2}dx}{4 a}+\frac {\sqrt {x}}{2 a (a+b x)^2}\right )}{6 a}+\frac {\sqrt {x}}{3 a (a+b x)^3}\right )}{8 a}+\frac {\sqrt {x}}{4 a (a+b x)^4}\right )}{10 a b}+\frac {\sqrt {x} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {x} (a+b x)}dx}{2 a}+\frac {\sqrt {x}}{a (a+b x)}\right )}{4 a}+\frac {\sqrt {x}}{2 a (a+b x)^2}\right )}{6 a}+\frac {\sqrt {x}}{3 a (a+b x)^3}\right )}{8 a}+\frac {\sqrt {x}}{4 a (a+b x)^4}\right )}{10 a b}+\frac {\sqrt {x} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{a+b x}d\sqrt {x}}{a}+\frac {\sqrt {x}}{a (a+b x)}\right )}{4 a}+\frac {\sqrt {x}}{2 a (a+b x)^2}\right )}{6 a}+\frac {\sqrt {x}}{3 a (a+b x)^3}\right )}{8 a}+\frac {\sqrt {x}}{4 a (a+b x)^4}\right )}{10 a b}+\frac {\sqrt {x} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a+b x)}\right )}{4 a}+\frac {\sqrt {x}}{2 a (a+b x)^2}\right )}{6 a}+\frac {\sqrt {x}}{3 a (a+b x)^3}\right )}{8 a}+\frac {\sqrt {x}}{4 a (a+b x)^4}\right )}{10 a b}+\frac {\sqrt {x} (A b-a B)}{5 a b (a+b x)^5}\) |
((A*b - a*B)*Sqrt[x])/(5*a*b*(a + b*x)^5) + ((9*A*b + a*B)*(Sqrt[x]/(4*a*( a + b*x)^4) + (7*(Sqrt[x]/(3*a*(a + b*x)^3) + (5*(Sqrt[x]/(2*a*(a + b*x)^2 ) + (3*(Sqrt[x]/(a*(a + b*x)) + ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]/(a^(3/2) *Sqrt[b])))/(4*a)))/(6*a)))/(8*a)))/(10*a*b)
3.8.79.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] :> 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]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.71
| method | result | size |
| derivativedivides | \(\frac {\frac {7 \left (9 A b +B a \right ) b^{3} x^{\frac {9}{2}}}{128 a^{5}}+\frac {49 b^{2} \left (9 A b +B a \right ) x^{\frac {7}{2}}}{192 a^{4}}+\frac {7 \left (9 A b +B a \right ) b \,x^{\frac {5}{2}}}{15 a^{3}}+\frac {79 \left (9 A b +B a \right ) x^{\frac {3}{2}}}{192 a^{2}}+\frac {\left (193 A b -7 B a \right ) \sqrt {x}}{128 b a}}{\left (b x +a \right )^{5}}+\frac {7 \left (9 A b +B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 a^{5} b \sqrt {b a}}\) | \(135\) |
| default | \(\frac {\frac {7 \left (9 A b +B a \right ) b^{3} x^{\frac {9}{2}}}{128 a^{5}}+\frac {49 b^{2} \left (9 A b +B a \right ) x^{\frac {7}{2}}}{192 a^{4}}+\frac {7 \left (9 A b +B a \right ) b \,x^{\frac {5}{2}}}{15 a^{3}}+\frac {79 \left (9 A b +B a \right ) x^{\frac {3}{2}}}{192 a^{2}}+\frac {\left (193 A b -7 B a \right ) \sqrt {x}}{128 b a}}{\left (b x +a \right )^{5}}+\frac {7 \left (9 A b +B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 a^{5} b \sqrt {b a}}\) | \(135\) |
2*(7/256*(9*A*b+B*a)/a^5*b^3*x^(9/2)+49/384/a^4*b^2*(9*A*b+B*a)*x^(7/2)+7/ 30/a^3*(9*A*b+B*a)*b*x^(5/2)+79/384/a^2*(9*A*b+B*a)*x^(3/2)+1/256*(193*A*b -7*B*a)/b/a*x^(1/2))/(b*x+a)^5+7/128*(9*A*b+B*a)/a^5/b/(b*a)^(1/2)*arctan( b*x^(1/2)/(b*a)^(1/2))
Time = 0.42 (sec) , antiderivative size = 637, normalized size of antiderivative = 3.35 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {105 \, {\left (B a^{6} + 9 \, A a^{5} b + {\left (B a b^{5} + 9 \, A b^{6}\right )} x^{5} + 5 \, {\left (B a^{2} b^{4} + 9 \, A a b^{5}\right )} x^{4} + 10 \, {\left (B a^{3} b^{3} + 9 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (B a^{4} b^{2} + 9 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (B a^{5} b + 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (105 \, B a^{6} b - 2895 \, A a^{5} b^{2} - 105 \, {\left (B a^{2} b^{5} + 9 \, A a b^{6}\right )} x^{4} - 490 \, {\left (B a^{3} b^{4} + 9 \, A a^{2} b^{5}\right )} x^{3} - 896 \, {\left (B a^{4} b^{3} + 9 \, A a^{3} b^{4}\right )} x^{2} - 790 \, {\left (B a^{5} b^{2} + 9 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{3840 \, {\left (a^{6} b^{7} x^{5} + 5 \, a^{7} b^{6} x^{4} + 10 \, a^{8} b^{5} x^{3} + 10 \, a^{9} b^{4} x^{2} + 5 \, a^{10} b^{3} x + a^{11} b^{2}\right )}}, -\frac {105 \, {\left (B a^{6} + 9 \, A a^{5} b + {\left (B a b^{5} + 9 \, A b^{6}\right )} x^{5} + 5 \, {\left (B a^{2} b^{4} + 9 \, A a b^{5}\right )} x^{4} + 10 \, {\left (B a^{3} b^{3} + 9 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (B a^{4} b^{2} + 9 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (B a^{5} b + 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (105 \, B a^{6} b - 2895 \, A a^{5} b^{2} - 105 \, {\left (B a^{2} b^{5} + 9 \, A a b^{6}\right )} x^{4} - 490 \, {\left (B a^{3} b^{4} + 9 \, A a^{2} b^{5}\right )} x^{3} - 896 \, {\left (B a^{4} b^{3} + 9 \, A a^{3} b^{4}\right )} x^{2} - 790 \, {\left (B a^{5} b^{2} + 9 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{6} b^{7} x^{5} + 5 \, a^{7} b^{6} x^{4} + 10 \, a^{8} b^{5} x^{3} + 10 \, a^{9} b^{4} x^{2} + 5 \, a^{10} b^{3} x + a^{11} b^{2}\right )}}\right ] \]
[-1/3840*(105*(B*a^6 + 9*A*a^5*b + (B*a*b^5 + 9*A*b^6)*x^5 + 5*(B*a^2*b^4 + 9*A*a*b^5)*x^4 + 10*(B*a^3*b^3 + 9*A*a^2*b^4)*x^3 + 10*(B*a^4*b^2 + 9*A* a^3*b^3)*x^2 + 5*(B*a^5*b + 9*A*a^4*b^2)*x)*sqrt(-a*b)*log((b*x - a - 2*sq rt(-a*b)*sqrt(x))/(b*x + a)) + 2*(105*B*a^6*b - 2895*A*a^5*b^2 - 105*(B*a^ 2*b^5 + 9*A*a*b^6)*x^4 - 490*(B*a^3*b^4 + 9*A*a^2*b^5)*x^3 - 896*(B*a^4*b^ 3 + 9*A*a^3*b^4)*x^2 - 790*(B*a^5*b^2 + 9*A*a^4*b^3)*x)*sqrt(x))/(a^6*b^7* x^5 + 5*a^7*b^6*x^4 + 10*a^8*b^5*x^3 + 10*a^9*b^4*x^2 + 5*a^10*b^3*x + a^1 1*b^2), -1/1920*(105*(B*a^6 + 9*A*a^5*b + (B*a*b^5 + 9*A*b^6)*x^5 + 5*(B*a ^2*b^4 + 9*A*a*b^5)*x^4 + 10*(B*a^3*b^3 + 9*A*a^2*b^4)*x^3 + 10*(B*a^4*b^2 + 9*A*a^3*b^3)*x^2 + 5*(B*a^5*b + 9*A*a^4*b^2)*x)*sqrt(a*b)*arctan(sqrt(a *b)/(b*sqrt(x))) + (105*B*a^6*b - 2895*A*a^5*b^2 - 105*(B*a^2*b^5 + 9*A*a* b^6)*x^4 - 490*(B*a^3*b^4 + 9*A*a^2*b^5)*x^3 - 896*(B*a^4*b^3 + 9*A*a^3*b^ 4)*x^2 - 790*(B*a^5*b^2 + 9*A*a^4*b^3)*x)*sqrt(x))/(a^6*b^7*x^5 + 5*a^7*b^ 6*x^4 + 10*a^8*b^5*x^3 + 10*a^9*b^4*x^2 + 5*a^10*b^3*x + a^11*b^2)]
Timed out. \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {105 \, {\left (B a b^{4} + 9 \, A b^{5}\right )} x^{\frac {9}{2}} + 490 \, {\left (B a^{2} b^{3} + 9 \, A a b^{4}\right )} x^{\frac {7}{2}} + 896 \, {\left (B a^{3} b^{2} + 9 \, A a^{2} b^{3}\right )} x^{\frac {5}{2}} + 790 \, {\left (B a^{4} b + 9 \, A a^{3} b^{2}\right )} x^{\frac {3}{2}} - 15 \, {\left (7 \, B a^{5} - 193 \, A a^{4} b\right )} \sqrt {x}}{1920 \, {\left (a^{5} b^{6} x^{5} + 5 \, a^{6} b^{5} x^{4} + 10 \, a^{7} b^{4} x^{3} + 10 \, a^{8} b^{3} x^{2} + 5 \, a^{9} b^{2} x + a^{10} b\right )}} + \frac {7 \, {\left (B a + 9 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{5} b} \]
1/1920*(105*(B*a*b^4 + 9*A*b^5)*x^(9/2) + 490*(B*a^2*b^3 + 9*A*a*b^4)*x^(7 /2) + 896*(B*a^3*b^2 + 9*A*a^2*b^3)*x^(5/2) + 790*(B*a^4*b + 9*A*a^3*b^2)* x^(3/2) - 15*(7*B*a^5 - 193*A*a^4*b)*sqrt(x))/(a^5*b^6*x^5 + 5*a^6*b^5*x^4 + 10*a^7*b^4*x^3 + 10*a^8*b^3*x^2 + 5*a^9*b^2*x + a^10*b) + 7/128*(B*a + 9*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5*b)
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {7 \, {\left (B a + 9 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{5} b} + \frac {105 \, B a b^{4} x^{\frac {9}{2}} + 945 \, A b^{5} x^{\frac {9}{2}} + 490 \, B a^{2} b^{3} x^{\frac {7}{2}} + 4410 \, A a b^{4} x^{\frac {7}{2}} + 896 \, B a^{3} b^{2} x^{\frac {5}{2}} + 8064 \, A a^{2} b^{3} x^{\frac {5}{2}} + 790 \, B a^{4} b x^{\frac {3}{2}} + 7110 \, A a^{3} b^{2} x^{\frac {3}{2}} - 105 \, B a^{5} \sqrt {x} + 2895 \, A a^{4} b \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a^{5} b} \]
7/128*(B*a + 9*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5*b) + 1/1920 *(105*B*a*b^4*x^(9/2) + 945*A*b^5*x^(9/2) + 490*B*a^2*b^3*x^(7/2) + 4410*A *a*b^4*x^(7/2) + 896*B*a^3*b^2*x^(5/2) + 8064*A*a^2*b^3*x^(5/2) + 790*B*a^ 4*b*x^(3/2) + 7110*A*a^3*b^2*x^(3/2) - 105*B*a^5*sqrt(x) + 2895*A*a^4*b*sq rt(x))/((b*x + a)^5*a^5*b)
Time = 10.18 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {79\,x^{3/2}\,\left (9\,A\,b+B\,a\right )}{192\,a^2}+\frac {49\,b^2\,x^{7/2}\,\left (9\,A\,b+B\,a\right )}{192\,a^4}+\frac {7\,b^3\,x^{9/2}\,\left (9\,A\,b+B\,a\right )}{128\,a^5}+\frac {\sqrt {x}\,\left (193\,A\,b-7\,B\,a\right )}{128\,a\,b}+\frac {7\,b\,x^{5/2}\,\left (9\,A\,b+B\,a\right )}{15\,a^3}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5}+\frac {7\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (9\,A\,b+B\,a\right )}{128\,a^{11/2}\,b^{3/2}} \]
((79*x^(3/2)*(9*A*b + B*a))/(192*a^2) + (49*b^2*x^(7/2)*(9*A*b + B*a))/(19 2*a^4) + (7*b^3*x^(9/2)*(9*A*b + B*a))/(128*a^5) + (x^(1/2)*(193*A*b - 7*B *a))/(128*a*b) + (7*b*x^(5/2)*(9*A*b + B*a))/(15*a^3))/(a^5 + b^5*x^5 + 5* a*b^4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x) + (7*atan((b^(1/2 )*x^(1/2))/a^(1/2))*(9*A*b + B*a))/(128*a^(11/2)*b^(3/2))