Integrand size = 29, antiderivative size = 195 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {(A b-a B) x^{7/2}}{5 a b (a+b x)^5}-\frac {(3 A b+7 a B) x^{5/2}}{40 a b^2 (a+b x)^4}-\frac {(3 A b+7 a B) x^{3/2}}{48 a b^3 (a+b x)^3}-\frac {(3 A b+7 a B) \sqrt {x}}{64 a b^4 (a+b x)^2}+\frac {(3 A b+7 a B) \sqrt {x}}{128 a^2 b^4 (a+b x)}+\frac {(3 A b+7 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{5/2} b^{9/2}} \]
1/5*(A*b-B*a)*x^(7/2)/a/b/(b*x+a)^5-1/40*(3*A*b+7*B*a)*x^(5/2)/a/b^2/(b*x+ a)^4-1/48*(3*A*b+7*B*a)*x^(3/2)/a/b^3/(b*x+a)^3+1/128*(3*A*b+7*B*a)*arctan (b^(1/2)*x^(1/2)/a^(1/2))/a^(5/2)/b^(9/2)-1/64*(3*A*b+7*B*a)*x^(1/2)/a/b^4 /(b*x+a)^2+1/128*(3*A*b+7*B*a)*x^(1/2)/a^2/b^4/(b*x+a)
Time = 0.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.74 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\sqrt {x} \left (-105 a^5 B+45 A b^5 x^4+105 a b^4 x^3 (2 A+B x)-14 a^3 b^2 x (15 A+64 B x)-5 a^4 b (9 A+98 B x)-2 a^2 b^3 x^2 (192 A+395 B x)\right )}{1920 a^2 b^4 (a+b x)^5}+\frac {(3 A b+7 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{5/2} b^{9/2}} \]
(Sqrt[x]*(-105*a^5*B + 45*A*b^5*x^4 + 105*a*b^4*x^3*(2*A + B*x) - 14*a^3*b ^2*x*(15*A + 64*B*x) - 5*a^4*b*(9*A + 98*B*x) - 2*a^2*b^3*x^2*(192*A + 395 *B*x)))/(1920*a^2*b^4*(a + b*x)^5) + ((3*A*b + 7*a*B)*ArcTan[(Sqrt[b]*Sqrt [x])/Sqrt[a]])/(128*a^(5/2)*b^(9/2))
Time = 0.27 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.90, 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, 51, 51, 51, 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.
| \(\displaystyle \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {x^{5/2} (A+B x)}{b^6 (a+b x)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {x^{5/2} (A+B x)}{(a+b x)^6}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(7 a B+3 A b) \int \frac {x^{5/2}}{(a+b x)^5}dx}{10 a b}+\frac {x^{7/2} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(7 a B+3 A b) \left (\frac {5 \int \frac {x^{3/2}}{(a+b x)^4}dx}{8 b}-\frac {x^{5/2}}{4 b (a+b x)^4}\right )}{10 a b}+\frac {x^{7/2} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(7 a B+3 A b) \left (\frac {5 \left (\frac {\int \frac {\sqrt {x}}{(a+b x)^3}dx}{2 b}-\frac {x^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {x^{5/2}}{4 b (a+b x)^4}\right )}{10 a b}+\frac {x^{7/2} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(7 a B+3 A b) \left (\frac {5 \left (\frac {\frac {\int \frac {1}{\sqrt {x} (a+b x)^2}dx}{4 b}-\frac {\sqrt {x}}{2 b (a+b x)^2}}{2 b}-\frac {x^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {x^{5/2}}{4 b (a+b x)^4}\right )}{10 a b}+\frac {x^{7/2} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(7 a B+3 A b) \left (\frac {5 \left (\frac {\frac {\frac {\int \frac {1}{\sqrt {x} (a+b x)}dx}{2 a}+\frac {\sqrt {x}}{a (a+b x)}}{4 b}-\frac {\sqrt {x}}{2 b (a+b x)^2}}{2 b}-\frac {x^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {x^{5/2}}{4 b (a+b x)^4}\right )}{10 a b}+\frac {x^{7/2} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(7 a B+3 A b) \left (\frac {5 \left (\frac {\frac {\frac {\int \frac {1}{a+b x}d\sqrt {x}}{a}+\frac {\sqrt {x}}{a (a+b x)}}{4 b}-\frac {\sqrt {x}}{2 b (a+b x)^2}}{2 b}-\frac {x^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {x^{5/2}}{4 b (a+b x)^4}\right )}{10 a b}+\frac {x^{7/2} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(7 a B+3 A b) \left (\frac {5 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a+b x)}}{4 b}-\frac {\sqrt {x}}{2 b (a+b x)^2}}{2 b}-\frac {x^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {x^{5/2}}{4 b (a+b x)^4}\right )}{10 a b}+\frac {x^{7/2} (A b-a B)}{5 a b (a+b x)^5}\) |
((A*b - a*B)*x^(7/2))/(5*a*b*(a + b*x)^5) + ((3*A*b + 7*a*B)*(-1/4*x^(5/2) /(b*(a + b*x)^4) + (5*(-1/3*x^(3/2)/(b*(a + b*x)^3) + (-1/2*Sqrt[x]/(b*(a + b*x)^2) + (Sqrt[x]/(a*(a + b*x)) + ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]/(a^ (3/2)*Sqrt[b]))/(4*b))/(2*b)))/(8*b)))/(10*a*b)
3.8.76.3.1 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/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[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] && 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 = 1.98 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (3 A b +7 B a \right ) x^{\frac {9}{2}}}{128 a^{2}}+\frac {\left (21 A b -79 B a \right ) x^{\frac {7}{2}}}{192 b a}-\frac {\left (3 A b +7 B a \right ) x^{\frac {5}{2}}}{15 b^{2}}-\frac {7 a \left (3 A b +7 B a \right ) x^{\frac {3}{2}}}{192 b^{3}}-\frac {\left (3 A b +7 B a \right ) a^{2} \sqrt {x}}{128 b^{4}}}{\left (b x +a \right )^{5}}+\frac {\left (3 A b +7 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 a^{2} b^{4} \sqrt {b a}}\) | \(137\) |
| default | \(\frac {\frac {\left (3 A b +7 B a \right ) x^{\frac {9}{2}}}{128 a^{2}}+\frac {\left (21 A b -79 B a \right ) x^{\frac {7}{2}}}{192 b a}-\frac {\left (3 A b +7 B a \right ) x^{\frac {5}{2}}}{15 b^{2}}-\frac {7 a \left (3 A b +7 B a \right ) x^{\frac {3}{2}}}{192 b^{3}}-\frac {\left (3 A b +7 B a \right ) a^{2} \sqrt {x}}{128 b^{4}}}{\left (b x +a \right )^{5}}+\frac {\left (3 A b +7 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 a^{2} b^{4} \sqrt {b a}}\) | \(137\) |
2*(1/256*(3*A*b+7*B*a)/a^2*x^(9/2)+1/384*(21*A*b-79*B*a)/b/a*x^(7/2)-1/30/ b^2*(3*A*b+7*B*a)*x^(5/2)-7/384*a/b^3*(3*A*b+7*B*a)*x^(3/2)-1/256*(3*A*b+7 *B*a)*a^2/b^4*x^(1/2))/(b*x+a)^5+1/128*(3*A*b+7*B*a)/a^2/b^4/(b*a)^(1/2)*a rctan(b*x^(1/2)/(b*a)^(1/2))
Time = 0.42 (sec) , antiderivative size = 657, normalized size of antiderivative = 3.37 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {15 \, {\left (7 \, B a^{6} + 3 \, A a^{5} b + {\left (7 \, B a b^{5} + 3 \, A b^{6}\right )} x^{5} + 5 \, {\left (7 \, B a^{2} b^{4} + 3 \, A a b^{5}\right )} x^{4} + 10 \, {\left (7 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (7 \, B a^{4} b^{2} + 3 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (7 \, B a^{5} b + 3 \, 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 + 45 \, A a^{5} b^{2} - 15 \, {\left (7 \, B a^{2} b^{5} + 3 \, A a b^{6}\right )} x^{4} + 10 \, {\left (79 \, B a^{3} b^{4} - 21 \, A a^{2} b^{5}\right )} x^{3} + 128 \, {\left (7 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4}\right )} x^{2} + 70 \, {\left (7 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{3840 \, {\left (a^{3} b^{10} x^{5} + 5 \, a^{4} b^{9} x^{4} + 10 \, a^{5} b^{8} x^{3} + 10 \, a^{6} b^{7} x^{2} + 5 \, a^{7} b^{6} x + a^{8} b^{5}\right )}}, -\frac {15 \, {\left (7 \, B a^{6} + 3 \, A a^{5} b + {\left (7 \, B a b^{5} + 3 \, A b^{6}\right )} x^{5} + 5 \, {\left (7 \, B a^{2} b^{4} + 3 \, A a b^{5}\right )} x^{4} + 10 \, {\left (7 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (7 \, B a^{4} b^{2} + 3 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (7 \, B a^{5} b + 3 \, 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 + 45 \, A a^{5} b^{2} - 15 \, {\left (7 \, B a^{2} b^{5} + 3 \, A a b^{6}\right )} x^{4} + 10 \, {\left (79 \, B a^{3} b^{4} - 21 \, A a^{2} b^{5}\right )} x^{3} + 128 \, {\left (7 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4}\right )} x^{2} + 70 \, {\left (7 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{3} b^{10} x^{5} + 5 \, a^{4} b^{9} x^{4} + 10 \, a^{5} b^{8} x^{3} + 10 \, a^{6} b^{7} x^{2} + 5 \, a^{7} b^{6} x + a^{8} b^{5}\right )}}\right ] \]
[-1/3840*(15*(7*B*a^6 + 3*A*a^5*b + (7*B*a*b^5 + 3*A*b^6)*x^5 + 5*(7*B*a^2 *b^4 + 3*A*a*b^5)*x^4 + 10*(7*B*a^3*b^3 + 3*A*a^2*b^4)*x^3 + 10*(7*B*a^4*b ^2 + 3*A*a^3*b^3)*x^2 + 5*(7*B*a^5*b + 3*A*a^4*b^2)*x)*sqrt(-a*b)*log((b*x - a - 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) + 2*(105*B*a^6*b + 45*A*a^5*b^2 - 15*(7*B*a^2*b^5 + 3*A*a*b^6)*x^4 + 10*(79*B*a^3*b^4 - 21*A*a^2*b^5)*x^3 + 128*(7*B*a^4*b^3 + 3*A*a^3*b^4)*x^2 + 70*(7*B*a^5*b^2 + 3*A*a^4*b^3)*x)*sq rt(x))/(a^3*b^10*x^5 + 5*a^4*b^9*x^4 + 10*a^5*b^8*x^3 + 10*a^6*b^7*x^2 + 5 *a^7*b^6*x + a^8*b^5), -1/1920*(15*(7*B*a^6 + 3*A*a^5*b + (7*B*a*b^5 + 3*A *b^6)*x^5 + 5*(7*B*a^2*b^4 + 3*A*a*b^5)*x^4 + 10*(7*B*a^3*b^3 + 3*A*a^2*b^ 4)*x^3 + 10*(7*B*a^4*b^2 + 3*A*a^3*b^3)*x^2 + 5*(7*B*a^5*b + 3*A*a^4*b^2)* x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (105*B*a^6*b + 45*A*a^5*b^2 - 15*(7*B*a^2*b^5 + 3*A*a*b^6)*x^4 + 10*(79*B*a^3*b^4 - 21*A*a^2*b^5)*x^3 + 128*(7*B*a^4*b^3 + 3*A*a^3*b^4)*x^2 + 70*(7*B*a^5*b^2 + 3*A*a^4*b^3)*x)*s qrt(x))/(a^3*b^10*x^5 + 5*a^4*b^9*x^4 + 10*a^5*b^8*x^3 + 10*a^6*b^7*x^2 + 5*a^7*b^6*x + a^8*b^5)]
Timed out. \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.05 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {15 \, {\left (7 \, B a b^{4} + 3 \, A b^{5}\right )} x^{\frac {9}{2}} - 10 \, {\left (79 \, B a^{2} b^{3} - 21 \, A a b^{4}\right )} x^{\frac {7}{2}} - 128 \, {\left (7 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} x^{\frac {5}{2}} - 70 \, {\left (7 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x^{\frac {3}{2}} - 15 \, {\left (7 \, B a^{5} + 3 \, A a^{4} b\right )} \sqrt {x}}{1920 \, {\left (a^{2} b^{9} x^{5} + 5 \, a^{3} b^{8} x^{4} + 10 \, a^{4} b^{7} x^{3} + 10 \, a^{5} b^{6} x^{2} + 5 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} + \frac {{\left (7 \, B a + 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{2} b^{4}} \]
1/1920*(15*(7*B*a*b^4 + 3*A*b^5)*x^(9/2) - 10*(79*B*a^2*b^3 - 21*A*a*b^4)* x^(7/2) - 128*(7*B*a^3*b^2 + 3*A*a^2*b^3)*x^(5/2) - 70*(7*B*a^4*b + 3*A*a^ 3*b^2)*x^(3/2) - 15*(7*B*a^5 + 3*A*a^4*b)*sqrt(x))/(a^2*b^9*x^5 + 5*a^3*b^ 8*x^4 + 10*a^4*b^7*x^3 + 10*a^5*b^6*x^2 + 5*a^6*b^5*x + a^7*b^4) + 1/128*( 7*B*a + 3*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2*b^4)
Time = 0.27 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.80 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {{\left (7 \, B a + 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{2} b^{4}} + \frac {105 \, B a b^{4} x^{\frac {9}{2}} + 45 \, A b^{5} x^{\frac {9}{2}} - 790 \, B a^{2} b^{3} x^{\frac {7}{2}} + 210 \, A a b^{4} x^{\frac {7}{2}} - 896 \, B a^{3} b^{2} x^{\frac {5}{2}} - 384 \, A a^{2} b^{3} x^{\frac {5}{2}} - 490 \, B a^{4} b x^{\frac {3}{2}} - 210 \, A a^{3} b^{2} x^{\frac {3}{2}} - 105 \, B a^{5} \sqrt {x} - 45 \, A a^{4} b \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a^{2} b^{4}} \]
1/128*(7*B*a + 3*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2*b^4) + 1/ 1920*(105*B*a*b^4*x^(9/2) + 45*A*b^5*x^(9/2) - 790*B*a^2*b^3*x^(7/2) + 210 *A*a*b^4*x^(7/2) - 896*B*a^3*b^2*x^(5/2) - 384*A*a^2*b^3*x^(5/2) - 490*B*a ^4*b*x^(3/2) - 210*A*a^3*b^2*x^(3/2) - 105*B*a^5*sqrt(x) - 45*A*a^4*b*sqrt (x))/((b*x + a)^5*a^2*b^4)
Time = 10.11 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.90 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (3\,A\,b+7\,B\,a\right )}{128\,a^{5/2}\,b^{9/2}}-\frac {\frac {x^{5/2}\,\left (3\,A\,b+7\,B\,a\right )}{15\,b^2}-\frac {x^{9/2}\,\left (3\,A\,b+7\,B\,a\right )}{128\,a^2}+\frac {a^2\,\sqrt {x}\,\left (3\,A\,b+7\,B\,a\right )}{128\,b^4}-\frac {x^{7/2}\,\left (21\,A\,b-79\,B\,a\right )}{192\,a\,b}+\frac {7\,a\,x^{3/2}\,\left (3\,A\,b+7\,B\,a\right )}{192\,b^3}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \]
(atan((b^(1/2)*x^(1/2))/a^(1/2))*(3*A*b + 7*B*a))/(128*a^(5/2)*b^(9/2)) - ((x^(5/2)*(3*A*b + 7*B*a))/(15*b^2) - (x^(9/2)*(3*A*b + 7*B*a))/(128*a^2) + (a^2*x^(1/2)*(3*A*b + 7*B*a))/(128*b^4) - (x^(7/2)*(21*A*b - 79*B*a))/(1 92*a*b) + (7*a*x^(3/2)*(3*A*b + 7*B*a))/(192*b^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)