Integrand size = 29, antiderivative size = 240 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {231 \sqrt {b} (13 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{15/2}} \]
-77/128*(13*A*b-3*B*a)/a^6/b/x^(3/2)+1/5*(A*b-B*a)/a/b/x^(3/2)/(b*x+a)^5+1 /40*(13*A*b-3*B*a)/a^2/b/x^(3/2)/(b*x+a)^4+11/240*(13*A*b-3*B*a)/a^3/b/x^( 3/2)/(b*x+a)^3+33/320*(13*A*b-3*B*a)/a^4/b/x^(3/2)/(b*x+a)^2+231/640*(13*A *b-3*B*a)/a^5/b/x^(3/2)/(b*x+a)+231/128*(13*A*b-3*B*a)*arctan(b^(1/2)*x^(1 /2)/a^(1/2))*b^(1/2)/a^(15/2)+231/128*(13*A*b-3*B*a)/a^7/x^(1/2)
Time = 0.35 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.72 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-\frac {\sqrt {a} \left (-45045 A b^6 x^6+1280 a^6 (A+3 B x)+1155 a b^5 x^5 (-182 A+9 B x)+462 a^2 b^4 x^4 (-832 A+105 B x)+66 a^3 b^3 x^3 (-5135 A+1344 B x)+55 a^4 b^2 x^2 (-2509 A+1422 B x)+5 a^5 b x (-3328 A+6369 B x)\right )}{x^{3/2} (a+b x)^5}+3465 \sqrt {b} (13 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{1920 a^{15/2}} \]
(-((Sqrt[a]*(-45045*A*b^6*x^6 + 1280*a^6*(A + 3*B*x) + 1155*a*b^5*x^5*(-18 2*A + 9*B*x) + 462*a^2*b^4*x^4*(-832*A + 105*B*x) + 66*a^3*b^3*x^3*(-5135* A + 1344*B*x) + 55*a^4*b^2*x^2*(-2509*A + 1422*B*x) + 5*a^5*b*x*(-3328*A + 6369*B*x)))/(x^(3/2)*(a + b*x)^5)) + 3465*Sqrt[b]*(13*A*b - 3*a*B)*ArcTan [(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(1920*a^(15/2))
Time = 0.29 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.89, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1184, 27, 87, 52, 52, 52, 52, 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.
| \(\displaystyle \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {A+B x}{b^6 x^{5/2} (a+b x)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {A+B x}{x^{5/2} (a+b x)^6}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(13 A b-3 a B) \int \frac {1}{x^{5/2} (a+b x)^5}dx}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \int \frac {1}{x^{5/2} (a+b x)^4}dx}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \int \frac {1}{x^{5/2} (a+b x)^3}dx}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \left (\frac {7 \int \frac {1}{x^{5/2} (a+b x)^2}dx}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\right )}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \int \frac {1}{x^{5/2} (a+b x)}dx}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\right )}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \int \frac {1}{x^{3/2} (a+b x)}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\right )}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {b \int \frac {1}{\sqrt {x} (a+b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\right )}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {2 b \int \frac {1}{a+b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\right )}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\right )}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\) |
(A*b - a*B)/(5*a*b*x^(3/2)*(a + b*x)^5) + ((13*A*b - 3*a*B)*(1/(4*a*x^(3/2 )*(a + b*x)^4) + (11*(1/(3*a*x^(3/2)*(a + b*x)^3) + (3*(1/(2*a*x^(3/2)*(a + b*x)^2) + (7*(1/(a*x^(3/2)*(a + b*x)) + (5*(-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))/(2 *a)))/(4*a)))/(2*a)))/(8*a)))/(10*a*b)
3.8.81.3.1 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] :> 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]
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 = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(-\frac {2 \left (-18 A b x +3 a B x +a A \right )}{3 a^{7} x^{\frac {3}{2}}}+\frac {b \left (\frac {2 \left (\frac {1467}{256} A \,b^{5}-\frac {437}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {b^{3} a \left (9629 A b -2931 B a \right ) x^{\frac {7}{2}}}{192}+2 \left (\frac {1253}{30} A \,a^{2} b^{3}-\frac {131}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+2 \left (\frac {12131}{384} A \,a^{3} b^{2}-\frac {1327}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+2 \left (\frac {2373}{256} A \,a^{4} b -\frac {843}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (13 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 \sqrt {b a}}\right )}{a^{7}}\) | \(165\) |
| derivativedivides | \(\frac {2 b \left (\frac {\left (\frac {1467}{256} A \,b^{5}-\frac {437}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {b^{3} a \left (9629 A b -2931 B a \right ) x^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} A \,a^{2} b^{3}-\frac {131}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+\left (\frac {12131}{384} A \,a^{3} b^{2}-\frac {1327}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+\left (\frac {2373}{256} A \,a^{4} b -\frac {843}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (13 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{256 \sqrt {b a}}\right )}{a^{7}}-\frac {2 A}{3 a^{6} x^{\frac {3}{2}}}-\frac {2 \left (-6 A b +B a \right )}{a^{7} \sqrt {x}}\) | \(168\) |
| default | \(\frac {2 b \left (\frac {\left (\frac {1467}{256} A \,b^{5}-\frac {437}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {b^{3} a \left (9629 A b -2931 B a \right ) x^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} A \,a^{2} b^{3}-\frac {131}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+\left (\frac {12131}{384} A \,a^{3} b^{2}-\frac {1327}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+\left (\frac {2373}{256} A \,a^{4} b -\frac {843}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (13 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{256 \sqrt {b a}}\right )}{a^{7}}-\frac {2 A}{3 a^{6} x^{\frac {3}{2}}}-\frac {2 \left (-6 A b +B a \right )}{a^{7} \sqrt {x}}\) | \(168\) |
-2/3*(-18*A*b*x+3*B*a*x+A*a)/a^7/x^(3/2)+1/a^7*b*(2*((1467/256*A*b^5-437/2 56*B*a*b^4)*x^(9/2)+1/384*b^3*a*(9629*A*b-2931*B*a)*x^(7/2)+(1253/30*A*a^2 *b^3-131/10*B*a^3*b^2)*x^(5/2)+(12131/384*A*a^3*b^2-1327/128*B*a^4*b)*x^(3 /2)+(2373/256*A*a^4*b-843/256*a^5*B)*x^(1/2))/(b*x+a)^5+231/128*(13*A*b-3* B*a)/(b*a)^(1/2)*arctan(b*x^(1/2)/(b*a)^(1/2)))
Time = 0.44 (sec) , antiderivative size = 734, normalized size of antiderivative = 3.06 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {3465 \, {\left ({\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{7} + 5 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{6} + 10 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{5} + 10 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + 5 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{3} + {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x^{2}\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 (1280 \, A a^{6} + 3465 \, {\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} + 16170 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} + 29568 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} + 26070 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} + 10615 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 1280 \, {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x\right )} \sqrt {x}}{3840 \, {\left (a^{7} b^{5} x^{7} + 5 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{5} + 10 \, a^{10} b^{2} x^{4} + 5 \, a^{11} b x^{3} + a^{12} x^{2}\right )}}, \frac {3465 \, {\left ({\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{7} + 5 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{6} + 10 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{5} + 10 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + 5 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{3} + {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (1280 \, A a^{6} + 3465 \, {\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} + 16170 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} + 29568 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} + 26070 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} + 10615 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 1280 \, {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{7} b^{5} x^{7} + 5 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{5} + 10 \, a^{10} b^{2} x^{4} + 5 \, a^{11} b x^{3} + a^{12} x^{2}\right )}}\right ] \]
[-1/3840*(3465*((3*B*a*b^5 - 13*A*b^6)*x^7 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)* x^6 + 10*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^5 + 10*(3*B*a^4*b^2 - 13*A*a^3*b^3 )*x^4 + 5*(3*B*a^5*b - 13*A*a^4*b^2)*x^3 + (3*B*a^6 - 13*A*a^5*b)*x^2)*sqr t(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 2*(1280*A*a^6 + 3465*(3*B*a*b^5 - 13*A*b^6)*x^6 + 16170*(3*B*a^2*b^4 - 13*A*a*b^5)*x^5 + 29568*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 + 26070*(3*B*a^4*b^2 - 13*A*a^3*b^ 3)*x^3 + 10615*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + 1280*(3*B*a^6 - 13*A*a^5*b )*x)*sqrt(x))/(a^7*b^5*x^7 + 5*a^8*b^4*x^6 + 10*a^9*b^3*x^5 + 10*a^10*b^2* x^4 + 5*a^11*b*x^3 + a^12*x^2), 1/1920*(3465*((3*B*a*b^5 - 13*A*b^6)*x^7 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)*x^6 + 10*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^5 + 10*(3*B*a^4*b^2 - 13*A*a^3*b^3)*x^4 + 5*(3*B*a^5*b - 13*A*a^4*b^2)*x^3 + ( 3*B*a^6 - 13*A*a^5*b)*x^2)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) - (12 80*A*a^6 + 3465*(3*B*a*b^5 - 13*A*b^6)*x^6 + 16170*(3*B*a^2*b^4 - 13*A*a*b ^5)*x^5 + 29568*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 + 26070*(3*B*a^4*b^2 - 13 *A*a^3*b^3)*x^3 + 10615*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + 1280*(3*B*a^6 - 1 3*A*a^5*b)*x)*sqrt(x))/(a^7*b^5*x^7 + 5*a^8*b^4*x^6 + 10*a^9*b^3*x^5 + 10* a^10*b^2*x^4 + 5*a^11*b*x^3 + a^12*x^2)]
Timed out. \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {1280 \, A a^{6} + 3465 \, {\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} + 16170 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} + 29568 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} + 26070 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} + 10615 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 1280 \, {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x}{1920 \, {\left (a^{7} b^{5} x^{\frac {13}{2}} + 5 \, a^{8} b^{4} x^{\frac {11}{2}} + 10 \, a^{9} b^{3} x^{\frac {9}{2}} + 10 \, a^{10} b^{2} x^{\frac {7}{2}} + 5 \, a^{11} b x^{\frac {5}{2}} + a^{12} x^{\frac {3}{2}}\right )}} - \frac {231 \, {\left (3 \, B a b - 13 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{7}} \]
-1/1920*(1280*A*a^6 + 3465*(3*B*a*b^5 - 13*A*b^6)*x^6 + 16170*(3*B*a^2*b^4 - 13*A*a*b^5)*x^5 + 29568*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 + 26070*(3*B*a ^4*b^2 - 13*A*a^3*b^3)*x^3 + 10615*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + 1280*( 3*B*a^6 - 13*A*a^5*b)*x)/(a^7*b^5*x^(13/2) + 5*a^8*b^4*x^(11/2) + 10*a^9*b ^3*x^(9/2) + 10*a^10*b^2*x^(7/2) + 5*a^11*b*x^(5/2) + a^12*x^(3/2)) - 231/ 128*(3*B*a*b - 13*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^7)
Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.75 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {231 \, {\left (3 \, B a b - 13 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{7}} - \frac {2 \, {\left (3 \, B a x - 18 \, A b x + A a\right )}}{3 \, a^{7} x^{\frac {3}{2}}} - \frac {6555 \, B a b^{5} x^{\frac {9}{2}} - 22005 \, A b^{6} x^{\frac {9}{2}} + 29310 \, B a^{2} b^{4} x^{\frac {7}{2}} - 96290 \, A a b^{5} x^{\frac {7}{2}} + 50304 \, B a^{3} b^{3} x^{\frac {5}{2}} - 160384 \, A a^{2} b^{4} x^{\frac {5}{2}} + 39810 \, B a^{4} b^{2} x^{\frac {3}{2}} - 121310 \, A a^{3} b^{3} x^{\frac {3}{2}} + 12645 \, B a^{5} b \sqrt {x} - 35595 \, A a^{4} b^{2} \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a^{7}} \]
-231/128*(3*B*a*b - 13*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^7) - 2/3*(3*B*a*x - 18*A*b*x + A*a)/(a^7*x^(3/2)) - 1/1920*(6555*B*a*b^5*x^(9 /2) - 22005*A*b^6*x^(9/2) + 29310*B*a^2*b^4*x^(7/2) - 96290*A*a*b^5*x^(7/2 ) + 50304*B*a^3*b^3*x^(5/2) - 160384*A*a^2*b^4*x^(5/2) + 39810*B*a^4*b^2*x ^(3/2) - 121310*A*a^3*b^3*x^(3/2) + 12645*B*a^5*b*sqrt(x) - 35595*A*a^4*b^ 2*sqrt(x))/((b*x + a)^5*a^7)
Time = 10.24 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {2\,x\,\left (13\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {2\,A}{3\,a}+\frac {869\,b^2\,x^3\,\left (13\,A\,b-3\,B\,a\right )}{64\,a^4}+\frac {77\,b^3\,x^4\,\left (13\,A\,b-3\,B\,a\right )}{5\,a^5}+\frac {539\,b^4\,x^5\,\left (13\,A\,b-3\,B\,a\right )}{64\,a^6}+\frac {231\,b^5\,x^6\,\left (13\,A\,b-3\,B\,a\right )}{128\,a^7}+\frac {2123\,b\,x^2\,\left (13\,A\,b-3\,B\,a\right )}{384\,a^3}}{a^5\,x^{3/2}+b^5\,x^{13/2}+5\,a^4\,b\,x^{5/2}+5\,a\,b^4\,x^{11/2}+10\,a^3\,b^2\,x^{7/2}+10\,a^2\,b^3\,x^{9/2}}+\frac {231\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (13\,A\,b-3\,B\,a\right )}{128\,a^{15/2}} \]
((2*x*(13*A*b - 3*B*a))/(3*a^2) - (2*A)/(3*a) + (869*b^2*x^3*(13*A*b - 3*B *a))/(64*a^4) + (77*b^3*x^4*(13*A*b - 3*B*a))/(5*a^5) + (539*b^4*x^5*(13*A *b - 3*B*a))/(64*a^6) + (231*b^5*x^6*(13*A*b - 3*B*a))/(128*a^7) + (2123*b *x^2*(13*A*b - 3*B*a))/(384*a^3))/(a^5*x^(3/2) + b^5*x^(13/2) + 5*a^4*b*x^ (5/2) + 5*a*b^4*x^(11/2) + 10*a^3*b^2*x^(7/2) + 10*a^2*b^3*x^(9/2)) + (231 *b^(1/2)*atan((b^(1/2)*x^(1/2))/a^(1/2))*(13*A*b - 3*B*a))/(128*a^(15/2))