Integrand size = 31, antiderivative size = 404 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 a (5 A b-13 a B) \sqrt {x} (a+b x)}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 (5 A b-13 a B) x^{3/2} (a+b x)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (5 A b-13 a B) x^{5/2} (a+b x)}{320 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 a^{3/2} (5 A b-13 a B) (a+b x) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
33/64*(5*A*b-13*B*a)*x^(7/2)/a/b^4/((b*x+a)^2)^(1/2)+1/4*(A*b-B*a)*x^(13/2 )/a/b/(b*x+a)^3/((b*x+a)^2)^(1/2)+1/24*(5*A*b-13*B*a)*x^(11/2)/a/b^2/(b*x+ a)^2/((b*x+a)^2)^(1/2)+11/96*(5*A*b-13*B*a)*x^(9/2)/a/b^3/(b*x+a)/((b*x+a) ^2)^(1/2)+77/64*(5*A*b-13*B*a)*x^(3/2)*(b*x+a)/b^6/((b*x+a)^2)^(1/2)-231/3 20*(5*A*b-13*B*a)*x^(5/2)*(b*x+a)/a/b^5/((b*x+a)^2)^(1/2)+231/64*a^(3/2)*( 5*A*b-13*B*a)*(b*x+a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/b^(15/2)/((b*x+a)^2) ^(1/2)-231/64*a*(5*A*b-13*B*a)*(b*x+a)*x^(1/2)/b^7/((b*x+a)^2)^(1/2)
Time = 0.30 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.47 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {b} \sqrt {x} \left (45045 a^6 B-1155 a^5 b (15 A-143 B x)+128 b^6 x^5 (5 A+3 B x)-128 a b^5 x^4 (55 A+13 B x)+231 a^4 b^2 x (-275 A+949 B x)+11 a^2 b^4 x^3 (-4185 A+1664 B x)+33 a^3 b^3 x^2 (-2555 A+3627 B x)\right )-3465 a^{3/2} (-5 A b+13 a B) (a+b x)^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{960 b^{15/2} (a+b x)^3 \sqrt {(a+b x)^2}} \]
(Sqrt[b]*Sqrt[x]*(45045*a^6*B - 1155*a^5*b*(15*A - 143*B*x) + 128*b^6*x^5* (5*A + 3*B*x) - 128*a*b^5*x^4*(55*A + 13*B*x) + 231*a^4*b^2*x*(-275*A + 94 9*B*x) + 11*a^2*b^4*x^3*(-4185*A + 1664*B*x) + 33*a^3*b^3*x^2*(-2555*A + 3 627*B*x)) - 3465*a^(3/2)*(-5*A*b + 13*a*B)*(a + b*x)^4*ArcTan[(Sqrt[b]*Sqr t[x])/Sqrt[a]])/(960*b^(15/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])
Time = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.58, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {1187, 27, 87, 51, 51, 51, 60, 60, 60, 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.
| \(\displaystyle \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {x^{11/2} (A+B x)}{b^5 (a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {x^{11/2} (A+B x)}{(a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(a+b x) \left (\frac {x^{13/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(5 A b-13 a B) \int \frac {x^{11/2}}{(a+b x)^4}dx}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {x^{13/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(5 A b-13 a B) \left (\frac {11 \int \frac {x^{9/2}}{(a+b x)^3}dx}{6 b}-\frac {x^{11/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {x^{13/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(5 A b-13 a B) \left (\frac {11 \left (\frac {9 \int \frac {x^{7/2}}{(a+b x)^2}dx}{4 b}-\frac {x^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {x^{11/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {x^{13/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(5 A b-13 a B) \left (\frac {11 \left (\frac {9 \left (\frac {7 \int \frac {x^{5/2}}{a+b x}dx}{2 b}-\frac {x^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {x^{11/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {x^{13/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(5 A b-13 a B) \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {2 x^{5/2}}{5 b}-\frac {a \int \frac {x^{3/2}}{a+b x}dx}{b}\right )}{2 b}-\frac {x^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {x^{11/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {x^{13/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(5 A b-13 a B) \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 x^{3/2}}{3 b}-\frac {a \int \frac {\sqrt {x}}{a+b x}dx}{b}\right )}{b}\right )}{2 b}-\frac {x^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {x^{11/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {x^{13/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(5 A b-13 a B) \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 x^{3/2}}{3 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {a \int \frac {1}{\sqrt {x} (a+b x)}dx}{b}\right )}{b}\right )}{b}\right )}{2 b}-\frac {x^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {x^{11/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (\frac {x^{13/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(5 A b-13 a B) \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 x^{3/2}}{3 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \int \frac {1}{a+b x}d\sqrt {x}}{b}\right )}{b}\right )}{b}\right )}{2 b}-\frac {x^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {x^{11/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(a+b x) \left (\frac {x^{13/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(5 A b-13 a B) \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 x^{3/2}}{3 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}}\right )}{b}\right )}{b}\right )}{2 b}-\frac {x^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {x^{11/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(((A*b - a*B)*x^(13/2))/(4*a*b*(a + b*x)^4) - ((5*A*b - 13*a*B) *(-1/3*x^(11/2)/(b*(a + b*x)^3) + (11*(-1/2*x^(9/2)/(b*(a + b*x)^2) + (9*( -(x^(7/2)/(b*(a + b*x))) + (7*((2*x^(5/2))/(5*b) - (a*((2*x^(3/2))/(3*b) - (a*((2*Sqrt[x])/b - (2*Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(3/2) ))/b))/b))/(2*b)))/(4*b)))/(6*b)))/(8*a*b)))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.9.26.3.1 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/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 0.42 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(-\frac {2 \left (-3 b^{2} B \,x^{2}-5 A \,b^{2} x +25 B a b x +75 A b a -225 B \,a^{2}\right ) \sqrt {x}\, \sqrt {\left (b x +a \right )^{2}}}{15 b^{7} \left (b x +a \right )}+\frac {a^{2} \left (\frac {2 \left (-\frac {765}{128} A \,b^{4}+\frac {1477}{128} B \,b^{3} a \right ) x^{\frac {7}{2}}+2 \left (-\frac {5855}{384} A \,b^{3} a +\frac {11767}{384} B \,a^{2} b^{2}\right ) x^{\frac {5}{2}}-\frac {a^{2} b \left (5153 A b -10633 B a \right ) x^{\frac {3}{2}}}{192}+2 \left (-\frac {515}{128} A \,a^{3} b +\frac {1083}{128} B \,a^{4}\right ) \sqrt {x}}{\left (b x +a \right )^{4}}+\frac {231 \left (5 A b -13 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{64 \sqrt {b a}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{7} \left (b x +a \right )}\) | \(196\) |
| default | \(\frac {\left (384 B \,x^{\frac {13}{2}} \sqrt {b a}\, b^{6}+640 A \,x^{\frac {11}{2}} \sqrt {b a}\, b^{6}-1664 B \,x^{\frac {11}{2}} \sqrt {b a}\, a \,b^{5}-7040 A \,x^{\frac {9}{2}} \sqrt {b a}\, a \,b^{5}+18304 B \,x^{\frac {9}{2}} \sqrt {b a}\, a^{2} b^{4}-46035 A \,x^{\frac {7}{2}} \sqrt {b a}\, a^{2} b^{4}+119691 B \,x^{\frac {7}{2}} \sqrt {b a}\, a^{3} b^{3}-84315 A \,x^{\frac {5}{2}} \sqrt {b a}\, a^{3} b^{3}+17325 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{5} x^{4}+219219 B \,x^{\frac {5}{2}} \sqrt {b a}\, a^{4} b^{2}-45045 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b^{4} x^{4}+69300 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b^{4} x^{3}-180180 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{4} b^{3} x^{3}-63525 A \,x^{\frac {3}{2}} \sqrt {b a}\, a^{4} b^{2}+103950 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{4} b^{3} x^{2}+165165 B \,x^{\frac {3}{2}} \sqrt {b a}\, a^{5} b -270270 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{5} b^{2} x^{2}+69300 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{5} b^{2} x -180180 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{6} b x -17325 A \sqrt {x}\, \sqrt {b a}\, a^{5} b +17325 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{6} b +45045 B \sqrt {x}\, \sqrt {b a}\, a^{6}-45045 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{7}\right ) \left (b x +a \right )}{960 \sqrt {b a}\, b^{7} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(443\) |
-2/15*(-3*B*b^2*x^2-5*A*b^2*x+25*B*a*b*x+75*A*a*b-225*B*a^2)*x^(1/2)/b^7*( (b*x+a)^2)^(1/2)/(b*x+a)+a^2/b^7*(2*((-765/128*A*b^4+1477/128*B*b^3*a)*x^( 7/2)+(-5855/384*A*b^3*a+11767/384*B*a^2*b^2)*x^(5/2)-1/384*a^2*b*(5153*A*b -10633*B*a)*x^(3/2)+(-515/128*A*a^3*b+1083/128*B*a^4)*x^(1/2))/(b*x+a)^4+2 31/64*(5*A*b-13*B*a)/(b*a)^(1/2)*arctan(b*x^(1/2)/(b*a)^(1/2)))*((b*x+a)^2 )^(1/2)/(b*x+a)
Time = 0.34 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.59 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [-\frac {3465 \, {\left (13 \, B a^{6} - 5 \, A a^{5} b + {\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 4 \, {\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 6 \, {\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 4 \, {\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (384 \, B b^{6} x^{6} + 45045 \, B a^{6} - 17325 \, A a^{5} b - 128 \, {\left (13 \, B a b^{5} - 5 \, A b^{6}\right )} x^{5} + 1408 \, {\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 9207 \, {\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 16863 \, {\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 12705 \, {\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{1920 \, {\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}}, -\frac {3465 \, {\left (13 \, B a^{6} - 5 \, A a^{5} b + {\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 4 \, {\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 6 \, {\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 4 \, {\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (384 \, B b^{6} x^{6} + 45045 \, B a^{6} - 17325 \, A a^{5} b - 128 \, {\left (13 \, B a b^{5} - 5 \, A b^{6}\right )} x^{5} + 1408 \, {\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 9207 \, {\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 16863 \, {\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 12705 \, {\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{960 \, {\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}}\right ] \]
[-1/1920*(3465*(13*B*a^6 - 5*A*a^5*b + (13*B*a^2*b^4 - 5*A*a*b^5)*x^4 + 4* (13*B*a^3*b^3 - 5*A*a^2*b^4)*x^3 + 6*(13*B*a^4*b^2 - 5*A*a^3*b^3)*x^2 + 4* (13*B*a^5*b - 5*A*a^4*b^2)*x)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(384*B*b^6*x^6 + 45045*B*a^6 - 17325*A*a^5*b - 128*(1 3*B*a*b^5 - 5*A*b^6)*x^5 + 1408*(13*B*a^2*b^4 - 5*A*a*b^5)*x^4 + 9207*(13* B*a^3*b^3 - 5*A*a^2*b^4)*x^3 + 16863*(13*B*a^4*b^2 - 5*A*a^3*b^3)*x^2 + 12 705*(13*B*a^5*b - 5*A*a^4*b^2)*x)*sqrt(x))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^ 2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7), -1/960*(3465*(13*B*a^6 - 5*A*a^5*b + ( 13*B*a^2*b^4 - 5*A*a*b^5)*x^4 + 4*(13*B*a^3*b^3 - 5*A*a^2*b^4)*x^3 + 6*(13 *B*a^4*b^2 - 5*A*a^3*b^3)*x^2 + 4*(13*B*a^5*b - 5*A*a^4*b^2)*x)*sqrt(a/b)* arctan(b*sqrt(x)*sqrt(a/b)/a) - (384*B*b^6*x^6 + 45045*B*a^6 - 17325*A*a^5 *b - 128*(13*B*a*b^5 - 5*A*b^6)*x^5 + 1408*(13*B*a^2*b^4 - 5*A*a*b^5)*x^4 + 9207*(13*B*a^3*b^3 - 5*A*a^2*b^4)*x^3 + 16863*(13*B*a^4*b^2 - 5*A*a^3*b^ 3)*x^2 + 12705*(13*B*a^5*b - 5*A*a^4*b^2)*x)*sqrt(x))/(b^11*x^4 + 4*a*b^10 *x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7)]
Timed out. \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Time = 0.39 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.99 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {256 \, {\left (3 \, B b^{6} x^{2} + 5 \, B a b^{5} x\right )} x^{\frac {11}{2}} + 5 \, {\left (2747 \, {\left (3 \, B a b^{5} - A b^{6}\right )} x^{2} + 437 \, {\left (13 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} x\right )} x^{\frac {9}{2}} + 10 \, {\left (4667 \, {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} x^{2} + 671 \, {\left (13 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x\right )} x^{\frac {7}{2}} + 2860 \, {\left (22 \, {\left (3 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{2} + 3 \, {\left (13 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x\right )} x^{\frac {5}{2}} + 66 \, {\left (585 \, {\left (3 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 77 \, {\left (13 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} x\right )} x^{\frac {3}{2}} + 231 \, {\left (39 \, {\left (3 \, B a^{5} b - A a^{4} b^{2}\right )} x^{2} + 5 \, {\left (13 \, B a^{6} - 3 \, A a^{5} b\right )} x\right )} \sqrt {x}}{1920 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} - \frac {231 \, {\left (13 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} b^{7}} - \frac {77 \, {\left (13 \, {\left (3 \, B a b - A b^{2}\right )} x^{\frac {3}{2}} - 6 \, {\left (13 \, B a^{2} - 5 \, A a b\right )} \sqrt {x}\right )}}{128 \, b^{7}} \]
1/1920*(256*(3*B*b^6*x^2 + 5*B*a*b^5*x)*x^(11/2) + 5*(2747*(3*B*a*b^5 - A* b^6)*x^2 + 437*(13*B*a^2*b^4 - 3*A*a*b^5)*x)*x^(9/2) + 10*(4667*(3*B*a^2*b ^4 - A*a*b^5)*x^2 + 671*(13*B*a^3*b^3 - 3*A*a^2*b^4)*x)*x^(7/2) + 2860*(22 *(3*B*a^3*b^3 - A*a^2*b^4)*x^2 + 3*(13*B*a^4*b^2 - 3*A*a^3*b^3)*x)*x^(5/2) + 66*(585*(3*B*a^4*b^2 - A*a^3*b^3)*x^2 + 77*(13*B*a^5*b - 3*A*a^4*b^2)*x )*x^(3/2) + 231*(39*(3*B*a^5*b - A*a^4*b^2)*x^2 + 5*(13*B*a^6 - 3*A*a^5*b) *x)*sqrt(x))/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 + 5*a^4*b^7*x + a^5*b^6) - 231/64*(13*B*a^3 - 5*A*a^2*b)*arctan(b*sqrt(x)/sq rt(a*b))/(sqrt(a*b)*b^7) - 77/128*(13*(3*B*a*b - A*b^2)*x^(3/2) - 6*(13*B* a^2 - 5*A*a*b)*sqrt(x))/b^7
Time = 0.27 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.54 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {231 \, {\left (13 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} b^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {4431 \, B a^{3} b^{3} x^{\frac {7}{2}} - 2295 \, A a^{2} b^{4} x^{\frac {7}{2}} + 11767 \, B a^{4} b^{2} x^{\frac {5}{2}} - 5855 \, A a^{3} b^{3} x^{\frac {5}{2}} + 10633 \, B a^{5} b x^{\frac {3}{2}} - 5153 \, A a^{4} b^{2} x^{\frac {3}{2}} + 3249 \, B a^{6} \sqrt {x} - 1545 \, A a^{5} b \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} b^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left (3 \, B b^{20} x^{\frac {5}{2}} - 25 \, B a b^{19} x^{\frac {3}{2}} + 5 \, A b^{20} x^{\frac {3}{2}} + 225 \, B a^{2} b^{18} \sqrt {x} - 75 \, A a b^{19} \sqrt {x}\right )}}{15 \, b^{25} \mathrm {sgn}\left (b x + a\right )} \]
-231/64*(13*B*a^3 - 5*A*a^2*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^7* sgn(b*x + a)) + 1/192*(4431*B*a^3*b^3*x^(7/2) - 2295*A*a^2*b^4*x^(7/2) + 1 1767*B*a^4*b^2*x^(5/2) - 5855*A*a^3*b^3*x^(5/2) + 10633*B*a^5*b*x^(3/2) - 5153*A*a^4*b^2*x^(3/2) + 3249*B*a^6*sqrt(x) - 1545*A*a^5*b*sqrt(x))/((b*x + a)^4*b^7*sgn(b*x + a)) + 2/15*(3*B*b^20*x^(5/2) - 25*B*a*b^19*x^(3/2) + 5*A*b^20*x^(3/2) + 225*B*a^2*b^18*sqrt(x) - 75*A*a*b^19*sqrt(x))/(b^25*sgn (b*x + a))
Timed out. \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {x^{11/2}\,\left (A+B\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]