Integrand size = 33, antiderivative size = 302 \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {e^2 (B d-A e)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {B d-A e}{3 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e (B d-A e)}{2 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^3 (B d-A e) (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^3 (B d-A e) (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
-e^2*(-A*e+B*d)/(-a*e+b*d)^4/((b*x+a)^2)^(1/2)+1/4*(-A*b+B*a)/b/(-a*e+b*d) /(b*x+a)^3/((b*x+a)^2)^(1/2)+1/3*(A*e-B*d)/(-a*e+b*d)^2/(b*x+a)^2/((b*x+a) ^2)^(1/2)+1/2*e*(-A*e+B*d)/(-a*e+b*d)^3/(b*x+a)/((b*x+a)^2)^(1/2)-e^3*(-A* e+B*d)*(b*x+a)*ln(b*x+a)/(-a*e+b*d)^5/((b*x+a)^2)^(1/2)+e^3*(-A*e+B*d)*(b* x+a)*ln(e*x+d)/(-a*e+b*d)^5/((b*x+a)^2)^(1/2)
Time = 1.10 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.60 \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {4 (b d-a e)^3 (-B d+A e)+\frac {3 (-A b+a B) (b d-a e)^4}{b (a+b x)}-6 e (b d-a e)^2 (-B d+A e) (a+b x)+12 e^2 (b d-a e) (-B d+A e) (a+b x)^2+12 e^3 (-B d+A e) (a+b x)^3 \log (a+b x)+12 e^3 (B d-A e) (a+b x)^3 \log (d+e x)}{12 (b d-a e)^5 \left ((a+b x)^2\right )^{3/2}} \]
(4*(b*d - a*e)^3*(-(B*d) + A*e) + (3*(-(A*b) + a*B)*(b*d - a*e)^4)/(b*(a + b*x)) - 6*e*(b*d - a*e)^2*(-(B*d) + A*e)*(a + b*x) + 12*e^2*(b*d - a*e)*( -(B*d) + A*e)*(a + b*x)^2 + 12*e^3*(-(B*d) + A*e)*(a + b*x)^3*Log[a + b*x] + 12*e^3*(B*d - A*e)*(a + b*x)^3*Log[d + e*x])/(12*(b*d - a*e)^5*((a + b* x)^2)^(3/2))
Time = 0.44 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 86, 2009}
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.
| \(\displaystyle \int \frac {A+B x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {A+B x}{b^5 (a+b x)^5 (d+e x)}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {A+B x}{(a+b x)^5 (d+e x)}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {(a+b x) \int \left (-\frac {(A e-B d) e^4}{(b d-a e)^5 (d+e x)}+\frac {b (A e-B d) e^3}{(b d-a e)^5 (a+b x)}-\frac {b (A e-B d) e^2}{(b d-a e)^4 (a+b x)^2}+\frac {b (A e-B d) e}{(b d-a e)^3 (a+b x)^3}+\frac {b (B d-A e)}{(b d-a e)^2 (a+b x)^4}+\frac {A b-a B}{(b d-a e) (a+b x)^5}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {e^3 \log (a+b x) (B d-A e)}{(b d-a e)^5}+\frac {e^3 (B d-A e) \log (d+e x)}{(b d-a e)^5}-\frac {e^2 (B d-A e)}{(a+b x) (b d-a e)^4}+\frac {e (B d-A e)}{2 (a+b x)^2 (b d-a e)^3}-\frac {B d-A e}{3 (a+b x)^3 (b d-a e)^2}-\frac {A b-a B}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/4*(A*b - a*B)/(b*(b*d - a*e)*(a + b*x)^4) - (B*d - A*e)/(3* (b*d - a*e)^2*(a + b*x)^3) + (e*(B*d - A*e))/(2*(b*d - a*e)^3*(a + b*x)^2) - (e^2*(B*d - A*e))/((b*d - a*e)^4*(a + b*x)) - (e^3*(B*d - A*e)*Log[a + b*x])/(b*d - a*e)^5 + (e^3*(B*d - A*e)*Log[d + e*x])/(b*d - a*e)^5))/Sqrt[ a^2 + 2*a*b*x + b^2*x^2]
3.18.81.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(623\) vs. \(2(230)=460\).
Time = 0.34 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.07
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {b^{3} e^{2} \left (A e -B d \right ) x^{3}}{e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}}+\frac {\left (7 a e -b d \right ) b^{2} e \left (A e -B d \right ) x^{2}}{2 e^{4} a^{4}-8 b d \,e^{3} a^{3}+12 b^{2} d^{2} e^{2} a^{2}-8 b^{3} d^{3} e a +2 b^{4} d^{4}}+\frac {b \left (13 A \,a^{2} e^{3}-5 A a b d \,e^{2}+A \,b^{2} d^{2} e -13 B \,a^{2} d \,e^{2}+5 B a b \,d^{2} e -B \,b^{2} d^{3}\right ) x}{3 e^{4} a^{4}-12 b d \,e^{3} a^{3}+18 b^{2} d^{2} e^{2} a^{2}-12 b^{3} d^{3} e a +3 b^{4} d^{4}}+\frac {25 A \,a^{3} b \,e^{3}-23 A \,a^{2} b^{2} d \,e^{2}+13 A a \,b^{3} d^{2} e -3 A \,d^{3} b^{4}-3 B \,e^{3} a^{4}-13 B \,a^{3} b d \,e^{2}+5 B \,a^{2} b^{2} d^{2} e -B a \,b^{3} d^{3}}{12 b \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}\right )}{\left (b x +a \right )^{5}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{3} \left (A e -B d \right ) \ln \left (b x +a \right )}{\left (b x +a \right ) \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{3} \left (A e -B d \right ) \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}\) | \(624\) |
| default | \(-\frac {\left (52 B \,a^{3} b^{2} d \,e^{3} x -48 A \ln \left (e x +d \right ) a^{3} b^{2} e^{4} x +12 B \ln \left (e x +d \right ) a^{4} b d \,e^{3}+12 B \ln \left (e x +d \right ) b^{5} d \,e^{3} x^{4}-48 A \ln \left (e x +d \right ) a \,b^{4} e^{4} x^{3}-72 A \ln \left (e x +d \right ) a^{2} b^{3} e^{4} x^{2}-48 B \,x^{2} a \,b^{4} d^{2} e^{2}+12 B \,x^{3} a \,b^{4} d \,e^{3}+48 A \,x^{2} a \,b^{4} d \,e^{3}+42 B \,x^{2} a^{2} b^{3} d \,e^{3}-24 A x a \,b^{4} d^{2} e^{2}-72 B x \,a^{2} b^{3} d^{2} e^{2}+24 B x a \,b^{4} d^{3} e +72 A x \,a^{2} b^{3} d \,e^{3}+48 A \ln \left (b x +a \right ) x \,a^{3} b^{2} e^{4}-12 B \ln \left (b x +a \right ) a^{4} b d \,e^{3}+48 A \ln \left (b x +a \right ) x^{3} a \,b^{4} e^{4}+72 A \ln \left (b x +a \right ) x^{2} a^{2} b^{3} e^{4}-12 B \ln \left (b x +a \right ) b^{5} d \,e^{3} x^{4}-B \,b^{4} d^{4} a -25 A \,a^{4} b \,e^{4}-4 B \,b^{5} d^{4} x -12 A \ln \left (e x +d \right ) b^{5} e^{4} x^{4}-12 A \ln \left (e x +d \right ) a^{4} b \,e^{4}+3 B \,a^{5} e^{4}-3 A \,b^{5} d^{4}-48 B \ln \left (b x +a \right ) x \,a^{3} b^{2} d \,e^{3}+48 A \,a^{3} b^{2} d \,e^{3}-36 A \,a^{2} b^{3} d^{2} e^{2}+10 B \,a^{4} b d \,e^{3}+48 B \ln \left (e x +d \right ) a \,b^{4} d \,e^{3} x^{3}-52 A \,a^{3} b^{2} e^{4} x +4 A \,b^{5} d^{3} e x +72 B \ln \left (e x +d \right ) a^{2} b^{3} d \,e^{3} x^{2}+48 B \ln \left (e x +d \right ) a^{3} b^{2} d \,e^{3} x +12 A \ln \left (b x +a \right ) b^{5} e^{4} x^{4}-48 B \ln \left (b x +a \right ) x^{3} a \,b^{4} d \,e^{3}-72 B \ln \left (b x +a \right ) x^{2} a^{2} b^{3} d \,e^{3}-18 B \,a^{3} b^{2} d^{2} e^{2}+6 B \,a^{2} b^{3} d^{3} e -42 A \,x^{2} a^{2} b^{3} e^{4}-6 A \,x^{2} b^{5} d^{2} e^{2}+6 B \,x^{2} b^{5} d^{3} e -12 A \,x^{3} a \,b^{4} e^{4}+12 A \,x^{3} b^{5} d \,e^{3}-12 B \,x^{3} b^{5} d^{2} e^{2}+12 A \ln \left (b x +a \right ) a^{4} b \,e^{4}+16 A \,b^{4} d^{3} e a \right ) \left (b x +a \right )}{12 b \left (a e -b d \right )^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(777\) |
((b*x+a)^2)^(1/2)/(b*x+a)^5*(b^3*e^2*(A*e-B*d)/(a^4*e^4-4*a^3*b*d*e^3+6*a^ 2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*x^3+1/2*(7*a*e-b*d)*b^2*e*(A*e-B*d)/( a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*x^2+1/3*b*( 13*A*a^2*e^3-5*A*a*b*d*e^2+A*b^2*d^2*e-13*B*a^2*d*e^2+5*B*a*b*d^2*e-B*b^2* d^3)/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*x+1/1 2*(25*A*a^3*b*e^3-23*A*a^2*b^2*d*e^2+13*A*a*b^3*d^2*e-3*A*b^4*d^3-3*B*a^4* e^3-13*B*a^3*b*d*e^2+5*B*a^2*b^2*d^2*e-B*a*b^3*d^3)/b/(a^4*e^4-4*a^3*b*d*e ^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4))-((b*x+a)^2)^(1/2)/(b*x+a)*e^3 *(A*e-B*d)/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5* a*b^4*d^4*e-b^5*d^5)*ln(b*x+a)+((b*x+a)^2)^(1/2)/(b*x+a)*e^3*(A*e-B*d)/(a^ 5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^ 5*d^5)*ln(-e*x-d)
Leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (230) = 460\).
Time = 0.70 (sec) , antiderivative size = 969, normalized size of antiderivative = 3.21 \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {{\left (B a b^{4} + 3 \, A b^{5}\right )} d^{4} - 2 \, {\left (3 \, B a^{2} b^{3} + 8 \, A a b^{4}\right )} d^{3} e + 18 \, {\left (B a^{3} b^{2} + 2 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 2 \, {\left (5 \, B a^{4} b + 24 \, A a^{3} b^{2}\right )} d e^{3} - {\left (3 \, B a^{5} - 25 \, A a^{4} b\right )} e^{4} + 12 \, {\left (B b^{5} d^{2} e^{2} + A a b^{4} e^{4} - {\left (B a b^{4} + A b^{5}\right )} d e^{3}\right )} x^{3} - 6 \, {\left (B b^{5} d^{3} e - 7 \, A a^{2} b^{3} e^{4} - {\left (8 \, B a b^{4} + A b^{5}\right )} d^{2} e^{2} + {\left (7 \, B a^{2} b^{3} + 8 \, A a b^{4}\right )} d e^{3}\right )} x^{2} + 4 \, {\left (B b^{5} d^{4} + 13 \, A a^{3} b^{2} e^{4} - {\left (6 \, B a b^{4} + A b^{5}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{2} - {\left (13 \, B a^{3} b^{2} + 18 \, A a^{2} b^{3}\right )} d e^{3}\right )} x + 12 \, {\left (B a^{4} b d e^{3} - A a^{4} b e^{4} + {\left (B b^{5} d e^{3} - A b^{5} e^{4}\right )} x^{4} + 4 \, {\left (B a b^{4} d e^{3} - A a b^{4} e^{4}\right )} x^{3} + 6 \, {\left (B a^{2} b^{3} d e^{3} - A a^{2} b^{3} e^{4}\right )} x^{2} + 4 \, {\left (B a^{3} b^{2} d e^{3} - A a^{3} b^{2} e^{4}\right )} x\right )} \log \left (b x + a\right ) - 12 \, {\left (B a^{4} b d e^{3} - A a^{4} b e^{4} + {\left (B b^{5} d e^{3} - A b^{5} e^{4}\right )} x^{4} + 4 \, {\left (B a b^{4} d e^{3} - A a b^{4} e^{4}\right )} x^{3} + 6 \, {\left (B a^{2} b^{3} d e^{3} - A a^{2} b^{3} e^{4}\right )} x^{2} + 4 \, {\left (B a^{3} b^{2} d e^{3} - A a^{3} b^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (a^{4} b^{6} d^{5} - 5 \, a^{5} b^{5} d^{4} e + 10 \, a^{6} b^{4} d^{3} e^{2} - 10 \, a^{7} b^{3} d^{2} e^{3} + 5 \, a^{8} b^{2} d e^{4} - a^{9} b e^{5} + {\left (b^{10} d^{5} - 5 \, a b^{9} d^{4} e + 10 \, a^{2} b^{8} d^{3} e^{2} - 10 \, a^{3} b^{7} d^{2} e^{3} + 5 \, a^{4} b^{6} d e^{4} - a^{5} b^{5} e^{5}\right )} x^{4} + 4 \, {\left (a b^{9} d^{5} - 5 \, a^{2} b^{8} d^{4} e + 10 \, a^{3} b^{7} d^{3} e^{2} - 10 \, a^{4} b^{6} d^{2} e^{3} + 5 \, a^{5} b^{5} d e^{4} - a^{6} b^{4} e^{5}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d^{5} - 5 \, a^{3} b^{7} d^{4} e + 10 \, a^{4} b^{6} d^{3} e^{2} - 10 \, a^{5} b^{5} d^{2} e^{3} + 5 \, a^{6} b^{4} d e^{4} - a^{7} b^{3} e^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d^{5} - 5 \, a^{4} b^{6} d^{4} e + 10 \, a^{5} b^{5} d^{3} e^{2} - 10 \, a^{6} b^{4} d^{2} e^{3} + 5 \, a^{7} b^{3} d e^{4} - a^{8} b^{2} e^{5}\right )} x\right )}} \]
-1/12*((B*a*b^4 + 3*A*b^5)*d^4 - 2*(3*B*a^2*b^3 + 8*A*a*b^4)*d^3*e + 18*(B *a^3*b^2 + 2*A*a^2*b^3)*d^2*e^2 - 2*(5*B*a^4*b + 24*A*a^3*b^2)*d*e^3 - (3* B*a^5 - 25*A*a^4*b)*e^4 + 12*(B*b^5*d^2*e^2 + A*a*b^4*e^4 - (B*a*b^4 + A*b ^5)*d*e^3)*x^3 - 6*(B*b^5*d^3*e - 7*A*a^2*b^3*e^4 - (8*B*a*b^4 + A*b^5)*d^ 2*e^2 + (7*B*a^2*b^3 + 8*A*a*b^4)*d*e^3)*x^2 + 4*(B*b^5*d^4 + 13*A*a^3*b^2 *e^4 - (6*B*a*b^4 + A*b^5)*d^3*e + 6*(3*B*a^2*b^3 + A*a*b^4)*d^2*e^2 - (13 *B*a^3*b^2 + 18*A*a^2*b^3)*d*e^3)*x + 12*(B*a^4*b*d*e^3 - A*a^4*b*e^4 + (B *b^5*d*e^3 - A*b^5*e^4)*x^4 + 4*(B*a*b^4*d*e^3 - A*a*b^4*e^4)*x^3 + 6*(B*a ^2*b^3*d*e^3 - A*a^2*b^3*e^4)*x^2 + 4*(B*a^3*b^2*d*e^3 - A*a^3*b^2*e^4)*x) *log(b*x + a) - 12*(B*a^4*b*d*e^3 - A*a^4*b*e^4 + (B*b^5*d*e^3 - A*b^5*e^4 )*x^4 + 4*(B*a*b^4*d*e^3 - A*a*b^4*e^4)*x^3 + 6*(B*a^2*b^3*d*e^3 - A*a^2*b ^3*e^4)*x^2 + 4*(B*a^3*b^2*d*e^3 - A*a^3*b^2*e^4)*x)*log(e*x + d))/(a^4*b^ 6*d^5 - 5*a^5*b^5*d^4*e + 10*a^6*b^4*d^3*e^2 - 10*a^7*b^3*d^2*e^3 + 5*a^8* b^2*d*e^4 - a^9*b*e^5 + (b^10*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 1 0*a^3*b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^5*b^5*e^5)*x^4 + 4*(a*b^9*d^5 - 5* a^2*b^8*d^4*e + 10*a^3*b^7*d^3*e^2 - 10*a^4*b^6*d^2*e^3 + 5*a^5*b^5*d*e^4 - a^6*b^4*e^5)*x^3 + 6*(a^2*b^8*d^5 - 5*a^3*b^7*d^4*e + 10*a^4*b^6*d^3*e^2 - 10*a^5*b^5*d^2*e^3 + 5*a^6*b^4*d*e^4 - a^7*b^3*e^5)*x^2 + 4*(a^3*b^7*d^ 5 - 5*a^4*b^6*d^4*e + 10*a^5*b^5*d^3*e^2 - 10*a^6*b^4*d^2*e^3 + 5*a^7*b^3* d*e^4 - a^8*b^2*e^5)*x)
Timed out. \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(((2*a*b)/e>0)', see `assume?` fo r more det
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (230) = 460\).
Time = 0.34 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.07 \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {{\left (B b d e^{3} - A b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {{\left (B d e^{4} - A e^{5}\right )} \log \left ({\left | e x + d \right |}\right )}{b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {B a b^{4} d^{4} + 3 \, A b^{5} d^{4} - 6 \, B a^{2} b^{3} d^{3} e - 16 \, A a b^{4} d^{3} e + 18 \, B a^{3} b^{2} d^{2} e^{2} + 36 \, A a^{2} b^{3} d^{2} e^{2} - 10 \, B a^{4} b d e^{3} - 48 \, A a^{3} b^{2} d e^{3} - 3 \, B a^{5} e^{4} + 25 \, A a^{4} b e^{4} + 12 \, {\left (B b^{5} d^{2} e^{2} - B a b^{4} d e^{3} - A b^{5} d e^{3} + A a b^{4} e^{4}\right )} x^{3} - 6 \, {\left (B b^{5} d^{3} e - 8 \, B a b^{4} d^{2} e^{2} - A b^{5} d^{2} e^{2} + 7 \, B a^{2} b^{3} d e^{3} + 8 \, A a b^{4} d e^{3} - 7 \, A a^{2} b^{3} e^{4}\right )} x^{2} + 4 \, {\left (B b^{5} d^{4} - 6 \, B a b^{4} d^{3} e - A b^{5} d^{3} e + 18 \, B a^{2} b^{3} d^{2} e^{2} + 6 \, A a b^{4} d^{2} e^{2} - 13 \, B a^{3} b^{2} d e^{3} - 18 \, A a^{2} b^{3} d e^{3} + 13 \, A a^{3} b^{2} e^{4}\right )} x}{12 \, {\left (b d - a e\right )}^{5} {\left (b x + a\right )}^{4} b \mathrm {sgn}\left (b x + a\right )} \]
-(B*b*d*e^3 - A*b*e^4)*log(abs(b*x + a))/(b^6*d^5*sgn(b*x + a) - 5*a*b^5*d ^4*e*sgn(b*x + a) + 10*a^2*b^4*d^3*e^2*sgn(b*x + a) - 10*a^3*b^3*d^2*e^3*s gn(b*x + a) + 5*a^4*b^2*d*e^4*sgn(b*x + a) - a^5*b*e^5*sgn(b*x + a)) + (B* d*e^4 - A*e^5)*log(abs(e*x + d))/(b^5*d^5*e*sgn(b*x + a) - 5*a*b^4*d^4*e^2 *sgn(b*x + a) + 10*a^2*b^3*d^3*e^3*sgn(b*x + a) - 10*a^3*b^2*d^2*e^4*sgn(b *x + a) + 5*a^4*b*d*e^5*sgn(b*x + a) - a^5*e^6*sgn(b*x + a)) - 1/12*(B*a*b ^4*d^4 + 3*A*b^5*d^4 - 6*B*a^2*b^3*d^3*e - 16*A*a*b^4*d^3*e + 18*B*a^3*b^2 *d^2*e^2 + 36*A*a^2*b^3*d^2*e^2 - 10*B*a^4*b*d*e^3 - 48*A*a^3*b^2*d*e^3 - 3*B*a^5*e^4 + 25*A*a^4*b*e^4 + 12*(B*b^5*d^2*e^2 - B*a*b^4*d*e^3 - A*b^5*d *e^3 + A*a*b^4*e^4)*x^3 - 6*(B*b^5*d^3*e - 8*B*a*b^4*d^2*e^2 - A*b^5*d^2*e ^2 + 7*B*a^2*b^3*d*e^3 + 8*A*a*b^4*d*e^3 - 7*A*a^2*b^3*e^4)*x^2 + 4*(B*b^5 *d^4 - 6*B*a*b^4*d^3*e - A*b^5*d^3*e + 18*B*a^2*b^3*d^2*e^2 + 6*A*a*b^4*d^ 2*e^2 - 13*B*a^3*b^2*d*e^3 - 18*A*a^2*b^3*d*e^3 + 13*A*a^3*b^2*e^4)*x)/((b *d - a*e)^5*(b*x + a)^4*b*sgn(b*x + a))
Timed out. \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{\left (d+e\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]