Integrand size = 33, antiderivative size = 310 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^4}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e)}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (b d-a e)^2 (2 b B d+3 A b e-5 a B e)}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^4 x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \]
-2*e^2*(-a*e+b*d)*(2*A*b*e-5*B*a*e+3*B*b*d)/b^6/((b*x+a)^2)^(1/2)-1/4*(A*b -B*a)*(-a*e+b*d)^4/b^6/(b*x+a)^3/((b*x+a)^2)^(1/2)-1/3*(-a*e+b*d)^3*(4*A*b *e-5*B*a*e+B*b*d)/b^6/(b*x+a)^2/((b*x+a)^2)^(1/2)-e*(-a*e+b*d)^2*(3*A*b*e- 5*B*a*e+2*B*b*d)/b^6/(b*x+a)/((b*x+a)^2)^(1/2)+B*e^4*x*(b*x+a)/b^5/((b*x+a )^2)^(1/2)+e^3*(A*b*e-5*B*a*e+4*B*b*d)*(b*x+a)*ln(b*x+a)/b^6/((b*x+a)^2)^( 1/2)
Time = 1.14 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.07 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {-A b (b d-a e) \left (25 a^3 e^3+a^2 b e^2 (13 d+88 e x)+a b^2 e \left (7 d^2+40 d e x+108 e^2 x^2\right )+b^3 \left (3 d^3+16 d^2 e x+36 d e^2 x^2+48 e^3 x^3\right )\right )-B \left (77 a^5 e^4+4 a^4 b e^3 (-25 d+62 e x)+2 a^3 b^2 e^2 \left (9 d^2-176 d e x+126 e^2 x^2\right )+4 a^2 b^3 e \left (d^3+18 d^2 e x-108 d e^2 x^2+12 e^3 x^3\right )+a b^4 \left (d^4+16 d^3 e x+108 d^2 e^2 x^2-192 d e^3 x^3-48 e^4 x^4\right )+4 b^5 x \left (d^4+6 d^3 e x+18 d^2 e^2 x^2-3 e^4 x^4\right )\right )+12 e^3 (4 b B d+A b e-5 a B e) (a+b x)^4 \log (a+b x)}{12 b^6 (a+b x)^3 \sqrt {(a+b x)^2}} \]
(-(A*b*(b*d - a*e)*(25*a^3*e^3 + a^2*b*e^2*(13*d + 88*e*x) + a*b^2*e*(7*d^ 2 + 40*d*e*x + 108*e^2*x^2) + b^3*(3*d^3 + 16*d^2*e*x + 36*d*e^2*x^2 + 48* e^3*x^3))) - B*(77*a^5*e^4 + 4*a^4*b*e^3*(-25*d + 62*e*x) + 2*a^3*b^2*e^2* (9*d^2 - 176*d*e*x + 126*e^2*x^2) + 4*a^2*b^3*e*(d^3 + 18*d^2*e*x - 108*d* e^2*x^2 + 12*e^3*x^3) + a*b^4*(d^4 + 16*d^3*e*x + 108*d^2*e^2*x^2 - 192*d* e^3*x^3 - 48*e^4*x^4) + 4*b^5*x*(d^4 + 6*d^3*e*x + 18*d^2*e^2*x^2 - 3*e^4* x^4)) + 12*e^3*(4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)^4*Log[a + b*x])/(12*b ^6*(a + b*x)^3*Sqrt[(a + b*x)^2])
Time = 0.48 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.69, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {(A+B x) (d+e x)^4}{b^5 (a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {(A+B x) (d+e x)^4}{(a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {(a+b x) \int \left (\frac {B e^4}{b^5}+\frac {(4 b B d+A b e-5 a B e) e^3}{b^5 (a+b x)}+\frac {2 (b d-a e) (3 b B d+2 A b e-5 a B e) e^2}{b^5 (a+b x)^2}+\frac {2 (b d-a e)^2 (2 b B d+3 A b e-5 a B e) e}{b^5 (a+b x)^3}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^5 (a+b x)^4}+\frac {(A b-a B) (b d-a e)^4}{b^5 (a+b x)^5}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b x) \left (\frac {e^3 \log (a+b x) (-5 a B e+A b e+4 b B d)}{b^6}-\frac {2 e^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^6 (a+b x)}-\frac {e (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^6 (a+b x)^2}-\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{3 b^6 (a+b x)^3}-\frac {(A b-a B) (b d-a e)^4}{4 b^6 (a+b x)^4}+\frac {B e^4 x}{b^5}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*((B*e^4*x)/b^5 - ((A*b - a*B)*(b*d - a*e)^4)/(4*b^6*(a + b*x)^4 ) - ((b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a*B*e))/(3*b^6*(a + b*x)^3) - (e*( b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e))/(b^6*(a + b*x)^2) - (2*e^2*(b* d - a*e)*(3*b*B*d + 2*A*b*e - 5*a*B*e))/(b^6*(a + b*x)) + (e^3*(4*b*B*d + A*b*e - 5*a*B*e)*Log[a + b*x])/b^6))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.18.76.3.1 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_))*((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]
Time = 0.75 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.47
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, B \,e^{4} x}{\left (b x +a \right ) b^{5}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (4 A a \,b^{3} e^{4}-4 A \,b^{4} d \,e^{3}-10 B \,a^{2} b^{2} e^{4}+16 B a \,b^{3} d \,e^{3}-6 b^{4} B \,d^{2} e^{2}\right ) x^{3}+b e \left (9 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}-3 A \,b^{3} d^{2} e -25 B \,e^{3} a^{3}+36 B \,a^{2} b d \,e^{2}-9 B a \,b^{2} d^{2} e -2 B \,b^{3} d^{3}\right ) x^{2}+\left (\frac {22}{3} A \,a^{3} b \,e^{4}-4 A \,a^{2} b^{2} d \,e^{3}-2 A a \,b^{3} d^{2} e^{2}-\frac {4}{3} A \,b^{4} d^{3} e -\frac {65}{3} B \,a^{4} e^{4}+\frac {88}{3} B \,a^{3} b d \,e^{3}-6 B \,a^{2} b^{2} d^{2} e^{2}-\frac {4}{3} B a \,b^{3} d^{3} e -\frac {1}{3} b^{4} B \,d^{4}\right ) x +\frac {25 A \,a^{4} b \,e^{4}-12 A \,a^{3} b^{2} d \,e^{3}-6 A \,a^{2} b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e a -3 A \,b^{5} d^{4}-77 B \,a^{5} e^{4}+100 B \,a^{4} b d \,e^{3}-18 B \,a^{3} b^{2} d^{2} e^{2}-4 B \,a^{2} b^{3} d^{3} e -B \,b^{4} d^{4} a}{12 b}\right )}{\left (b x +a \right )^{5} b^{5}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{3} \left (A b e -5 B a e +4 B b d \right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{6}}\) | \(456\) |
| default | \(\frac {\left (352 B \,a^{3} b^{2} d \,e^{3} x -108 B \,x^{2} a \,b^{4} d^{2} e^{2}+192 B \,x^{3} a \,b^{4} d \,e^{3}-72 A \,x^{2} a \,b^{4} d \,e^{3}+432 B \,x^{2} a^{2} b^{3} d \,e^{3}-360 B \ln \left (b x +a \right ) x^{2} a^{3} b^{2} e^{4}-24 A x a \,b^{4} d^{2} e^{2}-72 B x \,a^{2} b^{3} d^{2} e^{2}-16 B x a \,b^{4} d^{3} e -48 A x \,a^{2} b^{3} d \,e^{3}+48 A \ln \left (b x +a \right ) x \,a^{3} b^{2} e^{4}-240 B \ln \left (b x +a \right ) x \,a^{4} b \,e^{4}+48 B \ln \left (b x +a \right ) a^{4} b d \,e^{3}+48 A \ln \left (b x +a \right ) x^{3} a \,b^{4} e^{4}-240 B \ln \left (b x +a \right ) x^{3} a^{2} b^{3} e^{4}+72 A \ln \left (b x +a \right ) x^{2} a^{2} b^{3} e^{4}-60 B \ln \left (b x +a \right ) a \,b^{4} e^{4} x^{4}+48 B \ln \left (b x +a \right ) b^{5} d \,e^{3} x^{4}+12 B \,x^{5} e^{4} b^{5}-60 B \ln \left (b x +a \right ) a^{5} e^{4}-B \,b^{4} d^{4} a +25 A \,a^{4} b \,e^{4}-4 B \,b^{5} d^{4} x -77 B \,a^{5} e^{4}-3 A \,b^{5} d^{4}+192 B \ln \left (b x +a \right ) x \,a^{3} b^{2} d \,e^{3}-12 A \,a^{3} b^{2} d \,e^{3}-6 A \,a^{2} b^{3} d^{2} e^{2}+100 B \,a^{4} b d \,e^{3}+88 A \,a^{3} b^{2} e^{4} x -16 A \,b^{5} d^{3} e x -248 B \,a^{4} b \,e^{4} x +12 A \ln \left (b x +a \right ) b^{5} e^{4} x^{4}+192 B \ln \left (b x +a \right ) x^{3} a \,b^{4} d \,e^{3}+288 B \ln \left (b x +a \right ) x^{2} a^{2} b^{3} d \,e^{3}-18 B \,a^{3} b^{2} d^{2} e^{2}-4 B \,a^{2} b^{3} d^{3} e +108 A \,x^{2} a^{2} b^{3} e^{4}-36 A \,x^{2} b^{5} d^{2} e^{2}-252 B \,x^{2} a^{3} b^{2} e^{4}-24 B \,x^{2} b^{5} d^{3} e +48 B \,x^{4} a \,b^{4} e^{4}+48 A \,x^{3} a \,b^{4} e^{4}-48 A \,x^{3} b^{5} d \,e^{3}-48 B \,x^{3} a^{2} b^{3} e^{4}-72 B \,x^{3} b^{5} d^{2} e^{2}+12 A \ln \left (b x +a \right ) a^{4} b \,e^{4}-4 A \,b^{4} d^{3} e a \right ) \left (b x +a \right )}{12 b^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(735\) |
((b*x+a)^2)^(1/2)/(b*x+a)*B*e^4/b^5*x+((b*x+a)^2)^(1/2)/(b*x+a)^5*((4*A*a* b^3*e^4-4*A*b^4*d*e^3-10*B*a^2*b^2*e^4+16*B*a*b^3*d*e^3-6*B*b^4*d^2*e^2)*x ^3+b*e*(9*A*a^2*b*e^3-6*A*a*b^2*d*e^2-3*A*b^3*d^2*e-25*B*a^3*e^3+36*B*a^2* b*d*e^2-9*B*a*b^2*d^2*e-2*B*b^3*d^3)*x^2+(22/3*A*a^3*b*e^4-4*A*a^2*b^2*d*e ^3-2*A*a*b^3*d^2*e^2-4/3*A*b^4*d^3*e-65/3*B*a^4*e^4+88/3*B*a^3*b*d*e^3-6*B *a^2*b^2*d^2*e^2-4/3*B*a*b^3*d^3*e-1/3*b^4*B*d^4)*x+1/12*(25*A*a^4*b*e^4-1 2*A*a^3*b^2*d*e^3-6*A*a^2*b^3*d^2*e^2-4*A*a*b^4*d^3*e-3*A*b^5*d^4-77*B*a^5 *e^4+100*B*a^4*b*d*e^3-18*B*a^3*b^2*d^2*e^2-4*B*a^2*b^3*d^3*e-B*a*b^4*d^4) /b)/b^5+((b*x+a)^2)^(1/2)/(b*x+a)/b^6*e^3*(A*b*e-5*B*a*e+4*B*b*d)*ln(b*x+a )
Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (240) = 480\).
Time = 0.44 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.00 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {12 \, B b^{5} e^{4} x^{5} + 48 \, B a b^{4} e^{4} x^{4} - {\left (B a b^{4} + 3 \, A b^{5}\right )} d^{4} - 4 \, {\left (B a^{2} b^{3} + A a b^{4}\right )} d^{3} e - 6 \, {\left (3 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{2} + 4 \, {\left (25 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} d e^{3} - {\left (77 \, B a^{5} - 25 \, A a^{4} b\right )} e^{4} - 24 \, {\left (3 \, B b^{5} d^{2} e^{2} - 2 \, {\left (4 \, B a b^{4} - A b^{5}\right )} d e^{3} + 2 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} e^{4}\right )} x^{3} - 12 \, {\left (2 \, B b^{5} d^{3} e + 3 \, {\left (3 \, B a b^{4} + A b^{5}\right )} d^{2} e^{2} - 6 \, {\left (6 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} + 3 \, {\left (7 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 4 \, {\left (B b^{5} d^{4} + 4 \, {\left (B a b^{4} + A b^{5}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{2} - 4 \, {\left (22 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d e^{3} + 2 \, {\left (31 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \, {\left (4 \, B a^{4} b d e^{3} - {\left (5 \, B a^{5} - A a^{4} b\right )} e^{4} + {\left (4 \, B b^{5} d e^{3} - {\left (5 \, B a b^{4} - A b^{5}\right )} e^{4}\right )} x^{4} + 4 \, {\left (4 \, B a b^{4} d e^{3} - {\left (5 \, B a^{2} b^{3} - A a b^{4}\right )} e^{4}\right )} x^{3} + 6 \, {\left (4 \, B a^{2} b^{3} d e^{3} - {\left (5 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 4 \, {\left (4 \, B a^{3} b^{2} d e^{3} - {\left (5 \, B a^{4} b - A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} \]
1/12*(12*B*b^5*e^4*x^5 + 48*B*a*b^4*e^4*x^4 - (B*a*b^4 + 3*A*b^5)*d^4 - 4* (B*a^2*b^3 + A*a*b^4)*d^3*e - 6*(3*B*a^3*b^2 + A*a^2*b^3)*d^2*e^2 + 4*(25* B*a^4*b - 3*A*a^3*b^2)*d*e^3 - (77*B*a^5 - 25*A*a^4*b)*e^4 - 24*(3*B*b^5*d ^2*e^2 - 2*(4*B*a*b^4 - A*b^5)*d*e^3 + 2*(B*a^2*b^3 - A*a*b^4)*e^4)*x^3 - 12*(2*B*b^5*d^3*e + 3*(3*B*a*b^4 + A*b^5)*d^2*e^2 - 6*(6*B*a^2*b^3 - A*a*b ^4)*d*e^3 + 3*(7*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*x^2 - 4*(B*b^5*d^4 + 4*(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 - 4*(22*B*a^3*b^2 - 3*A*a^2*b^3)*d*e^3 + 2*(31*B*a^4*b - 11*A*a^3*b^2)*e^4)*x + 12*(4*B*a^4* b*d*e^3 - (5*B*a^5 - A*a^4*b)*e^4 + (4*B*b^5*d*e^3 - (5*B*a*b^4 - A*b^5)*e ^4)*x^4 + 4*(4*B*a*b^4*d*e^3 - (5*B*a^2*b^3 - A*a*b^4)*e^4)*x^3 + 6*(4*B*a ^2*b^3*d*e^3 - (5*B*a^3*b^2 - A*a^2*b^3)*e^4)*x^2 + 4*(4*B*a^3*b^2*d*e^3 - (5*B*a^4*b - A*a^3*b^2)*e^4)*x)*log(b*x + a))/(b^10*x^4 + 4*a*b^9*x^3 + 6 *a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6)
\[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (240) = 480\).
Time = 0.25 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.44 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {1}{12} \, B e^{4} {\left (\frac {12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}} - \frac {60 \, a \log \left (b x + a\right )}{b^{6}}\right )} + \frac {1}{3} \, B d e^{3} {\left (\frac {48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}} + \frac {12 \, \log \left (b x + a\right )}{b^{5}}\right )} + \frac {1}{12} \, A e^{4} {\left (\frac {48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}} + \frac {12 \, \log \left (b x + a\right )}{b^{5}}\right )} - \frac {1}{2} \, B d^{2} e^{2} {\left (\frac {12 \, x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}} + \frac {6 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {3 \, a^{3}}{b^{8} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {1}{3} \, A d e^{3} {\left (\frac {12 \, x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}} + \frac {6 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {3 \, a^{3}}{b^{8} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {1}{12} \, B d^{4} {\left (\frac {4}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {3 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {1}{3} \, A d^{3} e {\left (\frac {4}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {3 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {1}{3} \, B d^{3} e {\left (\frac {6}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{3}} + \frac {3 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {1}{2} \, A d^{2} e^{2} {\left (\frac {6}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{3}} + \frac {3 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {A d^{4}}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} \]
1/12*B*e^4*((12*b^5*x^5 + 48*a*b^4*x^4 - 48*a^2*b^3*x^3 - 252*a^3*b^2*x^2 - 248*a^4*b*x - 77*a^5)/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^ 7*x + a^4*b^6) - 60*a*log(b*x + a)/b^6) + 1/3*B*d*e^3*((48*a*b^3*x^3 + 108 *a^2*b^2*x^2 + 88*a^3*b*x + 25*a^4)/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5) + 12*log(b*x + a)/b^5) + 1/12*A*e^4*((48*a*b^3*x ^3 + 108*a^2*b^2*x^2 + 88*a^3*b*x + 25*a^4)/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2 *b^7*x^2 + 4*a^3*b^6*x + a^4*b^5) + 12*log(b*x + a)/b^5) - 1/2*B*d^2*e^2*( 12*x^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) + 8*a^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^4) + 6*a/(b^6*(x + a/b)^2) - 8*a^2/(b^7*(x + a/b)^3) - 3*a^3 /(b^8*(x + a/b)^4)) - 1/3*A*d*e^3*(12*x^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2) *b^2) + 8*a^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^4) + 6*a/(b^6*(x + a/b)^2 ) - 8*a^2/(b^7*(x + a/b)^3) - 3*a^3/(b^8*(x + a/b)^4)) - 1/12*B*d^4*(4/((b ^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) - 3*a/(b^6*(x + a/b)^4)) - 1/3*A*d^3*e* (4/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) - 3*a/(b^6*(x + a/b)^4)) - 1/3*B* d^3*e*(6/(b^5*(x + a/b)^2) - 8*a/(b^6*(x + a/b)^3) + 3*a^2/(b^7*(x + a/b)^ 4)) - 1/2*A*d^2*e^2*(6/(b^5*(x + a/b)^2) - 8*a/(b^6*(x + a/b)^3) + 3*a^2/( b^7*(x + a/b)^4)) - 1/4*A*d^4/(b^5*(x + a/b)^4)
Time = 0.29 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {B e^{4} x}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {{\left (4 \, B b d e^{3} - 5 \, B a e^{4} + A b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {B a b^{4} d^{4} + 3 \, A b^{5} d^{4} + 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 18 \, B a^{3} b^{2} d^{2} e^{2} + 6 \, A a^{2} b^{3} d^{2} e^{2} - 100 \, B a^{4} b d e^{3} + 12 \, A a^{3} b^{2} d e^{3} + 77 \, B a^{5} e^{4} - 25 \, A a^{4} b e^{4} + 24 \, {\left (3 \, B b^{5} d^{2} e^{2} - 8 \, B a b^{4} d e^{3} + 2 \, A b^{5} d e^{3} + 5 \, B a^{2} b^{3} e^{4} - 2 \, A a b^{4} e^{4}\right )} x^{3} + 12 \, {\left (2 \, B b^{5} d^{3} e + 9 \, B a b^{4} d^{2} e^{2} + 3 \, A b^{5} d^{2} e^{2} - 36 \, B a^{2} b^{3} d e^{3} + 6 \, A a b^{4} d e^{3} + 25 \, B a^{3} b^{2} e^{4} - 9 \, A a^{2} b^{3} e^{4}\right )} x^{2} + 4 \, {\left (B b^{5} d^{4} + 4 \, B a b^{4} d^{3} e + 4 \, A b^{5} d^{3} e + 18 \, B a^{2} b^{3} d^{2} e^{2} + 6 \, A a b^{4} d^{2} e^{2} - 88 \, B a^{3} b^{2} d e^{3} + 12 \, A a^{2} b^{3} d e^{3} + 65 \, B a^{4} b e^{4} - 22 \, A a^{3} b^{2} e^{4}\right )} x}{12 \, {\left (b x + a\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} \]
B*e^4*x/(b^5*sgn(b*x + a)) + (4*B*b*d*e^3 - 5*B*a*e^4 + A*b*e^4)*log(abs(b *x + a))/(b^6*sgn(b*x + a)) - 1/12*(B*a*b^4*d^4 + 3*A*b^5*d^4 + 4*B*a^2*b^ 3*d^3*e + 4*A*a*b^4*d^3*e + 18*B*a^3*b^2*d^2*e^2 + 6*A*a^2*b^3*d^2*e^2 - 1 00*B*a^4*b*d*e^3 + 12*A*a^3*b^2*d*e^3 + 77*B*a^5*e^4 - 25*A*a^4*b*e^4 + 24 *(3*B*b^5*d^2*e^2 - 8*B*a*b^4*d*e^3 + 2*A*b^5*d*e^3 + 5*B*a^2*b^3*e^4 - 2* A*a*b^4*e^4)*x^3 + 12*(2*B*b^5*d^3*e + 9*B*a*b^4*d^2*e^2 + 3*A*b^5*d^2*e^2 - 36*B*a^2*b^3*d*e^3 + 6*A*a*b^4*d*e^3 + 25*B*a^3*b^2*e^4 - 9*A*a^2*b^3*e ^4)*x^2 + 4*(B*b^5*d^4 + 4*B*a*b^4*d^3*e + 4*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 - 88*B*a^3*b^2*d*e^3 + 12*A*a^2*b^3*d*e^3 + 65*B *a^4*b*e^4 - 22*A*a^3*b^2*e^4)*x)/((b*x + a)^4*b^6*sgn(b*x + a))
Timed out. \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^4}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]