Integrand size = 33, antiderivative size = 329 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^4}{2 b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 \left (6 a^2 B e^2-3 a b e (4 B d+A e)+2 b^2 d (3 B d+2 A e)\right ) x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^3 (4 b B d+A b e-3 a B e) x^2 (a+b x)}{2 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^4 x^3 (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \]
-(-a*e+b*d)^3*(4*A*b*e-5*B*a*e+B*b*d)/b^6/((b*x+a)^2)^(1/2)-1/2*(A*b-B*a)* (-a*e+b*d)^4/b^6/(b*x+a)/((b*x+a)^2)^(1/2)+e^2*(6*a^2*B*e^2-3*a*b*e*(A*e+4 *B*d)+2*b^2*d*(2*A*e+3*B*d))*x*(b*x+a)/b^5/((b*x+a)^2)^(1/2)+1/2*e^3*(A*b* e-3*B*a*e+4*B*b*d)*x^2*(b*x+a)/b^4/((b*x+a)^2)^(1/2)+1/3*B*e^4*x^3*(b*x+a) /b^3/((b*x+a)^2)^(1/2)+2*e*(-a*e+b*d)^2*(3*A*b*e-5*B*a*e+2*B*b*d)*(b*x+a)* ln(b*x+a)/b^6/((b*x+a)^2)^(1/2)
Time = 1.15 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.13 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {-3 A b \left (-7 a^4 e^4-2 a^3 b e^3 (-10 d+e x)+a^2 b^2 e^2 \left (-18 d^2+16 d e x+11 e^2 x^2\right )+4 a b^3 e \left (d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right )+b^4 \left (d^4+8 d^3 e x-8 d e^3 x^3-e^4 x^4\right )\right )+B \left (-27 a^5 e^4+6 a^4 b e^3 (14 d+e x)+3 a^3 b^2 e^2 \left (-30 d^2+8 d e x+21 e^2 x^2\right )+4 a^2 b^3 e \left (9 d^3-18 d^2 e x-33 d e^2 x^2+5 e^3 x^3\right )+a b^4 \left (-3 d^4+48 d^3 e x+72 d^2 e^2 x^2-48 d e^3 x^3-5 e^4 x^4\right )+2 b^5 x \left (-3 d^4+18 d^2 e^2 x^2+6 d e^3 x^3+e^4 x^4\right )\right )+12 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^2 \log (a+b x)}{6 b^6 (a+b x) \sqrt {(a+b x)^2}} \]
(-3*A*b*(-7*a^4*e^4 - 2*a^3*b*e^3*(-10*d + e*x) + a^2*b^2*e^2*(-18*d^2 + 1 6*d*e*x + 11*e^2*x^2) + 4*a*b^3*e*(d^3 - 6*d^2*e*x - 4*d*e^2*x^2 + e^3*x^3 ) + b^4*(d^4 + 8*d^3*e*x - 8*d*e^3*x^3 - e^4*x^4)) + B*(-27*a^5*e^4 + 6*a^ 4*b*e^3*(14*d + e*x) + 3*a^3*b^2*e^2*(-30*d^2 + 8*d*e*x + 21*e^2*x^2) + 4* a^2*b^3*e*(9*d^3 - 18*d^2*e*x - 33*d*e^2*x^2 + 5*e^3*x^3) + a*b^4*(-3*d^4 + 48*d^3*e*x + 72*d^2*e^2*x^2 - 48*d*e^3*x^3 - 5*e^4*x^4) + 2*b^5*x*(-3*d^ 4 + 18*d^2*e^2*x^2 + 6*d*e^3*x^3 + e^4*x^4)) + 12*e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e)*(a + b*x)^2*Log[a + b*x])/(6*b^6*(a + b*x)*Sqrt[(a + b*x)^2])
Time = 0.51 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.66, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^3 (a+b x) \int \frac {(A+B x) (d+e x)^4}{b^3 (a+b x)^3}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {(A+B x) (d+e x)^4}{(a+b x)^3}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {(a+b x) \int \left (\frac {B (a+b x)^2 e^4}{b^5}+\frac {(4 b B d+A b e-5 a B e) (a+b x) e^3}{b^5}+\frac {2 (b d-a e) (3 b B d+2 A b e-5 a B e) e^2}{b^5}+\frac {2 (b d-a e)^2 (2 b B d+3 A b e-5 a B e) e}{b^5 (a+b x)}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^5 (a+b x)^2}+\frac {(A b-a B) (b d-a e)^4}{b^5 (a+b x)^3}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b x) \left (\frac {e^3 (a+b x)^2 (-5 a B e+A b e+4 b B d)}{2 b^6}-\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 (a+b x)}-\frac {(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}+\frac {2 e (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6}+\frac {2 e^2 x (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^5}+\frac {B e^4 (a+b x)^3}{3 b^6}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*((2*e^2*(b*d - a*e)*(3*b*B*d + 2*A*b*e - 5*a*B*e)*x)/b^5 - ((A* b - a*B)*(b*d - a*e)^4)/(2*b^6*(a + b*x)^2) - ((b*d - a*e)^3*(b*B*d + 4*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)*(a + b* x)^2)/(2*b^6) + (B*e^4*(a + b*x)^3)/(3*b^6) + (2*e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e)*Log[a + b*x])/b^6))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.18.67.3.1 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 = 0.84 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.42
| method | result | size |
| risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{2} \left (-\frac {1}{3} b^{2} B \,x^{3} e^{2}-\frac {1}{2} A \,b^{2} e^{2} x^{2}+\frac {3}{2} B a b \,e^{2} x^{2}-2 B \,b^{2} d e \,x^{2}+3 A a b \,e^{2} x -4 A \,b^{2} d e x -6 B \,a^{2} e^{2} x +12 B a b d e x -6 B \,b^{2} d^{2} x \right )}{\left (b x +a \right ) b^{5}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e -5 B \,a^{4} e^{4}+16 B \,a^{3} b d \,e^{3}-18 B \,a^{2} b^{2} d^{2} e^{2}+8 B a \,b^{3} d^{3} e -b^{4} B \,d^{4}\right ) x +\frac {7 A \,a^{4} b \,e^{4}-20 A \,a^{3} b^{2} d \,e^{3}+18 A \,a^{2} b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e a -A \,b^{5} d^{4}-9 B \,a^{5} e^{4}+28 B \,a^{4} b d \,e^{3}-30 B \,a^{3} b^{2} d^{2} e^{2}+12 B \,a^{2} b^{3} d^{3} e -B \,b^{4} d^{4} a}{2 b}\right )}{\left (b x +a \right )^{3} b^{5}}+\frac {2 \sqrt {\left (b x +a \right )^{2}}\, e \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e -5 B \,e^{3} a^{3}+12 B \,a^{2} b d \,e^{2}-9 B a \,b^{2} d^{2} e +2 B \,b^{3} d^{3}\right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{6}}\) | \(466\) |
| default | \(\frac {\left (24 B \,a^{3} b^{2} d \,e^{3} x -108 B \ln \left (b x +a \right ) a^{3} b^{2} d^{2} e^{2}+24 B \ln \left (b x +a \right ) a^{2} b^{3} d^{3} e +72 B \,x^{2} a \,b^{4} d^{2} e^{2}-48 B \,x^{3} a \,b^{4} d \,e^{3}+48 A \,x^{2} a \,b^{4} d \,e^{3}-132 B \,x^{2} a^{2} b^{3} d \,e^{3}-60 B \ln \left (b x +a \right ) x^{2} a^{3} b^{2} e^{4}+72 A x a \,b^{4} d^{2} e^{2}-72 B x \,a^{2} b^{3} d^{2} e^{2}+48 B x a \,b^{4} d^{3} e -48 A x \,a^{2} b^{3} d \,e^{3}+72 A \ln \left (b x +a \right ) x \,a^{3} b^{2} e^{4}-120 B \ln \left (b x +a \right ) x \,a^{4} b \,e^{4}-72 A \ln \left (b x +a \right ) a^{3} b^{2} d \,e^{3}+36 A \ln \left (b x +a \right ) a^{2} b^{3} d^{2} e^{2}+144 B \ln \left (b x +a \right ) a^{4} b d \,e^{3}+36 A \ln \left (b x +a \right ) x^{2} a^{2} b^{3} e^{4}+24 B \ln \left (b x +a \right ) b^{5} d^{3} e \,x^{2}+36 A \ln \left (b x +a \right ) b^{5} d^{2} e^{2} x^{2}+2 B \,x^{5} e^{4} b^{5}+3 A \,x^{4} b^{5} e^{4}-60 B \ln \left (b x +a \right ) a^{5} e^{4}-3 B \,b^{4} d^{4} a +21 A \,a^{4} b \,e^{4}-6 B \,b^{5} d^{4} x -27 B \,a^{5} e^{4}-3 A \,b^{5} d^{4}+288 B \ln \left (b x +a \right ) x \,a^{3} b^{2} d \,e^{3}-216 B \ln \left (b x +a \right ) x \,a^{2} b^{3} d^{2} e^{2}+48 B \ln \left (b x +a \right ) x a \,b^{4} d^{3} e -60 A \,a^{3} b^{2} d \,e^{3}+54 A \,a^{2} b^{3} d^{2} e^{2}+84 B \,a^{4} b d \,e^{3}+6 A \,a^{3} b^{2} e^{4} x -24 A \,b^{5} d^{3} e x +6 B \,a^{4} b \,e^{4} x -72 A \ln \left (b x +a \right ) x^{2} a \,b^{4} d \,e^{3}+144 B \ln \left (b x +a \right ) x^{2} a^{2} b^{3} d \,e^{3}-108 B \ln \left (b x +a \right ) x^{2} a \,b^{4} d^{2} e^{2}-90 B \,a^{3} b^{2} d^{2} e^{2}+36 B \,a^{2} b^{3} d^{3} e -33 A \,x^{2} a^{2} b^{3} e^{4}+63 B \,x^{2} a^{3} b^{2} e^{4}-5 B \,x^{4} a \,b^{4} e^{4}+12 B \,x^{4} b^{5} d \,e^{3}-12 A \,x^{3} a \,b^{4} e^{4}+24 A \,x^{3} b^{5} d \,e^{3}+20 B \,x^{3} a^{2} b^{3} e^{4}+36 B \,x^{3} b^{5} d^{2} e^{2}+36 A \ln \left (b x +a \right ) a^{4} b \,e^{4}-12 A \,b^{4} d^{3} e a -144 A \ln \left (b x +a \right ) x \,a^{2} b^{3} d \,e^{3}+72 A \ln \left (b x +a \right ) x a \,b^{4} d^{2} e^{2}\right ) \left (b x +a \right )}{6 b^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(858\) |
-((b*x+a)^2)^(1/2)/(b*x+a)*e^2/b^5*(-1/3*b^2*B*x^3*e^2-1/2*A*b^2*e^2*x^2+3 /2*B*a*b*e^2*x^2-2*B*b^2*d*e*x^2+3*A*a*b*e^2*x-4*A*b^2*d*e*x-6*B*a^2*e^2*x +12*B*a*b*d*e*x-6*B*b^2*d^2*x)+((b*x+a)^2)^(1/2)/(b*x+a)^3*((4*A*a^3*b*e^4 -12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e-5*B*a^4*e^4+16*B*a^3* b*d*e^3-18*B*a^2*b^2*d^2*e^2+8*B*a*b^3*d^3*e-B*b^4*d^4)*x+1/2*(7*A*a^4*b*e ^4-20*A*a^3*b^2*d*e^3+18*A*a^2*b^3*d^2*e^2-4*A*a*b^4*d^3*e-A*b^5*d^4-9*B*a ^5*e^4+28*B*a^4*b*d*e^3-30*B*a^3*b^2*d^2*e^2+12*B*a^2*b^3*d^3*e-B*a*b^4*d^ 4)/b)/b^5+2*((b*x+a)^2)^(1/2)/(b*x+a)/b^6*e*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2 +3*A*b^3*d^2*e-5*B*a^3*e^3+12*B*a^2*b*d*e^2-9*B*a*b^2*d^2*e+2*B*b^3*d^3)*l n(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (257) = 514\).
Time = 0.39 (sec) , antiderivative size = 668, normalized size of antiderivative = 2.03 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {2 \, B b^{5} e^{4} x^{5} - 3 \, {\left (B a b^{4} + A b^{5}\right )} d^{4} + 12 \, {\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e - 18 \, {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{2} + 12 \, {\left (7 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} d e^{3} - 3 \, {\left (9 \, B a^{5} - 7 \, A a^{4} b\right )} e^{4} + {\left (12 \, B b^{5} d e^{3} - {\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} e^{4}\right )} x^{4} + 4 \, {\left (9 \, B b^{5} d^{2} e^{2} - 6 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d e^{3} + {\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 3 \, {\left (24 \, B a b^{4} d^{2} e^{2} - 4 \, {\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} d e^{3} + {\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 6 \, {\left (B b^{5} d^{4} - 4 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 12 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} - 4 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d e^{3} - {\left (B a^{4} b + A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \, {\left (2 \, B a^{2} b^{3} d^{3} e - 3 \, {\left (3 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{4} b - A a^{3} b^{2}\right )} d e^{3} - {\left (5 \, B a^{5} - 3 \, A a^{4} b\right )} e^{4} + {\left (2 \, B b^{5} d^{3} e - 3 \, {\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} - {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 2 \, {\left (2 \, B a b^{4} d^{3} e - 3 \, {\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} - {\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]
1/6*(2*B*b^5*e^4*x^5 - 3*(B*a*b^4 + A*b^5)*d^4 + 12*(3*B*a^2*b^3 - A*a*b^4 )*d^3*e - 18*(5*B*a^3*b^2 - 3*A*a^2*b^3)*d^2*e^2 + 12*(7*B*a^4*b - 5*A*a^3 *b^2)*d*e^3 - 3*(9*B*a^5 - 7*A*a^4*b)*e^4 + (12*B*b^5*d*e^3 - (5*B*a*b^4 - 3*A*b^5)*e^4)*x^4 + 4*(9*B*b^5*d^2*e^2 - 6*(2*B*a*b^4 - A*b^5)*d*e^3 + (5 *B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 + 3*(24*B*a*b^4*d^2*e^2 - 4*(11*B*a^2*b^3 - 4*A*a*b^4)*d*e^3 + (21*B*a^3*b^2 - 11*A*a^2*b^3)*e^4)*x^2 - 6*(B*b^5*d^ 4 - 4*(2*B*a*b^4 - A*b^5)*d^3*e + 12*(B*a^2*b^3 - A*a*b^4)*d^2*e^2 - 4*(B* a^3*b^2 - 2*A*a^2*b^3)*d*e^3 - (B*a^4*b + A*a^3*b^2)*e^4)*x + 12*(2*B*a^2* b^3*d^3*e - 3*(3*B*a^3*b^2 - A*a^2*b^3)*d^2*e^2 + 6*(2*B*a^4*b - A*a^3*b^2 )*d*e^3 - (5*B*a^5 - 3*A*a^4*b)*e^4 + (2*B*b^5*d^3*e - 3*(3*B*a*b^4 - A*b^ 5)*d^2*e^2 + 6*(2*B*a^2*b^3 - A*a*b^4)*d*e^3 - (5*B*a^3*b^2 - 3*A*a^2*b^3) *e^4)*x^2 + 2*(2*B*a*b^4*d^3*e - 3*(3*B*a^2*b^3 - A*a*b^4)*d^2*e^2 + 6*(2* B*a^3*b^2 - A*a^2*b^3)*d*e^3 - (5*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)*log(b*x + a))/(b^8*x^2 + 2*a*b^7*x + a^2*b^6)
\[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 761 vs. \(2 (257) = 514\).
Time = 0.20 (sec) , antiderivative size = 761, normalized size of antiderivative = 2.31 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {B e^{4} x^{4}}{3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {7 \, B a e^{4} x^{3}}{6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac {9 \, B a^{2} e^{4} x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac {10 \, B a^{3} e^{4} \log \left (x + \frac {a}{b}\right )}{b^{6}} + \frac {9 \, B a^{4} e^{4}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{6}} + \frac {{\left (4 \, B d e^{3} + A e^{4}\right )} x^{3}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {20 \, B a^{4} e^{4} x}{b^{7} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {5 \, {\left (4 \, B d e^{3} + A e^{4}\right )} a x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac {2 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {A d^{4}}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {39 \, B a^{5} e^{4}}{2 \, b^{8} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {6 \, {\left (4 \, B d e^{3} + A e^{4}\right )} a^{2} \log \left (x + \frac {a}{b}\right )}{b^{5}} - \frac {6 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} a \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {2 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {5 \, {\left (4 \, B d e^{3} + A e^{4}\right )} a^{3}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} + \frac {4 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} a^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac {B d^{4} + 4 \, A d^{3} e}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {12 \, {\left (4 \, B d e^{3} + A e^{4}\right )} a^{3} x}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {12 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} a^{2} x}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {4 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} a x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {23 \, {\left (4 \, B d e^{3} + A e^{4}\right )} a^{4}}{2 \, b^{7} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {11 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} a^{3}}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {3 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} a^{2}}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {{\left (B d^{4} + 4 \, A d^{3} e\right )} a}{2 \, b^{4} {\left (x + \frac {a}{b}\right )}^{2}} \]
1/3*B*e^4*x^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 7/6*B*a*e^4*x^3/(sqrt( b^2*x^2 + 2*a*b*x + a^2)*b^3) + 9/2*B*a^2*e^4*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^4) - 10*B*a^3*e^4*log(x + a/b)/b^6 + 9*B*a^4*e^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^6) + 1/2*(4*B*d*e^3 + A*e^4)*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 20*B*a^4*e^4*x/(b^7*(x + a/b)^2) - 5/2*(4*B*d*e^3 + A*e^4)*a *x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^3) + 2*(3*B*d^2*e^2 + 2*A*d*e^3)*x^2 /(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 1/2*A*d^4/(b^3*(x + a/b)^2) - 39/2* B*a^5*e^4/(b^8*(x + a/b)^2) + 6*(4*B*d*e^3 + A*e^4)*a^2*log(x + a/b)/b^5 - 6*(3*B*d^2*e^2 + 2*A*d*e^3)*a*log(x + a/b)/b^4 + 2*(2*B*d^3*e + 3*A*d^2*e ^2)*log(x + a/b)/b^3 - 5*(4*B*d*e^3 + A*e^4)*a^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^5) + 4*(3*B*d^2*e^2 + 2*A*d*e^3)*a^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 )*b^4) - (B*d^4 + 4*A*d^3*e)/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) + 12*(4*B *d*e^3 + A*e^4)*a^3*x/(b^6*(x + a/b)^2) - 12*(3*B*d^2*e^2 + 2*A*d*e^3)*a^2 *x/(b^5*(x + a/b)^2) + 4*(2*B*d^3*e + 3*A*d^2*e^2)*a*x/(b^4*(x + a/b)^2) + 23/2*(4*B*d*e^3 + A*e^4)*a^4/(b^7*(x + a/b)^2) - 11*(3*B*d^2*e^2 + 2*A*d* e^3)*a^3/(b^6*(x + a/b)^2) + 3*(2*B*d^3*e + 3*A*d^2*e^2)*a^2/(b^5*(x + a/b )^2) + 1/2*(B*d^4 + 4*A*d^3*e)*a/(b^4*(x + a/b)^2)
Time = 0.30 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (2 \, B b^{3} d^{3} e - 9 \, B a b^{2} d^{2} e^{2} + 3 \, A b^{3} d^{2} e^{2} + 12 \, B a^{2} b d e^{3} - 6 \, A a b^{2} d e^{3} - 5 \, B a^{3} e^{4} + 3 \, A a^{2} b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {B a b^{4} d^{4} + A b^{5} d^{4} - 12 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 30 \, B a^{3} b^{2} d^{2} e^{2} - 18 \, A a^{2} b^{3} d^{2} e^{2} - 28 \, B a^{4} b d e^{3} + 20 \, A a^{3} b^{2} d e^{3} + 9 \, B a^{5} e^{4} - 7 \, A a^{4} b e^{4} + 2 \, {\left (B b^{5} d^{4} - 8 \, B a b^{4} d^{3} e + 4 \, A b^{5} d^{3} e + 18 \, B a^{2} b^{3} d^{2} e^{2} - 12 \, A a b^{4} d^{2} e^{2} - 16 \, B a^{3} b^{2} d e^{3} + 12 \, A a^{2} b^{3} d e^{3} + 5 \, B a^{4} b e^{4} - 4 \, A a^{3} b^{2} e^{4}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, B b^{6} e^{4} x^{3} + 12 \, B b^{6} d e^{3} x^{2} - 9 \, B a b^{5} e^{4} x^{2} + 3 \, A b^{6} e^{4} x^{2} + 36 \, B b^{6} d^{2} e^{2} x - 72 \, B a b^{5} d e^{3} x + 24 \, A b^{6} d e^{3} x + 36 \, B a^{2} b^{4} e^{4} x - 18 \, A a b^{5} e^{4} x}{6 \, b^{9} \mathrm {sgn}\left (b x + a\right )} \]
2*(2*B*b^3*d^3*e - 9*B*a*b^2*d^2*e^2 + 3*A*b^3*d^2*e^2 + 12*B*a^2*b*d*e^3 - 6*A*a*b^2*d*e^3 - 5*B*a^3*e^4 + 3*A*a^2*b*e^4)*log(abs(b*x + a))/(b^6*sg n(b*x + a)) - 1/2*(B*a*b^4*d^4 + A*b^5*d^4 - 12*B*a^2*b^3*d^3*e + 4*A*a*b^ 4*d^3*e + 30*B*a^3*b^2*d^2*e^2 - 18*A*a^2*b^3*d^2*e^2 - 28*B*a^4*b*d*e^3 + 20*A*a^3*b^2*d*e^3 + 9*B*a^5*e^4 - 7*A*a^4*b*e^4 + 2*(B*b^5*d^4 - 8*B*a*b ^4*d^3*e + 4*A*b^5*d^3*e + 18*B*a^2*b^3*d^2*e^2 - 12*A*a*b^4*d^2*e^2 - 16* B*a^3*b^2*d*e^3 + 12*A*a^2*b^3*d*e^3 + 5*B*a^4*b*e^4 - 4*A*a^3*b^2*e^4)*x) /((b*x + a)^2*b^6*sgn(b*x + a)) + 1/6*(2*B*b^6*e^4*x^3 + 12*B*b^6*d*e^3*x^ 2 - 9*B*a*b^5*e^4*x^2 + 3*A*b^6*e^4*x^2 + 36*B*b^6*d^2*e^2*x - 72*B*a*b^5* d*e^3*x + 24*A*b^6*d*e^3*x + 36*B*a^2*b^4*e^4*x - 18*A*a*b^5*e^4*x)/(b^9*s gn(b*x + a))
Timed out. \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^4}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]