Integrand size = 28, antiderivative size = 302 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {20 e^3 (b d-a e)^3}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^5 (6 b d-5 a e) x (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^6 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 e^4 (b d-a e)^2 (a+b x) \log (a+b x)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}} \]
-20*e^3*(-a*e+b*d)^3/b^7/((b*x+a)^2)^(1/2)-1/4*(-a*e+b*d)^6/b^7/(b*x+a)^3/ ((b*x+a)^2)^(1/2)-2*e*(-a*e+b*d)^5/b^7/(b*x+a)^2/((b*x+a)^2)^(1/2)-15/2*e^ 2*(-a*e+b*d)^4/b^7/(b*x+a)/((b*x+a)^2)^(1/2)+e^5*(-5*a*e+6*b*d)*x*(b*x+a)/ b^6/((b*x+a)^2)^(1/2)+1/2*e^6*x^2*(b*x+a)/b^5/((b*x+a)^2)^(1/2)+15*e^4*(-a *e+b*d)^2*(b*x+a)*ln(b*x+a)/b^7/((b*x+a)^2)^(1/2)
Time = 1.10 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {57 a^6 e^6+14 a^5 b e^5 (-11 d+12 e x)+a^4 b^2 e^4 \left (125 d^2-496 d e x+132 e^2 x^2\right )-4 a^3 b^3 e^3 \left (5 d^3-110 d^2 e x+126 d e^2 x^2+8 e^3 x^3\right )-a^2 b^4 e^2 \left (5 d^4+80 d^3 e x-540 d^2 e^2 x^2+96 d e^3 x^3+68 e^4 x^4\right )-2 a b^5 e \left (d^5+10 d^4 e x+60 d^3 e^2 x^2-120 d^2 e^3 x^3-48 d e^4 x^4+6 e^5 x^5\right )-b^6 \left (d^6+8 d^5 e x+30 d^4 e^2 x^2+80 d^3 e^3 x^3-24 d e^5 x^5-2 e^6 x^6\right )+60 e^4 (b d-a e)^2 (a+b x)^4 \log (a+b x)}{4 b^7 (a+b x)^3 \sqrt {(a+b x)^2}} \]
(57*a^6*e^6 + 14*a^5*b*e^5*(-11*d + 12*e*x) + a^4*b^2*e^4*(125*d^2 - 496*d *e*x + 132*e^2*x^2) - 4*a^3*b^3*e^3*(5*d^3 - 110*d^2*e*x + 126*d*e^2*x^2 + 8*e^3*x^3) - a^2*b^4*e^2*(5*d^4 + 80*d^3*e*x - 540*d^2*e^2*x^2 + 96*d*e^3 *x^3 + 68*e^4*x^4) - 2*a*b^5*e*(d^5 + 10*d^4*e*x + 60*d^3*e^2*x^2 - 120*d^ 2*e^3*x^3 - 48*d*e^4*x^4 + 6*e^5*x^5) - b^6*(d^6 + 8*d^5*e*x + 30*d^4*e^2* x^2 + 80*d^3*e^3*x^3 - 24*d*e^5*x^5 - 2*e^6*x^6) + 60*e^4*(b*d - a*e)^2*(a + b*x)^4*Log[a + b*x])/(4*b^7*(a + b*x)^3*Sqrt[(a + b*x)^2])
Time = 0.42 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.60, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1102, 27, 49, 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.
| \(\displaystyle \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {(d+e x)^6}{b^5 (a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {(d+e x)^6}{(a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {(a+b x) \int \left (\frac {x e^6}{b^5}+\frac {(6 b d-5 a e) e^5}{b^6}+\frac {15 (b d-a e)^2 e^4}{b^6 (a+b x)}+\frac {20 (b d-a e)^3 e^3}{b^6 (a+b x)^2}+\frac {15 (b d-a e)^4 e^2}{b^6 (a+b x)^3}+\frac {6 (b d-a e)^5 e}{b^6 (a+b x)^4}+\frac {(b d-a e)^6}{b^6 (a+b x)^5}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b x) \left (\frac {15 e^4 (b d-a e)^2 \log (a+b x)}{b^7}-\frac {20 e^3 (b d-a e)^3}{b^7 (a+b x)}-\frac {15 e^2 (b d-a e)^4}{2 b^7 (a+b x)^2}-\frac {2 e (b d-a e)^5}{b^7 (a+b x)^3}-\frac {(b d-a e)^6}{4 b^7 (a+b x)^4}+\frac {e^5 x (6 b d-5 a e)}{b^6}+\frac {e^6 x^2}{2 b^5}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*((e^5*(6*b*d - 5*a*e)*x)/b^6 + (e^6*x^2)/(2*b^5) - (b*d - a*e)^ 6/(4*b^7*(a + b*x)^4) - (2*e*(b*d - a*e)^5)/(b^7*(a + b*x)^3) - (15*e^2*(b *d - a*e)^4)/(2*b^7*(a + b*x)^2) - (20*e^3*(b*d - a*e)^3)/(b^7*(a + b*x)) + (15*e^4*(b*d - a*e)^2*Log[a + b*x])/b^7))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.17.4.3.1 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] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Time = 2.85 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.25
| method | result | size |
| risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{5} \left (-\frac {1}{2} b e \,x^{2}+5 a e x -6 b d x \right )}{\left (b x +a \right ) b^{6}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (20 a^{3} b^{2} e^{6}-60 a^{2} d \,e^{5} b^{3}+60 a \,b^{4} d^{2} e^{4}-20 d^{3} e^{3} b^{5}\right ) x^{3}+\frac {15 b \,e^{2} \left (7 e^{4} a^{4}-20 b \,e^{3} d \,a^{3}+18 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e -b^{4} d^{4}\right ) x^{2}}{2}+e \left (47 a^{5} e^{5}-130 a^{4} b d \,e^{4}+110 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e -2 b^{5} d^{5}\right ) x +\frac {57 a^{6} e^{6}-154 a^{5} b d \,e^{5}+125 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}-5 a^{2} b^{4} d^{4} e^{2}-2 a \,b^{5} d^{5} e -b^{6} d^{6}}{4 b}\right )}{\left (b x +a \right )^{5} b^{6}}+\frac {15 \sqrt {\left (b x +a \right )^{2}}\, e^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{7}}\) | \(379\) |
| default | \(\frac {\left (168 x \,a^{5} b \,e^{6}-8 x \,b^{6} d^{5} e -12 x^{5} a \,b^{5} e^{6}+24 x^{5} b^{6} d \,e^{5}-68 x^{4} a^{2} b^{4} e^{6}-32 x^{3} a^{3} b^{3} e^{6}-80 x^{3} b^{6} d^{3} e^{3}+132 x^{2} a^{4} b^{2} e^{6}-120 \ln \left (b x +a \right ) a \,b^{5} d \,e^{5} x^{4}+240 \ln \left (b x +a \right ) a^{5} b \,e^{6} x +60 \ln \left (b x +a \right ) b^{6} d^{2} e^{4} x^{4}-120 \ln \left (b x +a \right ) a^{5} b d \,e^{5}+57 a^{6} e^{6}-b^{6} d^{6}-2 a \,b^{5} d^{5} e +125 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}-5 a^{2} b^{4} d^{4} e^{2}-154 a^{5} b d \,e^{5}+360 \ln \left (b x +a \right ) a^{4} b^{2} e^{6} x^{2}+60 \ln \left (b x +a \right ) a^{2} b^{4} e^{6} x^{4}+240 \ln \left (b x +a \right ) a^{3} b^{3} e^{6} x^{3}-480 \ln \left (b x +a \right ) a^{4} b^{2} d \,e^{5} x +240 \ln \left (b x +a \right ) a^{3} b^{3} d^{2} e^{4} x -480 \ln \left (b x +a \right ) a^{2} b^{4} d \,e^{5} x^{3}+240 \ln \left (b x +a \right ) a \,b^{5} d^{2} e^{4} x^{3}-720 \ln \left (b x +a \right ) a^{3} b^{3} d \,e^{5} x^{2}+360 \ln \left (b x +a \right ) a^{2} b^{4} d^{2} e^{4} x^{2}+60 \ln \left (b x +a \right ) a^{4} b^{2} d^{2} e^{4}+2 x^{6} b^{6} e^{6}+60 \ln \left (b x +a \right ) a^{6} e^{6}-30 x^{2} b^{6} d^{4} e^{2}+96 x^{4} a \,b^{5} d \,e^{5}-96 x^{3} a^{2} b^{4} d \,e^{5}+240 x^{3} a \,b^{5} d^{2} e^{4}-504 x^{2} a^{3} b^{3} d \,e^{5}+540 x^{2} a^{2} b^{4} d^{2} e^{4}-120 x^{2} a \,b^{5} d^{3} e^{3}-496 x \,a^{4} b^{2} d \,e^{5}+440 x \,a^{3} b^{3} d^{2} e^{4}-80 x \,a^{2} b^{4} d^{3} e^{3}-20 x a \,b^{5} d^{4} e^{2}\right ) \left (b x +a \right )}{4 b^{7} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(661\) |
-((b*x+a)^2)^(1/2)/(b*x+a)*e^5/b^6*(-1/2*b*e*x^2+5*a*e*x-6*b*d*x)+((b*x+a) ^2)^(1/2)/(b*x+a)^5*((20*a^3*b^2*e^6-60*a^2*b^3*d*e^5+60*a*b^4*d^2*e^4-20* b^5*d^3*e^3)*x^3+15/2*b*e^2*(7*a^4*e^4-20*a^3*b*d*e^3+18*a^2*b^2*d^2*e^2-4 *a*b^3*d^3*e-b^4*d^4)*x^2+e*(47*a^5*e^5-130*a^4*b*d*e^4+110*a^3*b^2*d^2*e^ 3-20*a^2*b^3*d^3*e^2-5*a*b^4*d^4*e-2*b^5*d^5)*x+1/4*(57*a^6*e^6-154*a^5*b* d*e^5+125*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3-5*a^2*b^4*d^4*e^2-2*a*b^5*d^5 *e-b^6*d^6)/b)/b^6+15*((b*x+a)^2)^(1/2)/(b*x+a)/b^7*e^4*(a^2*e^2-2*a*b*d*e +b^2*d^2)*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (219) = 438\).
Time = 0.27 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.89 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {2 \, b^{6} e^{6} x^{6} - b^{6} d^{6} - 2 \, a b^{5} d^{5} e - 5 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 125 \, a^{4} b^{2} d^{2} e^{4} - 154 \, a^{5} b d e^{5} + 57 \, a^{6} e^{6} + 12 \, {\left (2 \, b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 4 \, {\left (24 \, a b^{5} d e^{5} - 17 \, a^{2} b^{4} e^{6}\right )} x^{4} - 16 \, {\left (5 \, b^{6} d^{3} e^{3} - 15 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{3} - 6 \, {\left (5 \, b^{6} d^{4} e^{2} + 20 \, a b^{5} d^{3} e^{3} - 90 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} - 22 \, a^{4} b^{2} e^{6}\right )} x^{2} - 4 \, {\left (2 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 20 \, a^{2} b^{4} d^{3} e^{3} - 110 \, a^{3} b^{3} d^{2} e^{4} + 124 \, a^{4} b^{2} d e^{5} - 42 \, a^{5} b e^{6}\right )} x + 60 \, {\left (a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} + a^{6} e^{6} + {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \, {\left (a b^{5} d^{2} e^{4} - 2 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 6 \, {\left (a^{2} b^{4} d^{2} e^{4} - 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 4 \, {\left (a^{3} b^{3} d^{2} e^{4} - 2 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x\right )} \log \left (b x + a\right )}{4 \, {\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}} \]
1/4*(2*b^6*e^6*x^6 - b^6*d^6 - 2*a*b^5*d^5*e - 5*a^2*b^4*d^4*e^2 - 20*a^3* b^3*d^3*e^3 + 125*a^4*b^2*d^2*e^4 - 154*a^5*b*d*e^5 + 57*a^6*e^6 + 12*(2*b ^6*d*e^5 - a*b^5*e^6)*x^5 + 4*(24*a*b^5*d*e^5 - 17*a^2*b^4*e^6)*x^4 - 16*( 5*b^6*d^3*e^3 - 15*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 + 2*a^3*b^3*e^6)*x^3 - 6*(5*b^6*d^4*e^2 + 20*a*b^5*d^3*e^3 - 90*a^2*b^4*d^2*e^4 + 84*a^3*b^3*d*e^ 5 - 22*a^4*b^2*e^6)*x^2 - 4*(2*b^6*d^5*e + 5*a*b^5*d^4*e^2 + 20*a^2*b^4*d^ 3*e^3 - 110*a^3*b^3*d^2*e^4 + 124*a^4*b^2*d*e^5 - 42*a^5*b*e^6)*x + 60*(a^ 4*b^2*d^2*e^4 - 2*a^5*b*d*e^5 + a^6*e^6 + (b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a ^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 - 2*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 6*(a^2*b^4*d^2*e^4 - 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 4*(a^3*b^3*d^2*e ^4 - 2*a^4*b^2*d*e^5 + a^5*b*e^6)*x)*log(b*x + a))/(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)
\[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{6}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (219) = 438\).
Time = 0.25 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.91 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {1}{4} \, e^{6} {\left (\frac {2 \, b^{6} x^{6} - 12 \, a b^{5} x^{5} - 68 \, a^{2} b^{4} x^{4} - 32 \, a^{3} b^{3} x^{3} + 132 \, a^{4} b^{2} x^{2} + 168 \, a^{5} b x + 57 \, a^{6}}{b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}} + \frac {60 \, a^{2} \log \left (b x + a\right )}{b^{7}}\right )} + \frac {1}{2} \, d e^{5} {\left (\frac {12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}} - \frac {60 \, a \log \left (b x + a\right )}{b^{6}}\right )} + \frac {5}{4} \, d^{2} e^{4} {\left (\frac {48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}} + \frac {12 \, \log \left (b x + a\right )}{b^{5}}\right )} - \frac {5}{3} \, d^{3} e^{3} {\left (\frac {12 \, x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}} + \frac {6 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {3 \, a^{3}}{b^{8} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {1}{2} \, d^{5} e {\left (\frac {4}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {3 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {5}{4} \, d^{4} e^{2} {\left (\frac {6}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{3}} + \frac {3 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {d^{6}}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} \]
1/4*e^6*((2*b^6*x^6 - 12*a*b^5*x^5 - 68*a^2*b^4*x^4 - 32*a^3*b^3*x^3 + 132 *a^4*b^2*x^2 + 168*a^5*b*x + 57*a^6)/(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) + 60*a^2*log(b*x + a)/b^7) + 1/2*d*e^5*((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 - 7 7*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) + 5/4*d^2*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) - 5/3*d^3*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)) - 1/2*d ^5*e*(4/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) - 3*a/(b^6*(x + a/b)^4)) - 5 /4*d^4*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*d^6/(b^5*(x + a/b)^4)
Time = 0.27 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {15 \, {\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {b^{5} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, b^{5} d e^{5} x \mathrm {sgn}\left (b x + a\right ) - 10 \, a b^{4} e^{6} x \mathrm {sgn}\left (b x + a\right )}{2 \, b^{10}} - \frac {b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 5 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} - 125 \, a^{4} b^{2} d^{2} e^{4} + 154 \, a^{5} b d e^{5} - 57 \, a^{6} e^{6} + 80 \, {\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} - 18 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} - 7 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \, {\left (2 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 20 \, a^{2} b^{4} d^{3} e^{3} - 110 \, a^{3} b^{3} d^{2} e^{4} + 130 \, a^{4} b^{2} d e^{5} - 47 \, a^{5} b e^{6}\right )} x}{4 \, {\left (b x + a\right )}^{4} b^{7} \mathrm {sgn}\left (b x + a\right )} \]
15*(b^2*d^2*e^4 - 2*a*b*d*e^5 + a^2*e^6)*log(abs(b*x + a))/(b^7*sgn(b*x + a)) + 1/2*(b^5*e^6*x^2*sgn(b*x + a) + 12*b^5*d*e^5*x*sgn(b*x + a) - 10*a*b ^4*e^6*x*sgn(b*x + a))/b^10 - 1/4*(b^6*d^6 + 2*a*b^5*d^5*e + 5*a^2*b^4*d^4 *e^2 + 20*a^3*b^3*d^3*e^3 - 125*a^4*b^2*d^2*e^4 + 154*a^5*b*d*e^5 - 57*a^6 *e^6 + 80*(b^6*d^3*e^3 - 3*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 - a^3*b^3*e^6)* x^3 + 30*(b^6*d^4*e^2 + 4*a*b^5*d^3*e^3 - 18*a^2*b^4*d^2*e^4 + 20*a^3*b^3* d*e^5 - 7*a^4*b^2*e^6)*x^2 + 4*(2*b^6*d^5*e + 5*a*b^5*d^4*e^2 + 20*a^2*b^4 *d^3*e^3 - 110*a^3*b^3*d^2*e^4 + 130*a^4*b^2*d*e^5 - 47*a^5*b*e^6)*x)/((b* x + a)^4*b^7*sgn(b*x + a))
Timed out. \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^6}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]