Integrand size = 26, antiderivative size = 146 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=-\frac {b (b d-a e)^5 x}{e^6}+\frac {(b d-a e)^4 (a+b x)^2}{2 e^5}-\frac {(b d-a e)^3 (a+b x)^3}{3 e^4}+\frac {(b d-a e)^2 (a+b x)^4}{4 e^3}-\frac {(b d-a e) (a+b x)^5}{5 e^2}+\frac {(a+b x)^6}{6 e}+\frac {(b d-a e)^6 \log (d+e x)}{e^7} \]
-b*(-a*e+b*d)^5*x/e^6+1/2*(-a*e+b*d)^4*(b*x+a)^2/e^5-1/3*(-a*e+b*d)^3*(b*x +a)^3/e^4+1/4*(-a*e+b*d)^2*(b*x+a)^4/e^3-1/5*(-a*e+b*d)*(b*x+a)^5/e^2+1/6* (b*x+a)^6/e+(-a*e+b*d)^6*ln(e*x+d)/e^7
Time = 0.07 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.58 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {b e x \left (360 a^5 e^5+450 a^4 b e^4 (-2 d+e x)+200 a^3 b^2 e^3 \left (6 d^2-3 d e x+2 e^2 x^2\right )+75 a^2 b^3 e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+6 a b^4 e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+b^5 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 (b d-a e)^6 \log (d+e x)}{60 e^7} \]
(b*e*x*(360*a^5*e^5 + 450*a^4*b*e^4*(-2*d + e*x) + 200*a^3*b^2*e^3*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 75*a^2*b^3*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 6*a*b^4*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3 *x^3 + 12*e^4*x^4) + b^5*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e ^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + 60*(b*d - a*e)^6*Log[d + e*x])/(60* e^7)
Time = 0.30 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1098, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^6}{d+e x}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^6}{d+e x}dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {(a e-b d)^6}{e^6 (d+e x)}-\frac {b (b d-a e)^5}{e^6}+\frac {b (a+b x) (b d-a e)^4}{e^5}-\frac {b (a+b x)^2 (b d-a e)^3}{e^4}+\frac {b (a+b x)^3 (b d-a e)^2}{e^3}-\frac {b (a+b x)^4 (b d-a e)}{e^2}+\frac {b (a+b x)^5}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(b d-a e)^6 \log (d+e x)}{e^7}-\frac {b x (b d-a e)^5}{e^6}+\frac {(a+b x)^2 (b d-a e)^4}{2 e^5}-\frac {(a+b x)^3 (b d-a e)^3}{3 e^4}+\frac {(a+b x)^4 (b d-a e)^2}{4 e^3}-\frac {(a+b x)^5 (b d-a e)}{5 e^2}+\frac {(a+b x)^6}{6 e}\) |
-((b*(b*d - a*e)^5*x)/e^6) + ((b*d - a*e)^4*(a + b*x)^2)/(2*e^5) - ((b*d - a*e)^3*(a + b*x)^3)/(3*e^4) + ((b*d - a*e)^2*(a + b*x)^4)/(4*e^3) - ((b*d - a*e)*(a + b*x)^5)/(5*e^2) + (a + b*x)^6/(6*e) + ((b*d - a*e)^6*Log[d + e*x])/e^7
3.15.90.3.1 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_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(332\) vs. \(2(136)=272\).
Time = 2.28 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.28
| method | result | size |
| norman | \(\frac {b \left (6 a^{5} e^{5}-15 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}-15 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) x}{e^{6}}+\frac {b^{6} x^{6}}{6 e}+\frac {b^{2} \left (15 e^{4} a^{4}-20 b \,e^{3} d \,a^{3}+15 b^{2} e^{2} d^{2} a^{2}-6 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{2 e^{5}}+\frac {b^{5} \left (6 a e -b d \right ) x^{5}}{5 e^{2}}+\frac {b^{3} \left (20 a^{3} e^{3}-15 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{3}}{3 e^{4}}+\frac {b^{4} \left (15 a^{2} e^{2}-6 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}+\frac {\left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(333\) |
| risch | \(\frac {\ln \left (e x +d \right ) a^{6}}{e}+\frac {b^{6} x^{6}}{6 e}+\frac {6 b \,a^{5} x}{e}-\frac {b^{6} d^{5} x}{e^{6}}+\frac {\ln \left (e x +d \right ) b^{6} d^{6}}{e^{7}}+\frac {6 b^{5} x^{5} a}{5 e}-\frac {b^{6} x^{5} d}{5 e^{2}}+\frac {15 b^{4} x^{4} a^{2}}{4 e}+\frac {b^{6} x^{4} d^{2}}{4 e^{3}}+\frac {20 b^{3} x^{3} a^{3}}{3 e}-\frac {b^{6} x^{3} d^{3}}{3 e^{4}}+\frac {15 b^{2} x^{2} a^{4}}{2 e}+\frac {b^{6} x^{2} d^{4}}{2 e^{5}}-\frac {3 b^{5} x^{4} a d}{2 e^{2}}-\frac {5 b^{4} x^{3} a^{2} d}{e^{2}}+\frac {2 b^{5} x^{3} a \,d^{2}}{e^{3}}-\frac {10 b^{3} x^{2} a^{3} d}{e^{2}}+\frac {15 b^{4} x^{2} a^{2} d^{2}}{2 e^{3}}-\frac {3 b^{5} x^{2} a \,d^{3}}{e^{4}}-\frac {15 b^{2} a^{4} d x}{e^{2}}+\frac {20 b^{3} a^{3} d^{2} x}{e^{3}}-\frac {15 b^{4} a^{2} d^{3} x}{e^{4}}+\frac {6 b^{5} a \,d^{4} x}{e^{5}}-\frac {6 \ln \left (e x +d \right ) a^{5} b d}{e^{2}}+\frac {15 \ln \left (e x +d \right ) a^{4} b^{2} d^{2}}{e^{3}}-\frac {20 \ln \left (e x +d \right ) a^{3} b^{3} d^{3}}{e^{4}}+\frac {15 \ln \left (e x +d \right ) a^{2} b^{4} d^{4}}{e^{5}}-\frac {6 \ln \left (e x +d \right ) a \,b^{5} d^{5}}{e^{6}}\) | \(412\) |
| parallelrisch | \(\frac {360 x \,a^{5} b \,e^{6}-60 x \,b^{6} d^{5} e +72 x^{5} a \,b^{5} e^{6}-12 x^{5} b^{6} d \,e^{5}+225 x^{4} a^{2} b^{4} e^{6}+15 x^{4} b^{6} d^{2} e^{4}+400 x^{3} a^{3} b^{3} e^{6}-20 x^{3} b^{6} d^{3} e^{3}+450 x^{2} a^{4} b^{2} e^{6}+60 \ln \left (e x +d \right ) b^{6} d^{6}+10 x^{6} b^{6} e^{6}-1200 \ln \left (e x +d \right ) a^{3} b^{3} d^{3} e^{3}+900 \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{2}-360 \ln \left (e x +d \right ) a \,b^{5} d^{5} e +60 \ln \left (e x +d \right ) a^{6} e^{6}+30 x^{2} b^{6} d^{4} e^{2}-90 x^{4} a \,b^{5} d \,e^{5}-300 x^{3} a^{2} b^{4} d \,e^{5}+120 x^{3} a \,b^{5} d^{2} e^{4}-600 x^{2} a^{3} b^{3} d \,e^{5}+450 x^{2} a^{2} b^{4} d^{2} e^{4}-180 x^{2} a \,b^{5} d^{3} e^{3}-900 x \,a^{4} b^{2} d \,e^{5}+1200 x \,a^{3} b^{3} d^{2} e^{4}-900 x \,a^{2} b^{4} d^{3} e^{3}+360 x a \,b^{5} d^{4} e^{2}-360 \ln \left (e x +d \right ) a^{5} b d \,e^{5}+900 \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{4}}{60 e^{7}}\) | \(412\) |
| default | \(\frac {b \left (\frac {b^{5} x^{6} e^{5}}{6}+\frac {\left (\left (\left (2 a e -b d \right ) b^{2} e^{2}+b e \left (a b \,e^{2}+b^{2} d e \right )\right ) b^{2} e^{2}+b^{3} e^{3} \left (3 a b \,e^{2}-b^{2} d e \right )\right ) x^{5}}{5}+\frac {\left (\left (\left (2 a e -b d \right ) \left (a b \,e^{2}+b^{2} d e \right )+b e \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right )\right ) b^{2} e^{2}+\left (\left (2 a e -b d \right ) b^{2} e^{2}+b e \left (a b \,e^{2}+b^{2} d e \right )\right ) \left (3 a b \,e^{2}-b^{2} d e \right )+b^{3} e^{3} \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (2 a e -b d \right ) \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right ) b^{2} e^{2}+\left (\left (2 a e -b d \right ) \left (a b \,e^{2}+b^{2} d e \right )+b e \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right )\right ) \left (3 a b \,e^{2}-b^{2} d e \right )+\left (\left (2 a e -b d \right ) b^{2} e^{2}+b e \left (a b \,e^{2}+b^{2} d e \right )\right ) \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right )\right ) x^{3}}{3}+\frac {\left (\left (2 a e -b d \right ) \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right ) \left (3 a b \,e^{2}-b^{2} d e \right )+\left (\left (2 a e -b d \right ) \left (a b \,e^{2}+b^{2} d e \right )+b e \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right )\right ) \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right )\right ) x^{2}}{2}+\left (2 a e -b d \right ) \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right ) \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right ) x \right )}{e^{6}}+\frac {\left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(653\) |
b*(6*a^5*e^5-15*a^4*b*d*e^4+20*a^3*b^2*d^2*e^3-15*a^2*b^3*d^3*e^2+6*a*b^4* d^4*e-b^5*d^5)/e^6*x+1/6*b^6/e*x^6+1/2*b^2/e^5*(15*a^4*e^4-20*a^3*b*d*e^3+ 15*a^2*b^2*d^2*e^2-6*a*b^3*d^3*e+b^4*d^4)*x^2+1/5*b^5/e^2*(6*a*e-b*d)*x^5+ 1/3/e^4*b^3*(20*a^3*e^3-15*a^2*b*d*e^2+6*a*b^2*d^2*e-b^3*d^3)*x^3+1/4/e^3* b^4*(15*a^2*e^2-6*a*b*d*e+b^2*d^2)*x^4+(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d ^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d^5*e+b^6*d^6)/e^7*ln (e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (136) = 272\).
Time = 0.27 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.40 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {10 \, b^{6} e^{6} x^{6} - 12 \, {\left (b^{6} d e^{5} - 6 \, a b^{5} e^{6}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} - 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e^{2} - 6 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 60 \, {\left (b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \]
1/60*(10*b^6*e^6*x^6 - 12*(b^6*d*e^5 - 6*a*b^5*e^6)*x^5 + 15*(b^6*d^2*e^4 - 6*a*b^5*d*e^5 + 15*a^2*b^4*e^6)*x^4 - 20*(b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 + 15*a^2*b^4*d*e^5 - 20*a^3*b^3*e^6)*x^3 + 30*(b^6*d^4*e^2 - 6*a*b^5*d^3*e ^3 + 15*a^2*b^4*d^2*e^4 - 20*a^3*b^3*d*e^5 + 15*a^4*b^2*e^6)*x^2 - 60*(b^6 *d^5*e - 6*a*b^5*d^4*e^2 + 15*a^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 + 15*a^ 4*b^2*d*e^5 - 6*a^5*b*e^6)*x + 60*(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^ 4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6) *log(e*x + d))/e^7
Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (124) = 248\).
Time = 0.36 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {b^{6} x^{6}}{6 e} + x^{5} \cdot \left (\frac {6 a b^{5}}{5 e} - \frac {b^{6} d}{5 e^{2}}\right ) + x^{4} \cdot \left (\frac {15 a^{2} b^{4}}{4 e} - \frac {3 a b^{5} d}{2 e^{2}} + \frac {b^{6} d^{2}}{4 e^{3}}\right ) + x^{3} \cdot \left (\frac {20 a^{3} b^{3}}{3 e} - \frac {5 a^{2} b^{4} d}{e^{2}} + \frac {2 a b^{5} d^{2}}{e^{3}} - \frac {b^{6} d^{3}}{3 e^{4}}\right ) + x^{2} \cdot \left (\frac {15 a^{4} b^{2}}{2 e} - \frac {10 a^{3} b^{3} d}{e^{2}} + \frac {15 a^{2} b^{4} d^{2}}{2 e^{3}} - \frac {3 a b^{5} d^{3}}{e^{4}} + \frac {b^{6} d^{4}}{2 e^{5}}\right ) + x \left (\frac {6 a^{5} b}{e} - \frac {15 a^{4} b^{2} d}{e^{2}} + \frac {20 a^{3} b^{3} d^{2}}{e^{3}} - \frac {15 a^{2} b^{4} d^{3}}{e^{4}} + \frac {6 a b^{5} d^{4}}{e^{5}} - \frac {b^{6} d^{5}}{e^{6}}\right ) + \frac {\left (a e - b d\right )^{6} \log {\left (d + e x \right )}}{e^{7}} \]
b**6*x**6/(6*e) + x**5*(6*a*b**5/(5*e) - b**6*d/(5*e**2)) + x**4*(15*a**2* b**4/(4*e) - 3*a*b**5*d/(2*e**2) + b**6*d**2/(4*e**3)) + x**3*(20*a**3*b** 3/(3*e) - 5*a**2*b**4*d/e**2 + 2*a*b**5*d**2/e**3 - b**6*d**3/(3*e**4)) + x**2*(15*a**4*b**2/(2*e) - 10*a**3*b**3*d/e**2 + 15*a**2*b**4*d**2/(2*e**3 ) - 3*a*b**5*d**3/e**4 + b**6*d**4/(2*e**5)) + x*(6*a**5*b/e - 15*a**4*b** 2*d/e**2 + 20*a**3*b**3*d**2/e**3 - 15*a**2*b**4*d**3/e**4 + 6*a*b**5*d**4 /e**5 - b**6*d**5/e**6) + (a*e - b*d)**6*log(d + e*x)/e**7
Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (136) = 272\).
Time = 0.19 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.39 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {10 \, b^{6} e^{5} x^{6} - 12 \, {\left (b^{6} d e^{4} - 6 \, a b^{5} e^{5}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{3} - 6 \, a b^{5} d e^{4} + 15 \, a^{2} b^{4} e^{5}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{2} - 6 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} - 20 \, a^{3} b^{3} e^{5}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e - 6 \, a b^{5} d^{3} e^{2} + 15 \, a^{2} b^{4} d^{2} e^{3} - 20 \, a^{3} b^{3} d e^{4} + 15 \, a^{4} b^{2} e^{5}\right )} x^{2} - 60 \, {\left (b^{6} d^{5} - 6 \, a b^{5} d^{4} e + 15 \, a^{2} b^{4} d^{3} e^{2} - 20 \, a^{3} b^{3} d^{2} e^{3} + 15 \, a^{4} b^{2} d e^{4} - 6 \, a^{5} b e^{5}\right )} x}{60 \, e^{6}} + \frac {{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left (e x + d\right )}{e^{7}} \]
1/60*(10*b^6*e^5*x^6 - 12*(b^6*d*e^4 - 6*a*b^5*e^5)*x^5 + 15*(b^6*d^2*e^3 - 6*a*b^5*d*e^4 + 15*a^2*b^4*e^5)*x^4 - 20*(b^6*d^3*e^2 - 6*a*b^5*d^2*e^3 + 15*a^2*b^4*d*e^4 - 20*a^3*b^3*e^5)*x^3 + 30*(b^6*d^4*e - 6*a*b^5*d^3*e^2 + 15*a^2*b^4*d^2*e^3 - 20*a^3*b^3*d*e^4 + 15*a^4*b^2*e^5)*x^2 - 60*(b^6*d ^5 - 6*a*b^5*d^4*e + 15*a^2*b^4*d^3*e^2 - 20*a^3*b^3*d^2*e^3 + 15*a^4*b^2* d*e^4 - 6*a^5*b*e^5)*x)/e^6 + (b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^ 2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*log (e*x + d)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (136) = 272\).
Time = 0.26 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.57 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {10 \, b^{6} e^{5} x^{6} - 12 \, b^{6} d e^{4} x^{5} + 72 \, a b^{5} e^{5} x^{5} + 15 \, b^{6} d^{2} e^{3} x^{4} - 90 \, a b^{5} d e^{4} x^{4} + 225 \, a^{2} b^{4} e^{5} x^{4} - 20 \, b^{6} d^{3} e^{2} x^{3} + 120 \, a b^{5} d^{2} e^{3} x^{3} - 300 \, a^{2} b^{4} d e^{4} x^{3} + 400 \, a^{3} b^{3} e^{5} x^{3} + 30 \, b^{6} d^{4} e x^{2} - 180 \, a b^{5} d^{3} e^{2} x^{2} + 450 \, a^{2} b^{4} d^{2} e^{3} x^{2} - 600 \, a^{3} b^{3} d e^{4} x^{2} + 450 \, a^{4} b^{2} e^{5} x^{2} - 60 \, b^{6} d^{5} x + 360 \, a b^{5} d^{4} e x - 900 \, a^{2} b^{4} d^{3} e^{2} x + 1200 \, a^{3} b^{3} d^{2} e^{3} x - 900 \, a^{4} b^{2} d e^{4} x + 360 \, a^{5} b e^{5} x}{60 \, e^{6}} + \frac {{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} \]
1/60*(10*b^6*e^5*x^6 - 12*b^6*d*e^4*x^5 + 72*a*b^5*e^5*x^5 + 15*b^6*d^2*e^ 3*x^4 - 90*a*b^5*d*e^4*x^4 + 225*a^2*b^4*e^5*x^4 - 20*b^6*d^3*e^2*x^3 + 12 0*a*b^5*d^2*e^3*x^3 - 300*a^2*b^4*d*e^4*x^3 + 400*a^3*b^3*e^5*x^3 + 30*b^6 *d^4*e*x^2 - 180*a*b^5*d^3*e^2*x^2 + 450*a^2*b^4*d^2*e^3*x^2 - 600*a^3*b^3 *d*e^4*x^2 + 450*a^4*b^2*e^5*x^2 - 60*b^6*d^5*x + 360*a*b^5*d^4*e*x - 900* a^2*b^4*d^3*e^2*x + 1200*a^3*b^3*d^2*e^3*x - 900*a^4*b^2*d*e^4*x + 360*a^5 *b*e^5*x)/e^6 + (b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3 *d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*log(abs(e*x + d)) /e^7
Time = 10.17 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.64 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=x^5\,\left (\frac {6\,a\,b^5}{5\,e}-\frac {b^6\,d}{5\,e^2}\right )+x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{e}-\frac {15\,a^2\,b^4}{e}\right )}{3\,e}+\frac {20\,a^3\,b^3}{3\,e}\right )+x\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{e}-\frac {15\,a^2\,b^4}{e}\right )}{e}+\frac {20\,a^3\,b^3}{e}\right )}{e}-\frac {15\,a^4\,b^2}{e}\right )}{e}+\frac {6\,a^5\,b}{e}\right )-x^4\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{4\,e}-\frac {15\,a^2\,b^4}{4\,e}\right )-x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{e}-\frac {15\,a^2\,b^4}{e}\right )}{e}+\frac {20\,a^3\,b^3}{e}\right )}{2\,e}-\frac {15\,a^4\,b^2}{2\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{e^7}+\frac {b^6\,x^6}{6\,e} \]
x^5*((6*a*b^5)/(5*e) - (b^6*d)/(5*e^2)) + x^3*((d*((d*((6*a*b^5)/e - (b^6* d)/e^2))/e - (15*a^2*b^4)/e))/(3*e) + (20*a^3*b^3)/(3*e)) + x*((d*((d*((d* ((d*((6*a*b^5)/e - (b^6*d)/e^2))/e - (15*a^2*b^4)/e))/e + (20*a^3*b^3)/e)) /e - (15*a^4*b^2)/e))/e + (6*a^5*b)/e) - x^4*((d*((6*a*b^5)/e - (b^6*d)/e^ 2))/(4*e) - (15*a^2*b^4)/(4*e)) - x^2*((d*((d*((d*((6*a*b^5)/e - (b^6*d)/e ^2))/e - (15*a^2*b^4)/e))/e + (20*a^3*b^3)/e))/(2*e) - (15*a^4*b^2)/(2*e)) + (log(d + e*x)*(a^6*e^6 + b^6*d^6 + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3* e^3 + 15*a^4*b^2*d^2*e^4 - 6*a*b^5*d^5*e - 6*a^5*b*d*e^5))/e^7 + (b^6*x^6) /(6*e)