Integrand size = 31, antiderivative size = 170 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {b (b d-a e)^6 x}{e^7}-\frac {(b d-a e)^5 (a+b x)^2}{2 e^6}+\frac {(b d-a e)^4 (a+b x)^3}{3 e^5}-\frac {(b d-a e)^3 (a+b x)^4}{4 e^4}+\frac {(b d-a e)^2 (a+b x)^5}{5 e^3}-\frac {(b d-a e) (a+b x)^6}{6 e^2}+\frac {(a+b x)^7}{7 e}-\frac {(b d-a e)^7 \log (d+e x)}{e^8} \]
b*(-a*e+b*d)^6*x/e^7-1/2*(-a*e+b*d)^5*(b*x+a)^2/e^6+1/3*(-a*e+b*d)^4*(b*x+ a)^3/e^5-1/4*(-a*e+b*d)^3*(b*x+a)^4/e^4+1/5*(-a*e+b*d)^2*(b*x+a)^5/e^3-1/6 *(-a*e+b*d)*(b*x+a)^6/e^2+1/7*(b*x+a)^7/e-(-a*e+b*d)^7*ln(e*x+d)/e^8
Time = 0.07 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.79 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {b e x \left (2940 a^6 e^6+4410 a^5 b e^5 (-2 d+e x)+2450 a^4 b^2 e^4 \left (6 d^2-3 d e x+2 e^2 x^2\right )+1225 a^3 b^3 e^3 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+147 a^2 b^4 e^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+49 a b^5 e \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+b^6 \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )-420 (b d-a e)^7 \log (d+e x)}{420 e^8} \]
(b*e*x*(2940*a^6*e^6 + 4410*a^5*b*e^5*(-2*d + e*x) + 2450*a^4*b^2*e^4*(6*d ^2 - 3*d*e*x + 2*e^2*x^2) + 1225*a^3*b^3*e^3*(-12*d^3 + 6*d^2*e*x - 4*d*e^ 2*x^2 + 3*e^3*x^3) + 147*a^2*b^4*e^2*(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) + 49*a*b^5*e*(-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) + b^6*(420*d^6 - 210 *d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x ^5 + 60*e^6*x^6)) - 420*(b*d - a*e)^7*Log[d + e*x])/(420*e^8)
Time = 0.34 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^7}{d+e x}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^7}{d+e x}dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {(a e-b d)^7}{e^7 (d+e x)}+\frac {b (b d-a e)^6}{e^7}-\frac {b (a+b x) (b d-a e)^5}{e^6}+\frac {b (a+b x)^2 (b d-a e)^4}{e^5}-\frac {b (a+b x)^3 (b d-a e)^3}{e^4}+\frac {b (a+b x)^4 (b d-a e)^2}{e^3}-\frac {b (a+b x)^5 (b d-a e)}{e^2}+\frac {b (a+b x)^6}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(b d-a e)^7 \log (d+e x)}{e^8}+\frac {b x (b d-a e)^6}{e^7}-\frac {(a+b x)^2 (b d-a e)^5}{2 e^6}+\frac {(a+b x)^3 (b d-a e)^4}{3 e^5}-\frac {(a+b x)^4 (b d-a e)^3}{4 e^4}+\frac {(a+b x)^5 (b d-a e)^2}{5 e^3}-\frac {(a+b x)^6 (b d-a e)}{6 e^2}+\frac {(a+b x)^7}{7 e}\) |
(b*(b*d - a*e)^6*x)/e^7 - ((b*d - a*e)^5*(a + b*x)^2)/(2*e^6) + ((b*d - a* e)^4*(a + b*x)^3)/(3*e^5) - ((b*d - a*e)^3*(a + b*x)^4)/(4*e^4) + ((b*d - a*e)^2*(a + b*x)^5)/(5*e^3) - ((b*d - a*e)*(a + b*x)^6)/(6*e^2) + (a + b*x )^7/(7*e) - ((b*d - a*e)^7*Log[d + e*x])/e^8
3.20.23.3.1 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_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^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}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(436\) vs. \(2(158)=316\).
Time = 0.24 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.57
| method | result | size |
| norman | \(\frac {b \left (7 e^{6} a^{6}-21 b d \,e^{5} a^{5}+35 b^{2} d^{2} e^{4} a^{4}-35 b^{3} d^{3} e^{3} a^{3}+21 b^{4} d^{4} e^{2} a^{2}-7 b^{5} d^{5} e a +b^{6} d^{6}\right ) x}{e^{7}}+\frac {b^{7} x^{7}}{7 e}+\frac {b^{2} \left (21 e^{5} a^{5}-35 b d \,e^{4} a^{4}+35 b^{2} d^{2} e^{3} a^{3}-21 b^{3} d^{3} e^{2} a^{2}+7 b^{4} d^{4} e a -b^{5} d^{5}\right ) x^{2}}{2 e^{6}}+\frac {b^{3} \left (35 e^{4} a^{4}-35 b d \,e^{3} a^{3}+21 b^{2} d^{2} e^{2} a^{2}-7 b^{3} d^{3} e a +b^{4} d^{4}\right ) x^{3}}{3 e^{5}}+\frac {b^{4} \left (35 a^{3} e^{3}-21 a^{2} b d \,e^{2}+7 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{4}}{4 e^{4}}+\frac {b^{5} \left (21 e^{2} a^{2}-7 a b d e +b^{2} d^{2}\right ) x^{5}}{5 e^{3}}+\frac {b^{6} \left (7 a e -b d \right ) x^{6}}{6 e^{2}}+\frac {\left (e^{7} a^{7}-7 b d \,e^{6} a^{6}+21 b^{2} d^{2} e^{5} a^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(437\) |
| default | \(\frac {b \left (-\frac {7}{5} a \,b^{5} d \,e^{5} x^{5}-\frac {1}{2} b^{6} d^{5} e \,x^{2}+\frac {35}{3} a^{4} b^{2} e^{6} x^{3}+\frac {1}{3} b^{6} d^{4} e^{2} x^{3}+\frac {21}{2} a^{5} b \,e^{6} x^{2}-\frac {1}{4} b^{6} d^{3} e^{3} x^{4}+\frac {35}{4} a^{3} b^{3} e^{6} x^{4}+\frac {21}{5} a^{2} b^{4} e^{6} x^{5}+\frac {1}{5} b^{6} d^{2} e^{4} x^{5}+\frac {7}{6} a \,b^{5} e^{6} x^{6}-\frac {1}{6} b^{6} d \,e^{5} x^{6}-\frac {35}{3} a^{3} b^{3} d \,e^{5} x^{3}+7 a^{2} b^{4} d^{2} e^{4} x^{3}-\frac {7}{3} a \,b^{5} d^{3} e^{3} x^{3}-\frac {35}{2} a^{4} b^{2} d \,e^{5} x^{2}+b^{6} d^{6} x +\frac {7}{4} a \,b^{5} d^{2} e^{4} x^{4}+7 e^{6} a^{6} x -\frac {21}{4} a^{2} b^{4} d \,e^{5} x^{4}+\frac {1}{7} b^{6} x^{7} e^{6}+\frac {7}{2} a \,b^{5} d^{4} e^{2} x^{2}-21 b d \,e^{5} a^{5} x +\frac {35}{2} a^{3} b^{3} d^{2} e^{4} x^{2}-\frac {21}{2} a^{2} b^{4} d^{3} e^{3} x^{2}+21 b^{4} d^{4} e^{2} a^{2} x -7 b^{5} d^{5} e a x +35 b^{2} d^{2} e^{4} a^{4} x -35 b^{3} d^{3} e^{3} a^{3} x \right )}{e^{7}}+\frac {\left (e^{7} a^{7}-7 b d \,e^{6} a^{6}+21 b^{2} d^{2} e^{5} a^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(491\) |
| risch | \(-\frac {b^{7} d^{5} x^{2}}{2 e^{6}}+\frac {35 b^{3} a^{4} x^{3}}{3 e}+\frac {b^{7} d^{4} x^{3}}{3 e^{5}}+\frac {21 b^{2} a^{5} x^{2}}{2 e}-\frac {b^{7} d^{3} x^{4}}{4 e^{4}}+\frac {35 b^{4} a^{3} x^{4}}{4 e}+\frac {21 b^{5} a^{2} x^{5}}{5 e}+\frac {b^{7} d^{2} x^{5}}{5 e^{3}}+\frac {7 b^{6} a \,x^{6}}{6 e}-\frac {b^{7} d \,x^{6}}{6 e^{2}}+\frac {b^{7} d^{6} x}{e^{7}}+\frac {7 b \,a^{6} x}{e}-\frac {\ln \left (e x +d \right ) b^{7} d^{7}}{e^{8}}+\frac {21 \ln \left (e x +d \right ) b^{2} d^{2} a^{5}}{e^{3}}-\frac {35 \ln \left (e x +d \right ) b^{3} d^{3} a^{4}}{e^{4}}+\frac {35 \ln \left (e x +d \right ) b^{4} d^{4} a^{3}}{e^{5}}-\frac {21 \ln \left (e x +d \right ) b^{5} d^{5} a^{2}}{e^{6}}+\frac {7 \ln \left (e x +d \right ) b^{6} d^{6} a}{e^{7}}-\frac {7 \ln \left (e x +d \right ) b d \,a^{6}}{e^{2}}-\frac {7 b^{6} a d \,x^{5}}{5 e^{2}}-\frac {35 b^{4} a^{3} d \,x^{3}}{3 e^{2}}+\frac {b^{7} x^{7}}{7 e}+\frac {\ln \left (e x +d \right ) a^{7}}{e}+\frac {7 b^{5} a^{2} d^{2} x^{3}}{e^{3}}-\frac {7 b^{6} a \,d^{3} x^{3}}{3 e^{4}}-\frac {35 b^{3} a^{4} d \,x^{2}}{2 e^{2}}+\frac {7 b^{6} a \,d^{2} x^{4}}{4 e^{3}}-\frac {21 b^{5} a^{2} d \,x^{4}}{4 e^{2}}+\frac {7 b^{6} a \,d^{4} x^{2}}{2 e^{5}}-\frac {21 b^{2} d \,a^{5} x}{e^{2}}+\frac {35 b^{4} a^{3} d^{2} x^{2}}{2 e^{3}}-\frac {21 b^{5} a^{2} d^{3} x^{2}}{2 e^{4}}+\frac {21 b^{5} d^{4} a^{2} x}{e^{5}}-\frac {7 b^{6} d^{5} a x}{e^{6}}+\frac {35 b^{3} d^{2} a^{4} x}{e^{3}}-\frac {35 b^{4} d^{3} a^{3} x}{e^{4}}\) | \(539\) |
| parallelrisch | \(\frac {-8820 \ln \left (e x +d \right ) a^{2} b^{5} d^{5} e^{2}+2940 \ln \left (e x +d \right ) a \,b^{6} d^{6} e -8820 x \,a^{5} b^{2} d \,e^{6}+14700 x \,a^{4} b^{3} d^{2} e^{5}-14700 x \,a^{3} b^{4} d^{3} e^{4}+8820 x \,a^{2} b^{5} d^{4} e^{3}-2940 x a \,b^{6} d^{5} e^{2}+420 x \,b^{7} d^{6} e +4410 x^{2} a^{5} b^{2} e^{7}-210 x^{2} b^{7} d^{5} e^{2}+4900 x^{3} a^{4} b^{3} e^{7}+140 x^{3} b^{7} d^{4} e^{3}+3675 x^{4} a^{3} b^{4} e^{7}-105 x^{4} b^{7} d^{3} e^{4}+1764 x^{5} a^{2} b^{5} e^{7}+84 x^{5} b^{7} d^{2} e^{5}+490 x^{6} a \,b^{6} e^{7}-70 x^{6} b^{7} d \,e^{6}+2940 x \,a^{6} b \,e^{7}+420 \ln \left (e x +d \right ) a^{7} e^{7}-420 \ln \left (e x +d \right ) b^{7} d^{7}-7350 x^{2} a^{4} b^{3} d \,e^{6}+7350 x^{2} a^{3} b^{4} d^{2} e^{5}-4410 x^{2} a^{2} b^{5} d^{3} e^{4}+1470 x^{2} a \,b^{6} d^{4} e^{3}-4900 x^{3} a^{3} b^{4} d \,e^{6}+2940 x^{3} a^{2} b^{5} d^{2} e^{5}-980 x^{3} a \,b^{6} d^{3} e^{4}-2205 x^{4} a^{2} b^{5} d \,e^{6}+735 x^{4} a \,b^{6} d^{2} e^{5}-588 x^{5} a \,b^{6} d \,e^{6}-2940 \ln \left (e x +d \right ) a^{6} b d \,e^{6}+8820 \ln \left (e x +d \right ) a^{5} b^{2} d^{2} e^{5}-14700 \ln \left (e x +d \right ) a^{4} b^{3} d^{3} e^{4}+14700 \ln \left (e x +d \right ) a^{3} b^{4} d^{4} e^{3}+60 x^{7} b^{7} e^{7}}{420 e^{8}}\) | \(539\) |
b*(7*a^6*e^6-21*a^5*b*d*e^5+35*a^4*b^2*d^2*e^4-35*a^3*b^3*d^3*e^3+21*a^2*b ^4*d^4*e^2-7*a*b^5*d^5*e+b^6*d^6)/e^7*x+1/7/e*b^7*x^7+1/2*b^2/e^6*(21*a^5* e^5-35*a^4*b*d*e^4+35*a^3*b^2*d^2*e^3-21*a^2*b^3*d^3*e^2+7*a*b^4*d^4*e-b^5 *d^5)*x^2+1/3*b^3/e^5*(35*a^4*e^4-35*a^3*b*d*e^3+21*a^2*b^2*d^2*e^2-7*a*b^ 3*d^3*e+b^4*d^4)*x^3+1/4*b^4/e^4*(35*a^3*e^3-21*a^2*b*d*e^2+7*a*b^2*d^2*e- b^3*d^3)*x^4+1/5*b^5/e^3*(21*a^2*e^2-7*a*b*d*e+b^2*d^2)*x^5+1/6*b^6/e^2*(7 *a*e-b*d)*x^6+(a^7*e^7-7*a^6*b*d*e^6+21*a^5*b^2*d^2*e^5-35*a^4*b^3*d^3*e^4 +35*a^3*b^4*d^4*e^3-21*a^2*b^5*d^5*e^2+7*a*b^6*d^6*e-b^7*d^7)/e^8*ln(e*x+d )
Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (158) = 316\).
Time = 0.35 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.70 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {60 \, b^{7} e^{7} x^{7} - 70 \, {\left (b^{7} d e^{6} - 7 \, a b^{6} e^{7}\right )} x^{6} + 84 \, {\left (b^{7} d^{2} e^{5} - 7 \, a b^{6} d e^{6} + 21 \, a^{2} b^{5} e^{7}\right )} x^{5} - 105 \, {\left (b^{7} d^{3} e^{4} - 7 \, a b^{6} d^{2} e^{5} + 21 \, a^{2} b^{5} d e^{6} - 35 \, a^{3} b^{4} e^{7}\right )} x^{4} + 140 \, {\left (b^{7} d^{4} e^{3} - 7 \, a b^{6} d^{3} e^{4} + 21 \, a^{2} b^{5} d^{2} e^{5} - 35 \, a^{3} b^{4} d e^{6} + 35 \, a^{4} b^{3} e^{7}\right )} x^{3} - 210 \, {\left (b^{7} d^{5} e^{2} - 7 \, a b^{6} d^{4} e^{3} + 21 \, a^{2} b^{5} d^{3} e^{4} - 35 \, a^{3} b^{4} d^{2} e^{5} + 35 \, a^{4} b^{3} d e^{6} - 21 \, a^{5} b^{2} e^{7}\right )} x^{2} + 420 \, {\left (b^{7} d^{6} e - 7 \, a b^{6} d^{5} e^{2} + 21 \, a^{2} b^{5} d^{4} e^{3} - 35 \, a^{3} b^{4} d^{3} e^{4} + 35 \, a^{4} b^{3} d^{2} e^{5} - 21 \, a^{5} b^{2} d e^{6} + 7 \, a^{6} b e^{7}\right )} x - 420 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \]
1/420*(60*b^7*e^7*x^7 - 70*(b^7*d*e^6 - 7*a*b^6*e^7)*x^6 + 84*(b^7*d^2*e^5 - 7*a*b^6*d*e^6 + 21*a^2*b^5*e^7)*x^5 - 105*(b^7*d^3*e^4 - 7*a*b^6*d^2*e^ 5 + 21*a^2*b^5*d*e^6 - 35*a^3*b^4*e^7)*x^4 + 140*(b^7*d^4*e^3 - 7*a*b^6*d^ 3*e^4 + 21*a^2*b^5*d^2*e^5 - 35*a^3*b^4*d*e^6 + 35*a^4*b^3*e^7)*x^3 - 210* (b^7*d^5*e^2 - 7*a*b^6*d^4*e^3 + 21*a^2*b^5*d^3*e^4 - 35*a^3*b^4*d^2*e^5 + 35*a^4*b^3*d*e^6 - 21*a^5*b^2*e^7)*x^2 + 420*(b^7*d^6*e - 7*a*b^6*d^5*e^2 + 21*a^2*b^5*d^4*e^3 - 35*a^3*b^4*d^3*e^4 + 35*a^4*b^3*d^2*e^5 - 21*a^5*b ^2*d*e^6 + 7*a^6*b*e^7)*x - 420*(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5* e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6 *b*d*e^6 - a^7*e^7)*log(e*x + d))/e^8
Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (144) = 288\).
Time = 0.42 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.40 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {b^{7} x^{7}}{7 e} + x^{6} \cdot \left (\frac {7 a b^{6}}{6 e} - \frac {b^{7} d}{6 e^{2}}\right ) + x^{5} \cdot \left (\frac {21 a^{2} b^{5}}{5 e} - \frac {7 a b^{6} d}{5 e^{2}} + \frac {b^{7} d^{2}}{5 e^{3}}\right ) + x^{4} \cdot \left (\frac {35 a^{3} b^{4}}{4 e} - \frac {21 a^{2} b^{5} d}{4 e^{2}} + \frac {7 a b^{6} d^{2}}{4 e^{3}} - \frac {b^{7} d^{3}}{4 e^{4}}\right ) + x^{3} \cdot \left (\frac {35 a^{4} b^{3}}{3 e} - \frac {35 a^{3} b^{4} d}{3 e^{2}} + \frac {7 a^{2} b^{5} d^{2}}{e^{3}} - \frac {7 a b^{6} d^{3}}{3 e^{4}} + \frac {b^{7} d^{4}}{3 e^{5}}\right ) + x^{2} \cdot \left (\frac {21 a^{5} b^{2}}{2 e} - \frac {35 a^{4} b^{3} d}{2 e^{2}} + \frac {35 a^{3} b^{4} d^{2}}{2 e^{3}} - \frac {21 a^{2} b^{5} d^{3}}{2 e^{4}} + \frac {7 a b^{6} d^{4}}{2 e^{5}} - \frac {b^{7} d^{5}}{2 e^{6}}\right ) + x \left (\frac {7 a^{6} b}{e} - \frac {21 a^{5} b^{2} d}{e^{2}} + \frac {35 a^{4} b^{3} d^{2}}{e^{3}} - \frac {35 a^{3} b^{4} d^{3}}{e^{4}} + \frac {21 a^{2} b^{5} d^{4}}{e^{5}} - \frac {7 a b^{6} d^{5}}{e^{6}} + \frac {b^{7} d^{6}}{e^{7}}\right ) + \frac {\left (a e - b d\right )^{7} \log {\left (d + e x \right )}}{e^{8}} \]
b**7*x**7/(7*e) + x**6*(7*a*b**6/(6*e) - b**7*d/(6*e**2)) + x**5*(21*a**2* b**5/(5*e) - 7*a*b**6*d/(5*e**2) + b**7*d**2/(5*e**3)) + x**4*(35*a**3*b** 4/(4*e) - 21*a**2*b**5*d/(4*e**2) + 7*a*b**6*d**2/(4*e**3) - b**7*d**3/(4* e**4)) + x**3*(35*a**4*b**3/(3*e) - 35*a**3*b**4*d/(3*e**2) + 7*a**2*b**5* d**2/e**3 - 7*a*b**6*d**3/(3*e**4) + b**7*d**4/(3*e**5)) + x**2*(21*a**5*b **2/(2*e) - 35*a**4*b**3*d/(2*e**2) + 35*a**3*b**4*d**2/(2*e**3) - 21*a**2 *b**5*d**3/(2*e**4) + 7*a*b**6*d**4/(2*e**5) - b**7*d**5/(2*e**6)) + x*(7* a**6*b/e - 21*a**5*b**2*d/e**2 + 35*a**4*b**3*d**2/e**3 - 35*a**3*b**4*d** 3/e**4 + 21*a**2*b**5*d**4/e**5 - 7*a*b**6*d**5/e**6 + b**7*d**6/e**7) + ( a*e - b*d)**7*log(d + e*x)/e**8
Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (158) = 316\).
Time = 0.20 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.69 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {60 \, b^{7} e^{6} x^{7} - 70 \, {\left (b^{7} d e^{5} - 7 \, a b^{6} e^{6}\right )} x^{6} + 84 \, {\left (b^{7} d^{2} e^{4} - 7 \, a b^{6} d e^{5} + 21 \, a^{2} b^{5} e^{6}\right )} x^{5} - 105 \, {\left (b^{7} d^{3} e^{3} - 7 \, a b^{6} d^{2} e^{4} + 21 \, a^{2} b^{5} d e^{5} - 35 \, a^{3} b^{4} e^{6}\right )} x^{4} + 140 \, {\left (b^{7} d^{4} e^{2} - 7 \, a b^{6} d^{3} e^{3} + 21 \, a^{2} b^{5} d^{2} e^{4} - 35 \, a^{3} b^{4} d e^{5} + 35 \, a^{4} b^{3} e^{6}\right )} x^{3} - 210 \, {\left (b^{7} d^{5} e - 7 \, a b^{6} d^{4} e^{2} + 21 \, a^{2} b^{5} d^{3} e^{3} - 35 \, a^{3} b^{4} d^{2} e^{4} + 35 \, a^{4} b^{3} d e^{5} - 21 \, a^{5} b^{2} e^{6}\right )} x^{2} + 420 \, {\left (b^{7} d^{6} - 7 \, a b^{6} d^{5} e + 21 \, a^{2} b^{5} d^{4} e^{2} - 35 \, a^{3} b^{4} d^{3} e^{3} + 35 \, a^{4} b^{3} d^{2} e^{4} - 21 \, a^{5} b^{2} d e^{5} + 7 \, a^{6} b e^{6}\right )} x}{420 \, e^{7}} - \frac {{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \log \left (e x + d\right )}{e^{8}} \]
1/420*(60*b^7*e^6*x^7 - 70*(b^7*d*e^5 - 7*a*b^6*e^6)*x^6 + 84*(b^7*d^2*e^4 - 7*a*b^6*d*e^5 + 21*a^2*b^5*e^6)*x^5 - 105*(b^7*d^3*e^3 - 7*a*b^6*d^2*e^ 4 + 21*a^2*b^5*d*e^5 - 35*a^3*b^4*e^6)*x^4 + 140*(b^7*d^4*e^2 - 7*a*b^6*d^ 3*e^3 + 21*a^2*b^5*d^2*e^4 - 35*a^3*b^4*d*e^5 + 35*a^4*b^3*e^6)*x^3 - 210* (b^7*d^5*e - 7*a*b^6*d^4*e^2 + 21*a^2*b^5*d^3*e^3 - 35*a^3*b^4*d^2*e^4 + 3 5*a^4*b^3*d*e^5 - 21*a^5*b^2*e^6)*x^2 + 420*(b^7*d^6 - 7*a*b^6*d^5*e + 21* a^2*b^5*d^4*e^2 - 35*a^3*b^4*d^3*e^3 + 35*a^4*b^3*d^2*e^4 - 21*a^5*b^2*d*e ^5 + 7*a^6*b*e^6)*x)/e^7 - (b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d* e^6 - a^7*e^7)*log(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (158) = 316\).
Time = 0.26 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.93 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {60 \, b^{7} e^{6} x^{7} - 70 \, b^{7} d e^{5} x^{6} + 490 \, a b^{6} e^{6} x^{6} + 84 \, b^{7} d^{2} e^{4} x^{5} - 588 \, a b^{6} d e^{5} x^{5} + 1764 \, a^{2} b^{5} e^{6} x^{5} - 105 \, b^{7} d^{3} e^{3} x^{4} + 735 \, a b^{6} d^{2} e^{4} x^{4} - 2205 \, a^{2} b^{5} d e^{5} x^{4} + 3675 \, a^{3} b^{4} e^{6} x^{4} + 140 \, b^{7} d^{4} e^{2} x^{3} - 980 \, a b^{6} d^{3} e^{3} x^{3} + 2940 \, a^{2} b^{5} d^{2} e^{4} x^{3} - 4900 \, a^{3} b^{4} d e^{5} x^{3} + 4900 \, a^{4} b^{3} e^{6} x^{3} - 210 \, b^{7} d^{5} e x^{2} + 1470 \, a b^{6} d^{4} e^{2} x^{2} - 4410 \, a^{2} b^{5} d^{3} e^{3} x^{2} + 7350 \, a^{3} b^{4} d^{2} e^{4} x^{2} - 7350 \, a^{4} b^{3} d e^{5} x^{2} + 4410 \, a^{5} b^{2} e^{6} x^{2} + 420 \, b^{7} d^{6} x - 2940 \, a b^{6} d^{5} e x + 8820 \, a^{2} b^{5} d^{4} e^{2} x - 14700 \, a^{3} b^{4} d^{3} e^{3} x + 14700 \, a^{4} b^{3} d^{2} e^{4} x - 8820 \, a^{5} b^{2} d e^{5} x + 2940 \, a^{6} b e^{6} x}{420 \, e^{7}} - \frac {{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} \]
1/420*(60*b^7*e^6*x^7 - 70*b^7*d*e^5*x^6 + 490*a*b^6*e^6*x^6 + 84*b^7*d^2* e^4*x^5 - 588*a*b^6*d*e^5*x^5 + 1764*a^2*b^5*e^6*x^5 - 105*b^7*d^3*e^3*x^4 + 735*a*b^6*d^2*e^4*x^4 - 2205*a^2*b^5*d*e^5*x^4 + 3675*a^3*b^4*e^6*x^4 + 140*b^7*d^4*e^2*x^3 - 980*a*b^6*d^3*e^3*x^3 + 2940*a^2*b^5*d^2*e^4*x^3 - 4900*a^3*b^4*d*e^5*x^3 + 4900*a^4*b^3*e^6*x^3 - 210*b^7*d^5*e*x^2 + 1470*a *b^6*d^4*e^2*x^2 - 4410*a^2*b^5*d^3*e^3*x^2 + 7350*a^3*b^4*d^2*e^4*x^2 - 7 350*a^4*b^3*d*e^5*x^2 + 4410*a^5*b^2*e^6*x^2 + 420*b^7*d^6*x - 2940*a*b^6* d^5*e*x + 8820*a^2*b^5*d^4*e^2*x - 14700*a^3*b^4*d^3*e^3*x + 14700*a^4*b^3 *d^2*e^4*x - 8820*a^5*b^2*d*e^5*x + 2940*a^6*b*e^6*x)/e^7 - (b^7*d^7 - 7*a *b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*log(abs(e*x + d))/e^8
Time = 0.08 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.00 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=x^6\,\left (\frac {7\,a\,b^6}{6\,e}-\frac {b^7\,d}{6\,e^2}\right )+x\,\left (\frac {7\,a^6\,b}{e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e}-\frac {b^7\,d}{e^2}\right )}{e}-\frac {21\,a^2\,b^5}{e}\right )}{e}+\frac {35\,a^3\,b^4}{e}\right )}{e}-\frac {35\,a^4\,b^3}{e}\right )}{e}+\frac {21\,a^5\,b^2}{e}\right )}{e}\right )+x^4\,\left (\frac {d\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e}-\frac {b^7\,d}{e^2}\right )}{e}-\frac {21\,a^2\,b^5}{e}\right )}{4\,e}+\frac {35\,a^3\,b^4}{4\,e}\right )+x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e}-\frac {b^7\,d}{e^2}\right )}{e}-\frac {21\,a^2\,b^5}{e}\right )}{e}+\frac {35\,a^3\,b^4}{e}\right )}{e}-\frac {35\,a^4\,b^3}{e}\right )}{2\,e}+\frac {21\,a^5\,b^2}{2\,e}\right )-x^5\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e}-\frac {b^7\,d}{e^2}\right )}{5\,e}-\frac {21\,a^2\,b^5}{5\,e}\right )-x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e}-\frac {b^7\,d}{e^2}\right )}{e}-\frac {21\,a^2\,b^5}{e}\right )}{e}+\frac {35\,a^3\,b^4}{e}\right )}{3\,e}-\frac {35\,a^4\,b^3}{3\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}{e^8}+\frac {b^7\,x^7}{7\,e} \]
x^6*((7*a*b^6)/(6*e) - (b^7*d)/(6*e^2)) + x*((7*a^6*b)/e - (d*((d*((d*((d* ((d*((7*a*b^6)/e - (b^7*d)/e^2))/e - (21*a^2*b^5)/e))/e + (35*a^3*b^4)/e)) /e - (35*a^4*b^3)/e))/e + (21*a^5*b^2)/e))/e) + x^4*((d*((d*((7*a*b^6)/e - (b^7*d)/e^2))/e - (21*a^2*b^5)/e))/(4*e) + (35*a^3*b^4)/(4*e)) + x^2*((d* ((d*((d*((d*((7*a*b^6)/e - (b^7*d)/e^2))/e - (21*a^2*b^5)/e))/e + (35*a^3* b^4)/e))/e - (35*a^4*b^3)/e))/(2*e) + (21*a^5*b^2)/(2*e)) - x^5*((d*((7*a* b^6)/e - (b^7*d)/e^2))/(5*e) - (21*a^2*b^5)/(5*e)) - x^3*((d*((d*((d*((7*a *b^6)/e - (b^7*d)/e^2))/e - (21*a^2*b^5)/e))/e + (35*a^3*b^4)/e))/(3*e) - (35*a^4*b^3)/(3*e)) + (log(d + e*x)*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^ 2 + 35*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6 *d^6*e - 7*a^6*b*d*e^6))/e^8 + (b^7*x^7)/(7*e)