Integrand size = 31, antiderivative size = 187 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=-\frac {35 b^4 (b d-a e)^3 x}{e^7}+\frac {(b d-a e)^7}{3 e^8 (d+e x)^3}-\frac {7 b (b d-a e)^6}{2 e^8 (d+e x)^2}+\frac {21 b^2 (b d-a e)^5}{e^8 (d+e x)}+\frac {21 b^5 (b d-a e)^2 (d+e x)^2}{2 e^8}-\frac {7 b^6 (b d-a e) (d+e x)^3}{3 e^8}+\frac {b^7 (d+e x)^4}{4 e^8}+\frac {35 b^3 (b d-a e)^4 \log (d+e x)}{e^8} \]
-35*b^4*(-a*e+b*d)^3*x/e^7+1/3*(-a*e+b*d)^7/e^8/(e*x+d)^3-7/2*b*(-a*e+b*d) ^6/e^8/(e*x+d)^2+21*b^2*(-a*e+b*d)^5/e^8/(e*x+d)+21/2*b^5*(-a*e+b*d)^2*(e* x+d)^2/e^8-7/3*b^6*(-a*e+b*d)*(e*x+d)^3/e^8+1/4*b^7*(e*x+d)^4/e^8+35*b^3*( -a*e+b*d)^4*ln(e*x+d)/e^8
Time = 0.07 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=\frac {-12 b^4 e \left (20 b^3 d^3-70 a b^2 d^2 e+84 a^2 b d e^2-35 a^3 e^3\right ) x+6 b^5 e^2 \left (10 b^2 d^2-28 a b d e+21 a^2 e^2\right ) x^2-4 b^6 e^3 (4 b d-7 a e) x^3+3 b^7 e^4 x^4+\frac {4 (b d-a e)^7}{(d+e x)^3}-\frac {42 b (b d-a e)^6}{(d+e x)^2}+\frac {252 b^2 (b d-a e)^5}{d+e x}+420 b^3 (b d-a e)^4 \log (d+e x)}{12 e^8} \]
(-12*b^4*e*(20*b^3*d^3 - 70*a*b^2*d^2*e + 84*a^2*b*d*e^2 - 35*a^3*e^3)*x + 6*b^5*e^2*(10*b^2*d^2 - 28*a*b*d*e + 21*a^2*e^2)*x^2 - 4*b^6*e^3*(4*b*d - 7*a*e)*x^3 + 3*b^7*e^4*x^4 + (4*(b*d - a*e)^7)/(d + e*x)^3 - (42*b*(b*d - a*e)^6)/(d + e*x)^2 + (252*b^2*(b*d - a*e)^5)/(d + e*x) + 420*b^3*(b*d - a*e)^4*Log[d + e*x])/(12*e^8)
Time = 0.45 (sec) , antiderivative size = 187, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^7}{(d+e x)^4}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^7}{(d+e x)^4}dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {7 b^6 (d+e x)^2 (b d-a e)}{e^7}+\frac {21 b^5 (d+e x) (b d-a e)^2}{e^7}-\frac {35 b^4 (b d-a e)^3}{e^7}+\frac {35 b^3 (b d-a e)^4}{e^7 (d+e x)}-\frac {21 b^2 (b d-a e)^5}{e^7 (d+e x)^2}+\frac {7 b (b d-a e)^6}{e^7 (d+e x)^3}+\frac {(a e-b d)^7}{e^7 (d+e x)^4}+\frac {b^7 (d+e x)^3}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {7 b^6 (d+e x)^3 (b d-a e)}{3 e^8}+\frac {21 b^5 (d+e x)^2 (b d-a e)^2}{2 e^8}-\frac {35 b^4 x (b d-a e)^3}{e^7}+\frac {35 b^3 (b d-a e)^4 \log (d+e x)}{e^8}+\frac {21 b^2 (b d-a e)^5}{e^8 (d+e x)}-\frac {7 b (b d-a e)^6}{2 e^8 (d+e x)^2}+\frac {(b d-a e)^7}{3 e^8 (d+e x)^3}+\frac {b^7 (d+e x)^4}{4 e^8}\) |
(-35*b^4*(b*d - a*e)^3*x)/e^7 + (b*d - a*e)^7/(3*e^8*(d + e*x)^3) - (7*b*( b*d - a*e)^6)/(2*e^8*(d + e*x)^2) + (21*b^2*(b*d - a*e)^5)/(e^8*(d + e*x)) + (21*b^5*(b*d - a*e)^2*(d + e*x)^2)/(2*e^8) - (7*b^6*(b*d - a*e)*(d + e* x)^3)/(3*e^8) + (b^7*(d + e*x)^4)/(4*e^8) + (35*b^3*(b*d - a*e)^4*Log[d + e*x])/e^8
3.20.26.3.1 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. \(449\) vs. \(2(177)=354\).
Time = 0.26 (sec) , antiderivative size = 450, normalized size of antiderivative = 2.41
| method | result | size |
| norman | \(\frac {-\frac {2 e^{7} a^{7}+7 b d \,e^{6} a^{6}+42 b^{2} d^{2} e^{5} a^{5}-385 b^{3} d^{3} e^{4} a^{4}+1540 b^{4} d^{4} e^{3} a^{3}-2310 b^{5} d^{5} e^{2} a^{2}+1540 b^{6} d^{6} e a -385 b^{7} d^{7}}{6 e^{8}}+\frac {b^{7} x^{7}}{4 e}-\frac {3 \left (7 e^{5} a^{5} b^{2}-35 d \,e^{4} a^{4} b^{3}+140 d^{2} e^{3} a^{3} b^{4}-210 d^{3} e^{2} a^{2} b^{5}+140 d^{4} e a \,b^{6}-35 d^{5} b^{7}\right ) x^{2}}{e^{6}}-\frac {\left (7 e^{6} a^{6} b +42 d \,e^{5} a^{5} b^{2}-315 d^{2} e^{4} a^{4} b^{3}+1260 d^{3} e^{3} a^{3} b^{4}-1890 d^{4} e^{2} a^{2} b^{5}+1260 d^{5} e a \,b^{6}-315 d^{6} b^{7}\right ) x}{2 e^{7}}+\frac {35 b^{4} \left (4 a^{3} e^{3}-6 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{4}}{4 e^{4}}+\frac {7 b^{5} \left (6 e^{2} a^{2}-4 a b d e +b^{2} d^{2}\right ) x^{5}}{4 e^{3}}+\frac {7 b^{6} \left (4 a e -b d \right ) x^{6}}{12 e^{2}}}{\left (e x +d \right )^{3}}+\frac {35 b^{3} \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(450\) |
| default | \(\frac {b^{4} \left (\frac {1}{4} b^{3} x^{4} e^{3}+\frac {7}{3} x^{3} a \,b^{2} e^{3}-\frac {4}{3} x^{3} b^{3} d \,e^{2}+\frac {21}{2} x^{2} a^{2} b \,e^{3}-14 x^{2} a \,b^{2} d \,e^{2}+5 x^{2} b^{3} d^{2} e +35 a^{3} e^{3} x -84 a^{2} b d \,e^{2} x +70 a \,b^{2} d^{2} e x -20 b^{3} d^{3} x \right )}{e^{7}}-\frac {e^{7} a^{7}-7 b d \,e^{6} a^{6}+21 b^{2} d^{2} e^{5} a^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}}{3 e^{8} \left (e x +d \right )^{3}}-\frac {21 b^{2} \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}{e^{8} \left (e x +d \right )}+\frac {35 b^{3} \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {7 b \left (e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}\right )}{2 e^{8} \left (e x +d \right )^{2}}\) | \(458\) |
| risch | \(\frac {b^{7} x^{4}}{4 e^{4}}+\frac {7 b^{6} x^{3} a}{3 e^{4}}-\frac {4 b^{7} x^{3} d}{3 e^{5}}+\frac {21 b^{5} x^{2} a^{2}}{2 e^{4}}-\frac {14 b^{6} x^{2} a d}{e^{5}}+\frac {5 b^{7} x^{2} d^{2}}{e^{6}}+\frac {35 b^{4} a^{3} x}{e^{4}}-\frac {84 b^{5} a^{2} d x}{e^{5}}+\frac {70 b^{6} a \,d^{2} x}{e^{6}}-\frac {20 b^{7} d^{3} x}{e^{7}}+\frac {\left (-21 e^{6} a^{5} b^{2}+105 d \,e^{5} a^{4} b^{3}-210 d^{2} e^{4} a^{3} b^{4}+210 d^{3} e^{3} a^{2} b^{5}-105 d^{4} e^{2} a \,b^{6}+21 d^{5} e \,b^{7}\right ) x^{2}-\frac {7 b \left (e^{6} a^{6}+6 b d \,e^{5} a^{5}-45 b^{2} d^{2} e^{4} a^{4}+100 b^{3} d^{3} e^{3} a^{3}-105 b^{4} d^{4} e^{2} a^{2}+54 b^{5} d^{5} e a -11 b^{6} d^{6}\right ) x}{2}-\frac {2 e^{7} a^{7}+7 b d \,e^{6} a^{6}+42 b^{2} d^{2} e^{5} a^{5}-385 b^{3} d^{3} e^{4} a^{4}+910 b^{4} d^{4} e^{3} a^{3}-987 b^{5} d^{5} e^{2} a^{2}+518 b^{6} d^{6} e a -107 b^{7} d^{7}}{6 e}}{e^{7} \left (e x +d \right )^{3}}+\frac {35 b^{3} \ln \left (e x +d \right ) a^{4}}{e^{4}}-\frac {140 b^{4} \ln \left (e x +d \right ) d \,a^{3}}{e^{5}}+\frac {210 b^{5} \ln \left (e x +d \right ) d^{2} a^{2}}{e^{6}}-\frac {140 b^{6} \ln \left (e x +d \right ) d^{3} a}{e^{7}}+\frac {35 b^{7} \ln \left (e x +d \right ) d^{4}}{e^{8}}\) | \(487\) |
| parallelrisch | \(\frac {1260 \ln \left (e x +d \right ) x \,b^{7} d^{6} e +1260 \ln \left (e x +d \right ) x^{2} b^{7} d^{5} e^{2}+2520 \ln \left (e x +d \right ) a^{2} b^{5} d^{5} e^{2}-1680 \ln \left (e x +d \right ) a \,b^{6} d^{6} e -252 x \,a^{5} b^{2} d \,e^{6}+1890 x \,a^{4} b^{3} d^{2} e^{5}-7560 x \,a^{3} b^{4} d^{3} e^{4}+11340 x \,a^{2} b^{5} d^{4} e^{3}-7560 x a \,b^{6} d^{5} e^{2}-3080 b^{6} d^{6} e a +4620 b^{5} d^{5} e^{2} a^{2}-3080 b^{4} d^{4} e^{3} a^{3}+770 b^{3} d^{3} e^{4} a^{4}-84 b^{2} d^{2} e^{5} a^{5}-14 b d \,e^{6} a^{6}+1890 x \,b^{7} d^{6} e -252 x^{2} a^{5} b^{2} e^{7}+1260 x^{2} b^{7} d^{5} e^{2}+420 x^{4} a^{3} b^{4} e^{7}-105 x^{4} b^{7} d^{3} e^{4}+126 x^{5} a^{2} b^{5} e^{7}+21 x^{5} b^{7} d^{2} e^{5}+28 x^{6} a \,b^{6} e^{7}-7 x^{6} b^{7} d \,e^{6}-42 x \,a^{6} b \,e^{7}+770 b^{7} d^{7}-4 e^{7} a^{7}+1260 \ln \left (e x +d \right ) x^{2} a^{4} b^{3} d \,e^{6}-5040 \ln \left (e x +d \right ) x^{2} a^{3} b^{4} d^{2} e^{5}+7560 \ln \left (e x +d \right ) x^{2} a^{2} b^{5} d^{3} e^{4}-5040 \ln \left (e x +d \right ) x^{2} a \,b^{6} d^{4} e^{3}+420 \ln \left (e x +d \right ) b^{7} d^{7}+1260 x^{2} a^{4} b^{3} d \,e^{6}-5040 x^{2} a^{3} b^{4} d^{2} e^{5}+7560 x^{2} a^{2} b^{5} d^{3} e^{4}-5040 x^{2} a \,b^{6} d^{4} e^{3}-630 x^{4} a^{2} b^{5} d \,e^{6}+420 x^{4} a \,b^{6} d^{2} e^{5}-84 x^{5} a \,b^{6} d \,e^{6}+420 \ln \left (e x +d \right ) a^{4} b^{3} d^{3} e^{4}-1680 \ln \left (e x +d \right ) a^{3} b^{4} d^{4} e^{3}+1260 \ln \left (e x +d \right ) x \,a^{4} b^{3} d^{2} e^{5}+3 x^{7} b^{7} e^{7}-1680 \ln \left (e x +d \right ) x^{3} a^{3} b^{4} d \,e^{6}+2520 \ln \left (e x +d \right ) x^{3} a^{2} b^{5} d^{2} e^{5}-5040 \ln \left (e x +d \right ) x \,a^{3} b^{4} d^{3} e^{4}+7560 \ln \left (e x +d \right ) x \,a^{2} b^{5} d^{4} e^{3}-5040 \ln \left (e x +d \right ) x a \,b^{6} d^{5} e^{2}+420 \ln \left (e x +d \right ) x^{3} a^{4} b^{3} e^{7}+420 \ln \left (e x +d \right ) x^{3} b^{7} d^{4} e^{3}-1680 \ln \left (e x +d \right ) x^{3} a \,b^{6} d^{3} e^{4}}{12 e^{8} \left (e x +d \right )^{3}}\) | \(824\) |
(-1/6*(2*a^7*e^7+7*a^6*b*d*e^6+42*a^5*b^2*d^2*e^5-385*a^4*b^3*d^3*e^4+1540 *a^3*b^4*d^4*e^3-2310*a^2*b^5*d^5*e^2+1540*a*b^6*d^6*e-385*b^7*d^7)/e^8+1/ 4/e*b^7*x^7-3*(7*a^5*b^2*e^5-35*a^4*b^3*d*e^4+140*a^3*b^4*d^2*e^3-210*a^2* b^5*d^3*e^2+140*a*b^6*d^4*e-35*b^7*d^5)/e^6*x^2-1/2*(7*a^6*b*e^6+42*a^5*b^ 2*d*e^5-315*a^4*b^3*d^2*e^4+1260*a^3*b^4*d^3*e^3-1890*a^2*b^5*d^4*e^2+1260 *a*b^6*d^5*e-315*b^7*d^6)/e^7*x+35/4*b^4*(4*a^3*e^3-6*a^2*b*d*e^2+4*a*b^2* d^2*e-b^3*d^3)/e^4*x^4+7/4*b^5*(6*a^2*e^2-4*a*b*d*e+b^2*d^2)/e^3*x^5+7/12* b^6*(4*a*e-b*d)/e^2*x^6)/(e*x+d)^3+35*b^3/e^8*(a^4*e^4-4*a^3*b*d*e^3+6*a^2 *b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 737 vs. \(2 (177) = 354\).
Time = 0.30 (sec) , antiderivative size = 737, normalized size of antiderivative = 3.94 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=\frac {3 \, b^{7} e^{7} x^{7} + 214 \, b^{7} d^{7} - 1036 \, a b^{6} d^{6} e + 1974 \, a^{2} b^{5} d^{5} e^{2} - 1820 \, a^{3} b^{4} d^{4} e^{3} + 770 \, a^{4} b^{3} d^{3} e^{4} - 84 \, a^{5} b^{2} d^{2} e^{5} - 14 \, a^{6} b d e^{6} - 4 \, a^{7} e^{7} - 7 \, {\left (b^{7} d e^{6} - 4 \, a b^{6} e^{7}\right )} x^{6} + 21 \, {\left (b^{7} d^{2} e^{5} - 4 \, a b^{6} d e^{6} + 6 \, a^{2} b^{5} e^{7}\right )} x^{5} - 105 \, {\left (b^{7} d^{3} e^{4} - 4 \, a b^{6} d^{2} e^{5} + 6 \, a^{2} b^{5} d e^{6} - 4 \, a^{3} b^{4} e^{7}\right )} x^{4} - 2 \, {\left (278 \, b^{7} d^{4} e^{3} - 1022 \, a b^{6} d^{3} e^{4} + 1323 \, a^{2} b^{5} d^{2} e^{5} - 630 \, a^{3} b^{4} d e^{6}\right )} x^{3} - 6 \, {\left (68 \, b^{7} d^{5} e^{2} - 182 \, a b^{6} d^{4} e^{3} + 63 \, a^{2} b^{5} d^{3} e^{4} + 210 \, a^{3} b^{4} d^{2} e^{5} - 210 \, a^{4} b^{3} d e^{6} + 42 \, a^{5} b^{2} e^{7}\right )} x^{2} + 6 \, {\left (37 \, b^{7} d^{6} e - 238 \, a b^{6} d^{5} e^{2} + 567 \, a^{2} b^{5} d^{4} e^{3} - 630 \, a^{3} b^{4} d^{3} e^{4} + 315 \, a^{4} b^{3} d^{2} e^{5} - 42 \, a^{5} b^{2} d e^{6} - 7 \, a^{6} b e^{7}\right )} x + 420 \, {\left (b^{7} d^{7} - 4 \, a b^{6} d^{6} e + 6 \, a^{2} b^{5} d^{5} e^{2} - 4 \, a^{3} b^{4} d^{4} e^{3} + a^{4} b^{3} d^{3} e^{4} + {\left (b^{7} d^{4} e^{3} - 4 \, a b^{6} d^{3} e^{4} + 6 \, a^{2} b^{5} d^{2} e^{5} - 4 \, a^{3} b^{4} d e^{6} + a^{4} b^{3} e^{7}\right )} x^{3} + 3 \, {\left (b^{7} d^{5} e^{2} - 4 \, a b^{6} d^{4} e^{3} + 6 \, a^{2} b^{5} d^{3} e^{4} - 4 \, a^{3} b^{4} d^{2} e^{5} + a^{4} b^{3} d e^{6}\right )} x^{2} + 3 \, {\left (b^{7} d^{6} e - 4 \, a b^{6} d^{5} e^{2} + 6 \, a^{2} b^{5} d^{4} e^{3} - 4 \, a^{3} b^{4} d^{3} e^{4} + a^{4} b^{3} d^{2} e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} \]
1/12*(3*b^7*e^7*x^7 + 214*b^7*d^7 - 1036*a*b^6*d^6*e + 1974*a^2*b^5*d^5*e^ 2 - 1820*a^3*b^4*d^4*e^3 + 770*a^4*b^3*d^3*e^4 - 84*a^5*b^2*d^2*e^5 - 14*a ^6*b*d*e^6 - 4*a^7*e^7 - 7*(b^7*d*e^6 - 4*a*b^6*e^7)*x^6 + 21*(b^7*d^2*e^5 - 4*a*b^6*d*e^6 + 6*a^2*b^5*e^7)*x^5 - 105*(b^7*d^3*e^4 - 4*a*b^6*d^2*e^5 + 6*a^2*b^5*d*e^6 - 4*a^3*b^4*e^7)*x^4 - 2*(278*b^7*d^4*e^3 - 1022*a*b^6* d^3*e^4 + 1323*a^2*b^5*d^2*e^5 - 630*a^3*b^4*d*e^6)*x^3 - 6*(68*b^7*d^5*e^ 2 - 182*a*b^6*d^4*e^3 + 63*a^2*b^5*d^3*e^4 + 210*a^3*b^4*d^2*e^5 - 210*a^4 *b^3*d*e^6 + 42*a^5*b^2*e^7)*x^2 + 6*(37*b^7*d^6*e - 238*a*b^6*d^5*e^2 + 5 67*a^2*b^5*d^4*e^3 - 630*a^3*b^4*d^3*e^4 + 315*a^4*b^3*d^2*e^5 - 42*a^5*b^ 2*d*e^6 - 7*a^6*b*e^7)*x + 420*(b^7*d^7 - 4*a*b^6*d^6*e + 6*a^2*b^5*d^5*e^ 2 - 4*a^3*b^4*d^4*e^3 + a^4*b^3*d^3*e^4 + (b^7*d^4*e^3 - 4*a*b^6*d^3*e^4 + 6*a^2*b^5*d^2*e^5 - 4*a^3*b^4*d*e^6 + a^4*b^3*e^7)*x^3 + 3*(b^7*d^5*e^2 - 4*a*b^6*d^4*e^3 + 6*a^2*b^5*d^3*e^4 - 4*a^3*b^4*d^2*e^5 + a^4*b^3*d*e^6)* x^2 + 3*(b^7*d^6*e - 4*a*b^6*d^5*e^2 + 6*a^2*b^5*d^4*e^3 - 4*a^3*b^4*d^3*e ^4 + a^4*b^3*d^2*e^5)*x)*log(e*x + d))/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^ 9*x + d^3*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (172) = 344\).
Time = 23.24 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.53 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=\frac {b^{7} x^{4}}{4 e^{4}} + \frac {35 b^{3} \left (a e - b d\right )^{4} \log {\left (d + e x \right )}}{e^{8}} + x^{3} \cdot \left (\frac {7 a b^{6}}{3 e^{4}} - \frac {4 b^{7} d}{3 e^{5}}\right ) + x^{2} \cdot \left (\frac {21 a^{2} b^{5}}{2 e^{4}} - \frac {14 a b^{6} d}{e^{5}} + \frac {5 b^{7} d^{2}}{e^{6}}\right ) + x \left (\frac {35 a^{3} b^{4}}{e^{4}} - \frac {84 a^{2} b^{5} d}{e^{5}} + \frac {70 a b^{6} d^{2}}{e^{6}} - \frac {20 b^{7} d^{3}}{e^{7}}\right ) + \frac {- 2 a^{7} e^{7} - 7 a^{6} b d e^{6} - 42 a^{5} b^{2} d^{2} e^{5} + 385 a^{4} b^{3} d^{3} e^{4} - 910 a^{3} b^{4} d^{4} e^{3} + 987 a^{2} b^{5} d^{5} e^{2} - 518 a b^{6} d^{6} e + 107 b^{7} d^{7} + x^{2} \left (- 126 a^{5} b^{2} e^{7} + 630 a^{4} b^{3} d e^{6} - 1260 a^{3} b^{4} d^{2} e^{5} + 1260 a^{2} b^{5} d^{3} e^{4} - 630 a b^{6} d^{4} e^{3} + 126 b^{7} d^{5} e^{2}\right ) + x \left (- 21 a^{6} b e^{7} - 126 a^{5} b^{2} d e^{6} + 945 a^{4} b^{3} d^{2} e^{5} - 2100 a^{3} b^{4} d^{3} e^{4} + 2205 a^{2} b^{5} d^{4} e^{3} - 1134 a b^{6} d^{5} e^{2} + 231 b^{7} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} \]
b**7*x**4/(4*e**4) + 35*b**3*(a*e - b*d)**4*log(d + e*x)/e**8 + x**3*(7*a* b**6/(3*e**4) - 4*b**7*d/(3*e**5)) + x**2*(21*a**2*b**5/(2*e**4) - 14*a*b* *6*d/e**5 + 5*b**7*d**2/e**6) + x*(35*a**3*b**4/e**4 - 84*a**2*b**5*d/e**5 + 70*a*b**6*d**2/e**6 - 20*b**7*d**3/e**7) + (-2*a**7*e**7 - 7*a**6*b*d*e **6 - 42*a**5*b**2*d**2*e**5 + 385*a**4*b**3*d**3*e**4 - 910*a**3*b**4*d** 4*e**3 + 987*a**2*b**5*d**5*e**2 - 518*a*b**6*d**6*e + 107*b**7*d**7 + x** 2*(-126*a**5*b**2*e**7 + 630*a**4*b**3*d*e**6 - 1260*a**3*b**4*d**2*e**5 + 1260*a**2*b**5*d**3*e**4 - 630*a*b**6*d**4*e**3 + 126*b**7*d**5*e**2) + x *(-21*a**6*b*e**7 - 126*a**5*b**2*d*e**6 + 945*a**4*b**3*d**2*e**5 - 2100* a**3*b**4*d**3*e**4 + 2205*a**2*b**5*d**4*e**3 - 1134*a*b**6*d**5*e**2 + 2 31*b**7*d**6*e))/(6*d**3*e**8 + 18*d**2*e**9*x + 18*d*e**10*x**2 + 6*e**11 *x**3)
Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (177) = 354\).
Time = 0.22 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.59 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=\frac {107 \, b^{7} d^{7} - 518 \, a b^{6} d^{6} e + 987 \, a^{2} b^{5} d^{5} e^{2} - 910 \, a^{3} b^{4} d^{4} e^{3} + 385 \, a^{4} b^{3} d^{3} e^{4} - 42 \, a^{5} b^{2} d^{2} e^{5} - 7 \, a^{6} b d e^{6} - 2 \, a^{7} e^{7} + 126 \, {\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{2} + 21 \, {\left (11 \, b^{7} d^{6} e - 54 \, a b^{6} d^{5} e^{2} + 105 \, a^{2} b^{5} d^{4} e^{3} - 100 \, a^{3} b^{4} d^{3} e^{4} + 45 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x}{6 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac {3 \, b^{7} e^{3} x^{4} - 4 \, {\left (4 \, b^{7} d e^{2} - 7 \, a b^{6} e^{3}\right )} x^{3} + 6 \, {\left (10 \, b^{7} d^{2} e - 28 \, a b^{6} d e^{2} + 21 \, a^{2} b^{5} e^{3}\right )} x^{2} - 12 \, {\left (20 \, b^{7} d^{3} - 70 \, a b^{6} d^{2} e + 84 \, a^{2} b^{5} d e^{2} - 35 \, a^{3} b^{4} e^{3}\right )} x}{12 \, e^{7}} + \frac {35 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} \log \left (e x + d\right )}{e^{8}} \]
1/6*(107*b^7*d^7 - 518*a*b^6*d^6*e + 987*a^2*b^5*d^5*e^2 - 910*a^3*b^4*d^4 *e^3 + 385*a^4*b^3*d^3*e^4 - 42*a^5*b^2*d^2*e^5 - 7*a^6*b*d*e^6 - 2*a^7*e^ 7 + 126*(b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d ^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^2 + 21*(11*b^7*d^6*e - 54*a*b^6* d^5*e^2 + 105*a^2*b^5*d^4*e^3 - 100*a^3*b^4*d^3*e^4 + 45*a^4*b^3*d^2*e^5 - 6*a^5*b^2*d*e^6 - a^6*b*e^7)*x)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8) + 1/12*(3*b^7*e^3*x^4 - 4*(4*b^7*d*e^2 - 7*a*b^6*e^3)*x^3 + 6*(10 *b^7*d^2*e - 28*a*b^6*d*e^2 + 21*a^2*b^5*e^3)*x^2 - 12*(20*b^7*d^3 - 70*a* b^6*d^2*e + 84*a^2*b^5*d*e^2 - 35*a^3*b^4*e^3)*x)/e^7 + 35*(b^7*d^4 - 4*a* b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)*log(e*x + d )/e^8
Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (177) = 354\).
Time = 0.27 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.52 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=\frac {35 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} + \frac {107 \, b^{7} d^{7} - 518 \, a b^{6} d^{6} e + 987 \, a^{2} b^{5} d^{5} e^{2} - 910 \, a^{3} b^{4} d^{4} e^{3} + 385 \, a^{4} b^{3} d^{3} e^{4} - 42 \, a^{5} b^{2} d^{2} e^{5} - 7 \, a^{6} b d e^{6} - 2 \, a^{7} e^{7} + 126 \, {\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{2} + 21 \, {\left (11 \, b^{7} d^{6} e - 54 \, a b^{6} d^{5} e^{2} + 105 \, a^{2} b^{5} d^{4} e^{3} - 100 \, a^{3} b^{4} d^{3} e^{4} + 45 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x}{6 \, {\left (e x + d\right )}^{3} e^{8}} + \frac {3 \, b^{7} e^{12} x^{4} - 16 \, b^{7} d e^{11} x^{3} + 28 \, a b^{6} e^{12} x^{3} + 60 \, b^{7} d^{2} e^{10} x^{2} - 168 \, a b^{6} d e^{11} x^{2} + 126 \, a^{2} b^{5} e^{12} x^{2} - 240 \, b^{7} d^{3} e^{9} x + 840 \, a b^{6} d^{2} e^{10} x - 1008 \, a^{2} b^{5} d e^{11} x + 420 \, a^{3} b^{4} e^{12} x}{12 \, e^{16}} \]
35*(b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^ 3*e^4)*log(abs(e*x + d))/e^8 + 1/6*(107*b^7*d^7 - 518*a*b^6*d^6*e + 987*a^ 2*b^5*d^5*e^2 - 910*a^3*b^4*d^4*e^3 + 385*a^4*b^3*d^3*e^4 - 42*a^5*b^2*d^2 *e^5 - 7*a^6*b*d*e^6 - 2*a^7*e^7 + 126*(b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10 *a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^2 + 21*(11*b^7*d^6*e - 54*a*b^6*d^5*e^2 + 105*a^2*b^5*d^4*e^3 - 100*a^3*b^4 *d^3*e^4 + 45*a^4*b^3*d^2*e^5 - 6*a^5*b^2*d*e^6 - a^6*b*e^7)*x)/((e*x + d) ^3*e^8) + 1/12*(3*b^7*e^12*x^4 - 16*b^7*d*e^11*x^3 + 28*a*b^6*e^12*x^3 + 6 0*b^7*d^2*e^10*x^2 - 168*a*b^6*d*e^11*x^2 + 126*a^2*b^5*e^12*x^2 - 240*b^7 *d^3*e^9*x + 840*a*b^6*d^2*e^10*x - 1008*a^2*b^5*d*e^11*x + 420*a^3*b^4*e^ 12*x)/e^16
Time = 10.92 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.98 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=x\,\left (\frac {35\,a^3\,b^4}{e^4}-\frac {4\,b^7\,d^3}{e^7}+\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {7\,a\,b^6}{e^4}-\frac {4\,b^7\,d}{e^5}\right )}{e}-\frac {21\,a^2\,b^5}{e^4}+\frac {6\,b^7\,d^2}{e^6}\right )}{e}-\frac {6\,d^2\,\left (\frac {7\,a\,b^6}{e^4}-\frac {4\,b^7\,d}{e^5}\right )}{e^2}\right )-\frac {\frac {2\,a^7\,e^7+7\,a^6\,b\,d\,e^6+42\,a^5\,b^2\,d^2\,e^5-385\,a^4\,b^3\,d^3\,e^4+910\,a^3\,b^4\,d^4\,e^3-987\,a^2\,b^5\,d^5\,e^2+518\,a\,b^6\,d^6\,e-107\,b^7\,d^7}{6\,e}+x\,\left (\frac {7\,a^6\,b\,e^6}{2}+21\,a^5\,b^2\,d\,e^5-\frac {315\,a^4\,b^3\,d^2\,e^4}{2}+350\,a^3\,b^4\,d^3\,e^3-\frac {735\,a^2\,b^5\,d^4\,e^2}{2}+189\,a\,b^6\,d^5\,e-\frac {77\,b^7\,d^6}{2}\right )-x^2\,\left (-21\,a^5\,b^2\,e^6+105\,a^4\,b^3\,d\,e^5-210\,a^3\,b^4\,d^2\,e^4+210\,a^2\,b^5\,d^3\,e^3-105\,a\,b^6\,d^4\,e^2+21\,b^7\,d^5\,e\right )}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}+x^3\,\left (\frac {7\,a\,b^6}{3\,e^4}-\frac {4\,b^7\,d}{3\,e^5}\right )-x^2\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^4}-\frac {4\,b^7\,d}{e^5}\right )}{e}-\frac {21\,a^2\,b^5}{2\,e^4}+\frac {3\,b^7\,d^2}{e^6}\right )+\frac {\ln \left (d+e\,x\right )\,\left (35\,a^4\,b^3\,e^4-140\,a^3\,b^4\,d\,e^3+210\,a^2\,b^5\,d^2\,e^2-140\,a\,b^6\,d^3\,e+35\,b^7\,d^4\right )}{e^8}+\frac {b^7\,x^4}{4\,e^4} \]
x*((35*a^3*b^4)/e^4 - (4*b^7*d^3)/e^7 + (4*d*((4*d*((7*a*b^6)/e^4 - (4*b^7 *d)/e^5))/e - (21*a^2*b^5)/e^4 + (6*b^7*d^2)/e^6))/e - (6*d^2*((7*a*b^6)/e ^4 - (4*b^7*d)/e^5))/e^2) - ((2*a^7*e^7 - 107*b^7*d^7 - 987*a^2*b^5*d^5*e^ 2 + 910*a^3*b^4*d^4*e^3 - 385*a^4*b^3*d^3*e^4 + 42*a^5*b^2*d^2*e^5 + 518*a *b^6*d^6*e + 7*a^6*b*d*e^6)/(6*e) + x*((7*a^6*b*e^6)/2 - (77*b^7*d^6)/2 + 21*a^5*b^2*d*e^5 - (735*a^2*b^5*d^4*e^2)/2 + 350*a^3*b^4*d^3*e^3 - (315*a^ 4*b^3*d^2*e^4)/2 + 189*a*b^6*d^5*e) - x^2*(21*b^7*d^5*e - 21*a^5*b^2*e^6 - 105*a*b^6*d^4*e^2 + 105*a^4*b^3*d*e^5 + 210*a^2*b^5*d^3*e^3 - 210*a^3*b^4 *d^2*e^4))/(d^3*e^7 + e^10*x^3 + 3*d^2*e^8*x + 3*d*e^9*x^2) + x^3*((7*a*b^ 6)/(3*e^4) - (4*b^7*d)/(3*e^5)) - x^2*((2*d*((7*a*b^6)/e^4 - (4*b^7*d)/e^5 ))/e - (21*a^2*b^5)/(2*e^4) + (3*b^7*d^2)/e^6) + (log(d + e*x)*(35*b^7*d^4 + 35*a^4*b^3*e^4 - 140*a^3*b^4*d*e^3 + 210*a^2*b^5*d^2*e^2 - 140*a*b^6*d^ 3*e))/e^8 + (b^7*x^4)/(4*e^4)