Integrand size = 33, antiderivative size = 208 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 (b d-a e)^7}{3 e^8 (d+e x)^{3/2}}-\frac {14 b (b d-a e)^6}{e^8 \sqrt {d+e x}}-\frac {42 b^2 (b d-a e)^5 \sqrt {d+e x}}{e^8}+\frac {70 b^3 (b d-a e)^4 (d+e x)^{3/2}}{3 e^8}-\frac {14 b^4 (b d-a e)^3 (d+e x)^{5/2}}{e^8}+\frac {6 b^5 (b d-a e)^2 (d+e x)^{7/2}}{e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{9/2}}{9 e^8}+\frac {2 b^7 (d+e x)^{11/2}}{11 e^8} \]
2/3*(-a*e+b*d)^7/e^8/(e*x+d)^(3/2)+70/3*b^3*(-a*e+b*d)^4*(e*x+d)^(3/2)/e^8 -14*b^4*(-a*e+b*d)^3*(e*x+d)^(5/2)/e^8+6*b^5*(-a*e+b*d)^2*(e*x+d)^(7/2)/e^ 8-14/9*b^6*(-a*e+b*d)*(e*x+d)^(9/2)/e^8+2/11*b^7*(e*x+d)^(11/2)/e^8-14*b*( -a*e+b*d)^6/e^8/(e*x+d)^(1/2)-42*b^2*(-a*e+b*d)^5*(e*x+d)^(1/2)/e^8
Time = 0.17 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.81 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \left (-33 a^7 e^7-231 a^6 b e^6 (2 d+3 e x)+693 a^5 b^2 e^5 \left (8 d^2+12 d e x+3 e^2 x^2\right )+1155 a^4 b^3 e^4 \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+231 a^3 b^4 e^3 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )-99 a^2 b^5 e^2 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )+11 a b^6 e \left (1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right )-b^7 \left (2048 d^7+3072 d^6 e x+768 d^5 e^2 x^2-128 d^4 e^3 x^3+48 d^3 e^4 x^4-24 d^2 e^5 x^5+14 d e^6 x^6-9 e^7 x^7\right )\right )}{99 e^8 (d+e x)^{3/2}} \]
(2*(-33*a^7*e^7 - 231*a^6*b*e^6*(2*d + 3*e*x) + 693*a^5*b^2*e^5*(8*d^2 + 1 2*d*e*x + 3*e^2*x^2) + 1155*a^4*b^3*e^4*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^ 2 + e^3*x^3) + 231*a^3*b^4*e^3*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8 *d*e^3*x^3 + 3*e^4*x^4) - 99*a^2*b^5*e^2*(256*d^5 + 384*d^4*e*x + 96*d^3*e ^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5) + 11*a*b^6*e*(1024*d^6 + 1536*d^5*e*x + 384*d^4*e^2*x^2 - 64*d^3*e^3*x^3 + 24*d^2*e^4*x^4 - 12*d* e^5*x^5 + 7*e^6*x^6) - b^7*(2048*d^7 + 3072*d^6*e*x + 768*d^5*e^2*x^2 - 12 8*d^4*e^3*x^3 + 48*d^3*e^4*x^4 - 24*d^2*e^5*x^5 + 14*d*e^6*x^6 - 9*e^7*x^7 )))/(99*e^8*(d + e*x)^(3/2))
Time = 0.35 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1184, 27, 53, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^7}{(d+e x)^{5/2}}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^7}{(d+e x)^{5/2}}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {7 b^6 (d+e x)^{7/2} (b d-a e)}{e^7}+\frac {21 b^5 (d+e x)^{5/2} (b d-a e)^2}{e^7}-\frac {35 b^4 (d+e x)^{3/2} (b d-a e)^3}{e^7}+\frac {35 b^3 \sqrt {d+e x} (b d-a e)^4}{e^7}-\frac {21 b^2 (b d-a e)^5}{e^7 \sqrt {d+e x}}+\frac {7 b (b d-a e)^6}{e^7 (d+e x)^{3/2}}+\frac {(a e-b d)^7}{e^7 (d+e x)^{5/2}}+\frac {b^7 (d+e x)^{9/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {14 b^6 (d+e x)^{9/2} (b d-a e)}{9 e^8}+\frac {6 b^5 (d+e x)^{7/2} (b d-a e)^2}{e^8}-\frac {14 b^4 (d+e x)^{5/2} (b d-a e)^3}{e^8}+\frac {70 b^3 (d+e x)^{3/2} (b d-a e)^4}{3 e^8}-\frac {42 b^2 \sqrt {d+e x} (b d-a e)^5}{e^8}-\frac {14 b (b d-a e)^6}{e^8 \sqrt {d+e x}}+\frac {2 (b d-a e)^7}{3 e^8 (d+e x)^{3/2}}+\frac {2 b^7 (d+e x)^{11/2}}{11 e^8}\) |
(2*(b*d - a*e)^7)/(3*e^8*(d + e*x)^(3/2)) - (14*b*(b*d - a*e)^6)/(e^8*Sqrt [d + e*x]) - (42*b^2*(b*d - a*e)^5*Sqrt[d + e*x])/e^8 + (70*b^3*(b*d - a*e )^4*(d + e*x)^(3/2))/(3*e^8) - (14*b^4*(b*d - a*e)^3*(d + e*x)^(5/2))/e^8 + (6*b^5*(b*d - a*e)^2*(d + e*x)^(7/2))/e^8 - (14*b^6*(b*d - a*e)*(d + e*x )^(9/2))/(9*e^8) + (2*b^7*(d + e*x)^(11/2))/(11*e^8)
3.21.64.3.1 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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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]
Time = 0.38 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.73
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (-\frac {3}{11} b^{7} x^{7}+a^{7}-\frac {7}{3} a \,b^{6} x^{6}-9 a^{2} b^{5} x^{5}-21 a^{3} b^{4} x^{4}-35 a^{4} b^{3} x^{3}-63 a^{5} b^{2} x^{2}+21 a^{6} b x \right ) e^{7}+14 b \left (\frac {1}{33} b^{6} x^{6}+\frac {2}{7} a \,b^{5} x^{5}+\frac {9}{7} a^{2} b^{4} x^{4}+4 a^{3} b^{3} x^{3}+15 a^{4} b^{2} x^{2}-18 a^{5} b x +a^{6}\right ) d \,e^{6}-168 b^{2} \left (\frac {1}{231} b^{5} x^{5}+\frac {1}{21} a \,b^{4} x^{4}+\frac {2}{7} a^{2} b^{3} x^{3}+2 a^{3} b^{2} x^{2}-5 a^{4} b x +a^{5}\right ) d^{2} e^{5}+560 \left (\frac {1}{385} x^{4} b^{4}+\frac {4}{105} a \,b^{3} x^{3}+\frac {18}{35} x^{2} b^{2} a^{2}-\frac {12}{5} b \,a^{3} x +a^{4}\right ) b^{3} d^{3} e^{4}-896 b^{4} \left (\frac {1}{231} x^{3} b^{3}+\frac {1}{7} a \,b^{2} x^{2}-\frac {9}{7} b \,a^{2} x +a^{3}\right ) d^{4} e^{3}+768 b^{5} \left (\frac {1}{33} b^{2} x^{2}-\frac {2}{3} a b x +a^{2}\right ) d^{5} e^{2}-\frac {1024 \left (-\frac {3 b x}{11}+a \right ) b^{6} d^{6} e}{3}+\frac {2048 b^{7} d^{7}}{33}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{8}}\) | \(359\) |
| risch | \(\frac {2 b^{2} \left (9 x^{5} b^{5} e^{5}+77 x^{4} a \,b^{4} e^{5}-32 x^{4} b^{5} d \,e^{4}+297 x^{3} a^{2} b^{3} e^{5}-286 x^{3} a \,b^{4} d \,e^{4}+79 x^{3} b^{5} d^{2} e^{3}+693 x^{2} a^{3} b^{2} e^{5}-1188 x^{2} a^{2} b^{3} d \,e^{4}+759 x^{2} a \,b^{4} d^{2} e^{3}-174 x^{2} b^{5} d^{3} e^{2}+1155 x \,a^{4} b \,e^{5}-3234 x \,a^{3} b^{2} d \,e^{4}+3663 x \,a^{2} b^{3} d^{2} e^{3}-1936 x a \,b^{4} d^{3} e^{2}+397 x \,b^{5} d^{4} e +2079 e^{5} a^{5}-9240 b d \,e^{4} a^{4}+16863 b^{2} d^{2} e^{3} a^{3}-15642 b^{3} d^{3} e^{2} a^{2}+7337 b^{4} d^{4} e a -1388 b^{5} d^{5}\right ) \sqrt {e x +d}}{99 e^{8}}-\frac {2 \left (21 b e x +a e +20 b d \right ) \left (e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}\right )}{3 e^{8} \left (e x +d \right )^{\frac {3}{2}}}\) | \(379\) |
| gosper | \(-\frac {2 \left (-9 x^{7} b^{7} e^{7}-77 x^{6} a \,b^{6} e^{7}+14 x^{6} b^{7} d \,e^{6}-297 x^{5} a^{2} b^{5} e^{7}+132 x^{5} a \,b^{6} d \,e^{6}-24 x^{5} b^{7} d^{2} e^{5}-693 x^{4} a^{3} b^{4} e^{7}+594 x^{4} a^{2} b^{5} d \,e^{6}-264 x^{4} a \,b^{6} d^{2} e^{5}+48 x^{4} b^{7} d^{3} e^{4}-1155 x^{3} a^{4} b^{3} e^{7}+1848 x^{3} a^{3} b^{4} d \,e^{6}-1584 x^{3} a^{2} b^{5} d^{2} e^{5}+704 x^{3} a \,b^{6} d^{3} e^{4}-128 x^{3} b^{7} d^{4} e^{3}-2079 x^{2} a^{5} b^{2} e^{7}+6930 x^{2} a^{4} b^{3} d \,e^{6}-11088 x^{2} a^{3} b^{4} d^{2} e^{5}+9504 x^{2} a^{2} b^{5} d^{3} e^{4}-4224 x^{2} a \,b^{6} d^{4} e^{3}+768 x^{2} b^{7} d^{5} e^{2}+693 x \,a^{6} b \,e^{7}-8316 x \,a^{5} b^{2} d \,e^{6}+27720 x \,a^{4} b^{3} d^{2} e^{5}-44352 x \,a^{3} b^{4} d^{3} e^{4}+38016 x \,a^{2} b^{5} d^{4} e^{3}-16896 x a \,b^{6} d^{5} e^{2}+3072 x \,b^{7} d^{6} e +33 e^{7} a^{7}+462 b d \,e^{6} a^{6}-5544 b^{2} d^{2} e^{5} a^{5}+18480 b^{3} d^{3} e^{4} a^{4}-29568 b^{4} d^{4} e^{3} a^{3}+25344 b^{5} d^{5} e^{2} a^{2}-11264 b^{6} d^{6} e a +2048 b^{7} d^{7}\right )}{99 \left (e x +d \right )^{\frac {3}{2}} e^{8}}\) | \(498\) |
| trager | \(-\frac {2 \left (-9 x^{7} b^{7} e^{7}-77 x^{6} a \,b^{6} e^{7}+14 x^{6} b^{7} d \,e^{6}-297 x^{5} a^{2} b^{5} e^{7}+132 x^{5} a \,b^{6} d \,e^{6}-24 x^{5} b^{7} d^{2} e^{5}-693 x^{4} a^{3} b^{4} e^{7}+594 x^{4} a^{2} b^{5} d \,e^{6}-264 x^{4} a \,b^{6} d^{2} e^{5}+48 x^{4} b^{7} d^{3} e^{4}-1155 x^{3} a^{4} b^{3} e^{7}+1848 x^{3} a^{3} b^{4} d \,e^{6}-1584 x^{3} a^{2} b^{5} d^{2} e^{5}+704 x^{3} a \,b^{6} d^{3} e^{4}-128 x^{3} b^{7} d^{4} e^{3}-2079 x^{2} a^{5} b^{2} e^{7}+6930 x^{2} a^{4} b^{3} d \,e^{6}-11088 x^{2} a^{3} b^{4} d^{2} e^{5}+9504 x^{2} a^{2} b^{5} d^{3} e^{4}-4224 x^{2} a \,b^{6} d^{4} e^{3}+768 x^{2} b^{7} d^{5} e^{2}+693 x \,a^{6} b \,e^{7}-8316 x \,a^{5} b^{2} d \,e^{6}+27720 x \,a^{4} b^{3} d^{2} e^{5}-44352 x \,a^{3} b^{4} d^{3} e^{4}+38016 x \,a^{2} b^{5} d^{4} e^{3}-16896 x a \,b^{6} d^{5} e^{2}+3072 x \,b^{7} d^{6} e +33 e^{7} a^{7}+462 b d \,e^{6} a^{6}-5544 b^{2} d^{2} e^{5} a^{5}+18480 b^{3} d^{3} e^{4} a^{4}-29568 b^{4} d^{4} e^{3} a^{3}+25344 b^{5} d^{5} e^{2} a^{2}-11264 b^{6} d^{6} e a +2048 b^{7} d^{7}\right )}{99 \left (e x +d \right )^{\frac {3}{2}} e^{8}}\) | \(498\) |
| derivativedivides | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {14 a \,b^{6} e \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {14 b^{7} d \left (e x +d \right )^{\frac {9}{2}}}{9}+6 a^{2} b^{5} e^{2} \left (e x +d \right )^{\frac {7}{2}}-12 a \,b^{6} d e \left (e x +d \right )^{\frac {7}{2}}+6 b^{7} d^{2} \left (e x +d \right )^{\frac {7}{2}}+14 a^{3} b^{4} e^{3} \left (e x +d \right )^{\frac {5}{2}}-42 a^{2} b^{5} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}+42 a \,b^{6} d^{2} e \left (e x +d \right )^{\frac {5}{2}}-14 b^{7} d^{3} \left (e x +d \right )^{\frac {5}{2}}+\frac {70 a^{4} b^{3} e^{4} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {280 a^{3} b^{4} d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+140 a^{2} b^{5} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {280 a \,b^{6} d^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {70 b^{7} d^{4} \left (e x +d \right )^{\frac {3}{2}}}{3}+42 a^{5} b^{2} e^{5} \sqrt {e x +d}-210 a^{4} b^{3} d \,e^{4} \sqrt {e x +d}+420 a^{3} b^{4} d^{2} e^{3} \sqrt {e x +d}-420 a^{2} b^{5} d^{3} e^{2} \sqrt {e x +d}+210 a \,b^{6} d^{4} e \sqrt {e x +d}-42 b^{7} d^{5} \sqrt {e x +d}-\frac {2 \left (e^{7} a^{7}-7 b d \,e^{6} a^{6}+21 b^{2} d^{2} e^{5} a^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {14 b \left (e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}\right )}{\sqrt {e x +d}}}{e^{8}}\) | \(552\) |
| default | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {14 a \,b^{6} e \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {14 b^{7} d \left (e x +d \right )^{\frac {9}{2}}}{9}+6 a^{2} b^{5} e^{2} \left (e x +d \right )^{\frac {7}{2}}-12 a \,b^{6} d e \left (e x +d \right )^{\frac {7}{2}}+6 b^{7} d^{2} \left (e x +d \right )^{\frac {7}{2}}+14 a^{3} b^{4} e^{3} \left (e x +d \right )^{\frac {5}{2}}-42 a^{2} b^{5} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}+42 a \,b^{6} d^{2} e \left (e x +d \right )^{\frac {5}{2}}-14 b^{7} d^{3} \left (e x +d \right )^{\frac {5}{2}}+\frac {70 a^{4} b^{3} e^{4} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {280 a^{3} b^{4} d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+140 a^{2} b^{5} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {280 a \,b^{6} d^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {70 b^{7} d^{4} \left (e x +d \right )^{\frac {3}{2}}}{3}+42 a^{5} b^{2} e^{5} \sqrt {e x +d}-210 a^{4} b^{3} d \,e^{4} \sqrt {e x +d}+420 a^{3} b^{4} d^{2} e^{3} \sqrt {e x +d}-420 a^{2} b^{5} d^{3} e^{2} \sqrt {e x +d}+210 a \,b^{6} d^{4} e \sqrt {e x +d}-42 b^{7} d^{5} \sqrt {e x +d}-\frac {2 \left (e^{7} a^{7}-7 b d \,e^{6} a^{6}+21 b^{2} d^{2} e^{5} a^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {14 b \left (e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}\right )}{\sqrt {e x +d}}}{e^{8}}\) | \(552\) |
-2/3*((-3/11*b^7*x^7+a^7-7/3*a*b^6*x^6-9*a^2*b^5*x^5-21*a^3*b^4*x^4-35*a^4 *b^3*x^3-63*a^5*b^2*x^2+21*a^6*b*x)*e^7+14*b*(1/33*b^6*x^6+2/7*a*b^5*x^5+9 /7*a^2*b^4*x^4+4*a^3*b^3*x^3+15*a^4*b^2*x^2-18*a^5*b*x+a^6)*d*e^6-168*b^2* (1/231*b^5*x^5+1/21*a*b^4*x^4+2/7*a^2*b^3*x^3+2*a^3*b^2*x^2-5*a^4*b*x+a^5) *d^2*e^5+560*(1/385*x^4*b^4+4/105*a*b^3*x^3+18/35*x^2*b^2*a^2-12/5*b*a^3*x +a^4)*b^3*d^3*e^4-896*b^4*(1/231*x^3*b^3+1/7*a*b^2*x^2-9/7*b*a^2*x+a^3)*d^ 4*e^3+768*b^5*(1/33*b^2*x^2-2/3*a*b*x+a^2)*d^5*e^2-1024/3*(-3/11*b*x+a)*b^ 6*d^6*e+2048/33*b^7*d^7)/(e*x+d)^(3/2)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (184) = 368\).
Time = 0.32 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.33 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (9 \, b^{7} e^{7} x^{7} - 2048 \, b^{7} d^{7} + 11264 \, a b^{6} d^{6} e - 25344 \, a^{2} b^{5} d^{5} e^{2} + 29568 \, a^{3} b^{4} d^{4} e^{3} - 18480 \, a^{4} b^{3} d^{3} e^{4} + 5544 \, a^{5} b^{2} d^{2} e^{5} - 462 \, a^{6} b d e^{6} - 33 \, a^{7} e^{7} - 7 \, {\left (2 \, b^{7} d e^{6} - 11 \, a b^{6} e^{7}\right )} x^{6} + 3 \, {\left (8 \, b^{7} d^{2} e^{5} - 44 \, a b^{6} d e^{6} + 99 \, a^{2} b^{5} e^{7}\right )} x^{5} - 3 \, {\left (16 \, b^{7} d^{3} e^{4} - 88 \, a b^{6} d^{2} e^{5} + 198 \, a^{2} b^{5} d e^{6} - 231 \, a^{3} b^{4} e^{7}\right )} x^{4} + {\left (128 \, b^{7} d^{4} e^{3} - 704 \, a b^{6} d^{3} e^{4} + 1584 \, a^{2} b^{5} d^{2} e^{5} - 1848 \, a^{3} b^{4} d e^{6} + 1155 \, a^{4} b^{3} e^{7}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{5} e^{2} - 1408 \, a b^{6} d^{4} e^{3} + 3168 \, a^{2} b^{5} d^{3} e^{4} - 3696 \, a^{3} b^{4} d^{2} e^{5} + 2310 \, a^{4} b^{3} d e^{6} - 693 \, a^{5} b^{2} e^{7}\right )} x^{2} - 3 \, {\left (1024 \, b^{7} d^{6} e - 5632 \, a b^{6} d^{5} e^{2} + 12672 \, a^{2} b^{5} d^{4} e^{3} - 14784 \, a^{3} b^{4} d^{3} e^{4} + 9240 \, a^{4} b^{3} d^{2} e^{5} - 2772 \, a^{5} b^{2} d e^{6} + 231 \, a^{6} b e^{7}\right )} x\right )} \sqrt {e x + d}}{99 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} \]
2/99*(9*b^7*e^7*x^7 - 2048*b^7*d^7 + 11264*a*b^6*d^6*e - 25344*a^2*b^5*d^5 *e^2 + 29568*a^3*b^4*d^4*e^3 - 18480*a^4*b^3*d^3*e^4 + 5544*a^5*b^2*d^2*e^ 5 - 462*a^6*b*d*e^6 - 33*a^7*e^7 - 7*(2*b^7*d*e^6 - 11*a*b^6*e^7)*x^6 + 3* (8*b^7*d^2*e^5 - 44*a*b^6*d*e^6 + 99*a^2*b^5*e^7)*x^5 - 3*(16*b^7*d^3*e^4 - 88*a*b^6*d^2*e^5 + 198*a^2*b^5*d*e^6 - 231*a^3*b^4*e^7)*x^4 + (128*b^7*d ^4*e^3 - 704*a*b^6*d^3*e^4 + 1584*a^2*b^5*d^2*e^5 - 1848*a^3*b^4*d*e^6 + 1 155*a^4*b^3*e^7)*x^3 - 3*(256*b^7*d^5*e^2 - 1408*a*b^6*d^4*e^3 + 3168*a^2* b^5*d^3*e^4 - 3696*a^3*b^4*d^2*e^5 + 2310*a^4*b^3*d*e^6 - 693*a^5*b^2*e^7) *x^2 - 3*(1024*b^7*d^6*e - 5632*a*b^6*d^5*e^2 + 12672*a^2*b^5*d^4*e^3 - 14 784*a^3*b^4*d^3*e^4 + 9240*a^4*b^3*d^2*e^5 - 2772*a^5*b^2*d*e^6 + 231*a^6* b*e^7)*x)*sqrt(e*x + d)/(e^10*x^2 + 2*d*e^9*x + d^2*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (194) = 388\).
Time = 17.62 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{7} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{7}} - \frac {7 b \left (a e - b d\right )^{6}}{e^{7} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (7 a b^{6} e - 7 b^{7} d\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (21 a^{5} b^{2} e^{5} - 105 a^{4} b^{3} d e^{4} + 210 a^{3} b^{4} d^{2} e^{3} - 210 a^{2} b^{5} d^{3} e^{2} + 105 a b^{6} d^{4} e - 21 b^{7} d^{5}\right )}{e^{7}} - \frac {\left (a e - b d\right )^{7}}{3 e^{7} \left (d + e x\right )^{\frac {3}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\begin {cases} a^{7} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{4}}{8 b} & \text {otherwise} \end {cases}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(b**7*(d + e*x)**(11/2)/(11*e**7) - 7*b*(a*e - b*d)**6/(e**7* sqrt(d + e*x)) + (d + e*x)**(9/2)*(7*a*b**6*e - 7*b**7*d)/(9*e**7) + (d + e*x)**(7/2)*(21*a**2*b**5*e**2 - 42*a*b**6*d*e + 21*b**7*d**2)/(7*e**7) + (d + e*x)**(5/2)*(35*a**3*b**4*e**3 - 105*a**2*b**5*d*e**2 + 105*a*b**6*d* *2*e - 35*b**7*d**3)/(5*e**7) + (d + e*x)**(3/2)*(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)/(3*e**7) + sqrt(d + e*x)*(21*a**5*b**2*e**5 - 105*a**4*b**3*d*e**4 + 210*a**3*b**4*d**2*e**3 - 210*a**2*b**5*d**3*e**2 + 105*a*b**6*d**4*e - 21 *b**7*d**5)/e**7 - (a*e - b*d)**7/(3*e**7*(d + e*x)**(3/2)))/e, Ne(e, 0)), (Piecewise((a**7*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**4/(8*b), Tr ue))/d**(5/2), True))
Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (184) = 368\).
Time = 0.20 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.22 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {9 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{7} - 77 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 297 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 2079 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} \sqrt {e x + d}}{e^{7}} + \frac {33 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7} - 21 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{7}}\right )}}{99 \, e} \]
2/99*((9*(e*x + d)^(11/2)*b^7 - 77*(b^7*d - a*b^6*e)*(e*x + d)^(9/2) + 297 *(b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*(e*x + d)^(7/2) - 693*(b^7*d^3 - 3* a*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*(e*x + d)^(5/2) + 1155*(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)*(e *x + d)^(3/2) - 2079*(b^7*d^5 - 5*a*b^6*d^4*e + 10*a^2*b^5*d^3*e^2 - 10*a^ 3*b^4*d^2*e^3 + 5*a^4*b^3*d*e^4 - a^5*b^2*e^5)*sqrt(e*x + d))/e^7 + 33*(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 - 21*(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*d^2*e ^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6)*(e*x + d))/((e*x + d)^(3/2)*e^7))/e
Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (184) = 368\).
Time = 0.30 (sec) , antiderivative size = 612, normalized size of antiderivative = 2.94 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (21 \, {\left (e x + d\right )} b^{7} d^{6} - b^{7} d^{7} - 126 \, {\left (e x + d\right )} a b^{6} d^{5} e + 7 \, a b^{6} d^{6} e + 315 \, {\left (e x + d\right )} a^{2} b^{5} d^{4} e^{2} - 21 \, a^{2} b^{5} d^{5} e^{2} - 420 \, {\left (e x + d\right )} a^{3} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{4} d^{4} e^{3} + 315 \, {\left (e x + d\right )} a^{4} b^{3} d^{2} e^{4} - 35 \, a^{4} b^{3} d^{3} e^{4} - 126 \, {\left (e x + d\right )} a^{5} b^{2} d e^{5} + 21 \, a^{5} b^{2} d^{2} e^{5} + 21 \, {\left (e x + d\right )} a^{6} b e^{6} - 7 \, a^{6} b d e^{6} + a^{7} e^{7}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{8}} + \frac {2 \, {\left (9 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{7} e^{80} - 77 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{7} d e^{80} + 297 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{7} d^{2} e^{80} - 693 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{7} d^{3} e^{80} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{7} d^{4} e^{80} - 2079 \, \sqrt {e x + d} b^{7} d^{5} e^{80} + 77 \, {\left (e x + d\right )}^{\frac {9}{2}} a b^{6} e^{81} - 594 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{6} d e^{81} + 2079 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{6} d^{2} e^{81} - 4620 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{6} d^{3} e^{81} + 10395 \, \sqrt {e x + d} a b^{6} d^{4} e^{81} + 297 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{2} b^{5} e^{82} - 2079 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{5} d e^{82} + 6930 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{5} d^{2} e^{82} - 20790 \, \sqrt {e x + d} a^{2} b^{5} d^{3} e^{82} + 693 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{3} b^{4} e^{83} - 4620 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{4} d e^{83} + 20790 \, \sqrt {e x + d} a^{3} b^{4} d^{2} e^{83} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{4} b^{3} e^{84} - 10395 \, \sqrt {e x + d} a^{4} b^{3} d e^{84} + 2079 \, \sqrt {e x + d} a^{5} b^{2} e^{85}\right )}}{99 \, e^{88}} \]
-2/3*(21*(e*x + d)*b^7*d^6 - b^7*d^7 - 126*(e*x + d)*a*b^6*d^5*e + 7*a*b^6 *d^6*e + 315*(e*x + d)*a^2*b^5*d^4*e^2 - 21*a^2*b^5*d^5*e^2 - 420*(e*x + d )*a^3*b^4*d^3*e^3 + 35*a^3*b^4*d^4*e^3 + 315*(e*x + d)*a^4*b^3*d^2*e^4 - 3 5*a^4*b^3*d^3*e^4 - 126*(e*x + d)*a^5*b^2*d*e^5 + 21*a^5*b^2*d^2*e^5 + 21* (e*x + d)*a^6*b*e^6 - 7*a^6*b*d*e^6 + a^7*e^7)/((e*x + d)^(3/2)*e^8) + 2/9 9*(9*(e*x + d)^(11/2)*b^7*e^80 - 77*(e*x + d)^(9/2)*b^7*d*e^80 + 297*(e*x + d)^(7/2)*b^7*d^2*e^80 - 693*(e*x + d)^(5/2)*b^7*d^3*e^80 + 1155*(e*x + d )^(3/2)*b^7*d^4*e^80 - 2079*sqrt(e*x + d)*b^7*d^5*e^80 + 77*(e*x + d)^(9/2 )*a*b^6*e^81 - 594*(e*x + d)^(7/2)*a*b^6*d*e^81 + 2079*(e*x + d)^(5/2)*a*b ^6*d^2*e^81 - 4620*(e*x + d)^(3/2)*a*b^6*d^3*e^81 + 10395*sqrt(e*x + d)*a* b^6*d^4*e^81 + 297*(e*x + d)^(7/2)*a^2*b^5*e^82 - 2079*(e*x + d)^(5/2)*a^2 *b^5*d*e^82 + 6930*(e*x + d)^(3/2)*a^2*b^5*d^2*e^82 - 20790*sqrt(e*x + d)* a^2*b^5*d^3*e^82 + 693*(e*x + d)^(5/2)*a^3*b^4*e^83 - 4620*(e*x + d)^(3/2) *a^3*b^4*d*e^83 + 20790*sqrt(e*x + d)*a^3*b^4*d^2*e^83 + 1155*(e*x + d)^(3 /2)*a^4*b^3*e^84 - 10395*sqrt(e*x + d)*a^4*b^3*d*e^84 + 2079*sqrt(e*x + d) *a^5*b^2*e^85)/e^88
Time = 10.93 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.61 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2\,b^7\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}-\frac {\left (d+e\,x\right )\,\left (14\,a^6\,b\,e^6-84\,a^5\,b^2\,d\,e^5+210\,a^4\,b^3\,d^2\,e^4-280\,a^3\,b^4\,d^3\,e^3+210\,a^2\,b^5\,d^4\,e^2-84\,a\,b^6\,d^5\,e+14\,b^7\,d^6\right )+\frac {2\,a^7\,e^7}{3}-\frac {2\,b^7\,d^7}{3}-14\,a^2\,b^5\,d^5\,e^2+\frac {70\,a^3\,b^4\,d^4\,e^3}{3}-\frac {70\,a^4\,b^3\,d^3\,e^4}{3}+14\,a^5\,b^2\,d^2\,e^5+\frac {14\,a\,b^6\,d^6\,e}{3}-\frac {14\,a^6\,b\,d\,e^6}{3}}{e^8\,{\left (d+e\,x\right )}^{3/2}}+\frac {42\,b^2\,{\left (a\,e-b\,d\right )}^5\,\sqrt {d+e\,x}}{e^8}+\frac {70\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8}+\frac {14\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{e^8}+\frac {6\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{e^8} \]
(2*b^7*(d + e*x)^(11/2))/(11*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(9/ 2))/(9*e^8) - ((d + e*x)*(14*b^7*d^6 + 14*a^6*b*e^6 - 84*a^5*b^2*d*e^5 + 2 10*a^2*b^5*d^4*e^2 - 280*a^3*b^4*d^3*e^3 + 210*a^4*b^3*d^2*e^4 - 84*a*b^6* d^5*e) + (2*a^7*e^7)/3 - (2*b^7*d^7)/3 - 14*a^2*b^5*d^5*e^2 + (70*a^3*b^4* d^4*e^3)/3 - (70*a^4*b^3*d^3*e^4)/3 + 14*a^5*b^2*d^2*e^5 + (14*a*b^6*d^6*e )/3 - (14*a^6*b*d*e^6)/3)/(e^8*(d + e*x)^(3/2)) + (42*b^2*(a*e - b*d)^5*(d + e*x)^(1/2))/e^8 + (70*b^3*(a*e - b*d)^4*(d + e*x)^(3/2))/(3*e^8) + (14* b^4*(a*e - b*d)^3*(d + e*x)^(5/2))/e^8 + (6*b^5*(a*e - b*d)^2*(d + e*x)^(7 /2))/e^8