Integrand size = 33, antiderivative size = 210 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 (b d-a e)^7}{5 e^8 (d+e x)^{5/2}}-\frac {14 b (b d-a e)^6}{3 e^8 (d+e x)^{3/2}}+\frac {42 b^2 (b d-a e)^5}{e^8 \sqrt {d+e x}}+\frac {70 b^3 (b d-a e)^4 \sqrt {d+e x}}{e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{3/2}}{3 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{5/2}}{5 e^8}-\frac {2 b^6 (b d-a e) (d+e x)^{7/2}}{e^8}+\frac {2 b^7 (d+e x)^{9/2}}{9 e^8} \]
2/5*(-a*e+b*d)^7/e^8/(e*x+d)^(5/2)-14/3*b*(-a*e+b*d)^6/e^8/(e*x+d)^(3/2)-7 0/3*b^4*(-a*e+b*d)^3*(e*x+d)^(3/2)/e^8+42/5*b^5*(-a*e+b*d)^2*(e*x+d)^(5/2) /e^8-2*b^6*(-a*e+b*d)*(e*x+d)^(7/2)/e^8+2/9*b^7*(e*x+d)^(9/2)/e^8+42*b^2*( -a*e+b*d)^5/e^8/(e*x+d)^(1/2)+70*b^3*(-a*e+b*d)^4*(e*x+d)^(1/2)/e^8
Time = 0.18 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.79 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \left (-9 a^7 e^7-21 a^6 b e^6 (2 d+5 e x)-63 a^5 b^2 e^5 \left (8 d^2+20 d e x+15 e^2 x^2\right )+315 a^4 b^3 e^4 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )-105 a^3 b^4 e^3 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )+63 a^2 b^5 e^2 \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )-9 a b^6 e \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )+b^7 \left (2048 d^7+5120 d^6 e x+3840 d^5 e^2 x^2+640 d^4 e^3 x^3-80 d^3 e^4 x^4+24 d^2 e^5 x^5-10 d e^6 x^6+5 e^7 x^7\right )\right )}{45 e^8 (d+e x)^{5/2}} \]
(2*(-9*a^7*e^7 - 21*a^6*b*e^6*(2*d + 5*e*x) - 63*a^5*b^2*e^5*(8*d^2 + 20*d *e*x + 15*e^2*x^2) + 315*a^4*b^3*e^4*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) - 105*a^3*b^4*e^3*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 4 0*d*e^3*x^3 - 5*e^4*x^4) + 63*a^2*b^5*e^2*(256*d^5 + 640*d^4*e*x + 480*d^3 *e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5) - 9*a*b^6*e*(1024*d^ 6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 1 2*d*e^5*x^5 - 5*e^6*x^6) + b^7*(2048*d^7 + 5120*d^6*e*x + 3840*d^5*e^2*x^2 + 640*d^4*e^3*x^3 - 80*d^3*e^4*x^4 + 24*d^2*e^5*x^5 - 10*d*e^6*x^6 + 5*e^ 7*x^7)))/(45*e^8*(d + e*x)^(5/2))
Time = 0.35 (sec) , antiderivative size = 210, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^7}{(d+e x)^{7/2}}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^7}{(d+e x)^{7/2}}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {7 b^6 (d+e x)^{5/2} (b d-a e)}{e^7}+\frac {21 b^5 (d+e x)^{3/2} (b d-a e)^2}{e^7}-\frac {35 b^4 \sqrt {d+e x} (b d-a e)^3}{e^7}+\frac {35 b^3 (b d-a e)^4}{e^7 \sqrt {d+e x}}-\frac {21 b^2 (b d-a e)^5}{e^7 (d+e x)^{3/2}}+\frac {7 b (b d-a e)^6}{e^7 (d+e x)^{5/2}}+\frac {(a e-b d)^7}{e^7 (d+e x)^{7/2}}+\frac {b^7 (d+e x)^{7/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^6 (d+e x)^{7/2} (b d-a e)}{e^8}+\frac {42 b^5 (d+e x)^{5/2} (b d-a e)^2}{5 e^8}-\frac {70 b^4 (d+e x)^{3/2} (b d-a e)^3}{3 e^8}+\frac {70 b^3 \sqrt {d+e x} (b d-a e)^4}{e^8}+\frac {42 b^2 (b d-a e)^5}{e^8 \sqrt {d+e x}}-\frac {14 b (b d-a e)^6}{3 e^8 (d+e x)^{3/2}}+\frac {2 (b d-a e)^7}{5 e^8 (d+e x)^{5/2}}+\frac {2 b^7 (d+e x)^{9/2}}{9 e^8}\) |
(2*(b*d - a*e)^7)/(5*e^8*(d + e*x)^(5/2)) - (14*b*(b*d - a*e)^6)/(3*e^8*(d + e*x)^(3/2)) + (42*b^2*(b*d - a*e)^5)/(e^8*Sqrt[d + e*x]) + (70*b^3*(b*d - a*e)^4*Sqrt[d + e*x])/e^8 - (70*b^4*(b*d - a*e)^3*(d + e*x)^(3/2))/(3*e ^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(5/2))/(5*e^8) - (2*b^6*(b*d - a*e)* (d + e*x)^(7/2))/e^8 + (2*b^7*(d + e*x)^(9/2))/(9*e^8)
3.21.65.3.1 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 = 0.40 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.59
| method | result | size |
| risch | \(\frac {2 b^{3} \left (5 e^{4} x^{4} b^{4}+45 x^{3} a \,b^{3} e^{4}-25 x^{3} b^{4} d \,e^{3}+189 x^{2} a^{2} b^{2} e^{4}-243 x^{2} a \,b^{3} d \,e^{3}+84 x^{2} b^{4} d^{2} e^{2}+525 x \,a^{3} b \,e^{4}-1197 x \,a^{2} b^{2} d \,e^{3}+954 x a \,b^{3} d^{2} e^{2}-262 x \,b^{4} d^{3} e +1575 e^{4} a^{4}-5775 b d \,e^{3} a^{3}+8064 b^{2} d^{2} e^{2} a^{2}-5058 b^{3} d^{3} e a +1199 b^{4} d^{4}\right ) \sqrt {e x +d}}{45 e^{8}}-\frac {2 \left (315 b^{2} e^{2} x^{2}+35 a b \,e^{2} x +595 b^{2} d e x +3 e^{2} a^{2}+29 a b d e +283 b^{2} d^{2}\right ) \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}{15 e^{8} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d e x +d^{2}\right )}\) | \(334\) |
| pseudoelliptic | \(-\frac {2 \left (\left (-\frac {5}{9} b^{7} x^{7}+a^{7}-5 a \,b^{6} x^{6}-21 a^{2} b^{5} x^{5}-\frac {175}{3} a^{3} b^{4} x^{4}-175 a^{4} b^{3} x^{3}+105 a^{5} b^{2} x^{2}+\frac {35}{3} a^{6} b x \right ) e^{7}+\frac {14 \left (\frac {5}{21} b^{6} x^{6}+\frac {18}{7} a \,b^{5} x^{5}+15 a^{2} b^{4} x^{4}+100 a^{3} b^{3} x^{3}-225 a^{4} b^{2} x^{2}+30 a^{5} b x +a^{6}\right ) b d \,e^{6}}{3}+56 b^{2} \left (-\frac {1}{21} b^{5} x^{5}-\frac {5}{7} a \,b^{4} x^{4}-10 a^{2} b^{3} x^{3}+50 a^{3} b^{2} x^{2}-25 a^{4} b x +a^{5}\right ) d^{2} e^{5}-560 \left (-\frac {1}{63} x^{4} b^{4}-\frac {4}{7} a \,b^{3} x^{3}+6 x^{2} b^{2} a^{2}-\frac {20}{3} b \,a^{3} x +a^{4}\right ) b^{3} d^{3} e^{4}+\frac {4480 \left (-\frac {1}{21} x^{3} b^{3}+\frac {9}{7} a \,b^{2} x^{2}-3 b \,a^{2} x +a^{3}\right ) b^{4} d^{4} e^{3}}{3}-1792 b^{5} \left (\frac {5}{21} b^{2} x^{2}-\frac {10}{7} a b x +a^{2}\right ) d^{5} e^{2}+1024 b^{6} \left (-\frac {5 b x}{9}+a \right ) d^{6} e -\frac {2048 b^{7} d^{7}}{9}\right )}{5 \left (e x +d \right )^{\frac {5}{2}} e^{8}}\) | \(359\) |
| gosper | \(-\frac {2 \left (-5 x^{7} b^{7} e^{7}-45 x^{6} a \,b^{6} e^{7}+10 x^{6} b^{7} d \,e^{6}-189 x^{5} a^{2} b^{5} e^{7}+108 x^{5} a \,b^{6} d \,e^{6}-24 x^{5} b^{7} d^{2} e^{5}-525 x^{4} a^{3} b^{4} e^{7}+630 x^{4} a^{2} b^{5} d \,e^{6}-360 x^{4} a \,b^{6} d^{2} e^{5}+80 x^{4} b^{7} d^{3} e^{4}-1575 x^{3} a^{4} b^{3} e^{7}+4200 x^{3} a^{3} b^{4} d \,e^{6}-5040 x^{3} a^{2} b^{5} d^{2} e^{5}+2880 x^{3} a \,b^{6} d^{3} e^{4}-640 x^{3} b^{7} d^{4} e^{3}+945 x^{2} a^{5} b^{2} e^{7}-9450 x^{2} a^{4} b^{3} d \,e^{6}+25200 x^{2} a^{3} b^{4} d^{2} e^{5}-30240 x^{2} a^{2} b^{5} d^{3} e^{4}+17280 x^{2} a \,b^{6} d^{4} e^{3}-3840 x^{2} b^{7} d^{5} e^{2}+105 x \,a^{6} b \,e^{7}+1260 x \,a^{5} b^{2} d \,e^{6}-12600 x \,a^{4} b^{3} d^{2} e^{5}+33600 x \,a^{3} b^{4} d^{3} e^{4}-40320 x \,a^{2} b^{5} d^{4} e^{3}+23040 x a \,b^{6} d^{5} e^{2}-5120 x \,b^{7} d^{6} e +9 e^{7} a^{7}+42 b d \,e^{6} a^{6}+504 b^{2} d^{2} e^{5} a^{5}-5040 b^{3} d^{3} e^{4} a^{4}+13440 b^{4} d^{4} e^{3} a^{3}-16128 b^{5} d^{5} e^{2} a^{2}+9216 b^{6} d^{6} e a -2048 b^{7} d^{7}\right )}{45 \left (e x +d \right )^{\frac {5}{2}} e^{8}}\) | \(498\) |
| trager | \(-\frac {2 \left (-5 x^{7} b^{7} e^{7}-45 x^{6} a \,b^{6} e^{7}+10 x^{6} b^{7} d \,e^{6}-189 x^{5} a^{2} b^{5} e^{7}+108 x^{5} a \,b^{6} d \,e^{6}-24 x^{5} b^{7} d^{2} e^{5}-525 x^{4} a^{3} b^{4} e^{7}+630 x^{4} a^{2} b^{5} d \,e^{6}-360 x^{4} a \,b^{6} d^{2} e^{5}+80 x^{4} b^{7} d^{3} e^{4}-1575 x^{3} a^{4} b^{3} e^{7}+4200 x^{3} a^{3} b^{4} d \,e^{6}-5040 x^{3} a^{2} b^{5} d^{2} e^{5}+2880 x^{3} a \,b^{6} d^{3} e^{4}-640 x^{3} b^{7} d^{4} e^{3}+945 x^{2} a^{5} b^{2} e^{7}-9450 x^{2} a^{4} b^{3} d \,e^{6}+25200 x^{2} a^{3} b^{4} d^{2} e^{5}-30240 x^{2} a^{2} b^{5} d^{3} e^{4}+17280 x^{2} a \,b^{6} d^{4} e^{3}-3840 x^{2} b^{7} d^{5} e^{2}+105 x \,a^{6} b \,e^{7}+1260 x \,a^{5} b^{2} d \,e^{6}-12600 x \,a^{4} b^{3} d^{2} e^{5}+33600 x \,a^{3} b^{4} d^{3} e^{4}-40320 x \,a^{2} b^{5} d^{4} e^{3}+23040 x a \,b^{6} d^{5} e^{2}-5120 x \,b^{7} d^{6} e +9 e^{7} a^{7}+42 b d \,e^{6} a^{6}+504 b^{2} d^{2} e^{5} a^{5}-5040 b^{3} d^{3} e^{4} a^{4}+13440 b^{4} d^{4} e^{3} a^{3}-16128 b^{5} d^{5} e^{2} a^{2}+9216 b^{6} d^{6} e a -2048 b^{7} d^{7}\right )}{45 \left (e x +d \right )^{\frac {5}{2}} e^{8}}\) | \(498\) |
| derivativedivides | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {9}{2}}}{9}+2 a \,b^{6} e \left (e x +d \right )^{\frac {7}{2}}-2 b^{7} d \left (e x +d \right )^{\frac {7}{2}}+\frac {42 a^{2} b^{5} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {84 a \,b^{6} d e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {42 b^{7} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {70 a^{3} b^{4} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-70 a^{2} b^{5} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+70 a \,b^{6} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {70 b^{7} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+70 a^{4} b^{3} e^{4} \sqrt {e x +d}-280 a^{3} b^{4} d \,e^{3} \sqrt {e x +d}+420 a^{2} b^{5} d^{2} e^{2} \sqrt {e x +d}-280 a \,b^{6} d^{3} e \sqrt {e x +d}+70 b^{7} d^{4} \sqrt {e x +d}-\frac {2 \left (e^{7} a^{7}-7 b d \,e^{6} a^{6}+21 b^{2} d^{2} e^{5} a^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {14 b \left (e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {42 b^{2} \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}{\sqrt {e x +d}}}{e^{8}}\) | \(516\) |
| default | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {9}{2}}}{9}+2 a \,b^{6} e \left (e x +d \right )^{\frac {7}{2}}-2 b^{7} d \left (e x +d \right )^{\frac {7}{2}}+\frac {42 a^{2} b^{5} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {84 a \,b^{6} d e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {42 b^{7} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {70 a^{3} b^{4} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-70 a^{2} b^{5} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+70 a \,b^{6} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {70 b^{7} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+70 a^{4} b^{3} e^{4} \sqrt {e x +d}-280 a^{3} b^{4} d \,e^{3} \sqrt {e x +d}+420 a^{2} b^{5} d^{2} e^{2} \sqrt {e x +d}-280 a \,b^{6} d^{3} e \sqrt {e x +d}+70 b^{7} d^{4} \sqrt {e x +d}-\frac {2 \left (e^{7} a^{7}-7 b d \,e^{6} a^{6}+21 b^{2} d^{2} e^{5} a^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {14 b \left (e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {42 b^{2} \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}{\sqrt {e x +d}}}{e^{8}}\) | \(516\) |
2/45*b^3*(5*b^4*e^4*x^4+45*a*b^3*e^4*x^3-25*b^4*d*e^3*x^3+189*a^2*b^2*e^4* x^2-243*a*b^3*d*e^3*x^2+84*b^4*d^2*e^2*x^2+525*a^3*b*e^4*x-1197*a^2*b^2*d* e^3*x+954*a*b^3*d^2*e^2*x-262*b^4*d^3*e*x+1575*a^4*e^4-5775*a^3*b*d*e^3+80 64*a^2*b^2*d^2*e^2-5058*a*b^3*d^3*e+1199*b^4*d^4)*(e*x+d)^(1/2)/e^8-2/15*( 315*b^2*e^2*x^2+35*a*b*e^2*x+595*b^2*d*e*x+3*a^2*e^2+29*a*b*d*e+283*b^2*d^ 2)*(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^ 4*e-b^5*d^5)/e^8/(e*x+d)^(1/2)/(e^2*x^2+2*d*e*x+d^2)
Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (184) = 368\).
Time = 0.32 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.36 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (5 \, b^{7} e^{7} x^{7} + 2048 \, b^{7} d^{7} - 9216 \, a b^{6} d^{6} e + 16128 \, a^{2} b^{5} d^{5} e^{2} - 13440 \, a^{3} b^{4} d^{4} e^{3} + 5040 \, a^{4} b^{3} d^{3} e^{4} - 504 \, a^{5} b^{2} d^{2} e^{5} - 42 \, a^{6} b d e^{6} - 9 \, a^{7} e^{7} - 5 \, {\left (2 \, b^{7} d e^{6} - 9 \, a b^{6} e^{7}\right )} x^{6} + 3 \, {\left (8 \, b^{7} d^{2} e^{5} - 36 \, a b^{6} d e^{6} + 63 \, a^{2} b^{5} e^{7}\right )} x^{5} - 5 \, {\left (16 \, b^{7} d^{3} e^{4} - 72 \, a b^{6} d^{2} e^{5} + 126 \, a^{2} b^{5} d e^{6} - 105 \, a^{3} b^{4} e^{7}\right )} x^{4} + 5 \, {\left (128 \, b^{7} d^{4} e^{3} - 576 \, a b^{6} d^{3} e^{4} + 1008 \, a^{2} b^{5} d^{2} e^{5} - 840 \, a^{3} b^{4} d e^{6} + 315 \, a^{4} b^{3} e^{7}\right )} x^{3} + 15 \, {\left (256 \, b^{7} d^{5} e^{2} - 1152 \, a b^{6} d^{4} e^{3} + 2016 \, a^{2} b^{5} d^{3} e^{4} - 1680 \, a^{3} b^{4} d^{2} e^{5} + 630 \, a^{4} b^{3} d e^{6} - 63 \, a^{5} b^{2} e^{7}\right )} x^{2} + 5 \, {\left (1024 \, b^{7} d^{6} e - 4608 \, a b^{6} d^{5} e^{2} + 8064 \, a^{2} b^{5} d^{4} e^{3} - 6720 \, a^{3} b^{4} d^{3} e^{4} + 2520 \, a^{4} b^{3} d^{2} e^{5} - 252 \, a^{5} b^{2} d e^{6} - 21 \, a^{6} b e^{7}\right )} x\right )} \sqrt {e x + d}}{45 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} \]
2/45*(5*b^7*e^7*x^7 + 2048*b^7*d^7 - 9216*a*b^6*d^6*e + 16128*a^2*b^5*d^5* e^2 - 13440*a^3*b^4*d^4*e^3 + 5040*a^4*b^3*d^3*e^4 - 504*a^5*b^2*d^2*e^5 - 42*a^6*b*d*e^6 - 9*a^7*e^7 - 5*(2*b^7*d*e^6 - 9*a*b^6*e^7)*x^6 + 3*(8*b^7 *d^2*e^5 - 36*a*b^6*d*e^6 + 63*a^2*b^5*e^7)*x^5 - 5*(16*b^7*d^3*e^4 - 72*a *b^6*d^2*e^5 + 126*a^2*b^5*d*e^6 - 105*a^3*b^4*e^7)*x^4 + 5*(128*b^7*d^4*e ^3 - 576*a*b^6*d^3*e^4 + 1008*a^2*b^5*d^2*e^5 - 840*a^3*b^4*d*e^6 + 315*a^ 4*b^3*e^7)*x^3 + 15*(256*b^7*d^5*e^2 - 1152*a*b^6*d^4*e^3 + 2016*a^2*b^5*d ^3*e^4 - 1680*a^3*b^4*d^2*e^5 + 630*a^4*b^3*d*e^6 - 63*a^5*b^2*e^7)*x^2 + 5*(1024*b^7*d^6*e - 4608*a*b^6*d^5*e^2 + 8064*a^2*b^5*d^4*e^3 - 6720*a^3*b ^4*d^3*e^4 + 2520*a^4*b^3*d^2*e^5 - 252*a^5*b^2*d*e^6 - 21*a^6*b*e^7)*x)*s qrt(e*x + d)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8)
Time = 17.11 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.61 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{7/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{7} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{7}} - \frac {21 b^{2} \left (a e - b d\right )^{5}}{e^{7} \sqrt {d + e x}} - \frac {7 b \left (a e - b d\right )^{6}}{3 e^{7} \left (d + e x\right )^{\frac {3}{2}}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (7 a b^{6} e - 7 b^{7} d\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{e^{7}} - \frac {\left (a e - b d\right )^{7}}{5 e^{7} \left (d + e x\right )^{\frac {5}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\begin {cases} a^{7} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{4}}{8 b} & \text {otherwise} \end {cases}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(b**7*(d + e*x)**(9/2)/(9*e**7) - 21*b**2*(a*e - b*d)**5/(e** 7*sqrt(d + e*x)) - 7*b*(a*e - b*d)**6/(3*e**7*(d + e*x)**(3/2)) + (d + e*x )**(7/2)*(7*a*b**6*e - 7*b**7*d)/(7*e**7) + (d + e*x)**(5/2)*(21*a**2*b**5 *e**2 - 42*a*b**6*d*e + 21*b**7*d**2)/(5*e**7) + (d + e*x)**(3/2)*(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)/(3*e **7) + sqrt(d + e*x)*(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)/e**7 - (a*e - b*d)**7/( 5*e**7*(d + e*x)**(5/2)))/e, Ne(e, 0)), (Piecewise((a**7*x, Eq(b, 0)), ((a **2 + 2*a*b*x + b**2*x**2)**4/(8*b), True))/d**(7/2), True))
Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (184) = 368\).
Time = 0.20 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.20 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{7} - 45 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 189 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 525 \, {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 1575 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} \sqrt {e x + d}}{e^{7}} + \frac {3 \, {\left (3 \, b^{7} d^{7} - 21 \, a b^{6} d^{6} e + 63 \, a^{2} b^{5} d^{5} e^{2} - 105 \, a^{3} b^{4} d^{4} e^{3} + 105 \, a^{4} b^{3} d^{3} e^{4} - 63 \, a^{5} b^{2} d^{2} e^{5} + 21 \, a^{6} b d e^{6} - 3 \, a^{7} e^{7} + 315 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} {\left (e x + d\right )}^{2} - 35 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{7}}\right )}}{45 \, e} \]
2/45*((5*(e*x + d)^(9/2)*b^7 - 45*(b^7*d - a*b^6*e)*(e*x + d)^(7/2) + 189* (b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*(e*x + d)^(5/2) - 525*(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)^(3/2) + 1575*(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)*sqr t(e*x + d))/e^7 + 3*(3*b^7*d^7 - 21*a*b^6*d^6*e + 63*a^2*b^5*d^5*e^2 - 105 *a^3*b^4*d^4*e^3 + 105*a^4*b^3*d^3*e^4 - 63*a^5*b^2*d^2*e^5 + 21*a^6*b*d*e ^6 - 3*a^7*e^7 + 315*(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)*(e*x + d)^2 - 35*(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)^(5/2)*e^7))/e
Leaf count of result is larger than twice the leaf count of optimal. 608 vs. \(2 (184) = 368\).
Time = 0.29 (sec) , antiderivative size = 608, normalized size of antiderivative = 2.90 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (315 \, {\left (e x + d\right )}^{2} b^{7} d^{5} - 35 \, {\left (e x + d\right )} b^{7} d^{6} + 3 \, b^{7} d^{7} - 1575 \, {\left (e x + d\right )}^{2} a b^{6} d^{4} e + 210 \, {\left (e x + d\right )} a b^{6} d^{5} e - 21 \, a b^{6} d^{6} e + 3150 \, {\left (e x + d\right )}^{2} a^{2} b^{5} d^{3} e^{2} - 525 \, {\left (e x + d\right )} a^{2} b^{5} d^{4} e^{2} + 63 \, a^{2} b^{5} d^{5} e^{2} - 3150 \, {\left (e x + d\right )}^{2} a^{3} b^{4} d^{2} e^{3} + 700 \, {\left (e x + d\right )} a^{3} b^{4} d^{3} e^{3} - 105 \, a^{3} b^{4} d^{4} e^{3} + 1575 \, {\left (e x + d\right )}^{2} a^{4} b^{3} d e^{4} - 525 \, {\left (e x + d\right )} a^{4} b^{3} d^{2} e^{4} + 105 \, a^{4} b^{3} d^{3} e^{4} - 315 \, {\left (e x + d\right )}^{2} a^{5} b^{2} e^{5} + 210 \, {\left (e x + d\right )} a^{5} b^{2} d e^{5} - 63 \, a^{5} b^{2} d^{2} e^{5} - 35 \, {\left (e x + d\right )} a^{6} b e^{6} + 21 \, a^{6} b d e^{6} - 3 \, a^{7} e^{7}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{8}} + \frac {2 \, {\left (5 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{7} e^{64} - 45 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{7} d e^{64} + 189 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{7} d^{2} e^{64} - 525 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{7} d^{3} e^{64} + 1575 \, \sqrt {e x + d} b^{7} d^{4} e^{64} + 45 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{6} e^{65} - 378 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{6} d e^{65} + 1575 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{6} d^{2} e^{65} - 6300 \, \sqrt {e x + d} a b^{6} d^{3} e^{65} + 189 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{5} e^{66} - 1575 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{5} d e^{66} + 9450 \, \sqrt {e x + d} a^{2} b^{5} d^{2} e^{66} + 525 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{4} e^{67} - 6300 \, \sqrt {e x + d} a^{3} b^{4} d e^{67} + 1575 \, \sqrt {e x + d} a^{4} b^{3} e^{68}\right )}}{45 \, e^{72}} \]
2/15*(315*(e*x + d)^2*b^7*d^5 - 35*(e*x + d)*b^7*d^6 + 3*b^7*d^7 - 1575*(e *x + d)^2*a*b^6*d^4*e + 210*(e*x + d)*a*b^6*d^5*e - 21*a*b^6*d^6*e + 3150* (e*x + d)^2*a^2*b^5*d^3*e^2 - 525*(e*x + d)*a^2*b^5*d^4*e^2 + 63*a^2*b^5*d ^5*e^2 - 3150*(e*x + d)^2*a^3*b^4*d^2*e^3 + 700*(e*x + d)*a^3*b^4*d^3*e^3 - 105*a^3*b^4*d^4*e^3 + 1575*(e*x + d)^2*a^4*b^3*d*e^4 - 525*(e*x + d)*a^4 *b^3*d^2*e^4 + 105*a^4*b^3*d^3*e^4 - 315*(e*x + d)^2*a^5*b^2*e^5 + 210*(e* x + d)*a^5*b^2*d*e^5 - 63*a^5*b^2*d^2*e^5 - 35*(e*x + d)*a^6*b*e^6 + 21*a^ 6*b*d*e^6 - 3*a^7*e^7)/((e*x + d)^(5/2)*e^8) + 2/45*(5*(e*x + d)^(9/2)*b^7 *e^64 - 45*(e*x + d)^(7/2)*b^7*d*e^64 + 189*(e*x + d)^(5/2)*b^7*d^2*e^64 - 525*(e*x + d)^(3/2)*b^7*d^3*e^64 + 1575*sqrt(e*x + d)*b^7*d^4*e^64 + 45*( e*x + d)^(7/2)*a*b^6*e^65 - 378*(e*x + d)^(5/2)*a*b^6*d*e^65 + 1575*(e*x + d)^(3/2)*a*b^6*d^2*e^65 - 6300*sqrt(e*x + d)*a*b^6*d^3*e^65 + 189*(e*x + d)^(5/2)*a^2*b^5*e^66 - 1575*(e*x + d)^(3/2)*a^2*b^5*d*e^66 + 9450*sqrt(e* x + d)*a^2*b^5*d^2*e^66 + 525*(e*x + d)^(3/2)*a^3*b^4*e^67 - 6300*sqrt(e*x + d)*a^3*b^4*d*e^67 + 1575*sqrt(e*x + d)*a^4*b^3*e^68)/e^72
Time = 0.07 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.85 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2\,b^7\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^8}+\frac {{\left (d+e\,x\right )}^2\,\left (-42\,a^5\,b^2\,e^5+210\,a^4\,b^3\,d\,e^4-420\,a^3\,b^4\,d^2\,e^3+420\,a^2\,b^5\,d^3\,e^2-210\,a\,b^6\,d^4\,e+42\,b^7\,d^5\right )-\left (d+e\,x\right )\,\left (\frac {14\,a^6\,b\,e^6}{3}-28\,a^5\,b^2\,d\,e^5+70\,a^4\,b^3\,d^2\,e^4-\frac {280\,a^3\,b^4\,d^3\,e^3}{3}+70\,a^2\,b^5\,d^4\,e^2-28\,a\,b^6\,d^5\,e+\frac {14\,b^7\,d^6}{3}\right )-\frac {2\,a^7\,e^7}{5}+\frac {2\,b^7\,d^7}{5}+\frac {42\,a^2\,b^5\,d^5\,e^2}{5}-14\,a^3\,b^4\,d^4\,e^3+14\,a^4\,b^3\,d^3\,e^4-\frac {42\,a^5\,b^2\,d^2\,e^5}{5}-\frac {14\,a\,b^6\,d^6\,e}{5}+\frac {14\,a^6\,b\,d\,e^6}{5}}{e^8\,{\left (d+e\,x\right )}^{5/2}}+\frac {70\,b^3\,{\left (a\,e-b\,d\right )}^4\,\sqrt {d+e\,x}}{e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8} \]
(2*b^7*(d + e*x)^(9/2))/(9*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(7/2) )/(7*e^8) + ((d + e*x)^2*(42*b^7*d^5 - 42*a^5*b^2*e^5 + 210*a^4*b^3*d*e^4 + 420*a^2*b^5*d^3*e^2 - 420*a^3*b^4*d^2*e^3 - 210*a*b^6*d^4*e) - (d + e*x) *((14*b^7*d^6)/3 + (14*a^6*b*e^6)/3 - 28*a^5*b^2*d*e^5 + 70*a^2*b^5*d^4*e^ 2 - (280*a^3*b^4*d^3*e^3)/3 + 70*a^4*b^3*d^2*e^4 - 28*a*b^6*d^5*e) - (2*a^ 7*e^7)/5 + (2*b^7*d^7)/5 + (42*a^2*b^5*d^5*e^2)/5 - 14*a^3*b^4*d^4*e^3 + 1 4*a^4*b^3*d^3*e^4 - (42*a^5*b^2*d^2*e^5)/5 - (14*a*b^6*d^6*e)/5 + (14*a^6* b*d*e^6)/5)/(e^8*(d + e*x)^(5/2)) + (70*b^3*(a*e - b*d)^4*(d + e*x)^(1/2)) /e^8 + (70*b^4*(a*e - b*d)^3*(d + e*x)^(3/2))/(3*e^8) + (42*b^5*(a*e - b*d )^2*(d + e*x)^(5/2))/(5*e^8)