Integrand size = 33, antiderivative size = 216 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=-\frac {2 (b d-a e)^7 (d+e x)^{9/2}}{9 e^8}+\frac {14 b (b d-a e)^6 (d+e x)^{11/2}}{11 e^8}-\frac {42 b^2 (b d-a e)^5 (d+e x)^{13/2}}{13 e^8}+\frac {14 b^3 (b d-a e)^4 (d+e x)^{15/2}}{3 e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{17/2}}{17 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{19/2}}{19 e^8}-\frac {2 b^6 (b d-a e) (d+e x)^{21/2}}{3 e^8}+\frac {2 b^7 (d+e x)^{23/2}}{23 e^8} \]
-2/9*(-a*e+b*d)^7*(e*x+d)^(9/2)/e^8+14/11*b*(-a*e+b*d)^6*(e*x+d)^(11/2)/e^ 8-42/13*b^2*(-a*e+b*d)^5*(e*x+d)^(13/2)/e^8+14/3*b^3*(-a*e+b*d)^4*(e*x+d)^ (15/2)/e^8-70/17*b^4*(-a*e+b*d)^3*(e*x+d)^(17/2)/e^8+42/19*b^5*(-a*e+b*d)^ 2*(e*x+d)^(19/2)/e^8-2/3*b^6*(-a*e+b*d)*(e*x+d)^(21/2)/e^8+2/23*b^7*(e*x+d )^(23/2)/e^8
Time = 0.21 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.74 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (d+e x)^{9/2} \left (1062347 a^7 e^7+676039 a^6 b e^6 (-2 d+9 e x)+156009 a^5 b^2 e^5 \left (8 d^2-36 d e x+99 e^2 x^2\right )+52003 a^4 b^3 e^4 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+3059 a^3 b^4 e^3 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )+483 a^2 b^5 e^2 \left (-256 d^5+1152 d^4 e x-3168 d^3 e^2 x^2+6864 d^2 e^3 x^3-12870 d e^4 x^4+21879 e^5 x^5\right )+23 a b^6 e \left (1024 d^6-4608 d^5 e x+12672 d^4 e^2 x^2-27456 d^3 e^3 x^3+51480 d^2 e^4 x^4-87516 d e^5 x^5+138567 e^6 x^6\right )+b^7 \left (-2048 d^7+9216 d^6 e x-25344 d^5 e^2 x^2+54912 d^4 e^3 x^3-102960 d^3 e^4 x^4+175032 d^2 e^5 x^5-277134 d e^6 x^6+415701 e^7 x^7\right )\right )}{9561123 e^8} \]
(2*(d + e*x)^(9/2)*(1062347*a^7*e^7 + 676039*a^6*b*e^6*(-2*d + 9*e*x) + 15 6009*a^5*b^2*e^5*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 52003*a^4*b^3*e^4*(-16* d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + 3059*a^3*b^4*e^3*(128*d^ 4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4) + 483* a^2*b^5*e^2*(-256*d^5 + 1152*d^4*e*x - 3168*d^3*e^2*x^2 + 6864*d^2*e^3*x^3 - 12870*d*e^4*x^4 + 21879*e^5*x^5) + 23*a*b^6*e*(1024*d^6 - 4608*d^5*e*x + 12672*d^4*e^2*x^2 - 27456*d^3*e^3*x^3 + 51480*d^2*e^4*x^4 - 87516*d*e^5* x^5 + 138567*e^6*x^6) + b^7*(-2048*d^7 + 9216*d^6*e*x - 25344*d^5*e^2*x^2 + 54912*d^4*e^3*x^3 - 102960*d^3*e^4*x^4 + 175032*d^2*e^5*x^5 - 277134*d*e ^6*x^6 + 415701*e^7*x^7)))/(9561123*e^8)
Time = 0.39 (sec) , antiderivative size = 216, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^{7/2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^6 (a+b x)^7 (d+e x)^{7/2}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^7 (d+e x)^{7/2}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {7 b^6 (d+e x)^{19/2} (b d-a e)}{e^7}+\frac {21 b^5 (d+e x)^{17/2} (b d-a e)^2}{e^7}-\frac {35 b^4 (d+e x)^{15/2} (b d-a e)^3}{e^7}+\frac {35 b^3 (d+e x)^{13/2} (b d-a e)^4}{e^7}-\frac {21 b^2 (d+e x)^{11/2} (b d-a e)^5}{e^7}+\frac {7 b (d+e x)^{9/2} (b d-a e)^6}{e^7}+\frac {(d+e x)^{7/2} (a e-b d)^7}{e^7}+\frac {b^7 (d+e x)^{21/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^6 (d+e x)^{21/2} (b d-a e)}{3 e^8}+\frac {42 b^5 (d+e x)^{19/2} (b d-a e)^2}{19 e^8}-\frac {70 b^4 (d+e x)^{17/2} (b d-a e)^3}{17 e^8}+\frac {14 b^3 (d+e x)^{15/2} (b d-a e)^4}{3 e^8}-\frac {42 b^2 (d+e x)^{13/2} (b d-a e)^5}{13 e^8}+\frac {14 b (d+e x)^{11/2} (b d-a e)^6}{11 e^8}-\frac {2 (d+e x)^{9/2} (b d-a e)^7}{9 e^8}+\frac {2 b^7 (d+e x)^{23/2}}{23 e^8}\) |
(-2*(b*d - a*e)^7*(d + e*x)^(9/2))/(9*e^8) + (14*b*(b*d - a*e)^6*(d + e*x) ^(11/2))/(11*e^8) - (42*b^2*(b*d - a*e)^5*(d + e*x)^(13/2))/(13*e^8) + (14 *b^3*(b*d - a*e)^4*(d + e*x)^(15/2))/(3*e^8) - (70*b^4*(b*d - a*e)^3*(d + e*x)^(17/2))/(17*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(19/2))/(19*e^8) - (2*b^6*(b*d - a*e)*(d + e*x)^(21/2))/(3*e^8) + (2*b^7*(d + e*x)^(23/2))/( 23*e^8)
3.21.58.3.1 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 = 0.48 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.66
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {9}{23} b^{7} x^{7}+a^{7}+3 a \,b^{6} x^{6}+\frac {189}{19} a^{2} b^{5} x^{5}+\frac {315}{17} a^{3} b^{4} x^{4}+21 a^{4} b^{3} x^{3}+\frac {189}{13} a^{5} b^{2} x^{2}+\frac {63}{11} a^{6} b x \right ) e^{7}-\frac {14 b \left (\frac {33}{161} b^{6} x^{6}+\frac {198}{133} a \,b^{5} x^{5}+\frac {1485}{323} a^{2} b^{4} x^{4}+\frac {132}{17} a^{3} b^{3} x^{3}+\frac {99}{13} a^{4} b^{2} x^{2}+\frac {54}{13} a^{5} b x +a^{6}\right ) d \,e^{6}}{11}+\frac {168 b^{2} \left (\frac {429}{3059} b^{5} x^{5}+\frac {2145}{2261} a \,b^{4} x^{4}+\frac {858}{323} a^{2} b^{3} x^{3}+\frac {66}{17} a^{3} b^{2} x^{2}+3 a^{4} b x +a^{5}\right ) d^{2} e^{5}}{143}-\frac {112 \left (\frac {6435}{52003} x^{4} b^{4}+\frac {1716}{2261} a \,b^{3} x^{3}+\frac {594}{323} x^{2} b^{2} a^{2}+\frac {36}{17} b \,a^{3} x +a^{4}\right ) b^{3} d^{3} e^{4}}{143}+\frac {896 \left (\frac {429}{3059} x^{3} b^{3}+\frac {99}{133} a \,b^{2} x^{2}+\frac {27}{19} b \,a^{2} x +a^{3}\right ) b^{4} d^{4} e^{3}}{2431}-\frac {5376 b^{5} \left (\frac {33}{161} b^{2} x^{2}+\frac {6}{7} a b x +a^{2}\right ) d^{5} e^{2}}{46189}+\frac {1024 b^{6} \left (\frac {9 b x}{23}+a \right ) d^{6} e}{46189}-\frac {2048 b^{7} d^{7}}{1062347}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9 e^{8}}\) | \(359\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (415701 x^{7} b^{7} e^{7}+3187041 x^{6} a \,b^{6} e^{7}-277134 x^{6} b^{7} d \,e^{6}+10567557 x^{5} a^{2} b^{5} e^{7}-2012868 x^{5} a \,b^{6} d \,e^{6}+175032 x^{5} b^{7} d^{2} e^{5}+19684665 x^{4} a^{3} b^{4} e^{7}-6216210 x^{4} a^{2} b^{5} d \,e^{6}+1184040 x^{4} a \,b^{6} d^{2} e^{5}-102960 x^{4} b^{7} d^{3} e^{4}+22309287 x^{3} a^{4} b^{3} e^{7}-10498488 x^{3} a^{3} b^{4} d \,e^{6}+3315312 x^{3} a^{2} b^{5} d^{2} e^{5}-631488 x^{3} a \,b^{6} d^{3} e^{4}+54912 x^{3} b^{7} d^{4} e^{3}+15444891 x^{2} a^{5} b^{2} e^{7}-10296594 x^{2} a^{4} b^{3} d \,e^{6}+4845456 x^{2} a^{3} b^{4} d^{2} e^{5}-1530144 x^{2} a^{2} b^{5} d^{3} e^{4}+291456 x^{2} a \,b^{6} d^{4} e^{3}-25344 x^{2} b^{7} d^{5} e^{2}+6084351 x \,a^{6} b \,e^{7}-5616324 x \,a^{5} b^{2} d \,e^{6}+3744216 x \,a^{4} b^{3} d^{2} e^{5}-1761984 x \,a^{3} b^{4} d^{3} e^{4}+556416 x \,a^{2} b^{5} d^{4} e^{3}-105984 x a \,b^{6} d^{5} e^{2}+9216 x \,b^{7} d^{6} e +1062347 e^{7} a^{7}-1352078 b d \,e^{6} a^{6}+1248072 b^{2} d^{2} e^{5} a^{5}-832048 b^{3} d^{3} e^{4} a^{4}+391552 b^{4} d^{4} e^{3} a^{3}-123648 b^{5} d^{5} e^{2} a^{2}+23552 b^{6} d^{6} e a -2048 b^{7} d^{7}\right )}{9561123 e^{8}}\) | \(498\) |
| derivativedivides | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {23}{2}}}{23}+\frac {2 \left (\left (a e -b d \right ) b^{6}+3 b^{5} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {21}{2}}}{21}+\frac {2 \left (3 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{4}+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )+b \left (4 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+\left (2 a b e -2 b^{2} d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (a e -b d \right ) \left (4 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+\left (2 a b e -2 b^{2} d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )+3 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{8}}\) | \(936\) |
| default | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {23}{2}}}{23}+\frac {2 \left (\left (a e -b d \right ) b^{6}+3 b^{5} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {21}{2}}}{21}+\frac {2 \left (3 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{4}+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )+b \left (4 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+\left (2 a b e -2 b^{2} d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (a e -b d \right ) \left (4 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+\left (2 a b e -2 b^{2} d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )+3 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{8}}\) | \(936\) |
| trager | \(\frac {2 \left (415701 b^{7} e^{11} x^{11}+3187041 a \,b^{6} e^{11} x^{10}+1385670 b^{7} d \,e^{10} x^{10}+10567557 a^{2} b^{5} e^{11} x^{9}+10735296 a \,b^{6} d \,e^{10} x^{9}+1560702 b^{7} d^{2} e^{9} x^{9}+19684665 a^{3} b^{4} e^{11} x^{8}+36054018 a^{2} b^{5} d \,e^{10} x^{8}+12254814 a \,b^{6} d^{2} e^{9} x^{8}+597168 b^{7} d^{3} e^{8} x^{8}+22309287 a^{4} b^{3} e^{11} x^{7}+68240172 a^{3} b^{4} d \,e^{10} x^{7}+41855814 a^{2} b^{5} d^{2} e^{9} x^{7}+4775628 a \,b^{6} d^{3} e^{8} x^{7}+429 b^{7} d^{4} e^{7} x^{7}+15444891 a^{5} b^{2} e^{11} x^{6}+78940554 a^{4} b^{3} d \,e^{10} x^{6}+80959494 a^{3} b^{4} d^{2} e^{9} x^{6}+16704072 a^{2} b^{5} d^{3} e^{8} x^{6}+5313 a \,b^{6} d^{4} e^{7} x^{6}-462 b^{7} d^{5} e^{6} x^{6}+6084351 a^{6} b \,e^{11} x^{5}+56163240 a^{5} b^{2} d \,e^{10} x^{5}+96413562 a^{4} b^{3} d^{2} e^{9} x^{5}+33367572 a^{3} b^{4} d^{3} e^{8} x^{5}+30429 a^{2} b^{5} d^{4} e^{7} x^{5}-5796 a \,b^{6} d^{5} e^{6} x^{5}+504 b^{7} d^{6} e^{5} x^{5}+1062347 a^{7} e^{11} x^{4}+22985326 a^{6} b d \,e^{10} x^{4}+71452122 a^{5} b^{2} d^{2} e^{9} x^{4}+41602400 a^{4} b^{3} d^{3} e^{8} x^{4}+107065 a^{3} b^{4} d^{4} e^{7} x^{4}-33810 a^{2} b^{5} d^{5} e^{6} x^{4}+6440 a \,b^{6} d^{6} e^{5} x^{4}-560 b^{7} d^{7} e^{4} x^{4}+4249388 a^{7} d \,e^{10} x^{3}+31097794 a^{6} b \,d^{2} e^{9} x^{3}+33073908 a^{5} b^{2} d^{3} e^{8} x^{3}+260015 a^{4} b^{3} d^{4} e^{7} x^{3}-122360 a^{3} b^{4} d^{5} e^{6} x^{3}+38640 a^{2} b^{5} d^{6} e^{5} x^{3}-7360 a \,b^{6} d^{7} e^{4} x^{3}+640 b^{7} d^{8} e^{3} x^{3}+6374082 a^{7} d^{2} e^{9} x^{2}+16224936 a^{6} b \,d^{3} e^{8} x^{2}+468027 a^{5} b^{2} d^{4} e^{7} x^{2}-312018 a^{4} b^{3} d^{5} e^{6} x^{2}+146832 a^{3} b^{4} d^{6} e^{5} x^{2}-46368 a^{2} b^{5} d^{7} e^{4} x^{2}+8832 a \,b^{6} d^{8} e^{3} x^{2}-768 b^{7} d^{9} e^{2} x^{2}+4249388 a^{7} d^{3} e^{8} x +676039 a^{6} b \,d^{4} e^{7} x -624036 a^{5} b^{2} d^{5} e^{6} x +416024 a^{4} b^{3} d^{6} e^{5} x -195776 a^{3} b^{4} d^{7} e^{4} x +61824 a^{2} b^{5} d^{8} e^{3} x -11776 a \,b^{6} d^{9} e^{2} x +1024 b^{7} d^{10} e x +1062347 a^{7} d^{4} e^{7}-1352078 a^{6} b \,d^{5} e^{6}+1248072 a^{5} b^{2} d^{6} e^{5}-832048 a^{4} b^{3} d^{7} e^{4}+391552 a^{3} b^{4} d^{8} e^{3}-123648 a^{2} b^{5} d^{9} e^{2}+23552 a \,b^{6} d^{10} e -2048 b^{7} d^{11}\right ) \sqrt {e x +d}}{9561123 e^{8}}\) | \(998\) |
| risch | \(\frac {2 \left (415701 b^{7} e^{11} x^{11}+3187041 a \,b^{6} e^{11} x^{10}+1385670 b^{7} d \,e^{10} x^{10}+10567557 a^{2} b^{5} e^{11} x^{9}+10735296 a \,b^{6} d \,e^{10} x^{9}+1560702 b^{7} d^{2} e^{9} x^{9}+19684665 a^{3} b^{4} e^{11} x^{8}+36054018 a^{2} b^{5} d \,e^{10} x^{8}+12254814 a \,b^{6} d^{2} e^{9} x^{8}+597168 b^{7} d^{3} e^{8} x^{8}+22309287 a^{4} b^{3} e^{11} x^{7}+68240172 a^{3} b^{4} d \,e^{10} x^{7}+41855814 a^{2} b^{5} d^{2} e^{9} x^{7}+4775628 a \,b^{6} d^{3} e^{8} x^{7}+429 b^{7} d^{4} e^{7} x^{7}+15444891 a^{5} b^{2} e^{11} x^{6}+78940554 a^{4} b^{3} d \,e^{10} x^{6}+80959494 a^{3} b^{4} d^{2} e^{9} x^{6}+16704072 a^{2} b^{5} d^{3} e^{8} x^{6}+5313 a \,b^{6} d^{4} e^{7} x^{6}-462 b^{7} d^{5} e^{6} x^{6}+6084351 a^{6} b \,e^{11} x^{5}+56163240 a^{5} b^{2} d \,e^{10} x^{5}+96413562 a^{4} b^{3} d^{2} e^{9} x^{5}+33367572 a^{3} b^{4} d^{3} e^{8} x^{5}+30429 a^{2} b^{5} d^{4} e^{7} x^{5}-5796 a \,b^{6} d^{5} e^{6} x^{5}+504 b^{7} d^{6} e^{5} x^{5}+1062347 a^{7} e^{11} x^{4}+22985326 a^{6} b d \,e^{10} x^{4}+71452122 a^{5} b^{2} d^{2} e^{9} x^{4}+41602400 a^{4} b^{3} d^{3} e^{8} x^{4}+107065 a^{3} b^{4} d^{4} e^{7} x^{4}-33810 a^{2} b^{5} d^{5} e^{6} x^{4}+6440 a \,b^{6} d^{6} e^{5} x^{4}-560 b^{7} d^{7} e^{4} x^{4}+4249388 a^{7} d \,e^{10} x^{3}+31097794 a^{6} b \,d^{2} e^{9} x^{3}+33073908 a^{5} b^{2} d^{3} e^{8} x^{3}+260015 a^{4} b^{3} d^{4} e^{7} x^{3}-122360 a^{3} b^{4} d^{5} e^{6} x^{3}+38640 a^{2} b^{5} d^{6} e^{5} x^{3}-7360 a \,b^{6} d^{7} e^{4} x^{3}+640 b^{7} d^{8} e^{3} x^{3}+6374082 a^{7} d^{2} e^{9} x^{2}+16224936 a^{6} b \,d^{3} e^{8} x^{2}+468027 a^{5} b^{2} d^{4} e^{7} x^{2}-312018 a^{4} b^{3} d^{5} e^{6} x^{2}+146832 a^{3} b^{4} d^{6} e^{5} x^{2}-46368 a^{2} b^{5} d^{7} e^{4} x^{2}+8832 a \,b^{6} d^{8} e^{3} x^{2}-768 b^{7} d^{9} e^{2} x^{2}+4249388 a^{7} d^{3} e^{8} x +676039 a^{6} b \,d^{4} e^{7} x -624036 a^{5} b^{2} d^{5} e^{6} x +416024 a^{4} b^{3} d^{6} e^{5} x -195776 a^{3} b^{4} d^{7} e^{4} x +61824 a^{2} b^{5} d^{8} e^{3} x -11776 a \,b^{6} d^{9} e^{2} x +1024 b^{7} d^{10} e x +1062347 a^{7} d^{4} e^{7}-1352078 a^{6} b \,d^{5} e^{6}+1248072 a^{5} b^{2} d^{6} e^{5}-832048 a^{4} b^{3} d^{7} e^{4}+391552 a^{3} b^{4} d^{8} e^{3}-123648 a^{2} b^{5} d^{9} e^{2}+23552 a \,b^{6} d^{10} e -2048 b^{7} d^{11}\right ) \sqrt {e x +d}}{9561123 e^{8}}\) | \(998\) |
2/9*((9/23*b^7*x^7+a^7+3*a*b^6*x^6+189/19*a^2*b^5*x^5+315/17*a^3*b^4*x^4+2 1*a^4*b^3*x^3+189/13*a^5*b^2*x^2+63/11*a^6*b*x)*e^7-14/11*b*(33/161*b^6*x^ 6+198/133*a*b^5*x^5+1485/323*a^2*b^4*x^4+132/17*a^3*b^3*x^3+99/13*a^4*b^2* x^2+54/13*a^5*b*x+a^6)*d*e^6+168/143*b^2*(429/3059*b^5*x^5+2145/2261*a*b^4 *x^4+858/323*a^2*b^3*x^3+66/17*a^3*b^2*x^2+3*a^4*b*x+a^5)*d^2*e^5-112/143* (6435/52003*x^4*b^4+1716/2261*a*b^3*x^3+594/323*x^2*b^2*a^2+36/17*b*a^3*x+ a^4)*b^3*d^3*e^4+896/2431*(429/3059*x^3*b^3+99/133*a*b^2*x^2+27/19*b*a^2*x +a^3)*b^4*d^4*e^3-5376/46189*b^5*(33/161*b^2*x^2+6/7*a*b*x+a^2)*d^5*e^2+10 24/46189*b^6*(9/23*b*x+a)*d^6*e-2048/1062347*b^7*d^7)*(e*x+d)^(9/2)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 891 vs. \(2 (184) = 368\).
Time = 0.29 (sec) , antiderivative size = 891, normalized size of antiderivative = 4.12 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (415701 \, b^{7} e^{11} x^{11} - 2048 \, b^{7} d^{11} + 23552 \, a b^{6} d^{10} e - 123648 \, a^{2} b^{5} d^{9} e^{2} + 391552 \, a^{3} b^{4} d^{8} e^{3} - 832048 \, a^{4} b^{3} d^{7} e^{4} + 1248072 \, a^{5} b^{2} d^{6} e^{5} - 1352078 \, a^{6} b d^{5} e^{6} + 1062347 \, a^{7} d^{4} e^{7} + 138567 \, {\left (10 \, b^{7} d e^{10} + 23 \, a b^{6} e^{11}\right )} x^{10} + 7293 \, {\left (214 \, b^{7} d^{2} e^{9} + 1472 \, a b^{6} d e^{10} + 1449 \, a^{2} b^{5} e^{11}\right )} x^{9} + 1287 \, {\left (464 \, b^{7} d^{3} e^{8} + 9522 \, a b^{6} d^{2} e^{9} + 28014 \, a^{2} b^{5} d e^{10} + 15295 \, a^{3} b^{4} e^{11}\right )} x^{8} + 429 \, {\left (b^{7} d^{4} e^{7} + 11132 \, a b^{6} d^{3} e^{8} + 97566 \, a^{2} b^{5} d^{2} e^{9} + 159068 \, a^{3} b^{4} d e^{10} + 52003 \, a^{4} b^{3} e^{11}\right )} x^{7} - 231 \, {\left (2 \, b^{7} d^{5} e^{6} - 23 \, a b^{6} d^{4} e^{7} - 72312 \, a^{2} b^{5} d^{3} e^{8} - 350474 \, a^{3} b^{4} d^{2} e^{9} - 341734 \, a^{4} b^{3} d e^{10} - 66861 \, a^{5} b^{2} e^{11}\right )} x^{6} + 63 \, {\left (8 \, b^{7} d^{6} e^{5} - 92 \, a b^{6} d^{5} e^{6} + 483 \, a^{2} b^{5} d^{4} e^{7} + 529644 \, a^{3} b^{4} d^{3} e^{8} + 1530374 \, a^{4} b^{3} d^{2} e^{9} + 891480 \, a^{5} b^{2} d e^{10} + 96577 \, a^{6} b e^{11}\right )} x^{5} - {\left (560 \, b^{7} d^{7} e^{4} - 6440 \, a b^{6} d^{6} e^{5} + 33810 \, a^{2} b^{5} d^{5} e^{6} - 107065 \, a^{3} b^{4} d^{4} e^{7} - 41602400 \, a^{4} b^{3} d^{3} e^{8} - 71452122 \, a^{5} b^{2} d^{2} e^{9} - 22985326 \, a^{6} b d e^{10} - 1062347 \, a^{7} e^{11}\right )} x^{4} + {\left (640 \, b^{7} d^{8} e^{3} - 7360 \, a b^{6} d^{7} e^{4} + 38640 \, a^{2} b^{5} d^{6} e^{5} - 122360 \, a^{3} b^{4} d^{5} e^{6} + 260015 \, a^{4} b^{3} d^{4} e^{7} + 33073908 \, a^{5} b^{2} d^{3} e^{8} + 31097794 \, a^{6} b d^{2} e^{9} + 4249388 \, a^{7} d e^{10}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{9} e^{2} - 2944 \, a b^{6} d^{8} e^{3} + 15456 \, a^{2} b^{5} d^{7} e^{4} - 48944 \, a^{3} b^{4} d^{6} e^{5} + 104006 \, a^{4} b^{3} d^{5} e^{6} - 156009 \, a^{5} b^{2} d^{4} e^{7} - 5408312 \, a^{6} b d^{3} e^{8} - 2124694 \, a^{7} d^{2} e^{9}\right )} x^{2} + {\left (1024 \, b^{7} d^{10} e - 11776 \, a b^{6} d^{9} e^{2} + 61824 \, a^{2} b^{5} d^{8} e^{3} - 195776 \, a^{3} b^{4} d^{7} e^{4} + 416024 \, a^{4} b^{3} d^{6} e^{5} - 624036 \, a^{5} b^{2} d^{5} e^{6} + 676039 \, a^{6} b d^{4} e^{7} + 4249388 \, a^{7} d^{3} e^{8}\right )} x\right )} \sqrt {e x + d}}{9561123 \, e^{8}} \]
2/9561123*(415701*b^7*e^11*x^11 - 2048*b^7*d^11 + 23552*a*b^6*d^10*e - 123 648*a^2*b^5*d^9*e^2 + 391552*a^3*b^4*d^8*e^3 - 832048*a^4*b^3*d^7*e^4 + 12 48072*a^5*b^2*d^6*e^5 - 1352078*a^6*b*d^5*e^6 + 1062347*a^7*d^4*e^7 + 1385 67*(10*b^7*d*e^10 + 23*a*b^6*e^11)*x^10 + 7293*(214*b^7*d^2*e^9 + 1472*a*b ^6*d*e^10 + 1449*a^2*b^5*e^11)*x^9 + 1287*(464*b^7*d^3*e^8 + 9522*a*b^6*d^ 2*e^9 + 28014*a^2*b^5*d*e^10 + 15295*a^3*b^4*e^11)*x^8 + 429*(b^7*d^4*e^7 + 11132*a*b^6*d^3*e^8 + 97566*a^2*b^5*d^2*e^9 + 159068*a^3*b^4*d*e^10 + 52 003*a^4*b^3*e^11)*x^7 - 231*(2*b^7*d^5*e^6 - 23*a*b^6*d^4*e^7 - 72312*a^2* b^5*d^3*e^8 - 350474*a^3*b^4*d^2*e^9 - 341734*a^4*b^3*d*e^10 - 66861*a^5*b ^2*e^11)*x^6 + 63*(8*b^7*d^6*e^5 - 92*a*b^6*d^5*e^6 + 483*a^2*b^5*d^4*e^7 + 529644*a^3*b^4*d^3*e^8 + 1530374*a^4*b^3*d^2*e^9 + 891480*a^5*b^2*d*e^10 + 96577*a^6*b*e^11)*x^5 - (560*b^7*d^7*e^4 - 6440*a*b^6*d^6*e^5 + 33810*a ^2*b^5*d^5*e^6 - 107065*a^3*b^4*d^4*e^7 - 41602400*a^4*b^3*d^3*e^8 - 71452 122*a^5*b^2*d^2*e^9 - 22985326*a^6*b*d*e^10 - 1062347*a^7*e^11)*x^4 + (640 *b^7*d^8*e^3 - 7360*a*b^6*d^7*e^4 + 38640*a^2*b^5*d^6*e^5 - 122360*a^3*b^4 *d^5*e^6 + 260015*a^4*b^3*d^4*e^7 + 33073908*a^5*b^2*d^3*e^8 + 31097794*a^ 6*b*d^2*e^9 + 4249388*a^7*d*e^10)*x^3 - 3*(256*b^7*d^9*e^2 - 2944*a*b^6*d^ 8*e^3 + 15456*a^2*b^5*d^7*e^4 - 48944*a^3*b^4*d^6*e^5 + 104006*a^4*b^3*d^5 *e^6 - 156009*a^5*b^2*d^4*e^7 - 5408312*a^6*b*d^3*e^8 - 2124694*a^7*d^2*e^ 9)*x^2 + (1024*b^7*d^10*e - 11776*a*b^6*d^9*e^2 + 61824*a^2*b^5*d^8*e^3...
Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (201) = 402\).
Time = 2.16 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.67 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {b^{7} \left (d + e x\right )^{\frac {23}{2}}}{23 e^{7}} + \frac {\left (d + e x\right )^{\frac {21}{2}} \cdot \left (7 a b^{6} e - 7 b^{7} d\right )}{21 e^{7}} + \frac {\left (d + e x\right )^{\frac {19}{2}} \cdot \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{19 e^{7}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (21 a^{5} b^{2} e^{5} - 105 a^{4} b^{3} d e^{4} + 210 a^{3} b^{4} d^{2} e^{3} - 210 a^{2} b^{5} d^{3} e^{2} + 105 a b^{6} d^{4} e - 21 b^{7} d^{5}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (7 a^{6} b e^{6} - 42 a^{5} b^{2} d e^{5} + 105 a^{4} b^{3} d^{2} e^{4} - 140 a^{3} b^{4} d^{3} e^{3} + 105 a^{2} b^{5} d^{4} e^{2} - 42 a b^{6} d^{5} e + 7 b^{7} d^{6}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (a^{7} e^{7} - 7 a^{6} b d e^{6} + 21 a^{5} b^{2} d^{2} e^{5} - 35 a^{4} b^{3} d^{3} e^{4} + 35 a^{3} b^{4} d^{4} e^{3} - 21 a^{2} b^{5} d^{5} e^{2} + 7 a b^{6} d^{6} e - b^{7} d^{7}\right )}{9 e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (\begin {cases} a^{7} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{4}}{8 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**7*(d + e*x)**(23/2)/(23*e**7) + (d + e*x)**(21/2)*(7*a*b* *6*e - 7*b**7*d)/(21*e**7) + (d + e*x)**(19/2)*(21*a**2*b**5*e**2 - 42*a*b **6*d*e + 21*b**7*d**2)/(19*e**7) + (d + e*x)**(17/2)*(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)/(17*e**7) + (d + e*x)**(15/2)*(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)/(15*e**7) + (d + e*x)**(13/2)* (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)/(13*e**7) + (d + e *x)**(11/2)*(7*a**6*b*e**6 - 42*a**5*b**2*d*e**5 + 105*a**4*b**3*d**2*e**4 - 140*a**3*b**4*d**3*e**3 + 105*a**2*b**5*d**4*e**2 - 42*a*b**6*d**5*e + 7*b**7*d**6)/(11*e**7) + (d + e*x)**(9/2)*(a**7*e**7 - 7*a**6*b*d*e**6 + 2 1*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)/(9*e**7))/e, Ne(e, 0 )), (d**(7/2)*Piecewise((a**7*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)* *4/(8*b), True)), True))
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (184) = 368\).
Time = 0.21 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.11 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (415701 \, {\left (e x + d\right )}^{\frac {23}{2}} b^{7} - 3187041 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {21}{2}} + 10567557 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {19}{2}} - 19684665 \, {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 22309287 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 15444891 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 6084351 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 1062347 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{9561123 \, e^{8}} \]
2/9561123*(415701*(e*x + d)^(23/2)*b^7 - 3187041*(b^7*d - a*b^6*e)*(e*x + d)^(21/2) + 10567557*(b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*(e*x + d)^(19/2 ) - 19684665*(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)^(17/2) + 22309287*(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)^(15/2) - 15444891*(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)^(13/2) + 6084351*(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)^(11/2) - 1062347*(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)*(e*x + d)^(9/2))/e^8
Leaf count of result is larger than twice the leaf count of optimal. 3416 vs. \(2 (184) = 368\).
Time = 0.32 (sec) , antiderivative size = 3416, normalized size of antiderivative = 15.81 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
2/334639305*(334639305*sqrt(e*x + d)*a^7*d^4 + 446185740*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^7*d^3 + 780825045*((e*x + d)^(3/2) - 3*sqrt(e*x + d )*d)*a^6*b*d^4/e + 133855722*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 1 5*sqrt(e*x + d)*d^2)*a^7*d^2 + 468495027*(3*(e*x + d)^(5/2) - 10*(e*x + d) ^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^5*b^2*d^4/e^2 + 624660036*(3*(e*x + d)^ (5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^6*b*d^3/e + 3824449 2*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35* sqrt(e*x + d)*d^3)*a^7*d + 334639305*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/ 2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^4*b^3*d^4/e^3 + 80 3134332*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^5*b^2*d^3/e^2 + 401567166*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^6* b*d^2/e + 1062347*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^7 + 371 82145*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^ 2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^3*b^4*d^4/e^4 + 148 728580*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d ^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^4*b^3*d^3/e^3 + 13 3855722*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)* d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^5*b^2*d^2/e^2 ...
Time = 10.91 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2\,b^7\,{\left (d+e\,x\right )}^{23/2}}{23\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{21/2}}{21\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^7\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {42\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {14\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{15/2}}{3\,e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{19/2}}{19\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8} \]
(2*b^7*(d + e*x)^(23/2))/(23*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(21 /2))/(21*e^8) + (2*(a*e - b*d)^7*(d + e*x)^(9/2))/(9*e^8) + (42*b^2*(a*e - b*d)^5*(d + e*x)^(13/2))/(13*e^8) + (14*b^3*(a*e - b*d)^4*(d + e*x)^(15/2 ))/(3*e^8) + (70*b^4*(a*e - b*d)^3*(d + e*x)^(17/2))/(17*e^8) + (42*b^5*(a *e - b*d)^2*(d + e*x)^(19/2))/(19*e^8) + (14*b*(a*e - b*d)^6*(d + e*x)^(11 /2))/(11*e^8)