Integrand size = 33, antiderivative size = 216 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=-\frac {2 (b d-a e)^7 (d+e x)^{7/2}}{7 e^8}+\frac {14 b (b d-a e)^6 (d+e x)^{9/2}}{9 e^8}-\frac {42 b^2 (b d-a e)^5 (d+e x)^{11/2}}{11 e^8}+\frac {70 b^3 (b d-a e)^4 (d+e x)^{13/2}}{13 e^8}-\frac {14 b^4 (b d-a e)^3 (d+e x)^{15/2}}{3 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{17/2}}{17 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{19/2}}{19 e^8}+\frac {2 b^7 (d+e x)^{21/2}}{21 e^8} \]
-2/7*(-a*e+b*d)^7*(e*x+d)^(7/2)/e^8+14/9*b*(-a*e+b*d)^6*(e*x+d)^(9/2)/e^8- 42/11*b^2*(-a*e+b*d)^5*(e*x+d)^(11/2)/e^8+70/13*b^3*(-a*e+b*d)^4*(e*x+d)^( 13/2)/e^8-14/3*b^4*(-a*e+b*d)^3*(e*x+d)^(15/2)/e^8+42/17*b^5*(-a*e+b*d)^2* (e*x+d)^(17/2)/e^8-14/19*b^6*(-a*e+b*d)*(e*x+d)^(19/2)/e^8+2/21*b^7*(e*x+d )^(21/2)/e^8
Time = 0.21 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.74 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (d+e x)^{7/2} \left (415701 a^7 e^7+323323 a^6 b e^6 (-2 d+7 e x)+88179 a^5 b^2 e^5 \left (8 d^2-28 d e x+63 e^2 x^2\right )+33915 a^4 b^3 e^4 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+2261 a^3 b^4 e^3 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+399 a^2 b^5 e^2 \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+21 a b^6 e \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )+b^7 \left (-2048 d^7+7168 d^6 e x-16128 d^5 e^2 x^2+29568 d^4 e^3 x^3-48048 d^3 e^4 x^4+72072 d^2 e^5 x^5-102102 d e^6 x^6+138567 e^7 x^7\right )\right )}{2909907 e^8} \]
(2*(d + e*x)^(7/2)*(415701*a^7*e^7 + 323323*a^6*b*e^6*(-2*d + 7*e*x) + 881 79*a^5*b^2*e^5*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 33915*a^4*b^3*e^4*(-16*d^ 3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + 2261*a^3*b^4*e^3*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 399*a^ 2*b^5*e^2*(-256*d^5 + 896*d^4*e*x - 2016*d^3*e^2*x^2 + 3696*d^2*e^3*x^3 - 6006*d*e^4*x^4 + 9009*e^5*x^5) + 21*a*b^6*e*(1024*d^6 - 3584*d^5*e*x + 806 4*d^4*e^2*x^2 - 14784*d^3*e^3*x^3 + 24024*d^2*e^4*x^4 - 36036*d*e^5*x^5 + 51051*e^6*x^6) + b^7*(-2048*d^7 + 7168*d^6*e*x - 16128*d^5*e^2*x^2 + 29568 *d^4*e^3*x^3 - 48048*d^3*e^4*x^4 + 72072*d^2*e^5*x^5 - 102102*d*e^6*x^6 + 138567*e^7*x^7)))/(2909907*e^8)
Time = 0.36 (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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^6 (a+b x)^7 (d+e x)^{5/2}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^7 (d+e x)^{5/2}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {7 b^6 (d+e x)^{17/2} (b d-a e)}{e^7}+\frac {21 b^5 (d+e x)^{15/2} (b d-a e)^2}{e^7}-\frac {35 b^4 (d+e x)^{13/2} (b d-a e)^3}{e^7}+\frac {35 b^3 (d+e x)^{11/2} (b d-a e)^4}{e^7}-\frac {21 b^2 (d+e x)^{9/2} (b d-a e)^5}{e^7}+\frac {7 b (d+e x)^{7/2} (b d-a e)^6}{e^7}+\frac {(d+e x)^{5/2} (a e-b d)^7}{e^7}+\frac {b^7 (d+e x)^{19/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {14 b^6 (d+e x)^{19/2} (b d-a e)}{19 e^8}+\frac {42 b^5 (d+e x)^{17/2} (b d-a e)^2}{17 e^8}-\frac {14 b^4 (d+e x)^{15/2} (b d-a e)^3}{3 e^8}+\frac {70 b^3 (d+e x)^{13/2} (b d-a e)^4}{13 e^8}-\frac {42 b^2 (d+e x)^{11/2} (b d-a e)^5}{11 e^8}+\frac {14 b (d+e x)^{9/2} (b d-a e)^6}{9 e^8}-\frac {2 (d+e x)^{7/2} (b d-a e)^7}{7 e^8}+\frac {2 b^7 (d+e x)^{21/2}}{21 e^8}\) |
(-2*(b*d - a*e)^7*(d + e*x)^(7/2))/(7*e^8) + (14*b*(b*d - a*e)^6*(d + e*x) ^(9/2))/(9*e^8) - (42*b^2*(b*d - a*e)^5*(d + e*x)^(11/2))/(11*e^8) + (70*b ^3*(b*d - a*e)^4*(d + e*x)^(13/2))/(13*e^8) - (14*b^4*(b*d - a*e)^3*(d + e *x)^(15/2))/(3*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(17/2))/(17*e^8) - ( 14*b^6*(b*d - a*e)*(d + e*x)^(19/2))/(19*e^8) + (2*b^7*(d + e*x)^(21/2))/( 21*e^8)
3.21.59.3.1 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 = 0.46 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.66
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {1}{3} b^{7} x^{7}+a^{7}+\frac {49}{19} a \,b^{6} x^{6}+\frac {147}{17} a^{2} b^{5} x^{5}+\frac {49}{3} a^{3} b^{4} x^{4}+\frac {245}{13} a^{4} b^{3} x^{3}+\frac {147}{11} a^{5} b^{2} x^{2}+\frac {49}{9} a^{6} b x \right ) e^{7}-\frac {14 b \left (\frac {3}{19} b^{6} x^{6}+\frac {378}{323} a \,b^{5} x^{5}+\frac {63}{17} a^{2} b^{4} x^{4}+\frac {84}{13} a^{3} b^{3} x^{3}+\frac {945}{143} a^{4} b^{2} x^{2}+\frac {42}{11} a^{5} b x +a^{6}\right ) d \,e^{6}}{9}+\frac {56 b^{2} \left (\frac {33}{323} b^{5} x^{5}+\frac {231}{323} a \,b^{4} x^{4}+\frac {462}{221} a^{2} b^{3} x^{3}+\frac {42}{13} a^{3} b^{2} x^{2}+\frac {35}{13} a^{4} b x +a^{5}\right ) d^{2} e^{5}}{33}-\frac {560 b^{3} \left (\frac {143}{1615} x^{4} b^{4}+\frac {924}{1615} a \,b^{3} x^{3}+\frac {126}{85} x^{2} b^{2} a^{2}+\frac {28}{15} b \,a^{3} x +a^{4}\right ) d^{3} e^{4}}{429}+\frac {896 \left (\frac {33}{323} x^{3} b^{3}+\frac {189}{323} a \,b^{2} x^{2}+\frac {21}{17} b \,a^{2} x +a^{3}\right ) b^{4} d^{4} e^{3}}{1287}-\frac {1792 \left (\frac {3}{19} b^{2} x^{2}+\frac {14}{19} a b x +a^{2}\right ) b^{5} d^{5} e^{2}}{7293}+\frac {7168 b^{6} \left (\frac {b x}{3}+a \right ) d^{6} e}{138567}-\frac {2048 b^{7} d^{7}}{415701}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{8}}\) | \(359\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (138567 x^{7} b^{7} e^{7}+1072071 x^{6} a \,b^{6} e^{7}-102102 x^{6} b^{7} d \,e^{6}+3594591 x^{5} a^{2} b^{5} e^{7}-756756 x^{5} a \,b^{6} d \,e^{6}+72072 x^{5} b^{7} d^{2} e^{5}+6789783 x^{4} a^{3} b^{4} e^{7}-2396394 x^{4} a^{2} b^{5} d \,e^{6}+504504 x^{4} a \,b^{6} d^{2} e^{5}-48048 x^{4} b^{7} d^{3} e^{4}+7834365 x^{3} a^{4} b^{3} e^{7}-4178328 x^{3} a^{3} b^{4} d \,e^{6}+1474704 x^{3} a^{2} b^{5} d^{2} e^{5}-310464 x^{3} a \,b^{6} d^{3} e^{4}+29568 x^{3} b^{7} d^{4} e^{3}+5555277 x^{2} a^{5} b^{2} e^{7}-4273290 x^{2} a^{4} b^{3} d \,e^{6}+2279088 x^{2} a^{3} b^{4} d^{2} e^{5}-804384 x^{2} a^{2} b^{5} d^{3} e^{4}+169344 x^{2} a \,b^{6} d^{4} e^{3}-16128 x^{2} b^{7} d^{5} e^{2}+2263261 x \,a^{6} b \,e^{7}-2469012 x \,a^{5} b^{2} d \,e^{6}+1899240 x \,a^{4} b^{3} d^{2} e^{5}-1012928 x \,a^{3} b^{4} d^{3} e^{4}+357504 x \,a^{2} b^{5} d^{4} e^{3}-75264 x a \,b^{6} d^{5} e^{2}+7168 x \,b^{7} d^{6} e +415701 e^{7} a^{7}-646646 b d \,e^{6} a^{6}+705432 b^{2} d^{2} e^{5} a^{5}-542640 b^{3} d^{3} e^{4} a^{4}+289408 b^{4} d^{4} e^{3} a^{3}-102144 b^{5} d^{5} e^{2} a^{2}+21504 b^{6} d^{6} e a -2048 b^{7} d^{7}\right )}{2909907 e^{8}}\) | \(498\) |
| trager | \(\frac {2 \left (138567 e^{10} b^{7} x^{10}+1072071 a \,b^{6} e^{10} x^{9}+313599 b^{7} d \,e^{9} x^{9}+3594591 a^{2} b^{5} e^{10} x^{8}+2459457 a \,b^{6} d \,e^{9} x^{8}+181467 b^{7} d^{2} e^{8} x^{8}+6789783 a^{3} b^{4} e^{10} x^{7}+8387379 a^{2} b^{5} d \,e^{9} x^{7}+1450449 a \,b^{6} d^{2} e^{8} x^{7}+429 b^{7} d^{3} e^{7} x^{7}+7834365 a^{4} b^{3} e^{10} x^{6}+16191021 a^{3} b^{4} d \,e^{9} x^{6}+5069295 a^{2} b^{5} d^{2} e^{8} x^{6}+4851 a \,b^{6} d^{3} e^{7} x^{6}-462 b^{7} d^{4} e^{6} x^{6}+5555277 a^{5} b^{2} e^{10} x^{5}+19229805 a^{4} b^{3} d \,e^{9} x^{5}+10113453 a^{3} b^{4} d^{2} e^{8} x^{5}+25137 a^{2} b^{5} d^{3} e^{7} x^{5}-5292 a \,b^{6} d^{4} e^{6} x^{5}+504 b^{7} d^{5} e^{5} x^{5}+2263261 a^{6} b \,e^{10} x^{4}+14196819 a^{5} b^{2} d \,e^{9} x^{4}+12582465 a^{4} b^{3} d^{2} e^{8} x^{4}+79135 a^{3} b^{4} d^{3} e^{7} x^{4}-27930 a^{2} b^{5} d^{4} e^{6} x^{4}+5880 a \,b^{6} d^{5} e^{5} x^{4}-560 b^{7} d^{6} e^{4} x^{4}+415701 a^{7} e^{10} x^{3}+6143137 a^{6} b d \,e^{9} x^{3}+9964227 a^{5} b^{2} d^{2} e^{8} x^{3}+169575 a^{4} b^{3} d^{3} e^{7} x^{3}-90440 a^{3} b^{4} d^{4} e^{6} x^{3}+31920 a^{2} b^{5} d^{5} e^{5} x^{3}-6720 a \,b^{6} d^{6} e^{4} x^{3}+640 b^{7} d^{7} e^{3} x^{3}+1247103 a^{7} d \,e^{9} x^{2}+4849845 a^{6} b \,d^{2} e^{8} x^{2}+264537 a^{5} b^{2} d^{3} e^{7} x^{2}-203490 a^{4} b^{3} d^{4} e^{6} x^{2}+108528 a^{3} b^{4} d^{5} e^{5} x^{2}-38304 a^{2} b^{5} d^{6} e^{4} x^{2}+8064 a \,b^{6} d^{7} e^{3} x^{2}-768 b^{7} d^{8} e^{2} x^{2}+1247103 a^{7} d^{2} e^{8} x +323323 a^{6} b \,d^{3} e^{7} x -352716 a^{5} b^{2} d^{4} e^{6} x +271320 a^{4} b^{3} d^{5} e^{5} x -144704 a^{3} b^{4} d^{6} e^{4} x +51072 a^{2} b^{5} d^{7} e^{3} x -10752 a \,b^{6} d^{8} e^{2} x +1024 b^{7} d^{9} e x +415701 a^{7} d^{3} e^{7}-646646 a^{6} b \,d^{4} e^{6}+705432 a^{5} b^{2} d^{5} e^{5}-542640 a^{4} b^{3} d^{6} e^{4}+289408 a^{3} b^{4} d^{7} e^{3}-102144 a^{2} b^{5} d^{8} e^{2}+21504 a \,b^{6} d^{9} e -2048 b^{7} d^{10}\right ) \sqrt {e x +d}}{2909907 e^{8}}\) | \(872\) |
| risch | \(\frac {2 \left (138567 e^{10} b^{7} x^{10}+1072071 a \,b^{6} e^{10} x^{9}+313599 b^{7} d \,e^{9} x^{9}+3594591 a^{2} b^{5} e^{10} x^{8}+2459457 a \,b^{6} d \,e^{9} x^{8}+181467 b^{7} d^{2} e^{8} x^{8}+6789783 a^{3} b^{4} e^{10} x^{7}+8387379 a^{2} b^{5} d \,e^{9} x^{7}+1450449 a \,b^{6} d^{2} e^{8} x^{7}+429 b^{7} d^{3} e^{7} x^{7}+7834365 a^{4} b^{3} e^{10} x^{6}+16191021 a^{3} b^{4} d \,e^{9} x^{6}+5069295 a^{2} b^{5} d^{2} e^{8} x^{6}+4851 a \,b^{6} d^{3} e^{7} x^{6}-462 b^{7} d^{4} e^{6} x^{6}+5555277 a^{5} b^{2} e^{10} x^{5}+19229805 a^{4} b^{3} d \,e^{9} x^{5}+10113453 a^{3} b^{4} d^{2} e^{8} x^{5}+25137 a^{2} b^{5} d^{3} e^{7} x^{5}-5292 a \,b^{6} d^{4} e^{6} x^{5}+504 b^{7} d^{5} e^{5} x^{5}+2263261 a^{6} b \,e^{10} x^{4}+14196819 a^{5} b^{2} d \,e^{9} x^{4}+12582465 a^{4} b^{3} d^{2} e^{8} x^{4}+79135 a^{3} b^{4} d^{3} e^{7} x^{4}-27930 a^{2} b^{5} d^{4} e^{6} x^{4}+5880 a \,b^{6} d^{5} e^{5} x^{4}-560 b^{7} d^{6} e^{4} x^{4}+415701 a^{7} e^{10} x^{3}+6143137 a^{6} b d \,e^{9} x^{3}+9964227 a^{5} b^{2} d^{2} e^{8} x^{3}+169575 a^{4} b^{3} d^{3} e^{7} x^{3}-90440 a^{3} b^{4} d^{4} e^{6} x^{3}+31920 a^{2} b^{5} d^{5} e^{5} x^{3}-6720 a \,b^{6} d^{6} e^{4} x^{3}+640 b^{7} d^{7} e^{3} x^{3}+1247103 a^{7} d \,e^{9} x^{2}+4849845 a^{6} b \,d^{2} e^{8} x^{2}+264537 a^{5} b^{2} d^{3} e^{7} x^{2}-203490 a^{4} b^{3} d^{4} e^{6} x^{2}+108528 a^{3} b^{4} d^{5} e^{5} x^{2}-38304 a^{2} b^{5} d^{6} e^{4} x^{2}+8064 a \,b^{6} d^{7} e^{3} x^{2}-768 b^{7} d^{8} e^{2} x^{2}+1247103 a^{7} d^{2} e^{8} x +323323 a^{6} b \,d^{3} e^{7} x -352716 a^{5} b^{2} d^{4} e^{6} x +271320 a^{4} b^{3} d^{5} e^{5} x -144704 a^{3} b^{4} d^{6} e^{4} x +51072 a^{2} b^{5} d^{7} e^{3} x -10752 a \,b^{6} d^{8} e^{2} x +1024 b^{7} d^{9} e x +415701 a^{7} d^{3} e^{7}-646646 a^{6} b \,d^{4} e^{6}+705432 a^{5} b^{2} d^{5} e^{5}-542640 a^{4} b^{3} d^{6} e^{4}+289408 a^{3} b^{4} d^{7} e^{3}-102144 a^{2} b^{5} d^{8} e^{2}+21504 a \,b^{6} d^{9} e -2048 b^{7} d^{10}\right ) \sqrt {e x +d}}{2909907 e^{8}}\) | \(872\) |
| derivativedivides | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {21}{2}}}{21}+\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 {19}{2}}}{19}+\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 {17}{2}}}{17}+\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 {15}{2}}}{15}+\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 {13}{2}}}{13}+\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 {11}{2}}}{11}+\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 {9}{2}}}{9}+\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 {7}{2}}}{7}}{e^{8}}\) | \(936\) |
| default | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {21}{2}}}{21}+\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 {19}{2}}}{19}+\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 {17}{2}}}{17}+\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 {15}{2}}}{15}+\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 {13}{2}}}{13}+\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 {11}{2}}}{11}+\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 {9}{2}}}{9}+\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 {7}{2}}}{7}}{e^{8}}\) | \(936\) |
2/7*((1/3*b^7*x^7+a^7+49/19*a*b^6*x^6+147/17*a^2*b^5*x^5+49/3*a^3*b^4*x^4+ 245/13*a^4*b^3*x^3+147/11*a^5*b^2*x^2+49/9*a^6*b*x)*e^7-14/9*b*(3/19*b^6*x ^6+378/323*a*b^5*x^5+63/17*a^2*b^4*x^4+84/13*a^3*b^3*x^3+945/143*a^4*b^2*x ^2+42/11*a^5*b*x+a^6)*d*e^6+56/33*b^2*(33/323*b^5*x^5+231/323*a*b^4*x^4+46 2/221*a^2*b^3*x^3+42/13*a^3*b^2*x^2+35/13*a^4*b*x+a^5)*d^2*e^5-560/429*b^3 *(143/1615*x^4*b^4+924/1615*a*b^3*x^3+126/85*x^2*b^2*a^2+28/15*b*a^3*x+a^4 )*d^3*e^4+896/1287*(33/323*x^3*b^3+189/323*a*b^2*x^2+21/17*b*a^2*x+a^3)*b^ 4*d^4*e^3-1792/7293*(3/19*b^2*x^2+14/19*a*b*x+a^2)*b^5*d^5*e^2+7168/138567 *b^6*(1/3*b*x+a)*d^6*e-2048/415701*b^7*d^7)*(e*x+d)^(7/2)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 783 vs. \(2 (184) = 368\).
Time = 0.30 (sec) , antiderivative size = 783, normalized size of antiderivative = 3.62 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (138567 \, b^{7} e^{10} x^{10} - 2048 \, b^{7} d^{10} + 21504 \, a b^{6} d^{9} e - 102144 \, a^{2} b^{5} d^{8} e^{2} + 289408 \, a^{3} b^{4} d^{7} e^{3} - 542640 \, a^{4} b^{3} d^{6} e^{4} + 705432 \, a^{5} b^{2} d^{5} e^{5} - 646646 \, a^{6} b d^{4} e^{6} + 415701 \, a^{7} d^{3} e^{7} + 7293 \, {\left (43 \, b^{7} d e^{9} + 147 \, a b^{6} e^{10}\right )} x^{9} + 3861 \, {\left (47 \, b^{7} d^{2} e^{8} + 637 \, a b^{6} d e^{9} + 931 \, a^{2} b^{5} e^{10}\right )} x^{8} + 429 \, {\left (b^{7} d^{3} e^{7} + 3381 \, a b^{6} d^{2} e^{8} + 19551 \, a^{2} b^{5} d e^{9} + 15827 \, a^{3} b^{4} e^{10}\right )} x^{7} - 231 \, {\left (2 \, b^{7} d^{4} e^{6} - 21 \, a b^{6} d^{3} e^{7} - 21945 \, a^{2} b^{5} d^{2} e^{8} - 70091 \, a^{3} b^{4} d e^{9} - 33915 \, a^{4} b^{3} e^{10}\right )} x^{6} + 63 \, {\left (8 \, b^{7} d^{5} e^{5} - 84 \, a b^{6} d^{4} e^{6} + 399 \, a^{2} b^{5} d^{3} e^{7} + 160531 \, a^{3} b^{4} d^{2} e^{8} + 305235 \, a^{4} b^{3} d e^{9} + 88179 \, a^{5} b^{2} e^{10}\right )} x^{5} - 7 \, {\left (80 \, b^{7} d^{6} e^{4} - 840 \, a b^{6} d^{5} e^{5} + 3990 \, a^{2} b^{5} d^{4} e^{6} - 11305 \, a^{3} b^{4} d^{3} e^{7} - 1797495 \, a^{4} b^{3} d^{2} e^{8} - 2028117 \, a^{5} b^{2} d e^{9} - 323323 \, a^{6} b e^{10}\right )} x^{4} + {\left (640 \, b^{7} d^{7} e^{3} - 6720 \, a b^{6} d^{6} e^{4} + 31920 \, a^{2} b^{5} d^{5} e^{5} - 90440 \, a^{3} b^{4} d^{4} e^{6} + 169575 \, a^{4} b^{3} d^{3} e^{7} + 9964227 \, a^{5} b^{2} d^{2} e^{8} + 6143137 \, a^{6} b d e^{9} + 415701 \, a^{7} e^{10}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{8} e^{2} - 2688 \, a b^{6} d^{7} e^{3} + 12768 \, a^{2} b^{5} d^{6} e^{4} - 36176 \, a^{3} b^{4} d^{5} e^{5} + 67830 \, a^{4} b^{3} d^{4} e^{6} - 88179 \, a^{5} b^{2} d^{3} e^{7} - 1616615 \, a^{6} b d^{2} e^{8} - 415701 \, a^{7} d e^{9}\right )} x^{2} + {\left (1024 \, b^{7} d^{9} e - 10752 \, a b^{6} d^{8} e^{2} + 51072 \, a^{2} b^{5} d^{7} e^{3} - 144704 \, a^{3} b^{4} d^{6} e^{4} + 271320 \, a^{4} b^{3} d^{5} e^{5} - 352716 \, a^{5} b^{2} d^{4} e^{6} + 323323 \, a^{6} b d^{3} e^{7} + 1247103 \, a^{7} d^{2} e^{8}\right )} x\right )} \sqrt {e x + d}}{2909907 \, e^{8}} \]
2/2909907*(138567*b^7*e^10*x^10 - 2048*b^7*d^10 + 21504*a*b^6*d^9*e - 1021 44*a^2*b^5*d^8*e^2 + 289408*a^3*b^4*d^7*e^3 - 542640*a^4*b^3*d^6*e^4 + 705 432*a^5*b^2*d^5*e^5 - 646646*a^6*b*d^4*e^6 + 415701*a^7*d^3*e^7 + 7293*(43 *b^7*d*e^9 + 147*a*b^6*e^10)*x^9 + 3861*(47*b^7*d^2*e^8 + 637*a*b^6*d*e^9 + 931*a^2*b^5*e^10)*x^8 + 429*(b^7*d^3*e^7 + 3381*a*b^6*d^2*e^8 + 19551*a^ 2*b^5*d*e^9 + 15827*a^3*b^4*e^10)*x^7 - 231*(2*b^7*d^4*e^6 - 21*a*b^6*d^3* e^7 - 21945*a^2*b^5*d^2*e^8 - 70091*a^3*b^4*d*e^9 - 33915*a^4*b^3*e^10)*x^ 6 + 63*(8*b^7*d^5*e^5 - 84*a*b^6*d^4*e^6 + 399*a^2*b^5*d^3*e^7 + 160531*a^ 3*b^4*d^2*e^8 + 305235*a^4*b^3*d*e^9 + 88179*a^5*b^2*e^10)*x^5 - 7*(80*b^7 *d^6*e^4 - 840*a*b^6*d^5*e^5 + 3990*a^2*b^5*d^4*e^6 - 11305*a^3*b^4*d^3*e^ 7 - 1797495*a^4*b^3*d^2*e^8 - 2028117*a^5*b^2*d*e^9 - 323323*a^6*b*e^10)*x ^4 + (640*b^7*d^7*e^3 - 6720*a*b^6*d^6*e^4 + 31920*a^2*b^5*d^5*e^5 - 90440 *a^3*b^4*d^4*e^6 + 169575*a^4*b^3*d^3*e^7 + 9964227*a^5*b^2*d^2*e^8 + 6143 137*a^6*b*d*e^9 + 415701*a^7*e^10)*x^3 - 3*(256*b^7*d^8*e^2 - 2688*a*b^6*d ^7*e^3 + 12768*a^2*b^5*d^6*e^4 - 36176*a^3*b^4*d^5*e^5 + 67830*a^4*b^3*d^4 *e^6 - 88179*a^5*b^2*d^3*e^7 - 1616615*a^6*b*d^2*e^8 - 415701*a^7*d*e^9)*x ^2 + (1024*b^7*d^9*e - 10752*a*b^6*d^8*e^2 + 51072*a^2*b^5*d^7*e^3 - 14470 4*a^3*b^4*d^6*e^4 + 271320*a^4*b^3*d^5*e^5 - 352716*a^5*b^2*d^4*e^6 + 3233 23*a^6*b*d^3*e^7 + 1247103*a^7*d^2*e^8)*x)*sqrt(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (201) = 402\).
Time = 1.99 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.67 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {b^{7} \left (d + e x\right )^{\frac {21}{2}}}{21 e^{7}} + \frac {\left (d + e x\right )^{\frac {19}{2}} \cdot \left (7 a b^{6} e - 7 b^{7} d\right )}{19 e^{7}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (21 a^{5} b^{2} e^{5} - 105 a^{4} b^{3} d e^{4} + 210 a^{3} b^{4} d^{2} e^{3} - 210 a^{2} b^{5} d^{3} e^{2} + 105 a b^{6} d^{4} e - 21 b^{7} d^{5}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (7 a^{6} b e^{6} - 42 a^{5} b^{2} d e^{5} + 105 a^{4} b^{3} d^{2} e^{4} - 140 a^{3} b^{4} d^{3} e^{3} + 105 a^{2} b^{5} d^{4} e^{2} - 42 a b^{6} d^{5} e + 7 b^{7} d^{6}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{7} e^{7} - 7 a^{6} b d e^{6} + 21 a^{5} b^{2} d^{2} e^{5} - 35 a^{4} b^{3} d^{3} e^{4} + 35 a^{3} b^{4} d^{4} e^{3} - 21 a^{2} b^{5} d^{5} e^{2} + 7 a b^{6} d^{6} e - b^{7} d^{7}\right )}{7 e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (\begin {cases} a^{7} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{4}}{8 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**7*(d + e*x)**(21/2)/(21*e**7) + (d + e*x)**(19/2)*(7*a*b* *6*e - 7*b**7*d)/(19*e**7) + (d + e*x)**(17/2)*(21*a**2*b**5*e**2 - 42*a*b **6*d*e + 21*b**7*d**2)/(17*e**7) + (d + e*x)**(15/2)*(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)/(15*e**7) + (d + e*x)**(13/2)*(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)/(13*e**7) + (d + e*x)**(11/2)* (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)/(11*e**7) + (d + e *x)**(9/2)*(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)/(9*e**7) + (d + e*x)**(7/2)*(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)/(7*e**7))/e, Ne(e, 0)) , (d**(5/2)*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 = 0.19 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.11 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (138567 \, {\left (e x + d\right )}^{\frac {21}{2}} b^{7} - 1072071 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {19}{2}} + 3594591 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {17}{2}} - 6789783 \, {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 7834365 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 5555277 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 2263261 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 415701 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{2909907 \, e^{8}} \]
2/2909907*(138567*(e*x + d)^(21/2)*b^7 - 1072071*(b^7*d - a*b^6*e)*(e*x + d)^(19/2) + 3594591*(b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*(e*x + d)^(17/2) - 6789783*(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)^(15/2) + 7834365*(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)^(13/2) - 5555277*(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)^(11/2) + 2263261*(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)^(9/2) - 415701*(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)^(7/2))/e^8
Leaf count of result is larger than twice the leaf count of optimal. 2533 vs. \(2 (184) = 368\).
Time = 0.30 (sec) , antiderivative size = 2533, normalized size of antiderivative = 11.73 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
2/14549535*(14549535*sqrt(e*x + d)*a^7*d^3 + 14549535*((e*x + d)^(3/2) - 3 *sqrt(e*x + d)*d)*a^7*d^2 + 33948915*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d) *a^6*b*d^3/e + 2909907*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt (e*x + d)*d^2)*a^7*d + 20369349*(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^3/e^2 + 20369349*(3*(e*x + d)^(5/2) - 10 *(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^6*b*d^2/e + 415701*(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 + 14549535*(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^3/e^3 + 26189163*(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^2/e^2 + 8729721*(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/e + 1616615*(3 5*(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^3/e^4 + 4849845*(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^2/e^3 + 2909907*(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/e^2 + 323323*(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^6*b/e + 440895*(63*(e*x + d)^(...
Time = 0.06 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2\,b^7\,{\left (d+e\,x\right )}^{21/2}}{21\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{19/2}}{19\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^7\,{\left (d+e\,x\right )}^{7/2}}{7\,e^8}+\frac {42\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {70\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {14\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{15/2}}{3\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8} \]
(2*b^7*(d + e*x)^(21/2))/(21*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(19 /2))/(19*e^8) + (2*(a*e - b*d)^7*(d + e*x)^(7/2))/(7*e^8) + (42*b^2*(a*e - b*d)^5*(d + e*x)^(11/2))/(11*e^8) + (70*b^3*(a*e - b*d)^4*(d + e*x)^(13/2 ))/(13*e^8) + (14*b^4*(a*e - b*d)^3*(d + e*x)^(15/2))/(3*e^8) + (42*b^5*(a *e - b*d)^2*(d + e*x)^(17/2))/(17*e^8) + (14*b*(a*e - b*d)^6*(d + e*x)^(9/ 2))/(9*e^8)