Integrand size = 28, antiderivative size = 185 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (b d-a e)^6 (d+e x)^{7/2}}{7 e^7}-\frac {4 b (b d-a e)^5 (d+e x)^{9/2}}{3 e^7}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{11/2}}{11 e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{13/2}}{13 e^7}+\frac {2 b^4 (b d-a e)^2 (d+e x)^{15/2}}{e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{17/2}}{17 e^7}+\frac {2 b^6 (d+e x)^{19/2}}{19 e^7} \]
2/7*(-a*e+b*d)^6*(e*x+d)^(7/2)/e^7-4/3*b*(-a*e+b*d)^5*(e*x+d)^(9/2)/e^7+30 /11*b^2*(-a*e+b*d)^4*(e*x+d)^(11/2)/e^7-40/13*b^3*(-a*e+b*d)^3*(e*x+d)^(13 /2)/e^7+2*b^4*(-a*e+b*d)^2*(e*x+d)^(15/2)/e^7-12/17*b^5*(-a*e+b*d)*(e*x+d) ^(17/2)/e^7+2/19*b^6*(e*x+d)^(19/2)/e^7
Time = 0.17 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.57 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (d+e x)^{7/2} \left (138567 a^6 e^6+92378 a^5 b e^5 (-2 d+7 e x)+20995 a^4 b^2 e^4 \left (8 d^2-28 d e x+63 e^2 x^2\right )+6460 a^3 b^3 e^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+323 a^2 b^4 e^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+38 a b^5 e \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+b^6 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )}{969969 e^7} \]
(2*(d + e*x)^(7/2)*(138567*a^6*e^6 + 92378*a^5*b*e^5*(-2*d + 7*e*x) + 2099 5*a^4*b^2*e^4*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 6460*a^3*b^3*e^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + 323*a^2*b^4*e^2*(128*d^4 - 4 48*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 38*a*b^5* e*(-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) + b^6*(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)) )/(969969*e^7)
Time = 0.32 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1098, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int b^6 (a+b x)^6 (d+e x)^{5/2}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^6 (d+e x)^{5/2}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {6 b^5 (d+e x)^{15/2} (b d-a e)}{e^6}+\frac {15 b^4 (d+e x)^{13/2} (b d-a e)^2}{e^6}-\frac {20 b^3 (d+e x)^{11/2} (b d-a e)^3}{e^6}+\frac {15 b^2 (d+e x)^{9/2} (b d-a e)^4}{e^6}-\frac {6 b (d+e x)^{7/2} (b d-a e)^5}{e^6}+\frac {(d+e x)^{5/2} (a e-b d)^6}{e^6}+\frac {b^6 (d+e x)^{17/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {12 b^5 (d+e x)^{17/2} (b d-a e)}{17 e^7}+\frac {2 b^4 (d+e x)^{15/2} (b d-a e)^2}{e^7}-\frac {40 b^3 (d+e x)^{13/2} (b d-a e)^3}{13 e^7}+\frac {30 b^2 (d+e x)^{11/2} (b d-a e)^4}{11 e^7}-\frac {4 b (d+e x)^{9/2} (b d-a e)^5}{3 e^7}+\frac {2 (d+e x)^{7/2} (b d-a e)^6}{7 e^7}+\frac {2 b^6 (d+e x)^{19/2}}{19 e^7}\) |
(2*(b*d - a*e)^6*(d + e*x)^(7/2))/(7*e^7) - (4*b*(b*d - a*e)^5*(d + e*x)^( 9/2))/(3*e^7) + (30*b^2*(b*d - a*e)^4*(d + e*x)^(11/2))/(11*e^7) - (40*b^3 *(b*d - a*e)^3*(d + e*x)^(13/2))/(13*e^7) + (2*b^4*(b*d - a*e)^2*(d + e*x) ^(15/2))/e^7 - (12*b^5*(b*d - a*e)*(d + e*x)^(17/2))/(17*e^7) + (2*b^6*(d + e*x)^(19/2))/(19*e^7)
3.17.38.3.1 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_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 2.44 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.49
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (\left (\frac {7}{19} b^{6} x^{6}+a^{6}+\frac {42}{17} a \,x^{5} b^{5}+7 a^{2} x^{4} b^{4}+\frac {140}{13} a^{3} x^{3} b^{3}+\frac {105}{11} a^{4} x^{2} b^{2}+\frac {14}{3} a^{5} x b \right ) e^{6}-\frac {4 b \left (\frac {63}{323} b^{5} x^{5}+\frac {21}{17} a \,b^{4} x^{4}+\frac {42}{13} a^{2} b^{3} x^{3}+\frac {630}{143} a^{3} b^{2} x^{2}+\frac {35}{11} a^{4} b x +a^{5}\right ) d \,e^{5}}{3}+\frac {40 b^{2} d^{2} \left (\frac {231}{1615} b^{4} x^{4}+\frac {924}{1105} a \,b^{3} x^{3}+\frac {126}{65} a^{2} b^{2} x^{2}+\frac {28}{13} a^{3} b x +a^{4}\right ) e^{4}}{33}-\frac {320 b^{3} \left (\frac {231}{1615} b^{3} x^{3}+\frac {63}{85} a \,b^{2} x^{2}+\frac {7}{5} a^{2} b x +a^{3}\right ) d^{3} e^{3}}{429}+\frac {128 b^{4} \left (\frac {63}{323} b^{2} x^{2}+\frac {14}{17} a b x +a^{2}\right ) d^{4} e^{2}}{429}-\frac {512 b^{5} \left (\frac {7 b x}{19}+a \right ) d^{5} e}{7293}+\frac {1024 b^{6} d^{6}}{138567}\right )}{7 e^{7}}\) | \(276\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (51051 x^{6} b^{6} e^{6}+342342 x^{5} a \,b^{5} e^{6}-36036 x^{5} b^{6} d \,e^{5}+969969 x^{4} a^{2} b^{4} e^{6}-228228 x^{4} a \,b^{5} d \,e^{5}+24024 x^{4} b^{6} d^{2} e^{4}+1492260 x^{3} a^{3} b^{3} e^{6}-596904 x^{3} a^{2} b^{4} d \,e^{5}+140448 x^{3} a \,b^{5} d^{2} e^{4}-14784 x^{3} b^{6} d^{3} e^{3}+1322685 x^{2} a^{4} b^{2} e^{6}-813960 x^{2} a^{3} b^{3} d \,e^{5}+325584 x^{2} a^{2} b^{4} d^{2} e^{4}-76608 x^{2} a \,b^{5} d^{3} e^{3}+8064 x^{2} b^{6} d^{4} e^{2}+646646 x \,a^{5} b \,e^{6}-587860 x \,a^{4} b^{2} d \,e^{5}+361760 x \,a^{3} b^{3} d^{2} e^{4}-144704 x \,a^{2} b^{4} d^{3} e^{3}+34048 x a \,b^{5} d^{4} e^{2}-3584 x \,b^{6} d^{5} e +138567 a^{6} e^{6}-184756 a^{5} b d \,e^{5}+167960 a^{4} b^{2} d^{2} e^{4}-103360 a^{3} b^{3} d^{3} e^{3}+41344 a^{2} b^{4} d^{4} e^{2}-9728 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{969969 e^{7}}\) | \(377\) |
| derivativedivides | \(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {6 \left (2 a e b -2 b^{2} d \right ) b^{4} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a e b -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) b^{2}+\left (2 a e b -2 b^{2} d \right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )+2 \left (2 a e b -2 b^{2} d \right )^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{7}}\) | \(457\) |
| default | \(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {6 \left (2 a e b -2 b^{2} d \right ) b^{4} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a e b -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) b^{2}+\left (2 a e b -2 b^{2} d \right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )+2 \left (2 a e b -2 b^{2} d \right )^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{7}}\) | \(457\) |
| trager | \(\frac {2 \left (51051 b^{6} e^{9} x^{9}+342342 a \,b^{5} e^{9} x^{8}+117117 b^{6} d \,e^{8} x^{8}+969969 a^{2} b^{4} e^{9} x^{7}+798798 a \,b^{5} d \,e^{8} x^{7}+69069 b^{6} d^{2} e^{7} x^{7}+1492260 a^{3} b^{3} e^{9} x^{6}+2313003 a^{2} b^{4} d \,e^{8} x^{6}+482790 a \,b^{5} d^{2} e^{7} x^{6}+231 b^{6} d^{3} e^{6} x^{6}+1322685 a^{4} b^{2} e^{9} x^{5}+3662820 a^{3} b^{3} d \,e^{8} x^{5}+1444779 a^{2} b^{4} d^{2} e^{7} x^{5}+2394 a \,b^{5} d^{3} e^{6} x^{5}-252 b^{6} d^{4} e^{5} x^{5}+646646 a^{5} b \,e^{9} x^{4}+3380195 a^{4} b^{2} d \,e^{8} x^{4}+2396660 a^{3} b^{3} d^{2} e^{7} x^{4}+11305 a^{2} b^{4} d^{3} e^{6} x^{4}-2660 a \,b^{5} d^{4} e^{5} x^{4}+280 b^{6} d^{5} e^{4} x^{4}+138567 a^{6} e^{9} x^{3}+1755182 a^{5} b d \,e^{8} x^{3}+2372435 a^{4} b^{2} d^{2} e^{7} x^{3}+32300 a^{3} b^{3} d^{3} e^{6} x^{3}-12920 a^{2} b^{4} d^{4} e^{5} x^{3}+3040 a \,b^{5} d^{5} e^{4} x^{3}-320 b^{6} d^{6} e^{3} x^{3}+415701 a^{6} d \,e^{8} x^{2}+1385670 a^{5} b \,d^{2} e^{7} x^{2}+62985 a^{4} b^{2} d^{3} e^{6} x^{2}-38760 a^{3} b^{3} d^{4} e^{5} x^{2}+15504 a^{2} b^{4} d^{5} e^{4} x^{2}-3648 a \,b^{5} d^{6} e^{3} x^{2}+384 b^{6} d^{7} e^{2} x^{2}+415701 a^{6} d^{2} e^{7} x +92378 a^{5} b \,d^{3} e^{6} x -83980 a^{4} b^{2} d^{4} e^{5} x +51680 a^{3} b^{3} d^{5} e^{4} x -20672 a^{2} b^{4} d^{6} e^{3} x +4864 a \,b^{5} d^{7} e^{2} x -512 b^{6} d^{8} e x +138567 a^{6} d^{3} e^{6}-184756 a^{5} b \,d^{4} e^{5}+167960 a^{4} b^{2} d^{5} e^{4}-103360 a^{3} b^{3} d^{6} e^{3}+41344 a^{2} b^{4} d^{7} e^{2}-9728 a \,b^{5} d^{8} e +1024 b^{6} d^{9}\right ) \sqrt {e x +d}}{969969 e^{7}}\) | \(700\) |
| risch | \(\frac {2 \left (51051 b^{6} e^{9} x^{9}+342342 a \,b^{5} e^{9} x^{8}+117117 b^{6} d \,e^{8} x^{8}+969969 a^{2} b^{4} e^{9} x^{7}+798798 a \,b^{5} d \,e^{8} x^{7}+69069 b^{6} d^{2} e^{7} x^{7}+1492260 a^{3} b^{3} e^{9} x^{6}+2313003 a^{2} b^{4} d \,e^{8} x^{6}+482790 a \,b^{5} d^{2} e^{7} x^{6}+231 b^{6} d^{3} e^{6} x^{6}+1322685 a^{4} b^{2} e^{9} x^{5}+3662820 a^{3} b^{3} d \,e^{8} x^{5}+1444779 a^{2} b^{4} d^{2} e^{7} x^{5}+2394 a \,b^{5} d^{3} e^{6} x^{5}-252 b^{6} d^{4} e^{5} x^{5}+646646 a^{5} b \,e^{9} x^{4}+3380195 a^{4} b^{2} d \,e^{8} x^{4}+2396660 a^{3} b^{3} d^{2} e^{7} x^{4}+11305 a^{2} b^{4} d^{3} e^{6} x^{4}-2660 a \,b^{5} d^{4} e^{5} x^{4}+280 b^{6} d^{5} e^{4} x^{4}+138567 a^{6} e^{9} x^{3}+1755182 a^{5} b d \,e^{8} x^{3}+2372435 a^{4} b^{2} d^{2} e^{7} x^{3}+32300 a^{3} b^{3} d^{3} e^{6} x^{3}-12920 a^{2} b^{4} d^{4} e^{5} x^{3}+3040 a \,b^{5} d^{5} e^{4} x^{3}-320 b^{6} d^{6} e^{3} x^{3}+415701 a^{6} d \,e^{8} x^{2}+1385670 a^{5} b \,d^{2} e^{7} x^{2}+62985 a^{4} b^{2} d^{3} e^{6} x^{2}-38760 a^{3} b^{3} d^{4} e^{5} x^{2}+15504 a^{2} b^{4} d^{5} e^{4} x^{2}-3648 a \,b^{5} d^{6} e^{3} x^{2}+384 b^{6} d^{7} e^{2} x^{2}+415701 a^{6} d^{2} e^{7} x +92378 a^{5} b \,d^{3} e^{6} x -83980 a^{4} b^{2} d^{4} e^{5} x +51680 a^{3} b^{3} d^{5} e^{4} x -20672 a^{2} b^{4} d^{6} e^{3} x +4864 a \,b^{5} d^{7} e^{2} x -512 b^{6} d^{8} e x +138567 a^{6} d^{3} e^{6}-184756 a^{5} b \,d^{4} e^{5}+167960 a^{4} b^{2} d^{5} e^{4}-103360 a^{3} b^{3} d^{6} e^{3}+41344 a^{2} b^{4} d^{7} e^{2}-9728 a \,b^{5} d^{8} e +1024 b^{6} d^{9}\right ) \sqrt {e x +d}}{969969 e^{7}}\) | \(700\) |
2/7*(e*x+d)^(7/2)*((7/19*b^6*x^6+a^6+42/17*a*x^5*b^5+7*a^2*x^4*b^4+140/13* a^3*x^3*b^3+105/11*a^4*x^2*b^2+14/3*a^5*x*b)*e^6-4/3*b*(63/323*b^5*x^5+21/ 17*a*b^4*x^4+42/13*a^2*b^3*x^3+630/143*a^3*b^2*x^2+35/11*a^4*b*x+a^5)*d*e^ 5+40/33*b^2*d^2*(231/1615*b^4*x^4+924/1105*a*b^3*x^3+126/65*a^2*b^2*x^2+28 /13*a^3*b*x+a^4)*e^4-320/429*b^3*(231/1615*b^3*x^3+63/85*a*b^2*x^2+7/5*a^2 *b*x+a^3)*d^3*e^3+128/429*b^4*(63/323*b^2*x^2+14/17*a*b*x+a^2)*d^4*e^2-512 /7293*b^5*(7/19*b*x+a)*d^5*e+1024/138567*b^6*d^6)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (159) = 318\).
Time = 0.38 (sec) , antiderivative size = 635, normalized size of antiderivative = 3.43 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (51051 \, b^{6} e^{9} x^{9} + 1024 \, b^{6} d^{9} - 9728 \, a b^{5} d^{8} e + 41344 \, a^{2} b^{4} d^{7} e^{2} - 103360 \, a^{3} b^{3} d^{6} e^{3} + 167960 \, a^{4} b^{2} d^{5} e^{4} - 184756 \, a^{5} b d^{4} e^{5} + 138567 \, a^{6} d^{3} e^{6} + 9009 \, {\left (13 \, b^{6} d e^{8} + 38 \, a b^{5} e^{9}\right )} x^{8} + 3003 \, {\left (23 \, b^{6} d^{2} e^{7} + 266 \, a b^{5} d e^{8} + 323 \, a^{2} b^{4} e^{9}\right )} x^{7} + 231 \, {\left (b^{6} d^{3} e^{6} + 2090 \, a b^{5} d^{2} e^{7} + 10013 \, a^{2} b^{4} d e^{8} + 6460 \, a^{3} b^{3} e^{9}\right )} x^{6} - 63 \, {\left (4 \, b^{6} d^{4} e^{5} - 38 \, a b^{5} d^{3} e^{6} - 22933 \, a^{2} b^{4} d^{2} e^{7} - 58140 \, a^{3} b^{3} d e^{8} - 20995 \, a^{4} b^{2} e^{9}\right )} x^{5} + 7 \, {\left (40 \, b^{6} d^{5} e^{4} - 380 \, a b^{5} d^{4} e^{5} + 1615 \, a^{2} b^{4} d^{3} e^{6} + 342380 \, a^{3} b^{3} d^{2} e^{7} + 482885 \, a^{4} b^{2} d e^{8} + 92378 \, a^{5} b e^{9}\right )} x^{4} - {\left (320 \, b^{6} d^{6} e^{3} - 3040 \, a b^{5} d^{5} e^{4} + 12920 \, a^{2} b^{4} d^{4} e^{5} - 32300 \, a^{3} b^{3} d^{3} e^{6} - 2372435 \, a^{4} b^{2} d^{2} e^{7} - 1755182 \, a^{5} b d e^{8} - 138567 \, a^{6} e^{9}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{7} e^{2} - 1216 \, a b^{5} d^{6} e^{3} + 5168 \, a^{2} b^{4} d^{5} e^{4} - 12920 \, a^{3} b^{3} d^{4} e^{5} + 20995 \, a^{4} b^{2} d^{3} e^{6} + 461890 \, a^{5} b d^{2} e^{7} + 138567 \, a^{6} d e^{8}\right )} x^{2} - {\left (512 \, b^{6} d^{8} e - 4864 \, a b^{5} d^{7} e^{2} + 20672 \, a^{2} b^{4} d^{6} e^{3} - 51680 \, a^{3} b^{3} d^{5} e^{4} + 83980 \, a^{4} b^{2} d^{4} e^{5} - 92378 \, a^{5} b d^{3} e^{6} - 415701 \, a^{6} d^{2} e^{7}\right )} x\right )} \sqrt {e x + d}}{969969 \, e^{7}} \]
2/969969*(51051*b^6*e^9*x^9 + 1024*b^6*d^9 - 9728*a*b^5*d^8*e + 41344*a^2* b^4*d^7*e^2 - 103360*a^3*b^3*d^6*e^3 + 167960*a^4*b^2*d^5*e^4 - 184756*a^5 *b*d^4*e^5 + 138567*a^6*d^3*e^6 + 9009*(13*b^6*d*e^8 + 38*a*b^5*e^9)*x^8 + 3003*(23*b^6*d^2*e^7 + 266*a*b^5*d*e^8 + 323*a^2*b^4*e^9)*x^7 + 231*(b^6* d^3*e^6 + 2090*a*b^5*d^2*e^7 + 10013*a^2*b^4*d*e^8 + 6460*a^3*b^3*e^9)*x^6 - 63*(4*b^6*d^4*e^5 - 38*a*b^5*d^3*e^6 - 22933*a^2*b^4*d^2*e^7 - 58140*a^ 3*b^3*d*e^8 - 20995*a^4*b^2*e^9)*x^5 + 7*(40*b^6*d^5*e^4 - 380*a*b^5*d^4*e ^5 + 1615*a^2*b^4*d^3*e^6 + 342380*a^3*b^3*d^2*e^7 + 482885*a^4*b^2*d*e^8 + 92378*a^5*b*e^9)*x^4 - (320*b^6*d^6*e^3 - 3040*a*b^5*d^5*e^4 + 12920*a^2 *b^4*d^4*e^5 - 32300*a^3*b^3*d^3*e^6 - 2372435*a^4*b^2*d^2*e^7 - 1755182*a ^5*b*d*e^8 - 138567*a^6*e^9)*x^3 + 3*(128*b^6*d^7*e^2 - 1216*a*b^5*d^6*e^3 + 5168*a^2*b^4*d^5*e^4 - 12920*a^3*b^3*d^4*e^5 + 20995*a^4*b^2*d^3*e^6 + 461890*a^5*b*d^2*e^7 + 138567*a^6*d*e^8)*x^2 - (512*b^6*d^8*e - 4864*a*b^5 *d^7*e^2 + 20672*a^2*b^4*d^6*e^3 - 51680*a^3*b^3*d^5*e^4 + 83980*a^4*b^2*d ^4*e^5 - 92378*a^5*b*d^3*e^6 - 415701*a^6*d^2*e^7)*x)*sqrt(e*x + d)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (172) = 344\).
Time = 1.58 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.68 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {b^{6} \left (d + e x\right )^{\frac {19}{2}}}{19 e^{6}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (6 a b^{5} e - 6 b^{6} d\right )}{17 e^{6}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (15 a^{2} b^{4} e^{2} - 30 a b^{5} d e + 15 b^{6} d^{2}\right )}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (20 a^{3} b^{3} e^{3} - 60 a^{2} b^{4} d e^{2} + 60 a b^{5} d^{2} e - 20 b^{6} d^{3}\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (15 a^{4} b^{2} e^{4} - 60 a^{3} b^{3} d e^{3} + 90 a^{2} b^{4} d^{2} e^{2} - 60 a b^{5} d^{3} e + 15 b^{6} d^{4}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (6 a^{5} b e^{5} - 30 a^{4} b^{2} d e^{4} + 60 a^{3} b^{3} d^{2} e^{3} - 60 a^{2} b^{4} d^{3} e^{2} + 30 a b^{5} d^{4} e - 6 b^{6} d^{5}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{6} e^{6} - 6 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} - 6 a b^{5} d^{5} e + b^{6} d^{6}\right )}{7 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac {b^{6} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**6*(d + e*x)**(19/2)/(19*e**6) + (d + e*x)**(17/2)*(6*a*b* *5*e - 6*b**6*d)/(17*e**6) + (d + e*x)**(15/2)*(15*a**2*b**4*e**2 - 30*a*b **5*d*e + 15*b**6*d**2)/(15*e**6) + (d + e*x)**(13/2)*(20*a**3*b**3*e**3 - 60*a**2*b**4*d*e**2 + 60*a*b**5*d**2*e - 20*b**6*d**3)/(13*e**6) + (d + e *x)**(11/2)*(15*a**4*b**2*e**4 - 60*a**3*b**3*d*e**3 + 90*a**2*b**4*d**2*e **2 - 60*a*b**5*d**3*e + 15*b**6*d**4)/(11*e**6) + (d + e*x)**(9/2)*(6*a** 5*b*e**5 - 30*a**4*b**2*d*e**4 + 60*a**3*b**3*d**2*e**3 - 60*a**2*b**4*d** 3*e**2 + 30*a*b**5*d**4*e - 6*b**6*d**5)/(9*e**6) + (d + e*x)**(7/2)*(a**6 *e**6 - 6*a**5*b*d*e**5 + 15*a**4*b**2*d**2*e**4 - 20*a**3*b**3*d**3*e**3 + 15*a**2*b**4*d**4*e**2 - 6*a*b**5*d**5*e + b**6*d**6)/(7*e**6))/e, Ne(e, 0)), (d**(5/2)*(a**6*x + 3*a**5*b*x**2 + 5*a**4*b**2*x**3 + 5*a**3*b**3*x **4 + 3*a**2*b**4*x**5 + a*b**5*x**6 + b**6*x**7/7), True))
Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (159) = 318\).
Time = 0.22 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.89 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (51051 \, {\left (e x + d\right )}^{\frac {19}{2}} b^{6} - 342342 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 969969 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 1492260 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 1322685 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 646646 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 138567 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{969969 \, e^{7}} \]
2/969969*(51051*(e*x + d)^(19/2)*b^6 - 342342*(b^6*d - a*b^5*e)*(e*x + d)^ (17/2) + 969969*(b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2)*(e*x + d)^(15/2) - 1 492260*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*(e*x + d) ^(13/2) + 1322685*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3 *d*e^3 + a^4*b^2*e^4)*(e*x + d)^(11/2) - 646646*(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*(e *x + d)^(9/2) + 138567*(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20* a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*(e*x + d)^ (7/2))/e^7
Leaf count of result is larger than twice the leaf count of optimal. 2042 vs. \(2 (159) = 318\).
Time = 0.31 (sec) , antiderivative size = 2042, normalized size of antiderivative = 11.04 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
2/4849845*(4849845*sqrt(e*x + d)*a^6*d^3 + 4849845*((e*x + d)^(3/2) - 3*sq rt(e*x + d)*d)*a^6*d^2 + 9699690*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^5 *b*d^3/e + 969969*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^6*d + 4849845*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*s qrt(e*x + d)*d^2)*a^4*b^2*d^3/e^2 + 5819814*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^5*b*d^2/e + 138567*(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 + 2771340*(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^3*b^3*d^3/e^3 + 6235515*(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^2*d^2/e^2 + 2494206*(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*d/e + 230945*(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^2*b^4*d^3/e^4 + 923780*(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^3*d^2/e^3 + 692835*(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^2*d/e^2 + 92378*(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/e + 41990*(63*(e*x + d)^(11/2) - 385*(e...
Time = 0.06 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.88 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2\,b^6\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {30\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {2\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{15/2}}{e^7}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{9/2}}{3\,e^7} \]
(2*b^6*(d + e*x)^(19/2))/(19*e^7) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(17 /2))/(17*e^7) + (2*(a*e - b*d)^6*(d + e*x)^(7/2))/(7*e^7) + (30*b^2*(a*e - b*d)^4*(d + e*x)^(11/2))/(11*e^7) + (40*b^3*(a*e - b*d)^3*(d + e*x)^(13/2 ))/(13*e^7) + (2*b^4*(a*e - b*d)^2*(d + e*x)^(15/2))/e^7 + (4*b*(a*e - b*d )^5*(d + e*x)^(9/2))/(3*e^7)