Integrand size = 28, antiderivative size = 181 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 (b d-a e)^6 \sqrt {d+e x}}{e^7}-\frac {4 b (b d-a e)^5 (d+e x)^{3/2}}{e^7}+\frac {6 b^2 (b d-a e)^4 (d+e x)^{5/2}}{e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{7/2}}{7 e^7}+\frac {10 b^4 (b d-a e)^2 (d+e x)^{9/2}}{3 e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{11/2}}{11 e^7}+\frac {2 b^6 (d+e x)^{13/2}}{13 e^7} \]
-4*b*(-a*e+b*d)^5*(e*x+d)^(3/2)/e^7+6*b^2*(-a*e+b*d)^4*(e*x+d)^(5/2)/e^7-4 0/7*b^3*(-a*e+b*d)^3*(e*x+d)^(7/2)/e^7+10/3*b^4*(-a*e+b*d)^2*(e*x+d)^(9/2) /e^7-12/11*b^5*(-a*e+b*d)*(e*x+d)^(11/2)/e^7+2/13*b^6*(e*x+d)^(13/2)/e^7+2 *(-a*e+b*d)^6*(e*x+d)^(1/2)/e^7
Time = 0.13 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (3003 a^6 e^6+6006 a^5 b e^5 (-2 d+e x)+3003 a^4 b^2 e^4 \left (8 d^2-4 d e x+3 e^2 x^2\right )+1716 a^3 b^3 e^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+143 a^2 b^4 e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+26 a b^5 e \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+b^6 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )}{3003 e^7} \]
(2*Sqrt[d + e*x]*(3003*a^6*e^6 + 6006*a^5*b*e^5*(-2*d + e*x) + 3003*a^4*b^ 2*e^4*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 1716*a^3*b^3*e^3*(-16*d^3 + 8*d^2*e* x - 6*d*e^2*x^2 + 5*e^3*x^3) + 143*a^2*b^4*e^2*(128*d^4 - 64*d^3*e*x + 48* d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + 26*a*b^5*e*(-256*d^5 + 128*d^4* e*x - 96*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5) + b^6*( 1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2 - 320*d^3*e^3*x^3 + 280*d^2*e^4*x ^4 - 252*d*e^5*x^5 + 231*e^6*x^6)))/(3003*e^7)
Time = 0.31 (sec) , antiderivative size = 181, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^6}{\sqrt {d+e x}}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^6}{\sqrt {d+e x}}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {6 b^5 (d+e x)^{9/2} (b d-a e)}{e^6}+\frac {15 b^4 (d+e x)^{7/2} (b d-a e)^2}{e^6}-\frac {20 b^3 (d+e x)^{5/2} (b d-a e)^3}{e^6}+\frac {15 b^2 (d+e x)^{3/2} (b d-a e)^4}{e^6}-\frac {6 b \sqrt {d+e x} (b d-a e)^5}{e^6}+\frac {(a e-b d)^6}{e^6 \sqrt {d+e x}}+\frac {b^6 (d+e x)^{11/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {12 b^5 (d+e x)^{11/2} (b d-a e)}{11 e^7}+\frac {10 b^4 (d+e x)^{9/2} (b d-a e)^2}{3 e^7}-\frac {40 b^3 (d+e x)^{7/2} (b d-a e)^3}{7 e^7}+\frac {6 b^2 (d+e x)^{5/2} (b d-a e)^4}{e^7}-\frac {4 b (d+e x)^{3/2} (b d-a e)^5}{e^7}+\frac {2 \sqrt {d+e x} (b d-a e)^6}{e^7}+\frac {2 b^6 (d+e x)^{13/2}}{13 e^7}\) |
(2*(b*d - a*e)^6*Sqrt[d + e*x])/e^7 - (4*b*(b*d - a*e)^5*(d + e*x)^(3/2))/ e^7 + (6*b^2*(b*d - a*e)^4*(d + e*x)^(5/2))/e^7 - (40*b^3*(b*d - a*e)^3*(d + e*x)^(7/2))/(7*e^7) + (10*b^4*(b*d - a*e)^2*(d + e*x)^(9/2))/(3*e^7) - (12*b^5*(b*d - a*e)*(d + e*x)^(11/2))/(11*e^7) + (2*b^6*(d + e*x)^(13/2))/ (13*e^7)
3.17.41.3.1 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 = 2.36 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.52
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {1}{13} b^{6} x^{6}+a^{6}+\frac {6}{11} a \,x^{5} b^{5}+\frac {5}{3} a^{2} x^{4} b^{4}+\frac {20}{7} a^{3} x^{3} b^{3}+3 a^{4} x^{2} b^{2}+2 a^{5} x b \right ) e^{6}-4 \left (\frac {3}{143} b^{5} x^{5}+\frac {5}{33} a \,b^{4} x^{4}+\frac {10}{21} a^{2} b^{3} x^{3}+\frac {6}{7} a^{3} b^{2} x^{2}+a^{4} b x +a^{5}\right ) b d \,e^{5}+8 b^{2} d^{2} \left (\frac {5}{429} b^{4} x^{4}+\frac {20}{231} a \,b^{3} x^{3}+\frac {2}{7} a^{2} b^{2} x^{2}+\frac {4}{7} a^{3} b x +a^{4}\right ) e^{4}-\frac {64 b^{3} \left (\frac {5}{429} b^{3} x^{3}+\frac {1}{11} a \,b^{2} x^{2}+\frac {1}{3} a^{2} b x +a^{3}\right ) d^{3} e^{3}}{7}+\frac {128 b^{4} d^{4} \left (\frac {3}{143} b^{2} x^{2}+\frac {2}{11} a b x +a^{2}\right ) e^{2}}{21}-\frac {512 \left (\frac {b x}{13}+a \right ) b^{5} d^{5} e}{231}+\frac {1024 b^{6} d^{6}}{3003}\right ) \sqrt {e x +d}}{e^{7}}\) | \(275\) |
| gosper | \(\frac {2 \left (231 x^{6} b^{6} e^{6}+1638 x^{5} a \,b^{5} e^{6}-252 x^{5} b^{6} d \,e^{5}+5005 x^{4} a^{2} b^{4} e^{6}-1820 x^{4} a \,b^{5} d \,e^{5}+280 x^{4} b^{6} d^{2} e^{4}+8580 x^{3} a^{3} b^{3} e^{6}-5720 x^{3} a^{2} b^{4} d \,e^{5}+2080 x^{3} a \,b^{5} d^{2} e^{4}-320 x^{3} b^{6} d^{3} e^{3}+9009 x^{2} a^{4} b^{2} e^{6}-10296 x^{2} a^{3} b^{3} d \,e^{5}+6864 x^{2} a^{2} b^{4} d^{2} e^{4}-2496 x^{2} a \,b^{5} d^{3} e^{3}+384 x^{2} b^{6} d^{4} e^{2}+6006 x \,a^{5} b \,e^{6}-12012 x \,a^{4} b^{2} d \,e^{5}+13728 x \,a^{3} b^{3} d^{2} e^{4}-9152 x \,a^{2} b^{4} d^{3} e^{3}+3328 x a \,b^{5} d^{4} e^{2}-512 x \,b^{6} d^{5} e +3003 a^{6} e^{6}-12012 a^{5} b d \,e^{5}+24024 a^{4} b^{2} d^{2} e^{4}-27456 a^{3} b^{3} d^{3} e^{3}+18304 a^{2} b^{4} d^{4} e^{2}-6656 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \sqrt {e x +d}}{3003 e^{7}}\) | \(377\) |
| trager | \(\frac {2 \left (231 x^{6} b^{6} e^{6}+1638 x^{5} a \,b^{5} e^{6}-252 x^{5} b^{6} d \,e^{5}+5005 x^{4} a^{2} b^{4} e^{6}-1820 x^{4} a \,b^{5} d \,e^{5}+280 x^{4} b^{6} d^{2} e^{4}+8580 x^{3} a^{3} b^{3} e^{6}-5720 x^{3} a^{2} b^{4} d \,e^{5}+2080 x^{3} a \,b^{5} d^{2} e^{4}-320 x^{3} b^{6} d^{3} e^{3}+9009 x^{2} a^{4} b^{2} e^{6}-10296 x^{2} a^{3} b^{3} d \,e^{5}+6864 x^{2} a^{2} b^{4} d^{2} e^{4}-2496 x^{2} a \,b^{5} d^{3} e^{3}+384 x^{2} b^{6} d^{4} e^{2}+6006 x \,a^{5} b \,e^{6}-12012 x \,a^{4} b^{2} d \,e^{5}+13728 x \,a^{3} b^{3} d^{2} e^{4}-9152 x \,a^{2} b^{4} d^{3} e^{3}+3328 x a \,b^{5} d^{4} e^{2}-512 x \,b^{6} d^{5} e +3003 a^{6} e^{6}-12012 a^{5} b d \,e^{5}+24024 a^{4} b^{2} d^{2} e^{4}-27456 a^{3} b^{3} d^{3} e^{3}+18304 a^{2} b^{4} d^{4} e^{2}-6656 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \sqrt {e x +d}}{3003 e^{7}}\) | \(377\) |
| risch | \(\frac {2 \left (231 x^{6} b^{6} e^{6}+1638 x^{5} a \,b^{5} e^{6}-252 x^{5} b^{6} d \,e^{5}+5005 x^{4} a^{2} b^{4} e^{6}-1820 x^{4} a \,b^{5} d \,e^{5}+280 x^{4} b^{6} d^{2} e^{4}+8580 x^{3} a^{3} b^{3} e^{6}-5720 x^{3} a^{2} b^{4} d \,e^{5}+2080 x^{3} a \,b^{5} d^{2} e^{4}-320 x^{3} b^{6} d^{3} e^{3}+9009 x^{2} a^{4} b^{2} e^{6}-10296 x^{2} a^{3} b^{3} d \,e^{5}+6864 x^{2} a^{2} b^{4} d^{2} e^{4}-2496 x^{2} a \,b^{5} d^{3} e^{3}+384 x^{2} b^{6} d^{4} e^{2}+6006 x \,a^{5} b \,e^{6}-12012 x \,a^{4} b^{2} d \,e^{5}+13728 x \,a^{3} b^{3} d^{2} e^{4}-9152 x \,a^{2} b^{4} d^{3} e^{3}+3328 x a \,b^{5} d^{4} e^{2}-512 x \,b^{6} d^{5} e +3003 a^{6} e^{6}-12012 a^{5} b d \,e^{5}+24024 a^{4} b^{2} d^{2} e^{4}-27456 a^{3} b^{3} d^{3} e^{3}+18304 a^{2} b^{4} d^{4} e^{2}-6656 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \sqrt {e x +d}}{3003 e^{7}}\) | \(377\) |
| derivativedivides | \(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (2 a e b -2 b^{2} d \right ) b^{4} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a e b -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) b^{2}+\left (2 a e b -2 b^{2} d \right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )+2 \left (2 a e b -2 b^{2} d \right )^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {3}{2}}+2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \sqrt {e x +d}}{e^{7}}\) | \(455\) |
| default | \(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (2 a e b -2 b^{2} d \right ) b^{4} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a e b -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) b^{2}+\left (2 a e b -2 b^{2} d \right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )+2 \left (2 a e b -2 b^{2} d \right )^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {3}{2}}+2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \sqrt {e x +d}}{e^{7}}\) | \(455\) |
2*((1/13*b^6*x^6+a^6+6/11*a*x^5*b^5+5/3*a^2*x^4*b^4+20/7*a^3*x^3*b^3+3*a^4 *x^2*b^2+2*a^5*x*b)*e^6-4*(3/143*b^5*x^5+5/33*a*b^4*x^4+10/21*a^2*b^3*x^3+ 6/7*a^3*b^2*x^2+a^4*b*x+a^5)*b*d*e^5+8*b^2*d^2*(5/429*b^4*x^4+20/231*a*b^3 *x^3+2/7*a^2*b^2*x^2+4/7*a^3*b*x+a^4)*e^4-64/7*b^3*(5/429*b^3*x^3+1/11*a*b ^2*x^2+1/3*a^2*b*x+a^3)*d^3*e^3+128/21*b^4*d^4*(3/143*b^2*x^2+2/11*a*b*x+a ^2)*e^2-512/231*(1/13*b*x+a)*b^5*d^5*e+1024/3003*b^6*d^6)*(e*x+d)^(1/2)/e^ 7
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (159) = 318\).
Time = 0.34 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.97 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (231 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 6656 \, a b^{5} d^{5} e + 18304 \, a^{2} b^{4} d^{4} e^{2} - 27456 \, a^{3} b^{3} d^{3} e^{3} + 24024 \, a^{4} b^{2} d^{2} e^{4} - 12012 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} - 126 \, {\left (2 \, b^{6} d e^{5} - 13 \, a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{2} e^{4} - 52 \, a b^{5} d e^{5} + 143 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (16 \, b^{6} d^{3} e^{3} - 104 \, a b^{5} d^{2} e^{4} + 286 \, a^{2} b^{4} d e^{5} - 429 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{4} e^{2} - 832 \, a b^{5} d^{3} e^{3} + 2288 \, a^{2} b^{4} d^{2} e^{4} - 3432 \, a^{3} b^{3} d e^{5} + 3003 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{5} e - 1664 \, a b^{5} d^{4} e^{2} + 4576 \, a^{2} b^{4} d^{3} e^{3} - 6864 \, a^{3} b^{3} d^{2} e^{4} + 6006 \, a^{4} b^{2} d e^{5} - 3003 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{3003 \, e^{7}} \]
2/3003*(231*b^6*e^6*x^6 + 1024*b^6*d^6 - 6656*a*b^5*d^5*e + 18304*a^2*b^4* d^4*e^2 - 27456*a^3*b^3*d^3*e^3 + 24024*a^4*b^2*d^2*e^4 - 12012*a^5*b*d*e^ 5 + 3003*a^6*e^6 - 126*(2*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 + 35*(8*b^6*d^2*e^ 4 - 52*a*b^5*d*e^5 + 143*a^2*b^4*e^6)*x^4 - 20*(16*b^6*d^3*e^3 - 104*a*b^5 *d^2*e^4 + 286*a^2*b^4*d*e^5 - 429*a^3*b^3*e^6)*x^3 + 3*(128*b^6*d^4*e^2 - 832*a*b^5*d^3*e^3 + 2288*a^2*b^4*d^2*e^4 - 3432*a^3*b^3*d*e^5 + 3003*a^4* b^2*e^6)*x^2 - 2*(256*b^6*d^5*e - 1664*a*b^5*d^4*e^2 + 4576*a^2*b^4*d^3*e^ 3 - 6864*a^3*b^3*d^2*e^4 + 6006*a^4*b^2*d*e^5 - 3003*a^5*b*e^6)*x)*sqrt(e* x + d)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (168) = 336\).
Time = 1.23 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.72 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{6} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (6 a b^{5} e - 6 b^{6} d\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (15 a^{2} b^{4} e^{2} - 30 a b^{5} d e + 15 b^{6} d^{2}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (20 a^{3} b^{3} e^{3} - 60 a^{2} b^{4} d e^{2} + 60 a b^{5} d^{2} e - 20 b^{6} d^{3}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (15 a^{4} b^{2} e^{4} - 60 a^{3} b^{3} d e^{3} + 90 a^{2} b^{4} d^{2} e^{2} - 60 a b^{5} d^{3} e + 15 b^{6} d^{4}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (6 a^{5} b e^{5} - 30 a^{4} b^{2} d e^{4} + 60 a^{3} b^{3} d^{2} e^{3} - 60 a^{2} b^{4} d^{3} e^{2} + 30 a b^{5} d^{4} e - 6 b^{6} d^{5}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (a^{6} e^{6} - 6 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} - 6 a b^{5} d^{5} e + b^{6} d^{6}\right )}{e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac {b^{6} x^{7}}{7}}{\sqrt {d}} & \text {otherwise} \end {cases} \]
Piecewise((2*(b**6*(d + e*x)**(13/2)/(13*e**6) + (d + e*x)**(11/2)*(6*a*b* *5*e - 6*b**6*d)/(11*e**6) + (d + e*x)**(9/2)*(15*a**2*b**4*e**2 - 30*a*b* *5*d*e + 15*b**6*d**2)/(9*e**6) + (d + e*x)**(7/2)*(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)/(7*e**6) + (d + e*x)* *(5/2)*(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)/(5*e**6) + (d + e*x)**(3/2)*(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)/(3*e**6) + sqrt(d + e*x)*(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)/e**6)/e, Ne(e, 0)), ((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)/sqrt(d), True))
Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (159) = 318\).
Time = 0.22 (sec) , antiderivative size = 540, normalized size of antiderivative = 2.98 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15015 \, \sqrt {e x + d} a^{6} + 3003 \, {\left (\frac {10 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a b}{e} + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2}}{e^{2}}\right )} a^{4} + \frac {3432 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{3} b^{3}}{e^{3}} + 143 \, {\left (\frac {84 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} b^{2}}{e^{2}} + \frac {36 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a b^{3}}{e^{3}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{4}}{e^{4}}\right )} a^{2} + \frac {572 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a^{2} b^{4}}{e^{4}} + \frac {130 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} a b^{5}}{e^{5}} + \frac {5 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} b^{6}}{e^{6}}\right )}}{15015 \, e} \]
2/15015*(15015*sqrt(e*x + d)*a^6 + 3003*(10*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*b/e + (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2/e^2)*a^4 + 3432*(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/e^3 + 143*(84*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b^2/e^2 + 3 6*(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*b^3/e^3 + (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)*b^4/e^4)*a^2 + 572*(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/e^4 + 130*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d) ^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqr t(e*x + d)*d^5)*a*b^5/e^5 + 5*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2 )*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d) ^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*b^6/e^6)/e
Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (159) = 318\).
Time = 0.28 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.07 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (3003 \, \sqrt {e x + d} a^{6} + \frac {6006 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{5} b}{e} + \frac {3003 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{4} b^{2}}{e^{2}} + \frac {1716 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{3} b^{3}}{e^{3}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a^{2} b^{4}}{e^{4}} + \frac {26 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} a b^{5}}{e^{5}} + \frac {{\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} b^{6}}{e^{6}}\right )}}{3003 \, e} \]
2/3003*(3003*sqrt(e*x + d)*a^6 + 6006*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d )*a^5*b/e + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^4*b^2/e^2 + 1716*(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/e^3 + 143*(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/e^4 + 26*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d ^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*a*b^5/e^5 + (231*(e *x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580 *(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*b^6/e^6)/e
Time = 10.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2\,b^6\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,\sqrt {d+e\,x}}{e^7}+\frac {6\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{e^7}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {10\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^7}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{3/2}}{e^7} \]
(2*b^6*(d + e*x)^(13/2))/(13*e^7) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(11 /2))/(11*e^7) + (2*(a*e - b*d)^6*(d + e*x)^(1/2))/e^7 + (6*b^2*(a*e - b*d) ^4*(d + e*x)^(5/2))/e^7 + (40*b^3*(a*e - b*d)^3*(d + e*x)^(7/2))/(7*e^7) + (10*b^4*(a*e - b*d)^2*(d + e*x)^(9/2))/(3*e^7) + (4*b*(a*e - b*d)^5*(d + e*x)^(3/2))/e^7