Integrand size = 33, antiderivative size = 206 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 (b d-a e)^7}{e^8 \sqrt {d+e x}}+\frac {14 b (b d-a e)^6 \sqrt {d+e x}}{e^8}-\frac {14 b^2 (b d-a e)^5 (d+e x)^{3/2}}{e^8}+\frac {14 b^3 (b d-a e)^4 (d+e x)^{5/2}}{e^8}-\frac {10 b^4 (b d-a e)^3 (d+e x)^{7/2}}{e^8}+\frac {14 b^5 (b d-a e)^2 (d+e x)^{9/2}}{3 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{11/2}}{11 e^8}+\frac {2 b^7 (d+e x)^{13/2}}{13 e^8} \]
-14*b^2*(-a*e+b*d)^5*(e*x+d)^(3/2)/e^8+14*b^3*(-a*e+b*d)^4*(e*x+d)^(5/2)/e ^8-10*b^4*(-a*e+b*d)^3*(e*x+d)^(7/2)/e^8+14/3*b^5*(-a*e+b*d)^2*(e*x+d)^(9/ 2)/e^8-14/11*b^6*(-a*e+b*d)*(e*x+d)^(11/2)/e^8+2/13*b^7*(e*x+d)^(13/2)/e^8 +2*(-a*e+b*d)^7/e^8/(e*x+d)^(1/2)+14*b*(-a*e+b*d)^6*(e*x+d)^(1/2)/e^8
Time = 0.19 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.82 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {-858 a^7 e^7+6006 a^6 b e^6 (2 d+e x)+6006 a^5 b^2 e^5 \left (-8 d^2-4 d e x+e^2 x^2\right )+6006 a^4 b^3 e^4 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+858 a^3 b^4 e^3 \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+286 a^2 b^5 e^2 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )+26 a b^6 e \left (-1024 d^6-512 d^5 e x+128 d^4 e^2 x^2-64 d^3 e^3 x^3+40 d^2 e^4 x^4-28 d e^5 x^5+21 e^6 x^6\right )+2 b^7 \left (2048 d^7+1024 d^6 e x-256 d^5 e^2 x^2+128 d^4 e^3 x^3-80 d^3 e^4 x^4+56 d^2 e^5 x^5-42 d e^6 x^6+33 e^7 x^7\right )}{429 e^8 \sqrt {d+e x}} \]
(-858*a^7*e^7 + 6006*a^6*b*e^6*(2*d + e*x) + 6006*a^5*b^2*e^5*(-8*d^2 - 4* d*e*x + e^2*x^2) + 6006*a^4*b^3*e^4*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^ 3*x^3) + 858*a^3*b^4*e^3*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3 *x^3 + 5*e^4*x^4) + 286*a^2*b^5*e^2*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^ 2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5) + 26*a*b^6*e*(-1024*d^6 - 5 12*d^5*e*x + 128*d^4*e^2*x^2 - 64*d^3*e^3*x^3 + 40*d^2*e^4*x^4 - 28*d*e^5* x^5 + 21*e^6*x^6) + 2*b^7*(2048*d^7 + 1024*d^6*e*x - 256*d^5*e^2*x^2 + 128 *d^4*e^3*x^3 - 80*d^3*e^4*x^4 + 56*d^2*e^5*x^5 - 42*d*e^6*x^6 + 33*e^7*x^7 ))/(429*e^8*Sqrt[d + e*x])
Time = 0.35 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1184, 27, 53, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^7}{(d+e x)^{3/2}}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^7}{(d+e x)^{3/2}}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {7 b^6 (d+e x)^{9/2} (b d-a e)}{e^7}+\frac {21 b^5 (d+e x)^{7/2} (b d-a e)^2}{e^7}-\frac {35 b^4 (d+e x)^{5/2} (b d-a e)^3}{e^7}+\frac {35 b^3 (d+e x)^{3/2} (b d-a e)^4}{e^7}-\frac {21 b^2 \sqrt {d+e x} (b d-a e)^5}{e^7}+\frac {7 b (b d-a e)^6}{e^7 \sqrt {d+e x}}+\frac {(a e-b d)^7}{e^7 (d+e x)^{3/2}}+\frac {b^7 (d+e x)^{11/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {14 b^6 (d+e x)^{11/2} (b d-a e)}{11 e^8}+\frac {14 b^5 (d+e x)^{9/2} (b d-a e)^2}{3 e^8}-\frac {10 b^4 (d+e x)^{7/2} (b d-a e)^3}{e^8}+\frac {14 b^3 (d+e x)^{5/2} (b d-a e)^4}{e^8}-\frac {14 b^2 (d+e x)^{3/2} (b d-a e)^5}{e^8}+\frac {14 b \sqrt {d+e x} (b d-a e)^6}{e^8}+\frac {2 (b d-a e)^7}{e^8 \sqrt {d+e x}}+\frac {2 b^7 (d+e x)^{13/2}}{13 e^8}\) |
(2*(b*d - a*e)^7)/(e^8*Sqrt[d + e*x]) + (14*b*(b*d - a*e)^6*Sqrt[d + e*x]) /e^8 - (14*b^2*(b*d - a*e)^5*(d + e*x)^(3/2))/e^8 + (14*b^3*(b*d - a*e)^4* (d + e*x)^(5/2))/e^8 - (10*b^4*(b*d - a*e)^3*(d + e*x)^(7/2))/e^8 + (14*b^ 5*(b*d - a*e)^2*(d + e*x)^(9/2))/(3*e^8) - (14*b^6*(b*d - a*e)*(d + e*x)^( 11/2))/(11*e^8) + (2*b^7*(d + e*x)^(13/2))/(13*e^8)
3.21.63.3.1 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 = 0.36 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.74
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (-\frac {1}{13} b^{7} x^{7}+a^{7}-\frac {7}{11} a \,b^{6} x^{6}-\frac {7}{3} a^{2} b^{5} x^{5}-5 a^{3} b^{4} x^{4}-7 a^{4} b^{3} x^{3}-7 a^{5} b^{2} x^{2}-7 a^{6} b x \right ) e^{7}-14 \left (-\frac {1}{143} b^{6} x^{6}-\frac {2}{33} a \,b^{5} x^{5}-\frac {5}{21} a^{2} b^{4} x^{4}-\frac {4}{7} a^{3} b^{3} x^{3}-a^{4} b^{2} x^{2}-2 a^{5} b x +a^{6}\right ) b d \,e^{6}+56 b^{2} \left (-\frac {1}{429} b^{5} x^{5}-\frac {5}{231} a \,b^{4} x^{4}-\frac {2}{21} a^{2} b^{3} x^{3}-\frac {2}{7} a^{3} b^{2} x^{2}-a^{4} b x +a^{5}\right ) d^{2} e^{5}-112 b^{3} \left (-\frac {5}{3003} x^{4} b^{4}-\frac {4}{231} a \,b^{3} x^{3}-\frac {2}{21} x^{2} b^{2} a^{2}-\frac {4}{7} b \,a^{3} x +a^{4}\right ) d^{3} e^{4}+128 b^{4} \left (-\frac {1}{429} x^{3} b^{3}-\frac {1}{33} a \,b^{2} x^{2}-\frac {1}{3} b \,a^{2} x +a^{3}\right ) d^{4} e^{3}-\frac {256 b^{5} \left (-\frac {1}{143} b^{2} x^{2}-\frac {2}{11} a b x +a^{2}\right ) d^{5} e^{2}}{3}+\frac {1024 b^{6} \left (-\frac {b x}{13}+a \right ) d^{6} e}{33}-\frac {2048 b^{7} d^{7}}{429}\right )}{\sqrt {e x +d}\, e^{8}}\) | \(359\) |
| risch | \(\frac {2 b \left (33 b^{6} e^{6} x^{6}+273 a \,b^{5} e^{6} x^{5}-75 b^{6} d \,e^{5} x^{5}+1001 a^{2} b^{4} e^{6} x^{4}-637 a \,b^{5} d \,e^{5} x^{4}+131 b^{6} d^{2} e^{4} x^{4}+2145 a^{3} b^{3} e^{6} x^{3}-2431 a^{2} b^{4} d \,e^{5} x^{3}+1157 a \,b^{5} d^{2} e^{4} x^{3}-211 b^{6} d^{3} e^{3} x^{3}+3003 a^{4} b^{2} e^{6} x^{2}-5577 a^{3} b^{3} d \,e^{5} x^{2}+4719 a^{2} b^{4} d^{2} e^{4} x^{2}-1989 a \,b^{5} d^{3} e^{3} x^{2}+339 b^{6} d^{4} e^{2} x^{2}+3003 a^{5} b \,e^{6} x -9009 a^{4} b^{2} d \,e^{5} x +12441 a^{3} b^{3} d^{2} e^{4} x -9295 a^{2} b^{4} d^{3} e^{3} x +3653 a \,b^{5} d^{4} e^{2} x -595 b^{6} d^{5} e x +3003 e^{6} a^{6}-15015 b d \,e^{5} a^{5}+33033 b^{2} d^{2} e^{4} a^{4}-39897 b^{3} d^{3} e^{3} a^{3}+27599 b^{4} d^{4} e^{2} a^{2}-10309 b^{5} d^{5} e a +1619 b^{6} d^{6}\right ) \sqrt {e x +d}}{429 e^{8}}-\frac {2 \left (e^{7} a^{7}-7 b d \,e^{6} a^{6}+21 b^{2} d^{2} e^{5} a^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}{e^{8} \sqrt {e x +d}}\) | \(483\) |
| gosper | \(-\frac {2 \left (-33 x^{7} b^{7} e^{7}-273 x^{6} a \,b^{6} e^{7}+42 x^{6} b^{7} d \,e^{6}-1001 x^{5} a^{2} b^{5} e^{7}+364 x^{5} a \,b^{6} d \,e^{6}-56 x^{5} b^{7} d^{2} e^{5}-2145 x^{4} a^{3} b^{4} e^{7}+1430 x^{4} a^{2} b^{5} d \,e^{6}-520 x^{4} a \,b^{6} d^{2} e^{5}+80 x^{4} b^{7} d^{3} e^{4}-3003 x^{3} a^{4} b^{3} e^{7}+3432 x^{3} a^{3} b^{4} d \,e^{6}-2288 x^{3} a^{2} b^{5} d^{2} e^{5}+832 x^{3} a \,b^{6} d^{3} e^{4}-128 x^{3} b^{7} d^{4} e^{3}-3003 x^{2} a^{5} b^{2} e^{7}+6006 x^{2} a^{4} b^{3} d \,e^{6}-6864 x^{2} a^{3} b^{4} d^{2} e^{5}+4576 x^{2} a^{2} b^{5} d^{3} e^{4}-1664 x^{2} a \,b^{6} d^{4} e^{3}+256 x^{2} b^{7} d^{5} e^{2}-3003 x \,a^{6} b \,e^{7}+12012 x \,a^{5} b^{2} d \,e^{6}-24024 x \,a^{4} b^{3} d^{2} e^{5}+27456 x \,a^{3} b^{4} d^{3} e^{4}-18304 x \,a^{2} b^{5} d^{4} e^{3}+6656 x a \,b^{6} d^{5} e^{2}-1024 x \,b^{7} d^{6} e +429 e^{7} a^{7}-6006 b d \,e^{6} a^{6}+24024 b^{2} d^{2} e^{5} a^{5}-48048 b^{3} d^{3} e^{4} a^{4}+54912 b^{4} d^{4} e^{3} a^{3}-36608 b^{5} d^{5} e^{2} a^{2}+13312 b^{6} d^{6} e a -2048 b^{7} d^{7}\right )}{429 \sqrt {e x +d}\, e^{8}}\) | \(498\) |
| trager | \(-\frac {2 \left (-33 x^{7} b^{7} e^{7}-273 x^{6} a \,b^{6} e^{7}+42 x^{6} b^{7} d \,e^{6}-1001 x^{5} a^{2} b^{5} e^{7}+364 x^{5} a \,b^{6} d \,e^{6}-56 x^{5} b^{7} d^{2} e^{5}-2145 x^{4} a^{3} b^{4} e^{7}+1430 x^{4} a^{2} b^{5} d \,e^{6}-520 x^{4} a \,b^{6} d^{2} e^{5}+80 x^{4} b^{7} d^{3} e^{4}-3003 x^{3} a^{4} b^{3} e^{7}+3432 x^{3} a^{3} b^{4} d \,e^{6}-2288 x^{3} a^{2} b^{5} d^{2} e^{5}+832 x^{3} a \,b^{6} d^{3} e^{4}-128 x^{3} b^{7} d^{4} e^{3}-3003 x^{2} a^{5} b^{2} e^{7}+6006 x^{2} a^{4} b^{3} d \,e^{6}-6864 x^{2} a^{3} b^{4} d^{2} e^{5}+4576 x^{2} a^{2} b^{5} d^{3} e^{4}-1664 x^{2} a \,b^{6} d^{4} e^{3}+256 x^{2} b^{7} d^{5} e^{2}-3003 x \,a^{6} b \,e^{7}+12012 x \,a^{5} b^{2} d \,e^{6}-24024 x \,a^{4} b^{3} d^{2} e^{5}+27456 x \,a^{3} b^{4} d^{3} e^{4}-18304 x \,a^{2} b^{5} d^{4} e^{3}+6656 x a \,b^{6} d^{5} e^{2}-1024 x \,b^{7} d^{6} e +429 e^{7} a^{7}-6006 b d \,e^{6} a^{6}+24024 b^{2} d^{2} e^{5} a^{5}-48048 b^{3} d^{3} e^{4} a^{4}+54912 b^{4} d^{4} e^{3} a^{3}-36608 b^{5} d^{5} e^{2} a^{2}+13312 b^{6} d^{6} e a -2048 b^{7} d^{7}\right )}{429 \sqrt {e x +d}\, e^{8}}\) | \(498\) |
| derivativedivides | \(\frac {\frac {14 a \,b^{6} e \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {14 a^{2} b^{5} e^{2} \left (e x +d \right )^{\frac {9}{2}}}{3}+10 a^{3} b^{4} e^{3} \left (e x +d \right )^{\frac {7}{2}}+14 a^{4} b^{3} e^{4} \left (e x +d \right )^{\frac {5}{2}}+14 a^{5} b^{2} e^{5} \left (e x +d \right )^{\frac {3}{2}}+14 a^{6} b \,e^{6} \sqrt {e x +d}-\frac {2 \left (e^{7} a^{7}-7 b d \,e^{6} a^{6}+21 b^{2} d^{2} e^{5} a^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}{\sqrt {e x +d}}+\frac {2 b^{7} \left (e x +d \right )^{\frac {13}{2}}}{13}-84 a^{5} b^{2} d \,e^{5} \sqrt {e x +d}+30 a \,b^{6} d^{2} e \left (e x +d \right )^{\frac {7}{2}}-56 a^{3} b^{4} d \,e^{3} \left (e x +d \right )^{\frac {5}{2}}+84 a^{2} b^{5} d^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}-56 a \,b^{6} d^{3} e \left (e x +d \right )^{\frac {5}{2}}+210 a^{4} b^{3} d^{2} e^{4} \sqrt {e x +d}-84 a \,b^{6} d^{5} e \sqrt {e x +d}-280 a^{3} b^{4} d^{3} e^{3} \sqrt {e x +d}+210 a^{2} b^{5} d^{4} e^{2} \sqrt {e x +d}-\frac {14 b^{7} d \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {14 b^{7} d^{2} \left (e x +d \right )^{\frac {9}{2}}}{3}-10 b^{7} d^{3} \left (e x +d \right )^{\frac {7}{2}}+14 b^{7} d^{4} \left (e x +d \right )^{\frac {5}{2}}-14 b^{7} d^{5} \left (e x +d \right )^{\frac {3}{2}}-\frac {28 a \,b^{6} d e \left (e x +d \right )^{\frac {9}{2}}}{3}-30 a^{2} b^{5} d \,e^{2} \left (e x +d \right )^{\frac {7}{2}}-70 a^{4} b^{3} d \,e^{4} \left (e x +d \right )^{\frac {3}{2}}+140 a^{3} b^{4} d^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}-140 a^{2} b^{5} d^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}+70 a \,b^{6} d^{4} e \left (e x +d \right )^{\frac {3}{2}}+14 b^{7} d^{6} \sqrt {e x +d}}{e^{8}}\) | \(595\) |
| default | \(\frac {\frac {14 a \,b^{6} e \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {14 a^{2} b^{5} e^{2} \left (e x +d \right )^{\frac {9}{2}}}{3}+10 a^{3} b^{4} e^{3} \left (e x +d \right )^{\frac {7}{2}}+14 a^{4} b^{3} e^{4} \left (e x +d \right )^{\frac {5}{2}}+14 a^{5} b^{2} e^{5} \left (e x +d \right )^{\frac {3}{2}}+14 a^{6} b \,e^{6} \sqrt {e x +d}-\frac {2 \left (e^{7} a^{7}-7 b d \,e^{6} a^{6}+21 b^{2} d^{2} e^{5} a^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}{\sqrt {e x +d}}+\frac {2 b^{7} \left (e x +d \right )^{\frac {13}{2}}}{13}-84 a^{5} b^{2} d \,e^{5} \sqrt {e x +d}+30 a \,b^{6} d^{2} e \left (e x +d \right )^{\frac {7}{2}}-56 a^{3} b^{4} d \,e^{3} \left (e x +d \right )^{\frac {5}{2}}+84 a^{2} b^{5} d^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}-56 a \,b^{6} d^{3} e \left (e x +d \right )^{\frac {5}{2}}+210 a^{4} b^{3} d^{2} e^{4} \sqrt {e x +d}-84 a \,b^{6} d^{5} e \sqrt {e x +d}-280 a^{3} b^{4} d^{3} e^{3} \sqrt {e x +d}+210 a^{2} b^{5} d^{4} e^{2} \sqrt {e x +d}-\frac {14 b^{7} d \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {14 b^{7} d^{2} \left (e x +d \right )^{\frac {9}{2}}}{3}-10 b^{7} d^{3} \left (e x +d \right )^{\frac {7}{2}}+14 b^{7} d^{4} \left (e x +d \right )^{\frac {5}{2}}-14 b^{7} d^{5} \left (e x +d \right )^{\frac {3}{2}}-\frac {28 a \,b^{6} d e \left (e x +d \right )^{\frac {9}{2}}}{3}-30 a^{2} b^{5} d \,e^{2} \left (e x +d \right )^{\frac {7}{2}}-70 a^{4} b^{3} d \,e^{4} \left (e x +d \right )^{\frac {3}{2}}+140 a^{3} b^{4} d^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}-140 a^{2} b^{5} d^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}+70 a \,b^{6} d^{4} e \left (e x +d \right )^{\frac {3}{2}}+14 b^{7} d^{6} \sqrt {e x +d}}{e^{8}}\) | \(595\) |
-2*((-1/13*b^7*x^7+a^7-7/11*a*b^6*x^6-7/3*a^2*b^5*x^5-5*a^3*b^4*x^4-7*a^4* b^3*x^3-7*a^5*b^2*x^2-7*a^6*b*x)*e^7-14*(-1/143*b^6*x^6-2/33*a*b^5*x^5-5/2 1*a^2*b^4*x^4-4/7*a^3*b^3*x^3-a^4*b^2*x^2-2*a^5*b*x+a^6)*b*d*e^6+56*b^2*(- 1/429*b^5*x^5-5/231*a*b^4*x^4-2/21*a^2*b^3*x^3-2/7*a^3*b^2*x^2-a^4*b*x+a^5 )*d^2*e^5-112*b^3*(-5/3003*x^4*b^4-4/231*a*b^3*x^3-2/21*x^2*b^2*a^2-4/7*b* a^3*x+a^4)*d^3*e^4+128*b^4*(-1/429*x^3*b^3-1/33*a*b^2*x^2-1/3*b*a^2*x+a^3) *d^4*e^3-256/3*b^5*(-1/143*b^2*x^2-2/11*a*b*x+a^2)*d^5*e^2+1024/33*b^6*(-1 /13*b*x+a)*d^6*e-2048/429*b^7*d^7)/(e*x+d)^(1/2)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (184) = 368\).
Time = 0.31 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.29 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (33 \, b^{7} e^{7} x^{7} + 2048 \, b^{7} d^{7} - 13312 \, a b^{6} d^{6} e + 36608 \, a^{2} b^{5} d^{5} e^{2} - 54912 \, a^{3} b^{4} d^{4} e^{3} + 48048 \, a^{4} b^{3} d^{3} e^{4} - 24024 \, a^{5} b^{2} d^{2} e^{5} + 6006 \, a^{6} b d e^{6} - 429 \, a^{7} e^{7} - 21 \, {\left (2 \, b^{7} d e^{6} - 13 \, a b^{6} e^{7}\right )} x^{6} + 7 \, {\left (8 \, b^{7} d^{2} e^{5} - 52 \, a b^{6} d e^{6} + 143 \, a^{2} b^{5} e^{7}\right )} x^{5} - 5 \, {\left (16 \, b^{7} d^{3} e^{4} - 104 \, a b^{6} d^{2} e^{5} + 286 \, a^{2} b^{5} d e^{6} - 429 \, a^{3} b^{4} e^{7}\right )} x^{4} + {\left (128 \, b^{7} d^{4} e^{3} - 832 \, a b^{6} d^{3} e^{4} + 2288 \, a^{2} b^{5} d^{2} e^{5} - 3432 \, a^{3} b^{4} d e^{6} + 3003 \, a^{4} b^{3} e^{7}\right )} x^{3} - {\left (256 \, b^{7} d^{5} e^{2} - 1664 \, a b^{6} d^{4} e^{3} + 4576 \, a^{2} b^{5} d^{3} e^{4} - 6864 \, a^{3} b^{4} d^{2} e^{5} + 6006 \, a^{4} b^{3} d e^{6} - 3003 \, a^{5} b^{2} e^{7}\right )} x^{2} + {\left (1024 \, b^{7} d^{6} e - 6656 \, a b^{6} d^{5} e^{2} + 18304 \, a^{2} b^{5} d^{4} e^{3} - 27456 \, a^{3} b^{4} d^{3} e^{4} + 24024 \, a^{4} b^{3} d^{2} e^{5} - 12012 \, a^{5} b^{2} d e^{6} + 3003 \, a^{6} b e^{7}\right )} x\right )} \sqrt {e x + d}}{429 \, {\left (e^{9} x + d e^{8}\right )}} \]
2/429*(33*b^7*e^7*x^7 + 2048*b^7*d^7 - 13312*a*b^6*d^6*e + 36608*a^2*b^5*d ^5*e^2 - 54912*a^3*b^4*d^4*e^3 + 48048*a^4*b^3*d^3*e^4 - 24024*a^5*b^2*d^2 *e^5 + 6006*a^6*b*d*e^6 - 429*a^7*e^7 - 21*(2*b^7*d*e^6 - 13*a*b^6*e^7)*x^ 6 + 7*(8*b^7*d^2*e^5 - 52*a*b^6*d*e^6 + 143*a^2*b^5*e^7)*x^5 - 5*(16*b^7*d ^3*e^4 - 104*a*b^6*d^2*e^5 + 286*a^2*b^5*d*e^6 - 429*a^3*b^4*e^7)*x^4 + (1 28*b^7*d^4*e^3 - 832*a*b^6*d^3*e^4 + 2288*a^2*b^5*d^2*e^5 - 3432*a^3*b^4*d *e^6 + 3003*a^4*b^3*e^7)*x^3 - (256*b^7*d^5*e^2 - 1664*a*b^6*d^4*e^3 + 457 6*a^2*b^5*d^3*e^4 - 6864*a^3*b^4*d^2*e^5 + 6006*a^4*b^3*d*e^6 - 3003*a^5*b ^2*e^7)*x^2 + (1024*b^7*d^6*e - 6656*a*b^6*d^5*e^2 + 18304*a^2*b^5*d^4*e^3 - 27456*a^3*b^4*d^3*e^4 + 24024*a^4*b^3*d^2*e^5 - 12012*a^5*b^2*d*e^6 + 3 003*a^6*b*e^7)*x)*sqrt(e*x + d)/(e^9*x + d*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (192) = 384\).
Time = 16.52 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.34 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{7} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (7 a b^{6} e - 7 b^{7} d\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (21 a^{5} b^{2} e^{5} - 105 a^{4} b^{3} d e^{4} + 210 a^{3} b^{4} d^{2} e^{3} - 210 a^{2} b^{5} d^{3} e^{2} + 105 a b^{6} d^{4} e - 21 b^{7} d^{5}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (7 a^{6} b e^{6} - 42 a^{5} b^{2} d e^{5} + 105 a^{4} b^{3} d^{2} e^{4} - 140 a^{3} b^{4} d^{3} e^{3} + 105 a^{2} b^{5} d^{4} e^{2} - 42 a b^{6} d^{5} e + 7 b^{7} d^{6}\right )}{e^{7}} - \frac {\left (a e - b d\right )^{7}}{e^{7} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\begin {cases} a^{7} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{4}}{8 b} & \text {otherwise} \end {cases}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(b**7*(d + e*x)**(13/2)/(13*e**7) + (d + e*x)**(11/2)*(7*a*b* *6*e - 7*b**7*d)/(11*e**7) + (d + e*x)**(9/2)*(21*a**2*b**5*e**2 - 42*a*b* *6*d*e + 21*b**7*d**2)/(9*e**7) + (d + e*x)**(7/2)*(35*a**3*b**4*e**3 - 10 5*a**2*b**5*d*e**2 + 105*a*b**6*d**2*e - 35*b**7*d**3)/(7*e**7) + (d + e*x )**(5/2)*(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)/(5*e**7) + (d + e*x)**(3/2)*(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)/(3*e**7) + sqrt(d + e*x)* (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) /e**7 - (a*e - b*d)**7/(e**7*sqrt(d + e*x)))/e, Ne(e, 0)), (Piecewise((a** 7*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**4/(8*b), True))/d**(3/2), T rue))
Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (184) = 368\).
Time = 0.20 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.25 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {33 \, {\left (e x + d\right )}^{\frac {13}{2}} b^{7} - 273 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1001 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 2145 \, {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 3003 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 3003 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 3003 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} \sqrt {e x + d}}{e^{7}} + \frac {429 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )}}{\sqrt {e x + d} e^{7}}\right )}}{429 \, e} \]
2/429*((33*(e*x + d)^(13/2)*b^7 - 273*(b^7*d - a*b^6*e)*(e*x + d)^(11/2) + 1001*(b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*(e*x + d)^(9/2) - 2145*(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)^(7/2) + 3003* (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)^(5/2) - 3003*(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)^(3/2) + 300 3*(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)*sqrt(e*x + d))/e^7 + 429*(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)/(sqrt(e*x + d) *e^7))/e
Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (184) = 368\).
Time = 0.29 (sec) , antiderivative size = 631, normalized size of antiderivative = 3.06 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\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 )}}{\sqrt {e x + d} e^{8}} + \frac {2 \, {\left (33 \, {\left (e x + d\right )}^{\frac {13}{2}} b^{7} e^{96} - 273 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{7} d e^{96} + 1001 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{7} d^{2} e^{96} - 2145 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{7} d^{3} e^{96} + 3003 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{7} d^{4} e^{96} - 3003 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{7} d^{5} e^{96} + 3003 \, \sqrt {e x + d} b^{7} d^{6} e^{96} + 273 \, {\left (e x + d\right )}^{\frac {11}{2}} a b^{6} e^{97} - 2002 \, {\left (e x + d\right )}^{\frac {9}{2}} a b^{6} d e^{97} + 6435 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{6} d^{2} e^{97} - 12012 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{6} d^{3} e^{97} + 15015 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{6} d^{4} e^{97} - 18018 \, \sqrt {e x + d} a b^{6} d^{5} e^{97} + 1001 \, {\left (e x + d\right )}^{\frac {9}{2}} a^{2} b^{5} e^{98} - 6435 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{2} b^{5} d e^{98} + 18018 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{5} d^{2} e^{98} - 30030 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{5} d^{3} e^{98} + 45045 \, \sqrt {e x + d} a^{2} b^{5} d^{4} e^{98} + 2145 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{3} b^{4} e^{99} - 12012 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{3} b^{4} d e^{99} + 30030 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{4} d^{2} e^{99} - 60060 \, \sqrt {e x + d} a^{3} b^{4} d^{3} e^{99} + 3003 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{4} b^{3} e^{100} - 15015 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{4} b^{3} d e^{100} + 45045 \, \sqrt {e x + d} a^{4} b^{3} d^{2} e^{100} + 3003 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{5} b^{2} e^{101} - 18018 \, \sqrt {e x + d} a^{5} b^{2} d e^{101} + 3003 \, \sqrt {e x + d} a^{6} b e^{102}\right )}}{429 \, e^{104}} \]
2*(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)/(sqrt(e*x + d)*e^8) + 2/429*(33*(e*x + d)^(13/2)*b^7*e^96 - 273*(e*x + d)^(11/2)*b^7 *d*e^96 + 1001*(e*x + d)^(9/2)*b^7*d^2*e^96 - 2145*(e*x + d)^(7/2)*b^7*d^3 *e^96 + 3003*(e*x + d)^(5/2)*b^7*d^4*e^96 - 3003*(e*x + d)^(3/2)*b^7*d^5*e ^96 + 3003*sqrt(e*x + d)*b^7*d^6*e^96 + 273*(e*x + d)^(11/2)*a*b^6*e^97 - 2002*(e*x + d)^(9/2)*a*b^6*d*e^97 + 6435*(e*x + d)^(7/2)*a*b^6*d^2*e^97 - 12012*(e*x + d)^(5/2)*a*b^6*d^3*e^97 + 15015*(e*x + d)^(3/2)*a*b^6*d^4*e^9 7 - 18018*sqrt(e*x + d)*a*b^6*d^5*e^97 + 1001*(e*x + d)^(9/2)*a^2*b^5*e^98 - 6435*(e*x + d)^(7/2)*a^2*b^5*d*e^98 + 18018*(e*x + d)^(5/2)*a^2*b^5*d^2 *e^98 - 30030*(e*x + d)^(3/2)*a^2*b^5*d^3*e^98 + 45045*sqrt(e*x + d)*a^2*b ^5*d^4*e^98 + 2145*(e*x + d)^(7/2)*a^3*b^4*e^99 - 12012*(e*x + d)^(5/2)*a^ 3*b^4*d*e^99 + 30030*(e*x + d)^(3/2)*a^3*b^4*d^2*e^99 - 60060*sqrt(e*x + d )*a^3*b^4*d^3*e^99 + 3003*(e*x + d)^(5/2)*a^4*b^3*e^100 - 15015*(e*x + d)^ (3/2)*a^4*b^3*d*e^100 + 45045*sqrt(e*x + d)*a^4*b^3*d^2*e^100 + 3003*(e*x + d)^(3/2)*a^5*b^2*e^101 - 18018*sqrt(e*x + d)*a^5*b^2*d*e^101 + 3003*sqrt (e*x + d)*a^6*b*e^102)/e^104
Time = 11.01 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2\,b^7\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}-\frac {2\,a^7\,e^7-14\,a^6\,b\,d\,e^6+42\,a^5\,b^2\,d^2\,e^5-70\,a^4\,b^3\,d^3\,e^4+70\,a^3\,b^4\,d^4\,e^3-42\,a^2\,b^5\,d^5\,e^2+14\,a\,b^6\,d^6\,e-2\,b^7\,d^7}{e^8\,\sqrt {d+e\,x}}+\frac {14\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{3/2}}{e^8}+\frac {14\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{e^8}+\frac {10\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{e^8}+\frac {14\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,\sqrt {d+e\,x}}{e^8} \]
(2*b^7*(d + e*x)^(13/2))/(13*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(11 /2))/(11*e^8) - (2*a^7*e^7 - 2*b^7*d^7 - 42*a^2*b^5*d^5*e^2 + 70*a^3*b^4*d ^4*e^3 - 70*a^4*b^3*d^3*e^4 + 42*a^5*b^2*d^2*e^5 + 14*a*b^6*d^6*e - 14*a^6 *b*d*e^6)/(e^8*(d + e*x)^(1/2)) + (14*b^2*(a*e - b*d)^5*(d + e*x)^(3/2))/e ^8 + (14*b^3*(a*e - b*d)^4*(d + e*x)^(5/2))/e^8 + (10*b^4*(a*e - b*d)^3*(d + e*x)^(7/2))/e^8 + (14*b^5*(a*e - b*d)^2*(d + e*x)^(9/2))/(3*e^8) + (14* b*(a*e - b*d)^6*(d + e*x)^(1/2))/e^8