Integrand size = 33, antiderivative size = 212 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=-\frac {2 (b d-a e)^7 \sqrt {d+e x}}{e^8}+\frac {14 b (b d-a e)^6 (d+e x)^{3/2}}{3 e^8}-\frac {42 b^2 (b d-a e)^5 (d+e x)^{5/2}}{5 e^8}+\frac {10 b^3 (b d-a e)^4 (d+e x)^{7/2}}{e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{9/2}}{9 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{11/2}}{11 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{13/2}}{13 e^8}+\frac {2 b^7 (d+e x)^{15/2}}{15 e^8} \]
14/3*b*(-a*e+b*d)^6*(e*x+d)^(3/2)/e^8-42/5*b^2*(-a*e+b*d)^5*(e*x+d)^(5/2)/ e^8+10*b^3*(-a*e+b*d)^4*(e*x+d)^(7/2)/e^8-70/9*b^4*(-a*e+b*d)^3*(e*x+d)^(9 /2)/e^8+42/11*b^5*(-a*e+b*d)^2*(e*x+d)^(11/2)/e^8-14/13*b^6*(-a*e+b*d)*(e* x+d)^(13/2)/e^8+2/15*b^7*(e*x+d)^(15/2)/e^8-2*(-a*e+b*d)^7*(e*x+d)^(1/2)/e ^8
Time = 0.17 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.77 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (6435 a^7 e^7+15015 a^6 b e^6 (-2 d+e x)+9009 a^5 b^2 e^5 \left (8 d^2-4 d e x+3 e^2 x^2\right )+6435 a^4 b^3 e^4 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+715 a^3 b^4 e^3 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+195 a^2 b^5 e^2 \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+15 a b^6 e \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )+b^7 \left (-2048 d^7+1024 d^6 e x-768 d^5 e^2 x^2+640 d^4 e^3 x^3-560 d^3 e^4 x^4+504 d^2 e^5 x^5-462 d e^6 x^6+429 e^7 x^7\right )\right )}{6435 e^8} \]
(2*Sqrt[d + e*x]*(6435*a^7*e^7 + 15015*a^6*b*e^6*(-2*d + e*x) + 9009*a^5*b ^2*e^5*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 6435*a^4*b^3*e^4*(-16*d^3 + 8*d^2*e *x - 6*d*e^2*x^2 + 5*e^3*x^3) + 715*a^3*b^4*e^3*(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) + 195*a^2*b^5*e^2*(-256*d^5 + 12 8*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) + 15*a*b^6*e*(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) + b^7*(-2048*d^7 + 1024*d^6 *e*x - 768*d^5*e^2*x^2 + 640*d^4*e^3*x^3 - 560*d^3*e^4*x^4 + 504*d^2*e^5*x ^5 - 462*d*e^6*x^6 + 429*e^7*x^7)))/(6435*e^8)
Time = 0.35 (sec) , antiderivative size = 212, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^7}{\sqrt {d+e x}}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^7}{\sqrt {d+e x}}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {7 b^6 (d+e x)^{11/2} (b d-a e)}{e^7}+\frac {21 b^5 (d+e x)^{9/2} (b d-a e)^2}{e^7}-\frac {35 b^4 (d+e x)^{7/2} (b d-a e)^3}{e^7}+\frac {35 b^3 (d+e x)^{5/2} (b d-a e)^4}{e^7}-\frac {21 b^2 (d+e x)^{3/2} (b d-a e)^5}{e^7}+\frac {7 b \sqrt {d+e x} (b d-a e)^6}{e^7}+\frac {(a e-b d)^7}{e^7 \sqrt {d+e x}}+\frac {b^7 (d+e x)^{13/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {14 b^6 (d+e x)^{13/2} (b d-a e)}{13 e^8}+\frac {42 b^5 (d+e x)^{11/2} (b d-a e)^2}{11 e^8}-\frac {70 b^4 (d+e x)^{9/2} (b d-a e)^3}{9 e^8}+\frac {10 b^3 (d+e x)^{7/2} (b d-a e)^4}{e^8}-\frac {42 b^2 (d+e x)^{5/2} (b d-a e)^5}{5 e^8}+\frac {14 b (d+e x)^{3/2} (b d-a e)^6}{3 e^8}-\frac {2 \sqrt {d+e x} (b d-a e)^7}{e^8}+\frac {2 b^7 (d+e x)^{15/2}}{15 e^8}\) |
(-2*(b*d - a*e)^7*Sqrt[d + e*x])/e^8 + (14*b*(b*d - a*e)^6*(d + e*x)^(3/2) )/(3*e^8) - (42*b^2*(b*d - a*e)^5*(d + e*x)^(5/2))/(5*e^8) + (10*b^3*(b*d - a*e)^4*(d + e*x)^(7/2))/e^8 - (70*b^4*(b*d - a*e)^3*(d + e*x)^(9/2))/(9* e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(11/2))/(11*e^8) - (14*b^6*(b*d - a *e)*(d + e*x)^(13/2))/(13*e^8) + (2*b^7*(d + e*x)^(15/2))/(15*e^8)
3.21.62.3.1 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 = 0.35 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.69
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {1}{15} b^{7} x^{7}+a^{7}+\frac {7}{13} a \,b^{6} x^{6}+\frac {21}{11} a^{2} b^{5} x^{5}+\frac {35}{9} a^{3} b^{4} x^{4}+5 a^{4} b^{3} x^{3}+\frac {21}{5} a^{5} b^{2} x^{2}+\frac {7}{3} a^{6} b x \right ) e^{7}-\frac {14 b \left (\frac {1}{65} b^{6} x^{6}+\frac {18}{143} a \,b^{5} x^{5}+\frac {5}{11} a^{2} b^{4} x^{4}+\frac {20}{21} a^{3} b^{3} x^{3}+\frac {9}{7} a^{4} b^{2} x^{2}+\frac {6}{5} a^{5} b x +a^{6}\right ) d \,e^{6}}{3}+\frac {56 b^{2} \left (\frac {1}{143} b^{5} x^{5}+\frac {25}{429} a \,b^{4} x^{4}+\frac {50}{231} a^{2} b^{3} x^{3}+\frac {10}{21} a^{3} b^{2} x^{2}+\frac {5}{7} a^{4} b x +a^{5}\right ) d^{2} e^{5}}{5}-16 b^{3} d^{3} \left (\frac {7}{1287} x^{4} b^{4}+\frac {20}{429} a \,b^{3} x^{3}+\frac {2}{11} x^{2} b^{2} a^{2}+\frac {4}{9} b \,a^{3} x +a^{4}\right ) e^{4}+\frac {128 b^{4} \left (\frac {1}{143} x^{3} b^{3}+\frac {9}{143} a \,b^{2} x^{2}+\frac {3}{11} b \,a^{2} x +a^{3}\right ) d^{4} e^{3}}{9}-\frac {256 b^{5} \left (\frac {1}{65} b^{2} x^{2}+\frac {2}{13} a b x +a^{2}\right ) d^{5} e^{2}}{33}+\frac {1024 b^{6} \left (\frac {b x}{15}+a \right ) d^{6} e}{429}-\frac {2048 b^{7} d^{7}}{6435}\right ) \sqrt {e x +d}}{e^{8}}\) | \(359\) |
| gosper | \(\frac {2 \left (429 x^{7} b^{7} e^{7}+3465 x^{6} a \,b^{6} e^{7}-462 x^{6} b^{7} d \,e^{6}+12285 x^{5} a^{2} b^{5} e^{7}-3780 x^{5} a \,b^{6} d \,e^{6}+504 x^{5} b^{7} d^{2} e^{5}+25025 x^{4} a^{3} b^{4} e^{7}-13650 x^{4} a^{2} b^{5} d \,e^{6}+4200 x^{4} a \,b^{6} d^{2} e^{5}-560 x^{4} b^{7} d^{3} e^{4}+32175 x^{3} a^{4} b^{3} e^{7}-28600 x^{3} a^{3} b^{4} d \,e^{6}+15600 x^{3} a^{2} b^{5} d^{2} e^{5}-4800 x^{3} a \,b^{6} d^{3} e^{4}+640 x^{3} b^{7} d^{4} e^{3}+27027 x^{2} a^{5} b^{2} e^{7}-38610 x^{2} a^{4} b^{3} d \,e^{6}+34320 x^{2} a^{3} b^{4} d^{2} e^{5}-18720 x^{2} a^{2} b^{5} d^{3} e^{4}+5760 x^{2} a \,b^{6} d^{4} e^{3}-768 x^{2} b^{7} d^{5} e^{2}+15015 x \,a^{6} b \,e^{7}-36036 x \,a^{5} b^{2} d \,e^{6}+51480 x \,a^{4} b^{3} d^{2} e^{5}-45760 x \,a^{3} b^{4} d^{3} e^{4}+24960 x \,a^{2} b^{5} d^{4} e^{3}-7680 x a \,b^{6} d^{5} e^{2}+1024 x \,b^{7} d^{6} e +6435 e^{7} a^{7}-30030 b d \,e^{6} a^{6}+72072 b^{2} d^{2} e^{5} a^{5}-102960 b^{3} d^{3} e^{4} a^{4}+91520 b^{4} d^{4} e^{3} a^{3}-49920 b^{5} d^{5} e^{2} a^{2}+15360 b^{6} d^{6} e a -2048 b^{7} d^{7}\right ) \sqrt {e x +d}}{6435 e^{8}}\) | \(498\) |
| trager | \(\frac {2 \left (429 x^{7} b^{7} e^{7}+3465 x^{6} a \,b^{6} e^{7}-462 x^{6} b^{7} d \,e^{6}+12285 x^{5} a^{2} b^{5} e^{7}-3780 x^{5} a \,b^{6} d \,e^{6}+504 x^{5} b^{7} d^{2} e^{5}+25025 x^{4} a^{3} b^{4} e^{7}-13650 x^{4} a^{2} b^{5} d \,e^{6}+4200 x^{4} a \,b^{6} d^{2} e^{5}-560 x^{4} b^{7} d^{3} e^{4}+32175 x^{3} a^{4} b^{3} e^{7}-28600 x^{3} a^{3} b^{4} d \,e^{6}+15600 x^{3} a^{2} b^{5} d^{2} e^{5}-4800 x^{3} a \,b^{6} d^{3} e^{4}+640 x^{3} b^{7} d^{4} e^{3}+27027 x^{2} a^{5} b^{2} e^{7}-38610 x^{2} a^{4} b^{3} d \,e^{6}+34320 x^{2} a^{3} b^{4} d^{2} e^{5}-18720 x^{2} a^{2} b^{5} d^{3} e^{4}+5760 x^{2} a \,b^{6} d^{4} e^{3}-768 x^{2} b^{7} d^{5} e^{2}+15015 x \,a^{6} b \,e^{7}-36036 x \,a^{5} b^{2} d \,e^{6}+51480 x \,a^{4} b^{3} d^{2} e^{5}-45760 x \,a^{3} b^{4} d^{3} e^{4}+24960 x \,a^{2} b^{5} d^{4} e^{3}-7680 x a \,b^{6} d^{5} e^{2}+1024 x \,b^{7} d^{6} e +6435 e^{7} a^{7}-30030 b d \,e^{6} a^{6}+72072 b^{2} d^{2} e^{5} a^{5}-102960 b^{3} d^{3} e^{4} a^{4}+91520 b^{4} d^{4} e^{3} a^{3}-49920 b^{5} d^{5} e^{2} a^{2}+15360 b^{6} d^{6} e a -2048 b^{7} d^{7}\right ) \sqrt {e x +d}}{6435 e^{8}}\) | \(498\) |
| risch | \(\frac {2 \left (429 x^{7} b^{7} e^{7}+3465 x^{6} a \,b^{6} e^{7}-462 x^{6} b^{7} d \,e^{6}+12285 x^{5} a^{2} b^{5} e^{7}-3780 x^{5} a \,b^{6} d \,e^{6}+504 x^{5} b^{7} d^{2} e^{5}+25025 x^{4} a^{3} b^{4} e^{7}-13650 x^{4} a^{2} b^{5} d \,e^{6}+4200 x^{4} a \,b^{6} d^{2} e^{5}-560 x^{4} b^{7} d^{3} e^{4}+32175 x^{3} a^{4} b^{3} e^{7}-28600 x^{3} a^{3} b^{4} d \,e^{6}+15600 x^{3} a^{2} b^{5} d^{2} e^{5}-4800 x^{3} a \,b^{6} d^{3} e^{4}+640 x^{3} b^{7} d^{4} e^{3}+27027 x^{2} a^{5} b^{2} e^{7}-38610 x^{2} a^{4} b^{3} d \,e^{6}+34320 x^{2} a^{3} b^{4} d^{2} e^{5}-18720 x^{2} a^{2} b^{5} d^{3} e^{4}+5760 x^{2} a \,b^{6} d^{4} e^{3}-768 x^{2} b^{7} d^{5} e^{2}+15015 x \,a^{6} b \,e^{7}-36036 x \,a^{5} b^{2} d \,e^{6}+51480 x \,a^{4} b^{3} d^{2} e^{5}-45760 x \,a^{3} b^{4} d^{3} e^{4}+24960 x \,a^{2} b^{5} d^{4} e^{3}-7680 x a \,b^{6} d^{5} e^{2}+1024 x \,b^{7} d^{6} e +6435 e^{7} a^{7}-30030 b d \,e^{6} a^{6}+72072 b^{2} d^{2} e^{5} a^{5}-102960 b^{3} d^{3} e^{4} a^{4}+91520 b^{4} d^{4} e^{3} a^{3}-49920 b^{5} d^{5} e^{2} a^{2}+15360 b^{6} d^{6} e a -2048 b^{7} d^{7}\right ) \sqrt {e x +d}}{6435 e^{8}}\) | \(498\) |
| derivativedivides | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {15}{2}}}{15}+\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 {13}{2}}}{13}+\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 {11}{2}}}{11}+\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 {9}{2}}}{9}+\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 {7}{2}}}{7}+\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 {5}{2}}}{5}+\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 {3}{2}}}{3}+2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \sqrt {e x +d}}{e^{8}}\) | \(935\) |
| default | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {15}{2}}}{15}+\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 {13}{2}}}{13}+\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 {11}{2}}}{11}+\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 {9}{2}}}{9}+\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 {7}{2}}}{7}+\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 {5}{2}}}{5}+\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 {3}{2}}}{3}+2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \sqrt {e x +d}}{e^{8}}\) | \(935\) |
2*((1/15*b^7*x^7+a^7+7/13*a*b^6*x^6+21/11*a^2*b^5*x^5+35/9*a^3*b^4*x^4+5*a ^4*b^3*x^3+21/5*a^5*b^2*x^2+7/3*a^6*b*x)*e^7-14/3*b*(1/65*b^6*x^6+18/143*a *b^5*x^5+5/11*a^2*b^4*x^4+20/21*a^3*b^3*x^3+9/7*a^4*b^2*x^2+6/5*a^5*b*x+a^ 6)*d*e^6+56/5*b^2*(1/143*b^5*x^5+25/429*a*b^4*x^4+50/231*a^2*b^3*x^3+10/21 *a^3*b^2*x^2+5/7*a^4*b*x+a^5)*d^2*e^5-16*b^3*d^3*(7/1287*x^4*b^4+20/429*a* b^3*x^3+2/11*x^2*b^2*a^2+4/9*b*a^3*x+a^4)*e^4+128/9*b^4*(1/143*x^3*b^3+9/1 43*a*b^2*x^2+3/11*b*a^2*x+a^3)*d^4*e^3-256/33*b^5*(1/65*b^2*x^2+2/13*a*b*x +a^2)*d^5*e^2+1024/429*b^6*(1/15*b*x+a)*d^6*e-2048/6435*b^7*d^7)*(e*x+d)^( 1/2)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (184) = 368\).
Time = 0.28 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.18 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (429 \, b^{7} e^{7} x^{7} - 2048 \, b^{7} d^{7} + 15360 \, a b^{6} d^{6} e - 49920 \, a^{2} b^{5} d^{5} e^{2} + 91520 \, a^{3} b^{4} d^{4} e^{3} - 102960 \, a^{4} b^{3} d^{3} e^{4} + 72072 \, a^{5} b^{2} d^{2} e^{5} - 30030 \, a^{6} b d e^{6} + 6435 \, a^{7} e^{7} - 231 \, {\left (2 \, b^{7} d e^{6} - 15 \, a b^{6} e^{7}\right )} x^{6} + 63 \, {\left (8 \, b^{7} d^{2} e^{5} - 60 \, a b^{6} d e^{6} + 195 \, a^{2} b^{5} e^{7}\right )} x^{5} - 35 \, {\left (16 \, b^{7} d^{3} e^{4} - 120 \, a b^{6} d^{2} e^{5} + 390 \, a^{2} b^{5} d e^{6} - 715 \, a^{3} b^{4} e^{7}\right )} x^{4} + 5 \, {\left (128 \, b^{7} d^{4} e^{3} - 960 \, a b^{6} d^{3} e^{4} + 3120 \, a^{2} b^{5} d^{2} e^{5} - 5720 \, a^{3} b^{4} d e^{6} + 6435 \, a^{4} b^{3} e^{7}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{5} e^{2} - 1920 \, a b^{6} d^{4} e^{3} + 6240 \, a^{2} b^{5} d^{3} e^{4} - 11440 \, a^{3} b^{4} d^{2} e^{5} + 12870 \, a^{4} b^{3} d e^{6} - 9009 \, a^{5} b^{2} e^{7}\right )} x^{2} + {\left (1024 \, b^{7} d^{6} e - 7680 \, a b^{6} d^{5} e^{2} + 24960 \, a^{2} b^{5} d^{4} e^{3} - 45760 \, a^{3} b^{4} d^{3} e^{4} + 51480 \, a^{4} b^{3} d^{2} e^{5} - 36036 \, a^{5} b^{2} d e^{6} + 15015 \, a^{6} b e^{7}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{8}} \]
2/6435*(429*b^7*e^7*x^7 - 2048*b^7*d^7 + 15360*a*b^6*d^6*e - 49920*a^2*b^5 *d^5*e^2 + 91520*a^3*b^4*d^4*e^3 - 102960*a^4*b^3*d^3*e^4 + 72072*a^5*b^2* d^2*e^5 - 30030*a^6*b*d*e^6 + 6435*a^7*e^7 - 231*(2*b^7*d*e^6 - 15*a*b^6*e ^7)*x^6 + 63*(8*b^7*d^2*e^5 - 60*a*b^6*d*e^6 + 195*a^2*b^5*e^7)*x^5 - 35*( 16*b^7*d^3*e^4 - 120*a*b^6*d^2*e^5 + 390*a^2*b^5*d*e^6 - 715*a^3*b^4*e^7)* x^4 + 5*(128*b^7*d^4*e^3 - 960*a*b^6*d^3*e^4 + 3120*a^2*b^5*d^2*e^5 - 5720 *a^3*b^4*d*e^6 + 6435*a^4*b^3*e^7)*x^3 - 3*(256*b^7*d^5*e^2 - 1920*a*b^6*d ^4*e^3 + 6240*a^2*b^5*d^3*e^4 - 11440*a^3*b^4*d^2*e^5 + 12870*a^4*b^3*d*e^ 6 - 9009*a^5*b^2*e^7)*x^2 + (1024*b^7*d^6*e - 7680*a*b^6*d^5*e^2 + 24960*a ^2*b^5*d^4*e^3 - 45760*a^3*b^4*d^3*e^4 + 51480*a^4*b^3*d^2*e^5 - 36036*a^5 *b^2*d*e^6 + 15015*a^6*b*e^7)*x)*sqrt(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (197) = 394\).
Time = 1.67 (sec) , antiderivative size = 575, normalized size of antiderivative = 2.71 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{7} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (7 a b^{6} e - 7 b^{7} d\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (21 a^{5} b^{2} e^{5} - 105 a^{4} b^{3} d e^{4} + 210 a^{3} b^{4} d^{2} e^{3} - 210 a^{2} b^{5} d^{3} e^{2} + 105 a b^{6} d^{4} e - 21 b^{7} d^{5}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (7 a^{6} b e^{6} - 42 a^{5} b^{2} d e^{5} + 105 a^{4} b^{3} d^{2} e^{4} - 140 a^{3} b^{4} d^{3} e^{3} + 105 a^{2} b^{5} d^{4} e^{2} - 42 a b^{6} d^{5} e + 7 b^{7} d^{6}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (a^{7} e^{7} - 7 a^{6} b d e^{6} + 21 a^{5} b^{2} d^{2} e^{5} - 35 a^{4} b^{3} d^{3} e^{4} + 35 a^{3} b^{4} d^{4} e^{3} - 21 a^{2} b^{5} d^{5} e^{2} + 7 a b^{6} d^{6} e - b^{7} d^{7}\right )}{e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\begin {cases} a^{7} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{4}}{8 b} & \text {otherwise} \end {cases}}{\sqrt {d}} & \text {otherwise} \end {cases} \]
Piecewise((2*(b**7*(d + e*x)**(15/2)/(15*e**7) + (d + e*x)**(13/2)*(7*a*b* *6*e - 7*b**7*d)/(13*e**7) + (d + e*x)**(11/2)*(21*a**2*b**5*e**2 - 42*a*b **6*d*e + 21*b**7*d**2)/(11*e**7) + (d + e*x)**(9/2)*(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)/(9*e**7) + (d + e *x)**(7/2)*(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)/(7*e**7) + (d + e*x)**(5/2)*(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)/(5*e**7) + (d + e*x)**( 3/2)*(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)/(3*e**7) + sqrt(d + e*x)*(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)/e**7)/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))/sqrt(d) , True))
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (184) = 368\).
Time = 0.21 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.15 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} b^{7} - 3465 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 25025 \, {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 32175 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 27027 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 6435 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \sqrt {e x + d}\right )}}{6435 \, e^{8}} \]
2/6435*(429*(e*x + d)^(15/2)*b^7 - 3465*(b^7*d - a*b^6*e)*(e*x + d)^(13/2) + 12285*(b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*(e*x + d)^(11/2) - 25025*(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)^(9/2) + 32175*(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)^(7/2) - 27027*(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)^(5/2 ) + 15015*(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)^(3/2) - 6 435*(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 + 3 5*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
Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (184) = 368\).
Time = 0.28 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.25 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (6435 \, \sqrt {e x + d} a^{7} + \frac {15015 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{6} b}{e} + \frac {9009 \, {\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^{5} b^{2}}{e^{2}} + \frac {6435 \, {\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^{4} b^{3}}{e^{3}} + \frac {715 \, {\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^{3} b^{4}}{e^{4}} + \frac {195 \, {\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^{2} b^{5}}{e^{5}} + \frac {15 \, {\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 )} a b^{6}}{e^{6}} + \frac {{\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} - 3465 \, {\left (e x + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (e x + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {e x + d} d^{7}\right )} b^{7}}{e^{7}}\right )}}{6435 \, e} \]
2/6435*(6435*sqrt(e*x + d)*a^7 + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)* d)*a^6*b/e + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^5*b^2/e^2 + 6435*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 3 5*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^4*b^3/e^3 + 715*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d )^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^3*b^4/e^4 + 195*(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^2*b^5/e^5 + 15* (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)*a*b^6/e^6 + (429*(e*x + d)^(15/2) - 3465 *(e*x + d)^(13/2)*d + 12285*(e*x + d)^(11/2)*d^2 - 25025*(e*x + d)^(9/2)*d ^3 + 32175*(e*x + d)^(7/2)*d^4 - 27027*(e*x + d)^(5/2)*d^5 + 15015*(e*x + d)^(3/2)*d^6 - 6435*sqrt(e*x + d)*d^7)*b^7/e^7)/e
Time = 11.03 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2\,b^7\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^7\,\sqrt {d+e\,x}}{e^8}+\frac {42\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8}+\frac {10\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{7/2}}{e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8} \]
(2*b^7*(d + e*x)^(15/2))/(15*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(13 /2))/(13*e^8) + (2*(a*e - b*d)^7*(d + e*x)^(1/2))/e^8 + (42*b^2*(a*e - b*d )^5*(d + e*x)^(5/2))/(5*e^8) + (10*b^3*(a*e - b*d)^4*(d + e*x)^(7/2))/e^8 + (70*b^4*(a*e - b*d)^3*(d + e*x)^(9/2))/(9*e^8) + (42*b^5*(a*e - b*d)^2*( d + e*x)^(11/2))/(11*e^8) + (14*b*(a*e - b*d)^6*(d + e*x)^(3/2))/(3*e^8)