Integrand size = 33, antiderivative size = 362 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{16 e^7 (a+b x) (d+e x)^{16}}+\frac {2 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{15}}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x) (d+e x)^{14}}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}-\frac {5 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^{12}}+\frac {6 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}} \]
-1/16*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^16+2/5*b*(-a*e+b* d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^15-15/14*b^2*(-a*e+b*d)^4*((b*x +a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^14+20/13*b^3*(-a*e+b*d)^3*((b*x+a)^2)^(1/ 2)/e^7/(b*x+a)/(e*x+d)^13-5/4*b^4*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+ a)/(e*x+d)^12+6/11*b^5*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^11 -1/10*b^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^10
Time = 1.08 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (5005 a^6 e^6+2002 a^5 b e^5 (d+16 e x)+715 a^4 b^2 e^4 \left (d^2+16 d e x+120 e^2 x^2\right )+220 a^3 b^3 e^3 \left (d^3+16 d^2 e x+120 d e^2 x^2+560 e^3 x^3\right )+55 a^2 b^4 e^2 \left (d^4+16 d^3 e x+120 d^2 e^2 x^2+560 d e^3 x^3+1820 e^4 x^4\right )+10 a b^5 e \left (d^5+16 d^4 e x+120 d^3 e^2 x^2+560 d^2 e^3 x^3+1820 d e^4 x^4+4368 e^5 x^5\right )+b^6 \left (d^6+16 d^5 e x+120 d^4 e^2 x^2+560 d^3 e^3 x^3+1820 d^2 e^4 x^4+4368 d e^5 x^5+8008 e^6 x^6\right )\right )}{80080 e^7 (a+b x) (d+e x)^{16}} \]
-1/80080*(Sqrt[(a + b*x)^2]*(5005*a^6*e^6 + 2002*a^5*b*e^5*(d + 16*e*x) + 715*a^4*b^2*e^4*(d^2 + 16*d*e*x + 120*e^2*x^2) + 220*a^3*b^3*e^3*(d^3 + 16 *d^2*e*x + 120*d*e^2*x^2 + 560*e^3*x^3) + 55*a^2*b^4*e^2*(d^4 + 16*d^3*e*x + 120*d^2*e^2*x^2 + 560*d*e^3*x^3 + 1820*e^4*x^4) + 10*a*b^5*e*(d^5 + 16* d^4*e*x + 120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 1820*d*e^4*x^4 + 4368*e^5*x^ 5) + b^6*(d^6 + 16*d^5*e*x + 120*d^4*e^2*x^2 + 560*d^3*e^3*x^3 + 1820*d^2* e^4*x^4 + 4368*d*e^5*x^5 + 8008*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^16)
Time = 0.38 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.56, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{17}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{17}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6}{e^6 (d+e x)^{11}}-\frac {6 (b d-a e) b^5}{e^6 (d+e x)^{12}}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^{13}}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^{14}}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^{15}}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^{16}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{17}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {6 b^5 (b d-a e)}{11 e^7 (d+e x)^{11}}-\frac {5 b^4 (b d-a e)^2}{4 e^7 (d+e x)^{12}}+\frac {20 b^3 (b d-a e)^3}{13 e^7 (d+e x)^{13}}-\frac {15 b^2 (b d-a e)^4}{14 e^7 (d+e x)^{14}}+\frac {2 b (b d-a e)^5}{5 e^7 (d+e x)^{15}}-\frac {(b d-a e)^6}{16 e^7 (d+e x)^{16}}-\frac {b^6}{10 e^7 (d+e x)^{10}}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/16*(b*d - a*e)^6/(e^7*(d + e*x)^16) + ( 2*b*(b*d - a*e)^5)/(5*e^7*(d + e*x)^15) - (15*b^2*(b*d - a*e)^4)/(14*e^7*( d + e*x)^14) + (20*b^3*(b*d - a*e)^3)/(13*e^7*(d + e*x)^13) - (5*b^4*(b*d - a*e)^2)/(4*e^7*(d + e*x)^12) + (6*b^5*(b*d - a*e))/(11*e^7*(d + e*x)^11) - b^6/(10*e^7*(d + e*x)^10)))/(a + b*x)
3.21.14.3.1 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[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 86.88 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {5005 e^{6} a^{6}+2002 b d \,e^{5} a^{5}+715 b^{2} d^{2} e^{4} a^{4}+220 b^{3} d^{3} e^{3} a^{3}+55 b^{4} d^{4} e^{2} a^{2}+10 b^{5} d^{5} e a +b^{6} d^{6}}{80080 e^{7}}-\frac {b \left (2002 e^{5} a^{5}+715 b d \,e^{4} a^{4}+220 b^{2} d^{2} e^{3} a^{3}+55 b^{3} d^{3} e^{2} a^{2}+10 b^{4} d^{4} e a +b^{5} d^{5}\right ) x}{5005 e^{6}}-\frac {3 b^{2} \left (715 e^{4} a^{4}+220 b d \,e^{3} a^{3}+55 b^{2} d^{2} e^{2} a^{2}+10 b^{3} d^{3} e a +b^{4} d^{4}\right ) x^{2}}{2002 e^{5}}-\frac {b^{3} \left (220 a^{3} e^{3}+55 a^{2} b d \,e^{2}+10 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{143 e^{4}}-\frac {b^{4} \left (55 e^{2} a^{2}+10 a b d e +b^{2} d^{2}\right ) x^{4}}{44 e^{3}}-\frac {3 b^{5} \left (10 a e +b d \right ) x^{5}}{55 e^{2}}-\frac {b^{6} x^{6}}{10 e}\right )}{\left (b x +a \right ) \left (e x +d \right )^{16}}\) | \(351\) |
| gosper | \(-\frac {\left (8008 b^{6} e^{6} x^{6}+43680 a \,b^{5} e^{6} x^{5}+4368 b^{6} d \,e^{5} x^{5}+100100 a^{2} b^{4} e^{6} x^{4}+18200 a \,b^{5} d \,e^{5} x^{4}+1820 b^{6} d^{2} e^{4} x^{4}+123200 a^{3} b^{3} e^{6} x^{3}+30800 a^{2} b^{4} d \,e^{5} x^{3}+5600 a \,b^{5} d^{2} e^{4} x^{3}+560 b^{6} d^{3} e^{3} x^{3}+85800 a^{4} b^{2} e^{6} x^{2}+26400 a^{3} b^{3} d \,e^{5} x^{2}+6600 a^{2} b^{4} d^{2} e^{4} x^{2}+1200 a \,b^{5} d^{3} e^{3} x^{2}+120 b^{6} d^{4} e^{2} x^{2}+32032 a^{5} b \,e^{6} x +11440 a^{4} b^{2} d \,e^{5} x +3520 a^{3} b^{3} d^{2} e^{4} x +880 a^{2} b^{4} d^{3} e^{3} x +160 a \,b^{5} d^{4} e^{2} x +16 b^{6} d^{5} e x +5005 e^{6} a^{6}+2002 b d \,e^{5} a^{5}+715 b^{2} d^{2} e^{4} a^{4}+220 b^{3} d^{3} e^{3} a^{3}+55 b^{4} d^{4} e^{2} a^{2}+10 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{80080 e^{7} \left (e x +d \right )^{16} \left (b x +a \right )^{5}}\) | \(392\) |
| default | \(-\frac {\left (8008 b^{6} e^{6} x^{6}+43680 a \,b^{5} e^{6} x^{5}+4368 b^{6} d \,e^{5} x^{5}+100100 a^{2} b^{4} e^{6} x^{4}+18200 a \,b^{5} d \,e^{5} x^{4}+1820 b^{6} d^{2} e^{4} x^{4}+123200 a^{3} b^{3} e^{6} x^{3}+30800 a^{2} b^{4} d \,e^{5} x^{3}+5600 a \,b^{5} d^{2} e^{4} x^{3}+560 b^{6} d^{3} e^{3} x^{3}+85800 a^{4} b^{2} e^{6} x^{2}+26400 a^{3} b^{3} d \,e^{5} x^{2}+6600 a^{2} b^{4} d^{2} e^{4} x^{2}+1200 a \,b^{5} d^{3} e^{3} x^{2}+120 b^{6} d^{4} e^{2} x^{2}+32032 a^{5} b \,e^{6} x +11440 a^{4} b^{2} d \,e^{5} x +3520 a^{3} b^{3} d^{2} e^{4} x +880 a^{2} b^{4} d^{3} e^{3} x +160 a \,b^{5} d^{4} e^{2} x +16 b^{6} d^{5} e x +5005 e^{6} a^{6}+2002 b d \,e^{5} a^{5}+715 b^{2} d^{2} e^{4} a^{4}+220 b^{3} d^{3} e^{3} a^{3}+55 b^{4} d^{4} e^{2} a^{2}+10 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{80080 e^{7} \left (e x +d \right )^{16} \left (b x +a \right )^{5}}\) | \(392\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/80080/e^7*(5005*a^6*e^6+2002*a^5*b*d*e^5+715 *a^4*b^2*d^2*e^4+220*a^3*b^3*d^3*e^3+55*a^2*b^4*d^4*e^2+10*a*b^5*d^5*e+b^6 *d^6)-1/5005*b/e^6*(2002*a^5*e^5+715*a^4*b*d*e^4+220*a^3*b^2*d^2*e^3+55*a^ 2*b^3*d^3*e^2+10*a*b^4*d^4*e+b^5*d^5)*x-3/2002*b^2/e^5*(715*a^4*e^4+220*a^ 3*b*d*e^3+55*a^2*b^2*d^2*e^2+10*a*b^3*d^3*e+b^4*d^4)*x^2-1/143*b^3/e^4*(22 0*a^3*e^3+55*a^2*b*d*e^2+10*a*b^2*d^2*e+b^3*d^3)*x^3-1/44*b^4/e^3*(55*a^2* e^2+10*a*b*d*e+b^2*d^2)*x^4-3/55*b^5/e^2*(10*a*e+b*d)*x^5-1/10*b^6/e*x^6)/ (e*x+d)^16
Time = 0.37 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=-\frac {8008 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 10 \, a b^{5} d^{5} e + 55 \, a^{2} b^{4} d^{4} e^{2} + 220 \, a^{3} b^{3} d^{3} e^{3} + 715 \, a^{4} b^{2} d^{2} e^{4} + 2002 \, a^{5} b d e^{5} + 5005 \, a^{6} e^{6} + 4368 \, {\left (b^{6} d e^{5} + 10 \, a b^{5} e^{6}\right )} x^{5} + 1820 \, {\left (b^{6} d^{2} e^{4} + 10 \, a b^{5} d e^{5} + 55 \, a^{2} b^{4} e^{6}\right )} x^{4} + 560 \, {\left (b^{6} d^{3} e^{3} + 10 \, a b^{5} d^{2} e^{4} + 55 \, a^{2} b^{4} d e^{5} + 220 \, a^{3} b^{3} e^{6}\right )} x^{3} + 120 \, {\left (b^{6} d^{4} e^{2} + 10 \, a b^{5} d^{3} e^{3} + 55 \, a^{2} b^{4} d^{2} e^{4} + 220 \, a^{3} b^{3} d e^{5} + 715 \, a^{4} b^{2} e^{6}\right )} x^{2} + 16 \, {\left (b^{6} d^{5} e + 10 \, a b^{5} d^{4} e^{2} + 55 \, a^{2} b^{4} d^{3} e^{3} + 220 \, a^{3} b^{3} d^{2} e^{4} + 715 \, a^{4} b^{2} d e^{5} + 2002 \, a^{5} b e^{6}\right )} x}{80080 \, {\left (e^{23} x^{16} + 16 \, d e^{22} x^{15} + 120 \, d^{2} e^{21} x^{14} + 560 \, d^{3} e^{20} x^{13} + 1820 \, d^{4} e^{19} x^{12} + 4368 \, d^{5} e^{18} x^{11} + 8008 \, d^{6} e^{17} x^{10} + 11440 \, d^{7} e^{16} x^{9} + 12870 \, d^{8} e^{15} x^{8} + 11440 \, d^{9} e^{14} x^{7} + 8008 \, d^{10} e^{13} x^{6} + 4368 \, d^{11} e^{12} x^{5} + 1820 \, d^{12} e^{11} x^{4} + 560 \, d^{13} e^{10} x^{3} + 120 \, d^{14} e^{9} x^{2} + 16 \, d^{15} e^{8} x + d^{16} e^{7}\right )}} \]
-1/80080*(8008*b^6*e^6*x^6 + b^6*d^6 + 10*a*b^5*d^5*e + 55*a^2*b^4*d^4*e^2 + 220*a^3*b^3*d^3*e^3 + 715*a^4*b^2*d^2*e^4 + 2002*a^5*b*d*e^5 + 5005*a^6 *e^6 + 4368*(b^6*d*e^5 + 10*a*b^5*e^6)*x^5 + 1820*(b^6*d^2*e^4 + 10*a*b^5* d*e^5 + 55*a^2*b^4*e^6)*x^4 + 560*(b^6*d^3*e^3 + 10*a*b^5*d^2*e^4 + 55*a^2 *b^4*d*e^5 + 220*a^3*b^3*e^6)*x^3 + 120*(b^6*d^4*e^2 + 10*a*b^5*d^3*e^3 + 55*a^2*b^4*d^2*e^4 + 220*a^3*b^3*d*e^5 + 715*a^4*b^2*e^6)*x^2 + 16*(b^6*d^ 5*e + 10*a*b^5*d^4*e^2 + 55*a^2*b^4*d^3*e^3 + 220*a^3*b^3*d^2*e^4 + 715*a^ 4*b^2*d*e^5 + 2002*a^5*b*e^6)*x)/(e^23*x^16 + 16*d*e^22*x^15 + 120*d^2*e^2 1*x^14 + 560*d^3*e^20*x^13 + 1820*d^4*e^19*x^12 + 4368*d^5*e^18*x^11 + 800 8*d^6*e^17*x^10 + 11440*d^7*e^16*x^9 + 12870*d^8*e^15*x^8 + 11440*d^9*e^14 *x^7 + 8008*d^10*e^13*x^6 + 4368*d^11*e^12*x^5 + 1820*d^12*e^11*x^4 + 560* d^13*e^10*x^3 + 120*d^14*e^9*x^2 + 16*d^15*e^8*x + d^16*e^7)
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (271) = 542\).
Time = 0.28 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.92 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=\frac {b^{16} \mathrm {sgn}\left (b x + a\right )}{80080 \, {\left (b^{10} d^{10} e^{7} - 10 \, a b^{9} d^{9} e^{8} + 45 \, a^{2} b^{8} d^{8} e^{9} - 120 \, a^{3} b^{7} d^{7} e^{10} + 210 \, a^{4} b^{6} d^{6} e^{11} - 252 \, a^{5} b^{5} d^{5} e^{12} + 210 \, a^{6} b^{4} d^{4} e^{13} - 120 \, a^{7} b^{3} d^{3} e^{14} + 45 \, a^{8} b^{2} d^{2} e^{15} - 10 \, a^{9} b d e^{16} + a^{10} e^{17}\right )}} - \frac {8008 \, b^{6} e^{6} x^{6} \mathrm {sgn}\left (b x + a\right ) + 4368 \, b^{6} d e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 43680 \, a b^{5} e^{6} x^{5} \mathrm {sgn}\left (b x + a\right ) + 1820 \, b^{6} d^{2} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 18200 \, a b^{5} d e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 100100 \, a^{2} b^{4} e^{6} x^{4} \mathrm {sgn}\left (b x + a\right ) + 560 \, b^{6} d^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 5600 \, a b^{5} d^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 30800 \, a^{2} b^{4} d e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 123200 \, a^{3} b^{3} e^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) + 120 \, b^{6} d^{4} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1200 \, a b^{5} d^{3} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6600 \, a^{2} b^{4} d^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 26400 \, a^{3} b^{3} d e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 85800 \, a^{4} b^{2} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 16 \, b^{6} d^{5} e x \mathrm {sgn}\left (b x + a\right ) + 160 \, a b^{5} d^{4} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 880 \, a^{2} b^{4} d^{3} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 3520 \, a^{3} b^{3} d^{2} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 11440 \, a^{4} b^{2} d e^{5} x \mathrm {sgn}\left (b x + a\right ) + 32032 \, a^{5} b e^{6} x \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 10 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 55 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 220 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 715 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 2002 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 5005 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{80080 \, {\left (e x + d\right )}^{16} e^{7}} \]
1/80080*b^16*sgn(b*x + a)/(b^10*d^10*e^7 - 10*a*b^9*d^9*e^8 + 45*a^2*b^8*d ^8*e^9 - 120*a^3*b^7*d^7*e^10 + 210*a^4*b^6*d^6*e^11 - 252*a^5*b^5*d^5*e^1 2 + 210*a^6*b^4*d^4*e^13 - 120*a^7*b^3*d^3*e^14 + 45*a^8*b^2*d^2*e^15 - 10 *a^9*b*d*e^16 + a^10*e^17) - 1/80080*(8008*b^6*e^6*x^6*sgn(b*x + a) + 4368 *b^6*d*e^5*x^5*sgn(b*x + a) + 43680*a*b^5*e^6*x^5*sgn(b*x + a) + 1820*b^6* d^2*e^4*x^4*sgn(b*x + a) + 18200*a*b^5*d*e^5*x^4*sgn(b*x + a) + 100100*a^2 *b^4*e^6*x^4*sgn(b*x + a) + 560*b^6*d^3*e^3*x^3*sgn(b*x + a) + 5600*a*b^5* d^2*e^4*x^3*sgn(b*x + a) + 30800*a^2*b^4*d*e^5*x^3*sgn(b*x + a) + 123200*a ^3*b^3*e^6*x^3*sgn(b*x + a) + 120*b^6*d^4*e^2*x^2*sgn(b*x + a) + 1200*a*b^ 5*d^3*e^3*x^2*sgn(b*x + a) + 6600*a^2*b^4*d^2*e^4*x^2*sgn(b*x + a) + 26400 *a^3*b^3*d*e^5*x^2*sgn(b*x + a) + 85800*a^4*b^2*e^6*x^2*sgn(b*x + a) + 16* b^6*d^5*e*x*sgn(b*x + a) + 160*a*b^5*d^4*e^2*x*sgn(b*x + a) + 880*a^2*b^4* d^3*e^3*x*sgn(b*x + a) + 3520*a^3*b^3*d^2*e^4*x*sgn(b*x + a) + 11440*a^4*b ^2*d*e^5*x*sgn(b*x + a) + 32032*a^5*b*e^6*x*sgn(b*x + a) + b^6*d^6*sgn(b*x + a) + 10*a*b^5*d^5*e*sgn(b*x + a) + 55*a^2*b^4*d^4*e^2*sgn(b*x + a) + 22 0*a^3*b^3*d^3*e^3*sgn(b*x + a) + 715*a^4*b^2*d^2*e^4*sgn(b*x + a) + 2002*a ^5*b*d*e^5*sgn(b*x + a) + 5005*a^6*e^6*sgn(b*x + a))/((e*x + d)^16*e^7)
Time = 11.20 (sec) , antiderivative size = 1010, normalized size of antiderivative = 2.79 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=\frac {\left (\frac {-6\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-20\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-6\,a\,b^5\,d^4\,e+b^6\,d^5}{15\,e^7}+\frac {d\,\left (\frac {15\,a^4\,b^2\,e^5-20\,a^3\,b^3\,d\,e^4+15\,a^2\,b^4\,d^2\,e^3-6\,a\,b^5\,d^3\,e^2+b^6\,d^4\,e}{15\,e^7}-\frac {d\,\left (\frac {20\,a^3\,b^3\,e^5-15\,a^2\,b^4\,d\,e^4+6\,a\,b^5\,d^2\,e^3-b^6\,d^3\,e^2}{15\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{15\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{15\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{15\,e^4}\right )}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{15}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{12\,e^7}+\frac {d\,\left (\frac {b^6\,d}{12\,e^6}-\frac {b^5\,\left (3\,a\,e-2\,b\,d\right )}{6\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{12}}-\frac {\left (\frac {a^6}{16\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a\,b^5}{8\,e}-\frac {b^6\,d}{16\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{16\,e}\right )}{e}+\frac {5\,a^3\,b^3}{4\,e}\right )}{e}-\frac {15\,a^4\,b^2}{16\,e}\right )}{e}+\frac {3\,a^5\,b}{8\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{16}}-\frac {\left (\frac {15\,a^4\,b^2\,e^4-40\,a^3\,b^3\,d\,e^3+45\,a^2\,b^4\,d^2\,e^2-24\,a\,b^5\,d^3\,e+5\,b^6\,d^4}{14\,e^7}+\frac {d\,\left (\frac {-20\,a^3\,b^3\,e^4+30\,a^2\,b^4\,d\,e^3-18\,a\,b^5\,d^2\,e^2+4\,b^6\,d^3\,e}{14\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{14\,e^4}-\frac {b^5\,\left (3\,a\,e-b\,d\right )}{7\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{14\,e^5}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{14}}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{11\,e^7}+\frac {b^6\,d}{11\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{11}}+\frac {\left (\frac {-20\,a^3\,b^3\,e^3+45\,a^2\,b^4\,d\,e^2-36\,a\,b^5\,d^2\,e+10\,b^6\,d^3}{13\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{13\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{13\,e^5}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{13\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{13}}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{10\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}} \]
(((b^6*d^5 - 6*a^5*b*e^5 + 15*a^4*b^2*d*e^4 + 15*a^2*b^4*d^3*e^2 - 20*a^3* b^3*d^2*e^3 - 6*a*b^5*d^4*e)/(15*e^7) + (d*((b^6*d^4*e + 15*a^4*b^2*e^5 - 6*a*b^5*d^3*e^2 - 20*a^3*b^3*d*e^4 + 15*a^2*b^4*d^2*e^3)/(15*e^7) - (d*((2 0*a^3*b^3*e^5 - b^6*d^3*e^2 + 6*a*b^5*d^2*e^3 - 15*a^2*b^4*d*e^4)/(15*e^7) - (d*((d*((b^6*d)/(15*e^3) - (b^5*(6*a*e - b*d))/(15*e^3)))/e + (b^4*(15* a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(15*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a *b*x)^(1/2))/((a + b*x)*(d + e*x)^15) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 2 4*a*b^5*d*e)/(12*e^7) + (d*((b^6*d)/(12*e^6) - (b^5*(3*a*e - 2*b*d))/(6*e^ 6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^12) - ((a^6/ (16*e) - (d*((d*((d*((d*((d*((3*a*b^5)/(8*e) - (b^6*d)/(16*e^2)))/e - (15* a^2*b^4)/(16*e)))/e + (5*a^3*b^3)/(4*e)))/e - (15*a^4*b^2)/(16*e)))/e + (3 *a^5*b)/(8*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^1 6) - (((5*b^6*d^4 + 15*a^4*b^2*e^4 - 40*a^3*b^3*d*e^3 + 45*a^2*b^4*d^2*e^2 - 24*a*b^5*d^3*e)/(14*e^7) + (d*((4*b^6*d^3*e - 20*a^3*b^3*e^4 - 18*a*b^5 *d^2*e^2 + 30*a^2*b^4*d*e^3)/(14*e^7) + (d*((d*((b^6*d)/(14*e^4) - (b^5*(3 *a*e - b*d))/(7*e^4)))/e + (3*b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(14*e ^5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^14) + ( ((5*b^6*d - 6*a*b^5*e)/(11*e^7) + (b^6*d)/(11*e^7))*(a^2 + b^2*x^2 + 2*a*b *x)^(1/2))/((a + b*x)*(d + e*x)^11) + (((10*b^6*d^3 - 20*a^3*b^3*e^3 + 45* a^2*b^4*d*e^2 - 36*a*b^5*d^2*e)/(13*e^7) + (d*((d*((b^6*d)/(13*e^5) - (...