Integrand size = 33, antiderivative size = 360 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{14}} \, dx=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}+\frac {b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^{12}}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {2 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{10}}-\frac {5 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^9}+\frac {3 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^8}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7} \]
-1/13*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^13+1/2*b*(-a*e+b* d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^12-15/11*b^2*(-a*e+b*d)^4*((b*x +a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^11+2*b^3*(-a*e+b*d)^3*((b*x+a)^2)^(1/2)/e ^7/(b*x+a)/(e*x+d)^10-5/3*b^4*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/( e*x+d)^9+3/4*b^5*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^8-1/7*b^ 6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^7
Time = 1.08 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{14}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (924 a^6 e^6+462 a^5 b e^5 (d+13 e x)+210 a^4 b^2 e^4 \left (d^2+13 d e x+78 e^2 x^2\right )+84 a^3 b^3 e^3 \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )+28 a^2 b^4 e^2 \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )+7 a b^5 e \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )+b^6 \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )\right )}{12012 e^7 (a+b x) (d+e x)^{13}} \]
-1/12012*(Sqrt[(a + b*x)^2]*(924*a^6*e^6 + 462*a^5*b*e^5*(d + 13*e*x) + 21 0*a^4*b^2*e^4*(d^2 + 13*d*e*x + 78*e^2*x^2) + 84*a^3*b^3*e^3*(d^3 + 13*d^2 *e*x + 78*d*e^2*x^2 + 286*e^3*x^3) + 28*a^2*b^4*e^2*(d^4 + 13*d^3*e*x + 78 *d^2*e^2*x^2 + 286*d*e^3*x^3 + 715*e^4*x^4) + 7*a*b^5*e*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 + 1287*e^5*x^5) + b^6*( d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 12 87*d*e^5*x^5 + 1716*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^13)
Time = 0.39 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.55, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{14}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{14}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{14}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6}{e^6 (d+e x)^8}-\frac {6 (b d-a e) b^5}{e^6 (d+e x)^9}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^{10}}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^{11}}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^{12}}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^{13}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{14}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {3 b^5 (b d-a e)}{4 e^7 (d+e x)^8}-\frac {5 b^4 (b d-a e)^2}{3 e^7 (d+e x)^9}+\frac {2 b^3 (b d-a e)^3}{e^7 (d+e x)^{10}}-\frac {15 b^2 (b d-a e)^4}{11 e^7 (d+e x)^{11}}+\frac {b (b d-a e)^5}{2 e^7 (d+e x)^{12}}-\frac {(b d-a e)^6}{13 e^7 (d+e x)^{13}}-\frac {b^6}{7 e^7 (d+e x)^7}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/13*(b*d - a*e)^6/(e^7*(d + e*x)^13) + ( b*(b*d - a*e)^5)/(2*e^7*(d + e*x)^12) - (15*b^2*(b*d - a*e)^4)/(11*e^7*(d + e*x)^11) + (2*b^3*(b*d - a*e)^3)/(e^7*(d + e*x)^10) - (5*b^4*(b*d - a*e) ^2)/(3*e^7*(d + e*x)^9) + (3*b^5*(b*d - a*e))/(4*e^7*(d + e*x)^8) - b^6/(7 *e^7*(d + e*x)^7)))/(a + b*x)
3.21.11.3.1 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 = 20.54 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.98
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{7 e}-\frac {3 b^{5} \left (7 a e +b d \right ) x^{5}}{28 e^{2}}-\frac {5 b^{4} \left (28 e^{2} a^{2}+7 a b d e +b^{2} d^{2}\right ) x^{4}}{84 e^{3}}-\frac {b^{3} \left (84 a^{3} e^{3}+28 a^{2} b d \,e^{2}+7 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{42 e^{4}}-\frac {b^{2} \left (210 e^{4} a^{4}+84 b d \,e^{3} a^{3}+28 b^{2} d^{2} e^{2} a^{2}+7 b^{3} d^{3} e a +b^{4} d^{4}\right ) x^{2}}{154 e^{5}}-\frac {b \left (462 e^{5} a^{5}+210 b d \,e^{4} a^{4}+84 b^{2} d^{2} e^{3} a^{3}+28 b^{3} d^{3} e^{2} a^{2}+7 b^{4} d^{4} e a +b^{5} d^{5}\right ) x}{924 e^{6}}-\frac {924 e^{6} a^{6}+462 b d \,e^{5} a^{5}+210 b^{2} d^{2} e^{4} a^{4}+84 b^{3} d^{3} e^{3} a^{3}+28 b^{4} d^{4} e^{2} a^{2}+7 b^{5} d^{5} e a +b^{6} d^{6}}{12012 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{13}}\) | \(351\) |
| gosper | \(-\frac {\left (1716 b^{6} e^{6} x^{6}+9009 a \,b^{5} e^{6} x^{5}+1287 b^{6} d \,e^{5} x^{5}+20020 a^{2} b^{4} e^{6} x^{4}+5005 a \,b^{5} d \,e^{5} x^{4}+715 b^{6} d^{2} e^{4} x^{4}+24024 a^{3} b^{3} e^{6} x^{3}+8008 a^{2} b^{4} d \,e^{5} x^{3}+2002 a \,b^{5} d^{2} e^{4} x^{3}+286 b^{6} d^{3} e^{3} x^{3}+16380 a^{4} b^{2} e^{6} x^{2}+6552 a^{3} b^{3} d \,e^{5} x^{2}+2184 a^{2} b^{4} d^{2} e^{4} x^{2}+546 a \,b^{5} d^{3} e^{3} x^{2}+78 b^{6} d^{4} e^{2} x^{2}+6006 a^{5} b \,e^{6} x +2730 a^{4} b^{2} d \,e^{5} x +1092 a^{3} b^{3} d^{2} e^{4} x +364 a^{2} b^{4} d^{3} e^{3} x +91 a \,b^{5} d^{4} e^{2} x +13 b^{6} d^{5} e x +924 e^{6} a^{6}+462 b d \,e^{5} a^{5}+210 b^{2} d^{2} e^{4} a^{4}+84 b^{3} d^{3} e^{3} a^{3}+28 b^{4} d^{4} e^{2} a^{2}+7 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{12012 e^{7} \left (e x +d \right )^{13} \left (b x +a \right )^{5}}\) | \(392\) |
| default | \(-\frac {\left (1716 b^{6} e^{6} x^{6}+9009 a \,b^{5} e^{6} x^{5}+1287 b^{6} d \,e^{5} x^{5}+20020 a^{2} b^{4} e^{6} x^{4}+5005 a \,b^{5} d \,e^{5} x^{4}+715 b^{6} d^{2} e^{4} x^{4}+24024 a^{3} b^{3} e^{6} x^{3}+8008 a^{2} b^{4} d \,e^{5} x^{3}+2002 a \,b^{5} d^{2} e^{4} x^{3}+286 b^{6} d^{3} e^{3} x^{3}+16380 a^{4} b^{2} e^{6} x^{2}+6552 a^{3} b^{3} d \,e^{5} x^{2}+2184 a^{2} b^{4} d^{2} e^{4} x^{2}+546 a \,b^{5} d^{3} e^{3} x^{2}+78 b^{6} d^{4} e^{2} x^{2}+6006 a^{5} b \,e^{6} x +2730 a^{4} b^{2} d \,e^{5} x +1092 a^{3} b^{3} d^{2} e^{4} x +364 a^{2} b^{4} d^{3} e^{3} x +91 a \,b^{5} d^{4} e^{2} x +13 b^{6} d^{5} e x +924 e^{6} a^{6}+462 b d \,e^{5} a^{5}+210 b^{2} d^{2} e^{4} a^{4}+84 b^{3} d^{3} e^{3} a^{3}+28 b^{4} d^{4} e^{2} a^{2}+7 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{12012 e^{7} \left (e x +d \right )^{13} \left (b x +a \right )^{5}}\) | \(392\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/7*b^6/e*x^6-3/28*b^5/e^2*(7*a*e+b*d)*x^5-5/8 4*b^4/e^3*(28*a^2*e^2+7*a*b*d*e+b^2*d^2)*x^4-1/42*b^3/e^4*(84*a^3*e^3+28*a ^2*b*d*e^2+7*a*b^2*d^2*e+b^3*d^3)*x^3-1/154*b^2/e^5*(210*a^4*e^4+84*a^3*b* d*e^3+28*a^2*b^2*d^2*e^2+7*a*b^3*d^3*e+b^4*d^4)*x^2-1/924*b/e^6*(462*a^5*e ^5+210*a^4*b*d*e^4+84*a^3*b^2*d^2*e^3+28*a^2*b^3*d^3*e^2+7*a*b^4*d^4*e+b^5 *d^5)*x-1/12012/e^7*(924*a^6*e^6+462*a^5*b*d*e^5+210*a^4*b^2*d^2*e^4+84*a^ 3*b^3*d^3*e^3+28*a^2*b^4*d^4*e^2+7*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^13
Time = 0.27 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{14}} \, dx=-\frac {1716 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 7 \, a b^{5} d^{5} e + 28 \, a^{2} b^{4} d^{4} e^{2} + 84 \, a^{3} b^{3} d^{3} e^{3} + 210 \, a^{4} b^{2} d^{2} e^{4} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6} + 1287 \, {\left (b^{6} d e^{5} + 7 \, a b^{5} e^{6}\right )} x^{5} + 715 \, {\left (b^{6} d^{2} e^{4} + 7 \, a b^{5} d e^{5} + 28 \, a^{2} b^{4} e^{6}\right )} x^{4} + 286 \, {\left (b^{6} d^{3} e^{3} + 7 \, a b^{5} d^{2} e^{4} + 28 \, a^{2} b^{4} d e^{5} + 84 \, a^{3} b^{3} e^{6}\right )} x^{3} + 78 \, {\left (b^{6} d^{4} e^{2} + 7 \, a b^{5} d^{3} e^{3} + 28 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} + 210 \, a^{4} b^{2} e^{6}\right )} x^{2} + 13 \, {\left (b^{6} d^{5} e + 7 \, a b^{5} d^{4} e^{2} + 28 \, a^{2} b^{4} d^{3} e^{3} + 84 \, a^{3} b^{3} d^{2} e^{4} + 210 \, a^{4} b^{2} d e^{5} + 462 \, a^{5} b e^{6}\right )} x}{12012 \, {\left (e^{20} x^{13} + 13 \, d e^{19} x^{12} + 78 \, d^{2} e^{18} x^{11} + 286 \, d^{3} e^{17} x^{10} + 715 \, d^{4} e^{16} x^{9} + 1287 \, d^{5} e^{15} x^{8} + 1716 \, d^{6} e^{14} x^{7} + 1716 \, d^{7} e^{13} x^{6} + 1287 \, d^{8} e^{12} x^{5} + 715 \, d^{9} e^{11} x^{4} + 286 \, d^{10} e^{10} x^{3} + 78 \, d^{11} e^{9} x^{2} + 13 \, d^{12} e^{8} x + d^{13} e^{7}\right )}} \]
-1/12012*(1716*b^6*e^6*x^6 + b^6*d^6 + 7*a*b^5*d^5*e + 28*a^2*b^4*d^4*e^2 + 84*a^3*b^3*d^3*e^3 + 210*a^4*b^2*d^2*e^4 + 462*a^5*b*d*e^5 + 924*a^6*e^6 + 1287*(b^6*d*e^5 + 7*a*b^5*e^6)*x^5 + 715*(b^6*d^2*e^4 + 7*a*b^5*d*e^5 + 28*a^2*b^4*e^6)*x^4 + 286*(b^6*d^3*e^3 + 7*a*b^5*d^2*e^4 + 28*a^2*b^4*d*e ^5 + 84*a^3*b^3*e^6)*x^3 + 78*(b^6*d^4*e^2 + 7*a*b^5*d^3*e^3 + 28*a^2*b^4* d^2*e^4 + 84*a^3*b^3*d*e^5 + 210*a^4*b^2*e^6)*x^2 + 13*(b^6*d^5*e + 7*a*b^ 5*d^4*e^2 + 28*a^2*b^4*d^3*e^3 + 84*a^3*b^3*d^2*e^4 + 210*a^4*b^2*d*e^5 + 462*a^5*b*e^6)*x)/(e^20*x^13 + 13*d*e^19*x^12 + 78*d^2*e^18*x^11 + 286*d^3 *e^17*x^10 + 715*d^4*e^16*x^9 + 1287*d^5*e^15*x^8 + 1716*d^6*e^14*x^7 + 17 16*d^7*e^13*x^6 + 1287*d^8*e^12*x^5 + 715*d^9*e^11*x^4 + 286*d^10*e^10*x^3 + 78*d^11*e^9*x^2 + 13*d^12*e^8*x + d^13*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{14}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{14}} \, dx=\text {Exception raised: ValueError} \]
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. 654 vs. \(2 (271) = 542\).
Time = 0.29 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.82 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{14}} \, dx=\frac {b^{13} \mathrm {sgn}\left (b x + a\right )}{12012 \, {\left (b^{7} d^{7} e^{7} - 7 \, a b^{6} d^{6} e^{8} + 21 \, a^{2} b^{5} d^{5} e^{9} - 35 \, a^{3} b^{4} d^{4} e^{10} + 35 \, a^{4} b^{3} d^{3} e^{11} - 21 \, a^{5} b^{2} d^{2} e^{12} + 7 \, a^{6} b d e^{13} - a^{7} e^{14}\right )}} - \frac {1716 \, b^{6} e^{6} x^{6} \mathrm {sgn}\left (b x + a\right ) + 1287 \, b^{6} d e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 9009 \, a b^{5} e^{6} x^{5} \mathrm {sgn}\left (b x + a\right ) + 715 \, b^{6} d^{2} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 5005 \, a b^{5} d e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 20020 \, a^{2} b^{4} e^{6} x^{4} \mathrm {sgn}\left (b x + a\right ) + 286 \, b^{6} d^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2002 \, a b^{5} d^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 8008 \, a^{2} b^{4} d e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 24024 \, a^{3} b^{3} e^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) + 78 \, b^{6} d^{4} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 546 \, a b^{5} d^{3} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 2184 \, a^{2} b^{4} d^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6552 \, a^{3} b^{3} d e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 16380 \, a^{4} b^{2} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 13 \, b^{6} d^{5} e x \mathrm {sgn}\left (b x + a\right ) + 91 \, a b^{5} d^{4} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 364 \, a^{2} b^{4} d^{3} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 1092 \, a^{3} b^{3} d^{2} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 2730 \, a^{4} b^{2} d e^{5} x \mathrm {sgn}\left (b x + a\right ) + 6006 \, a^{5} b e^{6} x \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 7 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 28 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 210 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 462 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 924 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{12012 \, {\left (e x + d\right )}^{13} e^{7}} \]
1/12012*b^13*sgn(b*x + a)/(b^7*d^7*e^7 - 7*a*b^6*d^6*e^8 + 21*a^2*b^5*d^5* e^9 - 35*a^3*b^4*d^4*e^10 + 35*a^4*b^3*d^3*e^11 - 21*a^5*b^2*d^2*e^12 + 7* a^6*b*d*e^13 - a^7*e^14) - 1/12012*(1716*b^6*e^6*x^6*sgn(b*x + a) + 1287*b ^6*d*e^5*x^5*sgn(b*x + a) + 9009*a*b^5*e^6*x^5*sgn(b*x + a) + 715*b^6*d^2* e^4*x^4*sgn(b*x + a) + 5005*a*b^5*d*e^5*x^4*sgn(b*x + a) + 20020*a^2*b^4*e ^6*x^4*sgn(b*x + a) + 286*b^6*d^3*e^3*x^3*sgn(b*x + a) + 2002*a*b^5*d^2*e^ 4*x^3*sgn(b*x + a) + 8008*a^2*b^4*d*e^5*x^3*sgn(b*x + a) + 24024*a^3*b^3*e ^6*x^3*sgn(b*x + a) + 78*b^6*d^4*e^2*x^2*sgn(b*x + a) + 546*a*b^5*d^3*e^3* x^2*sgn(b*x + a) + 2184*a^2*b^4*d^2*e^4*x^2*sgn(b*x + a) + 6552*a^3*b^3*d* e^5*x^2*sgn(b*x + a) + 16380*a^4*b^2*e^6*x^2*sgn(b*x + a) + 13*b^6*d^5*e*x *sgn(b*x + a) + 91*a*b^5*d^4*e^2*x*sgn(b*x + a) + 364*a^2*b^4*d^3*e^3*x*sg n(b*x + a) + 1092*a^3*b^3*d^2*e^4*x*sgn(b*x + a) + 2730*a^4*b^2*d*e^5*x*sg n(b*x + a) + 6006*a^5*b*e^6*x*sgn(b*x + a) + b^6*d^6*sgn(b*x + a) + 7*a*b^ 5*d^5*e*sgn(b*x + a) + 28*a^2*b^4*d^4*e^2*sgn(b*x + a) + 84*a^3*b^3*d^3*e^ 3*sgn(b*x + a) + 210*a^4*b^2*d^2*e^4*sgn(b*x + a) + 462*a^5*b*d*e^5*sgn(b* x + a) + 924*a^6*e^6*sgn(b*x + a))/((e*x + d)^13*e^7)
Time = 11.13 (sec) , antiderivative size = 1010, normalized size of antiderivative = 2.81 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{14}} \, 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}{12\,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}{12\,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}{12\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{12\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{12\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{12\,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 )}^{12}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{9\,e^7}+\frac {d\,\left (\frac {b^6\,d}{9\,e^6}-\frac {2\,b^5\,\left (3\,a\,e-2\,b\,d\right )}{9\,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 )}^9}-\frac {\left (\frac {a^6}{13\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{13\,e}-\frac {b^6\,d}{13\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{13\,e}\right )}{e}+\frac {20\,a^3\,b^3}{13\,e}\right )}{e}-\frac {15\,a^4\,b^2}{13\,e}\right )}{e}+\frac {6\,a^5\,b}{13\,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 )}^{13}}-\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}{11\,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}{11\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{11\,e^4}-\frac {2\,b^5\,\left (3\,a\,e-b\,d\right )}{11\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{11\,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 )}^{11}}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{8\,e^7}+\frac {b^6\,d}{8\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}+\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}{10\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{10\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{10\,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 )}{10\,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 )}^{10}}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7} \]
(((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)/(12*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)/(12*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)/(12*e^7) - (d*((d*((b^6*d)/(12*e^3) - (b^5*(6*a*e - b*d))/(12*e^3)))/e + (b^4*(15* a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(12*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a *b*x)^(1/2))/((a + b*x)*(d + e*x)^12) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 2 4*a*b^5*d*e)/(9*e^7) + (d*((b^6*d)/(9*e^6) - (2*b^5*(3*a*e - 2*b*d))/(9*e^ 6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) - ((a^6/( 13*e) - (d*((d*((d*((d*((d*((6*a*b^5)/(13*e) - (b^6*d)/(13*e^2)))/e - (15* a^2*b^4)/(13*e)))/e + (20*a^3*b^3)/(13*e)))/e - (15*a^4*b^2)/(13*e)))/e + (6*a^5*b)/(13*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x )^13) - (((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)/(11*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)/(11*e^7) + (d*((d*((b^6*d)/(11*e^4) - (2*b ^5*(3*a*e - b*d))/(11*e^4)))/e + (3*b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e)) /(11*e^5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^1 1) + (((5*b^6*d - 6*a*b^5*e)/(8*e^7) + (b^6*d)/(8*e^7))*(a^2 + b^2*x^2 + 2 *a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) + (((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)/(10*e^7) + (d*((d*((b^6*d)/(10*e^5) ...