Integrand size = 33, antiderivative size = 200 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{120 (b d-a e)^3 (d+e x)^8}+\frac {b^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{840 (b d-a e)^4 (d+e x)^7} \]
1/10*(b*x+a)^6*((b*x+a)^2)^(1/2)/(-a*e+b*d)/(e*x+d)^10+1/30*b*(b*x+a)^6*(( b*x+a)^2)^(1/2)/(-a*e+b*d)^2/(e*x+d)^9+1/120*b^2*(b*x+a)^6*((b*x+a)^2)^(1/ 2)/(-a*e+b*d)^3/(e*x+d)^8+1/840*b^3*(b*x+a)^6*((b*x+a)^2)^(1/2)/(-a*e+b*d) ^4/(e*x+d)^7
Time = 1.08 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.48 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (84 a^6 e^6+56 a^5 b e^5 (d+10 e x)+35 a^4 b^2 e^4 \left (d^2+10 d e x+45 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+10 a^2 b^4 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+4 a b^5 e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+b^6 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )}{840 e^7 (a+b x) (d+e x)^{10}} \]
-1/840*(Sqrt[(a + b*x)^2]*(84*a^6*e^6 + 56*a^5*b*e^5*(d + 10*e*x) + 35*a^4 *b^2*e^4*(d^2 + 10*d*e*x + 45*e^2*x^2) + 20*a^3*b^3*e^3*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 10*a^2*b^4*e^2*(d^4 + 10*d^3*e*x + 45*d^2* e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + 4*a*b^5*e*(d^5 + 10*d^4*e*x + 45* d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) + b^6*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^ 5*x^5 + 210*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^10)
Time = 0.27 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.87, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1187, 27, 55, 55, 55, 48}
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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{11}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{11}}dx}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {3 b \int \frac {(a+b x)^6}{(d+e x)^{10}}dx}{10 (b d-a e)}+\frac {(a+b x)^7}{10 (d+e x)^{10} (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {3 b \left (\frac {2 b \int \frac {(a+b x)^6}{(d+e x)^9}dx}{9 (b d-a e)}+\frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}\right )}{10 (b d-a e)}+\frac {(a+b x)^7}{10 (d+e x)^{10} (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {3 b \left (\frac {2 b \left (\frac {b \int \frac {(a+b x)^6}{(d+e x)^8}dx}{8 (b d-a e)}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\right )}{9 (b d-a e)}+\frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}\right )}{10 (b d-a e)}+\frac {(a+b x)^7}{10 (d+e x)^{10} (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(a+b x)^7}{10 (d+e x)^{10} (b d-a e)}+\frac {3 b \left (\frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}+\frac {2 b \left (\frac {b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\right )}{9 (b d-a e)}\right )}{10 (b d-a e)}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((a + b*x)^7/(10*(b*d - a*e)*(d + e*x)^10) + (3*b*((a + b*x)^7/(9*(b*d - a*e)*(d + e*x)^9) + (2*b*((a + b*x)^7/(8*(b* d - a*e)*(d + e*x)^8) + (b*(a + b*x)^7)/(56*(b*d - a*e)^2*(d + e*x)^7)))/( 9*(b*d - a*e))))/(10*(b*d - a*e))))/(a + b*x)
3.21.8.3.1 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] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(350\) vs. \(2(148)=296\).
Time = 6.77 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.76
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{4 e}-\frac {3 b^{5} \left (4 a e +b d \right ) x^{5}}{10 e^{2}}-\frac {b^{4} \left (10 e^{2} a^{2}+4 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}-\frac {b^{3} \left (20 a^{3} e^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{7 e^{4}}-\frac {3 b^{2} \left (35 e^{4} a^{4}+20 b d \,e^{3} a^{3}+10 b^{2} d^{2} e^{2} a^{2}+4 b^{3} d^{3} e a +b^{4} d^{4}\right ) x^{2}}{56 e^{5}}-\frac {b \left (56 e^{5} a^{5}+35 b d \,e^{4} a^{4}+20 b^{2} d^{2} e^{3} a^{3}+10 b^{3} d^{3} e^{2} a^{2}+4 b^{4} d^{4} e a +b^{5} d^{5}\right ) x}{84 e^{6}}-\frac {84 e^{6} a^{6}+56 b d \,e^{5} a^{5}+35 b^{2} d^{2} e^{4} a^{4}+20 b^{3} d^{3} e^{3} a^{3}+10 b^{4} d^{4} e^{2} a^{2}+4 b^{5} d^{5} e a +b^{6} d^{6}}{840 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{10}}\) | \(351\) |
| gosper | \(-\frac {\left (210 b^{6} e^{6} x^{6}+1008 a \,b^{5} e^{6} x^{5}+252 b^{6} d \,e^{5} x^{5}+2100 a^{2} b^{4} e^{6} x^{4}+840 a \,b^{5} d \,e^{5} x^{4}+210 b^{6} d^{2} e^{4} x^{4}+2400 a^{3} b^{3} e^{6} x^{3}+1200 a^{2} b^{4} d \,e^{5} x^{3}+480 a \,b^{5} d^{2} e^{4} x^{3}+120 b^{6} d^{3} e^{3} x^{3}+1575 a^{4} b^{2} e^{6} x^{2}+900 a^{3} b^{3} d \,e^{5} x^{2}+450 a^{2} b^{4} d^{2} e^{4} x^{2}+180 a \,b^{5} d^{3} e^{3} x^{2}+45 b^{6} d^{4} e^{2} x^{2}+560 a^{5} b \,e^{6} x +350 a^{4} b^{2} d \,e^{5} x +200 a^{3} b^{3} d^{2} e^{4} x +100 a^{2} b^{4} d^{3} e^{3} x +40 a \,b^{5} d^{4} e^{2} x +10 b^{6} d^{5} e x +84 e^{6} a^{6}+56 b d \,e^{5} a^{5}+35 b^{2} d^{2} e^{4} a^{4}+20 b^{3} d^{3} e^{3} a^{3}+10 b^{4} d^{4} e^{2} a^{2}+4 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{840 e^{7} \left (e x +d \right )^{10} \left (b x +a \right )^{5}}\) | \(392\) |
| default | \(-\frac {\left (210 b^{6} e^{6} x^{6}+1008 a \,b^{5} e^{6} x^{5}+252 b^{6} d \,e^{5} x^{5}+2100 a^{2} b^{4} e^{6} x^{4}+840 a \,b^{5} d \,e^{5} x^{4}+210 b^{6} d^{2} e^{4} x^{4}+2400 a^{3} b^{3} e^{6} x^{3}+1200 a^{2} b^{4} d \,e^{5} x^{3}+480 a \,b^{5} d^{2} e^{4} x^{3}+120 b^{6} d^{3} e^{3} x^{3}+1575 a^{4} b^{2} e^{6} x^{2}+900 a^{3} b^{3} d \,e^{5} x^{2}+450 a^{2} b^{4} d^{2} e^{4} x^{2}+180 a \,b^{5} d^{3} e^{3} x^{2}+45 b^{6} d^{4} e^{2} x^{2}+560 a^{5} b \,e^{6} x +350 a^{4} b^{2} d \,e^{5} x +200 a^{3} b^{3} d^{2} e^{4} x +100 a^{2} b^{4} d^{3} e^{3} x +40 a \,b^{5} d^{4} e^{2} x +10 b^{6} d^{5} e x +84 e^{6} a^{6}+56 b d \,e^{5} a^{5}+35 b^{2} d^{2} e^{4} a^{4}+20 b^{3} d^{3} e^{3} a^{3}+10 b^{4} d^{4} e^{2} a^{2}+4 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{840 e^{7} \left (e x +d \right )^{10} \left (b x +a \right )^{5}}\) | \(392\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/4*b^6/e*x^6-3/10*b^5/e^2*(4*a*e+b*d)*x^5-1/4 *b^4/e^3*(10*a^2*e^2+4*a*b*d*e+b^2*d^2)*x^4-1/7*b^3/e^4*(20*a^3*e^3+10*a^2 *b*d*e^2+4*a*b^2*d^2*e+b^3*d^3)*x^3-3/56*b^2/e^5*(35*a^4*e^4+20*a^3*b*d*e^ 3+10*a^2*b^2*d^2*e^2+4*a*b^3*d^3*e+b^4*d^4)*x^2-1/84*b/e^6*(56*a^5*e^5+35* a^4*b*d*e^4+20*a^3*b^2*d^2*e^3+10*a^2*b^3*d^3*e^2+4*a*b^4*d^4*e+b^5*d^5)*x -1/840/e^7*(84*a^6*e^6+56*a^5*b*d*e^5+35*a^4*b^2*d^2*e^4+20*a^3*b^3*d^3*e^ 3+10*a^2*b^4*d^4*e^2+4*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^10
Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (148) = 296\).
Time = 0.31 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.26 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=-\frac {210 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 4 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6} + 252 \, {\left (b^{6} d e^{5} + 4 \, a b^{5} e^{6}\right )} x^{5} + 210 \, {\left (b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} + 120 \, {\left (b^{6} d^{3} e^{3} + 4 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} + 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 45 \, {\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \, {\left (b^{6} d^{5} e + 4 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 20 \, a^{3} b^{3} d^{2} e^{4} + 35 \, a^{4} b^{2} d e^{5} + 56 \, a^{5} b e^{6}\right )} x}{840 \, {\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} e^{7}\right )}} \]
-1/840*(210*b^6*e^6*x^6 + b^6*d^6 + 4*a*b^5*d^5*e + 10*a^2*b^4*d^4*e^2 + 2 0*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 56*a^5*b*d*e^5 + 84*a^6*e^6 + 252 *(b^6*d*e^5 + 4*a*b^5*e^6)*x^5 + 210*(b^6*d^2*e^4 + 4*a*b^5*d*e^5 + 10*a^2 *b^4*e^6)*x^4 + 120*(b^6*d^3*e^3 + 4*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 + 20 *a^3*b^3*e^6)*x^3 + 45*(b^6*d^4*e^2 + 4*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 + 20*a^3*b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 + 10*(b^6*d^5*e + 4*a*b^5*d^4*e^ 2 + 10*a^2*b^4*d^3*e^3 + 20*a^3*b^3*d^2*e^4 + 35*a^4*b^2*d*e^5 + 56*a^5*b* e^6)*x)/(e^17*x^10 + 10*d*e^16*x^9 + 45*d^2*e^15*x^8 + 120*d^3*e^14*x^7 + 210*d^4*e^13*x^6 + 252*d^5*e^12*x^5 + 210*d^6*e^11*x^4 + 120*d^7*e^10*x^3 + 45*d^8*e^9*x^2 + 10*d^9*e^8*x + d^10*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Exception raised: ValueError} \]
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. 611 vs. \(2 (148) = 296\).
Time = 0.29 (sec) , antiderivative size = 611, normalized size of antiderivative = 3.06 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\frac {b^{10} \mathrm {sgn}\left (b x + a\right )}{840 \, {\left (b^{4} d^{4} e^{7} - 4 \, a b^{3} d^{3} e^{8} + 6 \, a^{2} b^{2} d^{2} e^{9} - 4 \, a^{3} b d e^{10} + a^{4} e^{11}\right )}} - \frac {210 \, b^{6} e^{6} x^{6} \mathrm {sgn}\left (b x + a\right ) + 252 \, b^{6} d e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 1008 \, a b^{5} e^{6} x^{5} \mathrm {sgn}\left (b x + a\right ) + 210 \, b^{6} d^{2} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 840 \, a b^{5} d e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 2100 \, a^{2} b^{4} e^{6} x^{4} \mathrm {sgn}\left (b x + a\right ) + 120 \, b^{6} d^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 480 \, a b^{5} d^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 1200 \, a^{2} b^{4} d e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2400 \, a^{3} b^{3} e^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) + 45 \, b^{6} d^{4} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 180 \, a b^{5} d^{3} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 450 \, a^{2} b^{4} d^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 900 \, a^{3} b^{3} d e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1575 \, a^{4} b^{2} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, b^{6} d^{5} e x \mathrm {sgn}\left (b x + a\right ) + 40 \, a b^{5} d^{4} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 100 \, a^{2} b^{4} d^{3} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 200 \, a^{3} b^{3} d^{2} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 350 \, a^{4} b^{2} d e^{5} x \mathrm {sgn}\left (b x + a\right ) + 560 \, a^{5} b e^{6} x \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{840 \, {\left (e x + d\right )}^{10} e^{7}} \]
1/840*b^10*sgn(b*x + a)/(b^4*d^4*e^7 - 4*a*b^3*d^3*e^8 + 6*a^2*b^2*d^2*e^9 - 4*a^3*b*d*e^10 + a^4*e^11) - 1/840*(210*b^6*e^6*x^6*sgn(b*x + a) + 252* b^6*d*e^5*x^5*sgn(b*x + a) + 1008*a*b^5*e^6*x^5*sgn(b*x + a) + 210*b^6*d^2 *e^4*x^4*sgn(b*x + a) + 840*a*b^5*d*e^5*x^4*sgn(b*x + a) + 2100*a^2*b^4*e^ 6*x^4*sgn(b*x + a) + 120*b^6*d^3*e^3*x^3*sgn(b*x + a) + 480*a*b^5*d^2*e^4* x^3*sgn(b*x + a) + 1200*a^2*b^4*d*e^5*x^3*sgn(b*x + a) + 2400*a^3*b^3*e^6* x^3*sgn(b*x + a) + 45*b^6*d^4*e^2*x^2*sgn(b*x + a) + 180*a*b^5*d^3*e^3*x^2 *sgn(b*x + a) + 450*a^2*b^4*d^2*e^4*x^2*sgn(b*x + a) + 900*a^3*b^3*d*e^5*x ^2*sgn(b*x + a) + 1575*a^4*b^2*e^6*x^2*sgn(b*x + a) + 10*b^6*d^5*e*x*sgn(b *x + a) + 40*a*b^5*d^4*e^2*x*sgn(b*x + a) + 100*a^2*b^4*d^3*e^3*x*sgn(b*x + a) + 200*a^3*b^3*d^2*e^4*x*sgn(b*x + a) + 350*a^4*b^2*d*e^5*x*sgn(b*x + a) + 560*a^5*b*e^6*x*sgn(b*x + a) + b^6*d^6*sgn(b*x + a) + 4*a*b^5*d^5*e*s gn(b*x + a) + 10*a^2*b^4*d^4*e^2*sgn(b*x + a) + 20*a^3*b^3*d^3*e^3*sgn(b*x + a) + 35*a^4*b^2*d^2*e^4*sgn(b*x + a) + 56*a^5*b*d*e^5*sgn(b*x + a) + 84 *a^6*e^6*sgn(b*x + a))/((e*x + d)^10*e^7)
Time = 11.11 (sec) , antiderivative size = 1010, normalized size of antiderivative = 5.05 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, 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}{9\,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}{9\,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}{9\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{9\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{9\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{9\,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 )}^9}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{6\,e^7}+\frac {d\,\left (\frac {b^6\,d}{6\,e^6}-\frac {b^5\,\left (3\,a\,e-2\,b\,d\right )}{3\,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 )}^6}-\frac {\left (\frac {a^6}{10\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a\,b^5}{5\,e}-\frac {b^6\,d}{10\,e^2}\right )}{e}-\frac {3\,a^2\,b^4}{2\,e}\right )}{e}+\frac {2\,a^3\,b^3}{e}\right )}{e}-\frac {3\,a^4\,b^2}{2\,e}\right )}{e}+\frac {3\,a^5\,b}{5\,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 )}^{10}}-\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}{8\,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}{8\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{8\,e^4}-\frac {b^5\,\left (3\,a\,e-b\,d\right )}{4\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{8\,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 )}^8}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{5\,e^7}+\frac {b^6\,d}{5\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}+\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}{7\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{7\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{7\,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 )}{7\,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 )}^7}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \]
(((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)/(9*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)/(9*e^7) - (d*((20* 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)/(9*e^7) - (d*((d*((b^6*d)/(9*e^3) - (b^5*(6*a*e - b*d))/(9*e^3)))/e + (b^4*(15*a^2*e ^2 + b^2*d^2 - 6*a*b*d*e))/(9*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^ (1/2))/((a + b*x)*(d + e*x)^9) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 24*a*b^5 *d*e)/(6*e^7) + (d*((b^6*d)/(6*e^6) - (b^5*(3*a*e - 2*b*d))/(3*e^6)))/e)*( a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) - ((a^6/(10*e) - ( d*((d*((d*((d*((d*((3*a*b^5)/(5*e) - (b^6*d)/(10*e^2)))/e - (3*a^2*b^4)/(2 *e)))/e + (2*a^3*b^3)/e))/e - (3*a^4*b^2)/(2*e)))/e + (3*a^5*b)/(5*e)))/e) *(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (((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) /(8*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)/(8*e^7) + (d*((d*((b^6*d)/(8*e^4) - (b^5*(3*a*e - b*d))/(4*e^4)) )/e + (3*b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(8*e^5)))/e))/e)*(a^2 + b^ 2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) + (((5*b^6*d - 6*a*b^5*e)/ (5*e^7) + (b^6*d)/(5*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^5) + (((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)/(7*e^7) + (d*((d*((b^6*d)/(7*e^5) - (3*b^5*(2*a*e - b*d))/(7*e^5)...