Integrand size = 33, antiderivative size = 362 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15}} \, dx=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x) (d+e x)^{14}}+\frac {6 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}-\frac {5 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^{12}}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {3 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^{10}}+\frac {2 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^9}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8} \]
-1/14*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^14+6/13*b*(-a*e+b *d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^13-5/4*b^2*(-a*e+b*d)^4*((b*x+ a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^12+20/11*b^3*(-a*e+b*d)^3*((b*x+a)^2)^(1/2 )/e^7/(b*x+a)/(e*x+d)^11-3/2*b^4*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+a )/(e*x+d)^10+2/3*b^5*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^9-1/ 8*b^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^8
Time = 1.08 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (1716 a^6 e^6+792 a^5 b e^5 (d+14 e x)+330 a^4 b^2 e^4 \left (d^2+14 d e x+91 e^2 x^2\right )+120 a^3 b^3 e^3 \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )+36 a^2 b^4 e^2 \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )+8 a b^5 e \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )+b^6 \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )\right )}{24024 e^7 (a+b x) (d+e x)^{14}} \]
-1/24024*(Sqrt[(a + b*x)^2]*(1716*a^6*e^6 + 792*a^5*b*e^5*(d + 14*e*x) + 3 30*a^4*b^2*e^4*(d^2 + 14*d*e*x + 91*e^2*x^2) + 120*a^3*b^3*e^3*(d^3 + 14*d ^2*e*x + 91*d*e^2*x^2 + 364*e^3*x^3) + 36*a^2*b^4*e^2*(d^4 + 14*d^3*e*x + 91*d^2*e^2*x^2 + 364*d*e^3*x^3 + 1001*e^4*x^4) + 8*a*b^5*e*(d^5 + 14*d^4*e *x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^4 + 2002*e^5*x^5) + b ^6*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2002*d*e^5*x^5 + 3003*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^14)
Time = 0.39 (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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{15}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{15}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6}{e^6 (d+e x)^9}-\frac {6 (b d-a e) b^5}{e^6 (d+e x)^{10}}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^{11}}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^{12}}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^{13}}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^{14}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{15}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {2 b^5 (b d-a e)}{3 e^7 (d+e x)^9}-\frac {3 b^4 (b d-a e)^2}{2 e^7 (d+e x)^{10}}+\frac {20 b^3 (b d-a e)^3}{11 e^7 (d+e x)^{11}}-\frac {5 b^2 (b d-a e)^4}{4 e^7 (d+e x)^{12}}+\frac {6 b (b d-a e)^5}{13 e^7 (d+e x)^{13}}-\frac {(b d-a e)^6}{14 e^7 (d+e x)^{14}}-\frac {b^6}{8 e^7 (d+e x)^8}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/14*(b*d - a*e)^6/(e^7*(d + e*x)^14) + ( 6*b*(b*d - a*e)^5)/(13*e^7*(d + e*x)^13) - (5*b^2*(b*d - a*e)^4)/(4*e^7*(d + e*x)^12) + (20*b^3*(b*d - a*e)^3)/(11*e^7*(d + e*x)^11) - (3*b^4*(b*d - a*e)^2)/(2*e^7*(d + e*x)^10) + (2*b^5*(b*d - a*e))/(3*e^7*(d + e*x)^9) - b^6/(8*e^7*(d + e*x)^8)))/(a + b*x)
3.21.12.3.1 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 = 26.81 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{8 e}-\frac {b^{5} \left (8 a e +b d \right ) x^{5}}{12 e^{2}}-\frac {b^{4} \left (36 e^{2} a^{2}+8 a b d e +b^{2} d^{2}\right ) x^{4}}{24 e^{3}}-\frac {b^{3} \left (120 a^{3} e^{3}+36 a^{2} b d \,e^{2}+8 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{66 e^{4}}-\frac {b^{2} \left (330 e^{4} a^{4}+120 b d \,e^{3} a^{3}+36 b^{2} d^{2} e^{2} a^{2}+8 b^{3} d^{3} e a +b^{4} d^{4}\right ) x^{2}}{264 e^{5}}-\frac {b \left (792 e^{5} a^{5}+330 b d \,e^{4} a^{4}+120 b^{2} d^{2} e^{3} a^{3}+36 b^{3} d^{3} e^{2} a^{2}+8 b^{4} d^{4} e a +b^{5} d^{5}\right ) x}{1716 e^{6}}-\frac {1716 e^{6} a^{6}+792 b d \,e^{5} a^{5}+330 b^{2} d^{2} e^{4} a^{4}+120 b^{3} d^{3} e^{3} a^{3}+36 b^{4} d^{4} e^{2} a^{2}+8 b^{5} d^{5} e a +b^{6} d^{6}}{24024 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{14}}\) | \(351\) |
| gosper | \(-\frac {\left (3003 b^{6} e^{6} x^{6}+16016 a \,b^{5} e^{6} x^{5}+2002 b^{6} d \,e^{5} x^{5}+36036 a^{2} b^{4} e^{6} x^{4}+8008 a \,b^{5} d \,e^{5} x^{4}+1001 b^{6} d^{2} e^{4} x^{4}+43680 a^{3} b^{3} e^{6} x^{3}+13104 a^{2} b^{4} d \,e^{5} x^{3}+2912 a \,b^{5} d^{2} e^{4} x^{3}+364 b^{6} d^{3} e^{3} x^{3}+30030 a^{4} b^{2} e^{6} x^{2}+10920 a^{3} b^{3} d \,e^{5} x^{2}+3276 a^{2} b^{4} d^{2} e^{4} x^{2}+728 a \,b^{5} d^{3} e^{3} x^{2}+91 b^{6} d^{4} e^{2} x^{2}+11088 a^{5} b \,e^{6} x +4620 a^{4} b^{2} d \,e^{5} x +1680 a^{3} b^{3} d^{2} e^{4} x +504 a^{2} b^{4} d^{3} e^{3} x +112 a \,b^{5} d^{4} e^{2} x +14 b^{6} d^{5} e x +1716 e^{6} a^{6}+792 b d \,e^{5} a^{5}+330 b^{2} d^{2} e^{4} a^{4}+120 b^{3} d^{3} e^{3} a^{3}+36 b^{4} d^{4} e^{2} a^{2}+8 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{24024 e^{7} \left (e x +d \right )^{14} \left (b x +a \right )^{5}}\) | \(392\) |
| default | \(-\frac {\left (3003 b^{6} e^{6} x^{6}+16016 a \,b^{5} e^{6} x^{5}+2002 b^{6} d \,e^{5} x^{5}+36036 a^{2} b^{4} e^{6} x^{4}+8008 a \,b^{5} d \,e^{5} x^{4}+1001 b^{6} d^{2} e^{4} x^{4}+43680 a^{3} b^{3} e^{6} x^{3}+13104 a^{2} b^{4} d \,e^{5} x^{3}+2912 a \,b^{5} d^{2} e^{4} x^{3}+364 b^{6} d^{3} e^{3} x^{3}+30030 a^{4} b^{2} e^{6} x^{2}+10920 a^{3} b^{3} d \,e^{5} x^{2}+3276 a^{2} b^{4} d^{2} e^{4} x^{2}+728 a \,b^{5} d^{3} e^{3} x^{2}+91 b^{6} d^{4} e^{2} x^{2}+11088 a^{5} b \,e^{6} x +4620 a^{4} b^{2} d \,e^{5} x +1680 a^{3} b^{3} d^{2} e^{4} x +504 a^{2} b^{4} d^{3} e^{3} x +112 a \,b^{5} d^{4} e^{2} x +14 b^{6} d^{5} e x +1716 e^{6} a^{6}+792 b d \,e^{5} a^{5}+330 b^{2} d^{2} e^{4} a^{4}+120 b^{3} d^{3} e^{3} a^{3}+36 b^{4} d^{4} e^{2} a^{2}+8 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{24024 e^{7} \left (e x +d \right )^{14} \left (b x +a \right )^{5}}\) | \(392\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/8*b^6/e*x^6-1/12*b^5/e^2*(8*a*e+b*d)*x^5-1/2 4*b^4/e^3*(36*a^2*e^2+8*a*b*d*e+b^2*d^2)*x^4-1/66*b^3/e^4*(120*a^3*e^3+36* a^2*b*d*e^2+8*a*b^2*d^2*e+b^3*d^3)*x^3-1/264*b^2/e^5*(330*a^4*e^4+120*a^3* b*d*e^3+36*a^2*b^2*d^2*e^2+8*a*b^3*d^3*e+b^4*d^4)*x^2-1/1716*b/e^6*(792*a^ 5*e^5+330*a^4*b*d*e^4+120*a^3*b^2*d^2*e^3+36*a^2*b^3*d^3*e^2+8*a*b^4*d^4*e +b^5*d^5)*x-1/24024/e^7*(1716*a^6*e^6+792*a^5*b*d*e^5+330*a^4*b^2*d^2*e^4+ 120*a^3*b^3*d^3*e^3+36*a^2*b^4*d^4*e^2+8*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^14
Time = 0.28 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.37 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15}} \, dx=-\frac {3003 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 8 \, a b^{5} d^{5} e + 36 \, a^{2} b^{4} d^{4} e^{2} + 120 \, a^{3} b^{3} d^{3} e^{3} + 330 \, a^{4} b^{2} d^{2} e^{4} + 792 \, a^{5} b d e^{5} + 1716 \, a^{6} e^{6} + 2002 \, {\left (b^{6} d e^{5} + 8 \, a b^{5} e^{6}\right )} x^{5} + 1001 \, {\left (b^{6} d^{2} e^{4} + 8 \, a b^{5} d e^{5} + 36 \, a^{2} b^{4} e^{6}\right )} x^{4} + 364 \, {\left (b^{6} d^{3} e^{3} + 8 \, a b^{5} d^{2} e^{4} + 36 \, a^{2} b^{4} d e^{5} + 120 \, a^{3} b^{3} e^{6}\right )} x^{3} + 91 \, {\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 36 \, a^{2} b^{4} d^{2} e^{4} + 120 \, a^{3} b^{3} d e^{5} + 330 \, a^{4} b^{2} e^{6}\right )} x^{2} + 14 \, {\left (b^{6} d^{5} e + 8 \, a b^{5} d^{4} e^{2} + 36 \, a^{2} b^{4} d^{3} e^{3} + 120 \, a^{3} b^{3} d^{2} e^{4} + 330 \, a^{4} b^{2} d e^{5} + 792 \, a^{5} b e^{6}\right )} x}{24024 \, {\left (e^{21} x^{14} + 14 \, d e^{20} x^{13} + 91 \, d^{2} e^{19} x^{12} + 364 \, d^{3} e^{18} x^{11} + 1001 \, d^{4} e^{17} x^{10} + 2002 \, d^{5} e^{16} x^{9} + 3003 \, d^{6} e^{15} x^{8} + 3432 \, d^{7} e^{14} x^{7} + 3003 \, d^{8} e^{13} x^{6} + 2002 \, d^{9} e^{12} x^{5} + 1001 \, d^{10} e^{11} x^{4} + 364 \, d^{11} e^{10} x^{3} + 91 \, d^{12} e^{9} x^{2} + 14 \, d^{13} e^{8} x + d^{14} e^{7}\right )}} \]
-1/24024*(3003*b^6*e^6*x^6 + b^6*d^6 + 8*a*b^5*d^5*e + 36*a^2*b^4*d^4*e^2 + 120*a^3*b^3*d^3*e^3 + 330*a^4*b^2*d^2*e^4 + 792*a^5*b*d*e^5 + 1716*a^6*e ^6 + 2002*(b^6*d*e^5 + 8*a*b^5*e^6)*x^5 + 1001*(b^6*d^2*e^4 + 8*a*b^5*d*e^ 5 + 36*a^2*b^4*e^6)*x^4 + 364*(b^6*d^3*e^3 + 8*a*b^5*d^2*e^4 + 36*a^2*b^4* d*e^5 + 120*a^3*b^3*e^6)*x^3 + 91*(b^6*d^4*e^2 + 8*a*b^5*d^3*e^3 + 36*a^2* b^4*d^2*e^4 + 120*a^3*b^3*d*e^5 + 330*a^4*b^2*e^6)*x^2 + 14*(b^6*d^5*e + 8 *a*b^5*d^4*e^2 + 36*a^2*b^4*d^3*e^3 + 120*a^3*b^3*d^2*e^4 + 330*a^4*b^2*d* e^5 + 792*a^5*b*e^6)*x)/(e^21*x^14 + 14*d*e^20*x^13 + 91*d^2*e^19*x^12 + 3 64*d^3*e^18*x^11 + 1001*d^4*e^17*x^10 + 2002*d^5*e^16*x^9 + 3003*d^6*e^15* x^8 + 3432*d^7*e^14*x^7 + 3003*d^8*e^13*x^6 + 2002*d^9*e^12*x^5 + 1001*d^1 0*e^11*x^4 + 364*d^11*e^10*x^3 + 91*d^12*e^9*x^2 + 14*d^13*e^8*x + d^14*e^ 7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15}} \, dx=\text {Exception raised: ValueError} \]
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. 667 vs. \(2 (271) = 542\).
Time = 0.28 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15}} \, dx=\frac {b^{14} \mathrm {sgn}\left (b x + a\right )}{24024 \, {\left (b^{8} d^{8} e^{7} - 8 \, a b^{7} d^{7} e^{8} + 28 \, a^{2} b^{6} d^{6} e^{9} - 56 \, a^{3} b^{5} d^{5} e^{10} + 70 \, a^{4} b^{4} d^{4} e^{11} - 56 \, a^{5} b^{3} d^{3} e^{12} + 28 \, a^{6} b^{2} d^{2} e^{13} - 8 \, a^{7} b d e^{14} + a^{8} e^{15}\right )}} - \frac {3003 \, b^{6} e^{6} x^{6} \mathrm {sgn}\left (b x + a\right ) + 2002 \, b^{6} d e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 16016 \, a b^{5} e^{6} x^{5} \mathrm {sgn}\left (b x + a\right ) + 1001 \, b^{6} d^{2} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 8008 \, a b^{5} d e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 36036 \, a^{2} b^{4} e^{6} x^{4} \mathrm {sgn}\left (b x + a\right ) + 364 \, b^{6} d^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2912 \, a b^{5} d^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 13104 \, a^{2} b^{4} d e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 43680 \, a^{3} b^{3} e^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) + 91 \, b^{6} d^{4} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 728 \, a b^{5} d^{3} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3276 \, a^{2} b^{4} d^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 10920 \, a^{3} b^{3} d e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 30030 \, a^{4} b^{2} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 14 \, b^{6} d^{5} e x \mathrm {sgn}\left (b x + a\right ) + 112 \, a b^{5} d^{4} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 504 \, a^{2} b^{4} d^{3} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 1680 \, a^{3} b^{3} d^{2} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 4620 \, a^{4} b^{2} d e^{5} x \mathrm {sgn}\left (b x + a\right ) + 11088 \, a^{5} b e^{6} x \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 8 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 36 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 120 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 330 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 792 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 1716 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{24024 \, {\left (e x + d\right )}^{14} e^{7}} \]
1/24024*b^14*sgn(b*x + a)/(b^8*d^8*e^7 - 8*a*b^7*d^7*e^8 + 28*a^2*b^6*d^6* e^9 - 56*a^3*b^5*d^5*e^10 + 70*a^4*b^4*d^4*e^11 - 56*a^5*b^3*d^3*e^12 + 28 *a^6*b^2*d^2*e^13 - 8*a^7*b*d*e^14 + a^8*e^15) - 1/24024*(3003*b^6*e^6*x^6 *sgn(b*x + a) + 2002*b^6*d*e^5*x^5*sgn(b*x + a) + 16016*a*b^5*e^6*x^5*sgn( b*x + a) + 1001*b^6*d^2*e^4*x^4*sgn(b*x + a) + 8008*a*b^5*d*e^5*x^4*sgn(b* x + a) + 36036*a^2*b^4*e^6*x^4*sgn(b*x + a) + 364*b^6*d^3*e^3*x^3*sgn(b*x + a) + 2912*a*b^5*d^2*e^4*x^3*sgn(b*x + a) + 13104*a^2*b^4*d*e^5*x^3*sgn(b *x + a) + 43680*a^3*b^3*e^6*x^3*sgn(b*x + a) + 91*b^6*d^4*e^2*x^2*sgn(b*x + a) + 728*a*b^5*d^3*e^3*x^2*sgn(b*x + a) + 3276*a^2*b^4*d^2*e^4*x^2*sgn(b *x + a) + 10920*a^3*b^3*d*e^5*x^2*sgn(b*x + a) + 30030*a^4*b^2*e^6*x^2*sgn (b*x + a) + 14*b^6*d^5*e*x*sgn(b*x + a) + 112*a*b^5*d^4*e^2*x*sgn(b*x + a) + 504*a^2*b^4*d^3*e^3*x*sgn(b*x + a) + 1680*a^3*b^3*d^2*e^4*x*sgn(b*x + a ) + 4620*a^4*b^2*d*e^5*x*sgn(b*x + a) + 11088*a^5*b*e^6*x*sgn(b*x + a) + b ^6*d^6*sgn(b*x + a) + 8*a*b^5*d^5*e*sgn(b*x + a) + 36*a^2*b^4*d^4*e^2*sgn( b*x + a) + 120*a^3*b^3*d^3*e^3*sgn(b*x + a) + 330*a^4*b^2*d^2*e^4*sgn(b*x + a) + 792*a^5*b*d*e^5*sgn(b*x + a) + 1716*a^6*e^6*sgn(b*x + a))/((e*x + d )^14*e^7)
Time = 11.19 (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)^{15}} \, 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}{13\,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}{13\,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}{13\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{13\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{13\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{13\,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 )}^{13}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{10\,e^7}+\frac {d\,\left (\frac {b^6\,d}{10\,e^6}-\frac {b^5\,\left (3\,a\,e-2\,b\,d\right )}{5\,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 {\left (\frac {a^6}{14\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a\,b^5}{7\,e}-\frac {b^6\,d}{14\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{14\,e}\right )}{e}+\frac {10\,a^3\,b^3}{7\,e}\right )}{e}-\frac {15\,a^4\,b^2}{14\,e}\right )}{e}+\frac {3\,a^5\,b}{7\,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 {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}{12\,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}{12\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{12\,e^4}-\frac {b^5\,\left (3\,a\,e-b\,d\right )}{6\,e^4}\right )}{e}+\frac {b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{4\,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 )}^{12}}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{9\,e^7}+\frac {b^6\,d}{9\,e^7}\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 {-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}{11\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{11\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{11\,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 )}{11\,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 )}^{11}}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{8\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8} \]
(((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)/(13*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)/(13*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)/(13*e^7) - (d*((d*((b^6*d)/(13*e^3) - (b^5*(6*a*e - b*d))/(13*e^3)))/e + (b^4*(15* a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(13*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a *b*x)^(1/2))/((a + b*x)*(d + e*x)^13) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 2 4*a*b^5*d*e)/(10*e^7) + (d*((b^6*d)/(10*e^6) - (b^5*(3*a*e - 2*b*d))/(5*e^ 6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - ((a^6/ (14*e) - (d*((d*((d*((d*((d*((3*a*b^5)/(7*e) - (b^6*d)/(14*e^2)))/e - (15* a^2*b^4)/(14*e)))/e + (10*a^3*b^3)/(7*e)))/e - (15*a^4*b^2)/(14*e)))/e + ( 3*a^5*b)/(7*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^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)/(12*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)/(12*e^7) + (d*((d*((b^6*d)/(12*e^4) - (b^5*( 3*a*e - b*d))/(6*e^4)))/e + (b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(4*e^5 )))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^12) + ((( 5*b^6*d - 6*a*b^5*e)/(9*e^7) + (b^6*d)/(9*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^ (1/2))/((a + b*x)*(d + e*x)^9) + (((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)/(11*e^7) + (d*((d*((b^6*d)/(11*e^5) - (3*b^5...