Integrand size = 33, antiderivative size = 438 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}}+\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}-\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}+\frac {10 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}-\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6}+\frac {b^4 (6 b B d-A b e-5 a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4} \]
-1/10*(-a*e+b*d)^5*(-A*e+B*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^10+1/9 *(-a*e+b*d)^4*(-5*A*b*e-B*a*e+6*B*b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+ d)^9-5/8*b*(-a*e+b*d)^3*(-2*A*b*e-B*a*e+3*B*b*d)*((b*x+a)^2)^(1/2)/e^7/(b* x+a)/(e*x+d)^8+10/7*b^2*(-a*e+b*d)^2*(-A*b*e-B*a*e+2*B*b*d)*((b*x+a)^2)^(1 /2)/e^7/(b*x+a)/(e*x+d)^7-5/6*b^3*(-a*e+b*d)*(-A*b*e-2*B*a*e+3*B*b*d)*((b* x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^6+1/5*b^4*(-A*b*e-5*B*a*e+6*B*b*d)*((b*x +a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^5-1/4*b^5*B*((b*x+a)^2)^(1/2)/e^7/(b*x+a) /(e*x+d)^4
Time = 1.15 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.07 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (28 a^5 e^5 (9 A e+B (d+10 e x))+35 a^4 b e^4 \left (4 A e (d+10 e x)+B \left (d^2+10 d e x+45 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (7 A e \left (d^2+10 d e x+45 e^2 x^2\right )+3 B \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )\right )+10 a^2 b^3 e^2 \left (3 A e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+2 B \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )+10 a b^4 e \left (A e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+B \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )+b^5 \left (2 A e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+3 B \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )\right )}{2520 e^7 (a+b x) (d+e x)^{10}} \]
-1/2520*(Sqrt[(a + b*x)^2]*(28*a^5*e^5*(9*A*e + B*(d + 10*e*x)) + 35*a^4*b *e^4*(4*A*e*(d + 10*e*x) + B*(d^2 + 10*d*e*x + 45*e^2*x^2)) + 10*a^3*b^2*e ^3*(7*A*e*(d^2 + 10*d*e*x + 45*e^2*x^2) + 3*B*(d^3 + 10*d^2*e*x + 45*d*e^2 *x^2 + 120*e^3*x^3)) + 10*a^2*b^3*e^2*(3*A*e*(d^3 + 10*d^2*e*x + 45*d*e^2* x^2 + 120*e^3*x^3) + 2*B*(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)) + 10*a*b^4*e*(A*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 1 20*d*e^3*x^3 + 210*e^4*x^4) + B*(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^5*(2*A*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) + 3*B*(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 = 0.57 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.63, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 86, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{(d+e x)^{11}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{(d+e x)^{11}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{(d+e x)^{11}}dx}{a+b x}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B b^5}{e^6 (d+e x)^5}+\frac {(-6 b B d+A b e+5 a B e) b^4}{e^6 (d+e x)^6}-\frac {5 (b d-a e) (-3 b B d+A b e+2 a B e) b^3}{e^6 (d+e x)^7}+\frac {10 (b d-a e)^2 (-2 b B d+A b e+a B e) b^2}{e^6 (d+e x)^8}-\frac {5 (b d-a e)^3 (-3 b B d+2 A b e+a B e) b}{e^6 (d+e x)^9}+\frac {(a e-b d)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{10}}+\frac {(a e-b d)^5 (A e-B d)}{e^6 (d+e x)^{11}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {b^4 (-5 a B e-A b e+6 b B d)}{5 e^7 (d+e x)^5}-\frac {5 b^3 (b d-a e) (-2 a B e-A b e+3 b B d)}{6 e^7 (d+e x)^6}+\frac {10 b^2 (b d-a e)^2 (-a B e-A b e+2 b B d)}{7 e^7 (d+e x)^7}-\frac {5 b (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{8 e^7 (d+e x)^8}+\frac {(b d-a e)^4 (-a B e-5 A b e+6 b B d)}{9 e^7 (d+e x)^9}-\frac {(b d-a e)^5 (B d-A e)}{10 e^7 (d+e x)^{10}}-\frac {b^5 B}{4 e^7 (d+e x)^4}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/10*((b*d - a*e)^5*(B*d - A*e))/(e^7*(d + e*x)^10) + ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e))/(9*e^7*(d + e*x)^ 9) - (5*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e))/(8*e^7*(d + e*x)^8) + (10*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e))/(7*e^7*(d + e*x)^7) - (5 *b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e))/(6*e^7*(d + e*x)^6) + (b^4*( 6*b*B*d - A*b*e - 5*a*B*e))/(5*e^7*(d + e*x)^5) - (b^5*B)/(4*e^7*(d + e*x) ^4)))/(a + b*x)
3.18.55.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 = 7.18 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.37
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {B \,b^{5} x^{6}}{4 e}-\frac {b^{4} \left (2 A b e +10 B a e +3 B b d \right ) x^{5}}{10 e^{2}}-\frac {b^{3} \left (10 A a b \,e^{2}+2 A \,b^{2} d e +20 a^{2} B \,e^{2}+10 B a b d e +3 B \,b^{2} d^{2}\right ) x^{4}}{12 e^{3}}-\frac {b^{2} \left (30 A \,a^{2} b \,e^{3}+10 A a \,b^{2} d \,e^{2}+2 A \,b^{3} d^{2} e +30 B \,e^{3} a^{3}+20 B \,a^{2} b d \,e^{2}+10 B a \,b^{2} d^{2} e +3 B \,b^{3} d^{3}\right ) x^{3}}{21 e^{4}}-\frac {b \left (70 A \,a^{3} b \,e^{4}+30 A \,a^{2} b^{2} d \,e^{3}+10 A a \,b^{3} d^{2} e^{2}+2 A \,b^{4} d^{3} e +35 B \,a^{4} e^{4}+30 B \,a^{3} b d \,e^{3}+20 B \,a^{2} b^{2} d^{2} e^{2}+10 B a \,b^{3} d^{3} e +3 b^{4} B \,d^{4}\right ) x^{2}}{56 e^{5}}-\frac {\left (140 A \,a^{4} b \,e^{5}+70 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}+10 A a \,b^{4} d^{3} e^{2}+2 A \,b^{5} d^{4} e +28 B \,a^{5} e^{5}+35 B \,a^{4} b d \,e^{4}+30 B \,a^{3} b^{2} d^{2} e^{3}+20 B \,a^{2} b^{3} d^{3} e^{2}+10 B a \,b^{4} d^{4} e +3 B \,b^{5} d^{5}\right ) x}{252 e^{6}}-\frac {252 A \,a^{5} e^{6}+140 A \,a^{4} b d \,e^{5}+70 A \,a^{3} b^{2} d^{2} e^{4}+30 A \,a^{2} b^{3} d^{3} e^{3}+10 A a \,b^{4} d^{4} e^{2}+2 A \,b^{5} d^{5} e +28 B \,a^{5} d \,e^{5}+35 B \,a^{4} b \,d^{2} e^{4}+30 B \,a^{3} b^{2} d^{3} e^{3}+20 B \,a^{2} b^{3} d^{4} e^{2}+10 B a \,b^{4} d^{5} e +3 B \,b^{5} d^{6}}{2520 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{10}}\) | \(602\) |
| gosper | \(-\frac {\left (630 B \,b^{5} e^{6} x^{6}+504 A \,b^{5} e^{6} x^{5}+2520 B a \,b^{4} e^{6} x^{5}+756 B \,b^{5} d \,e^{5} x^{5}+2100 A a \,b^{4} e^{6} x^{4}+420 A \,b^{5} d \,e^{5} x^{4}+4200 B \,a^{2} b^{3} e^{6} x^{4}+2100 B a \,b^{4} d \,e^{5} x^{4}+630 B \,b^{5} d^{2} e^{4} x^{4}+3600 A \,a^{2} b^{3} e^{6} x^{3}+1200 A a \,b^{4} d \,e^{5} x^{3}+240 A \,b^{5} d^{2} e^{4} x^{3}+3600 B \,a^{3} b^{2} e^{6} x^{3}+2400 B \,a^{2} b^{3} d \,e^{5} x^{3}+1200 B a \,b^{4} d^{2} e^{4} x^{3}+360 B \,b^{5} d^{3} e^{3} x^{3}+3150 A \,a^{3} b^{2} e^{6} x^{2}+1350 A \,a^{2} b^{3} d \,e^{5} x^{2}+450 A a \,b^{4} d^{2} e^{4} x^{2}+90 A \,b^{5} d^{3} e^{3} x^{2}+1575 B \,a^{4} b \,e^{6} x^{2}+1350 B \,a^{3} b^{2} d \,e^{5} x^{2}+900 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+450 B a \,b^{4} d^{3} e^{3} x^{2}+135 B \,b^{5} d^{4} e^{2} x^{2}+1400 A \,a^{4} b \,e^{6} x +700 A \,a^{3} b^{2} d \,e^{5} x +300 A \,a^{2} b^{3} d^{2} e^{4} x +100 A a \,b^{4} d^{3} e^{3} x +20 A \,b^{5} d^{4} e^{2} x +280 B \,a^{5} e^{6} x +350 B \,a^{4} b d \,e^{5} x +300 B \,a^{3} b^{2} d^{2} e^{4} x +200 B \,a^{2} b^{3} d^{3} e^{3} x +100 B a \,b^{4} d^{4} e^{2} x +30 B \,b^{5} d^{5} e x +252 A \,a^{5} e^{6}+140 A \,a^{4} b d \,e^{5}+70 A \,a^{3} b^{2} d^{2} e^{4}+30 A \,a^{2} b^{3} d^{3} e^{3}+10 A a \,b^{4} d^{4} e^{2}+2 A \,b^{5} d^{5} e +28 B \,a^{5} d \,e^{5}+35 B \,a^{4} b \,d^{2} e^{4}+30 B \,a^{3} b^{2} d^{3} e^{3}+20 B \,a^{2} b^{3} d^{4} e^{2}+10 B a \,b^{4} d^{5} e +3 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2520 e^{7} \left (e x +d \right )^{10} \left (b x +a \right )^{5}}\) | \(689\) |
| default | \(-\frac {\left (630 B \,b^{5} e^{6} x^{6}+504 A \,b^{5} e^{6} x^{5}+2520 B a \,b^{4} e^{6} x^{5}+756 B \,b^{5} d \,e^{5} x^{5}+2100 A a \,b^{4} e^{6} x^{4}+420 A \,b^{5} d \,e^{5} x^{4}+4200 B \,a^{2} b^{3} e^{6} x^{4}+2100 B a \,b^{4} d \,e^{5} x^{4}+630 B \,b^{5} d^{2} e^{4} x^{4}+3600 A \,a^{2} b^{3} e^{6} x^{3}+1200 A a \,b^{4} d \,e^{5} x^{3}+240 A \,b^{5} d^{2} e^{4} x^{3}+3600 B \,a^{3} b^{2} e^{6} x^{3}+2400 B \,a^{2} b^{3} d \,e^{5} x^{3}+1200 B a \,b^{4} d^{2} e^{4} x^{3}+360 B \,b^{5} d^{3} e^{3} x^{3}+3150 A \,a^{3} b^{2} e^{6} x^{2}+1350 A \,a^{2} b^{3} d \,e^{5} x^{2}+450 A a \,b^{4} d^{2} e^{4} x^{2}+90 A \,b^{5} d^{3} e^{3} x^{2}+1575 B \,a^{4} b \,e^{6} x^{2}+1350 B \,a^{3} b^{2} d \,e^{5} x^{2}+900 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+450 B a \,b^{4} d^{3} e^{3} x^{2}+135 B \,b^{5} d^{4} e^{2} x^{2}+1400 A \,a^{4} b \,e^{6} x +700 A \,a^{3} b^{2} d \,e^{5} x +300 A \,a^{2} b^{3} d^{2} e^{4} x +100 A a \,b^{4} d^{3} e^{3} x +20 A \,b^{5} d^{4} e^{2} x +280 B \,a^{5} e^{6} x +350 B \,a^{4} b d \,e^{5} x +300 B \,a^{3} b^{2} d^{2} e^{4} x +200 B \,a^{2} b^{3} d^{3} e^{3} x +100 B a \,b^{4} d^{4} e^{2} x +30 B \,b^{5} d^{5} e x +252 A \,a^{5} e^{6}+140 A \,a^{4} b d \,e^{5}+70 A \,a^{3} b^{2} d^{2} e^{4}+30 A \,a^{2} b^{3} d^{3} e^{3}+10 A a \,b^{4} d^{4} e^{2}+2 A \,b^{5} d^{5} e +28 B \,a^{5} d \,e^{5}+35 B \,a^{4} b \,d^{2} e^{4}+30 B \,a^{3} b^{2} d^{3} e^{3}+20 B \,a^{2} b^{3} d^{4} e^{2}+10 B a \,b^{4} d^{5} e +3 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2520 e^{7} \left (e x +d \right )^{10} \left (b x +a \right )^{5}}\) | \(689\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/4*B*b^5/e*x^6-1/10*b^4/e^2*(2*A*b*e+10*B*a*e +3*B*b*d)*x^5-1/12*b^3/e^3*(10*A*a*b*e^2+2*A*b^2*d*e+20*B*a^2*e^2+10*B*a*b *d*e+3*B*b^2*d^2)*x^4-1/21*b^2/e^4*(30*A*a^2*b*e^3+10*A*a*b^2*d*e^2+2*A*b^ 3*d^2*e+30*B*a^3*e^3+20*B*a^2*b*d*e^2+10*B*a*b^2*d^2*e+3*B*b^3*d^3)*x^3-1/ 56*b/e^5*(70*A*a^3*b*e^4+30*A*a^2*b^2*d*e^3+10*A*a*b^3*d^2*e^2+2*A*b^4*d^3 *e+35*B*a^4*e^4+30*B*a^3*b*d*e^3+20*B*a^2*b^2*d^2*e^2+10*B*a*b^3*d^3*e+3*B *b^4*d^4)*x^2-1/252/e^6*(140*A*a^4*b*e^5+70*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d ^2*e^3+10*A*a*b^4*d^3*e^2+2*A*b^5*d^4*e+28*B*a^5*e^5+35*B*a^4*b*d*e^4+30*B *a^3*b^2*d^2*e^3+20*B*a^2*b^3*d^3*e^2+10*B*a*b^4*d^4*e+3*B*b^5*d^5)*x-1/25 20/e^7*(252*A*a^5*e^6+140*A*a^4*b*d*e^5+70*A*a^3*b^2*d^2*e^4+30*A*a^2*b^3* d^3*e^3+10*A*a*b^4*d^4*e^2+2*A*b^5*d^5*e+28*B*a^5*d*e^5+35*B*a^4*b*d^2*e^4 +30*B*a^3*b^2*d^3*e^3+20*B*a^2*b^3*d^4*e^2+10*B*a*b^4*d^5*e+3*B*b^5*d^6))/ (e*x+d)^10
Time = 0.42 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.51 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=-\frac {630 \, B b^{5} e^{6} x^{6} + 3 \, B b^{5} d^{6} + 252 \, A a^{5} e^{6} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 35 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 28 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 252 \, {\left (3 \, B b^{5} d e^{5} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 210 \, {\left (3 \, B b^{5} d^{2} e^{4} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 120 \, {\left (3 \, B b^{5} d^{3} e^{3} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 45 \, {\left (3 \, B b^{5} d^{4} e^{2} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 35 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 10 \, {\left (3 \, B b^{5} d^{5} e + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 35 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 28 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{2520 \, {\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} e^{7}\right )}} \]
-1/2520*(630*B*b^5*e^6*x^6 + 3*B*b^5*d^6 + 252*A*a^5*e^6 + 2*(5*B*a*b^4 + A*b^5)*d^5*e + 10*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 30*(B*a^3*b^2 + A*a^2* b^3)*d^3*e^3 + 35*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 28*(B*a^5 + 5*A*a^4*b) *d*e^5 + 252*(3*B*b^5*d*e^5 + 2*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 210*(3*B*b^ 5*d^2*e^4 + 2*(5*B*a*b^4 + A*b^5)*d*e^5 + 10*(2*B*a^2*b^3 + A*a*b^4)*e^6)* x^4 + 120*(3*B*b^5*d^3*e^3 + 2*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 10*(2*B*a^2*b ^3 + A*a*b^4)*d*e^5 + 30*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 45*(3*B*b^5*d^ 4*e^2 + 2*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 10*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 + 30*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 35*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 10*(3*B*b^5*d^5*e + 2*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 10*(2*B*a^2*b^3 + A* a*b^4)*d^3*e^3 + 30*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 35*(B*a^4*b + 2*A*a^ 3*b^2)*d*e^5 + 28*(B*a^5 + 5*A*a^4*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. 1063 vs. \(2 (347) = 694\).
Time = 0.29 (sec) , antiderivative size = 1063, normalized size of antiderivative = 2.43 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Too large to display} \]
1/2520*(3*B*b^10*d - 5*B*a*b^9*e + 2*A*b^10*e)*sgn(b*x + a)/(b^5*d^5*e^7 - 5*a*b^4*d^4*e^8 + 10*a^2*b^3*d^3*e^9 - 10*a^3*b^2*d^2*e^10 + 5*a^4*b*d*e^ 11 - a^5*e^12) - 1/2520*(630*B*b^5*e^6*x^6*sgn(b*x + a) + 756*B*b^5*d*e^5* x^5*sgn(b*x + a) + 2520*B*a*b^4*e^6*x^5*sgn(b*x + a) + 504*A*b^5*e^6*x^5*s gn(b*x + a) + 630*B*b^5*d^2*e^4*x^4*sgn(b*x + a) + 2100*B*a*b^4*d*e^5*x^4* sgn(b*x + a) + 420*A*b^5*d*e^5*x^4*sgn(b*x + a) + 4200*B*a^2*b^3*e^6*x^4*s gn(b*x + a) + 2100*A*a*b^4*e^6*x^4*sgn(b*x + a) + 360*B*b^5*d^3*e^3*x^3*sg n(b*x + a) + 1200*B*a*b^4*d^2*e^4*x^3*sgn(b*x + a) + 240*A*b^5*d^2*e^4*x^3 *sgn(b*x + a) + 2400*B*a^2*b^3*d*e^5*x^3*sgn(b*x + a) + 1200*A*a*b^4*d*e^5 *x^3*sgn(b*x + a) + 3600*B*a^3*b^2*e^6*x^3*sgn(b*x + a) + 3600*A*a^2*b^3*e ^6*x^3*sgn(b*x + a) + 135*B*b^5*d^4*e^2*x^2*sgn(b*x + a) + 450*B*a*b^4*d^3 *e^3*x^2*sgn(b*x + a) + 90*A*b^5*d^3*e^3*x^2*sgn(b*x + a) + 900*B*a^2*b^3* d^2*e^4*x^2*sgn(b*x + a) + 450*A*a*b^4*d^2*e^4*x^2*sgn(b*x + a) + 1350*B*a ^3*b^2*d*e^5*x^2*sgn(b*x + a) + 1350*A*a^2*b^3*d*e^5*x^2*sgn(b*x + a) + 15 75*B*a^4*b*e^6*x^2*sgn(b*x + a) + 3150*A*a^3*b^2*e^6*x^2*sgn(b*x + a) + 30 *B*b^5*d^5*e*x*sgn(b*x + a) + 100*B*a*b^4*d^4*e^2*x*sgn(b*x + a) + 20*A*b^ 5*d^4*e^2*x*sgn(b*x + a) + 200*B*a^2*b^3*d^3*e^3*x*sgn(b*x + a) + 100*A*a* b^4*d^3*e^3*x*sgn(b*x + a) + 300*B*a^3*b^2*d^2*e^4*x*sgn(b*x + a) + 300*A* a^2*b^3*d^2*e^4*x*sgn(b*x + a) + 350*B*a^4*b*d*e^5*x*sgn(b*x + a) + 700*A* a^3*b^2*d*e^5*x*sgn(b*x + a) + 280*B*a^5*e^6*x*sgn(b*x + a) + 1400*A*a^...
Time = 11.24 (sec) , antiderivative size = 1488, normalized size of antiderivative = 3.40 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Too large to display} \]
- (((10*B*b^5*d^2 - 4*A*b^5*d*e + 5*A*a*b^4*e^2 + 10*B*a^2*b^3*e^2 - 20*B* a*b^4*d*e)/(6*e^7) - (d*((b^4*(A*b*e + 5*B*a*e - 4*B*b*d))/(6*e^6) - (B*b^ 5*d)/(6*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) - (((A*b^5*e - 5*B*b^5*d + 5*B*a*b^4*e)/(5*e^7) - (B*b^5*d)/(5*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^5) - (((A*a^5)/(10*e) - (d*((B*a^5 + 5*A*a^4*b)/(10*e) + (d*((d*((d*((d*((A*b^5 + 5*B*a*b^4)/(10*e ) - (B*b^5*d)/(10*e^2)))/e - (a*b^3*(A*b + 2*B*a))/(2*e)))/e + (a^2*b^2*(A *b + B*a))/e))/e - (a^3*b*(2*A*b + B*a))/(2*e)))/e))/e)*(a^2 + b^2*x^2 + 2 *a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (((6*A*b^5*d^2*e - 10*B*b^5*d^3 + 10*A*a^2*b^3*e^3 + 10*B*a^3*b^2*e^3 - 30*B*a^2*b^3*d*e^2 - 15*A*a*b^4*d* e^2 + 30*B*a*b^4*d^2*e)/(7*e^7) - (d*((5*A*a*b^4*e^3 - 3*A*b^5*d*e^2 + 6*B *b^5*d^2*e + 10*B*a^2*b^3*e^3 - 15*B*a*b^4*d*e^2)/(7*e^7) - (d*((b^4*(A*b* e + 5*B*a*e - 3*B*b*d))/(7*e^5) - (B*b^5*d)/(7*e^5)))/e))/e)*(a^2 + b^2*x^ 2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) - (((B*a^5*e^5 - B*b^5*d^5 + 5 *A*a^4*b*e^5 + A*b^5*d^4*e - 5*A*a*b^4*d^3*e^2 - 10*A*a^3*b^2*d*e^4 + 10*A *a^2*b^3*d^2*e^3 - 10*B*a^2*b^3*d^3*e^2 + 10*B*a^3*b^2*d^2*e^3 + 5*B*a*b^4 *d^4*e - 5*B*a^4*b*d*e^4)/(9*e^7) - (d*((5*B*a^4*b*e^5 + B*b^5*d^4*e + 10* A*a^3*b^2*e^5 - A*b^5*d^3*e^2 + 5*A*a*b^4*d^2*e^3 - 10*A*a^2*b^3*d*e^4 - 5 *B*a*b^4*d^3*e^2 - 10*B*a^3*b^2*d*e^4 + 10*B*a^2*b^3*d^2*e^3)/(9*e^7) - (d *((10*A*a^2*b^3*e^5 + 10*B*a^3*b^2*e^5 + A*b^5*d^2*e^3 - B*b^5*d^3*e^2 ...