Integrand size = 28, antiderivative size = 266 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e)^5 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^6}+\frac {5 e^2 (b d-a e)^3 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^6}+\frac {10 e^3 (b d-a e)^2 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^6}+\frac {e^4 (b d-a e) (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^6}+\frac {e^5 (a+b x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{11 b^6} \]
1/6*(-a*e+b*d)^5*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^6+5/7*e*(-a*e+b*d)^4*(b*x+a )^6*((b*x+a)^2)^(1/2)/b^6+5/4*e^2*(-a*e+b*d)^3*(b*x+a)^7*((b*x+a)^2)^(1/2) /b^6+10/9*e^3*(-a*e+b*d)^2*(b*x+a)^8*((b*x+a)^2)^(1/2)/b^6+1/2*e^4*(-a*e+b *d)*(b*x+a)^9*((b*x+a)^2)^(1/2)/b^6+1/11*e^5*(b*x+a)^10*((b*x+a)^2)^(1/2)/ b^6
Time = 1.08 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.45 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (462 a^5 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+330 a^4 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+165 a^3 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+55 a^2 b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+11 a b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+b^5 x^5 \left (462 d^5+1980 d^4 e x+3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1386 d e^4 x^4+252 e^5 x^5\right )\right )}{2772 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(462*a^5*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^ 2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + 330*a^4*b*x*(21*d^5 + 70*d^4*e*x + 10 5*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 165*a^3*b^2*x ^2*(56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x ^4 + 21*e^5*x^5) + 55*a^2*b^3*x^3*(126*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5) + 11*a*b^4*x^4*(252*d^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126* e^5*x^5) + b^5*x^5*(462*d^5 + 1980*d^4*e*x + 3465*d^3*e^2*x^2 + 3080*d^2*e ^3*x^3 + 1386*d*e^4*x^4 + 252*e^5*x^5)))/(2772*(a + b*x))
Time = 0.47 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.65, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1102, 27, 49, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^5 \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^5 (d+e x)^5dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^5 (d+e x)^5dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {e^5 (a+b x)^{10}}{b^5}+\frac {5 e^4 (b d-a e) (a+b x)^9}{b^5}+\frac {10 e^3 (b d-a e)^2 (a+b x)^8}{b^5}+\frac {10 e^2 (b d-a e)^3 (a+b x)^7}{b^5}+\frac {5 e (b d-a e)^4 (a+b x)^6}{b^5}+\frac {(b d-a e)^5 (a+b x)^5}{b^5}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {e^4 (a+b x)^{10} (b d-a e)}{2 b^6}+\frac {10 e^3 (a+b x)^9 (b d-a e)^2}{9 b^6}+\frac {5 e^2 (a+b x)^8 (b d-a e)^3}{4 b^6}+\frac {5 e (a+b x)^7 (b d-a e)^4}{7 b^6}+\frac {(a+b x)^6 (b d-a e)^5}{6 b^6}+\frac {e^5 (a+b x)^{11}}{11 b^6}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^5*(a + b*x)^6)/(6*b^6) + (5*e *(b*d - a*e)^4*(a + b*x)^7)/(7*b^6) + (5*e^2*(b*d - a*e)^3*(a + b*x)^8)/(4 *b^6) + (10*e^3*(b*d - a*e)^2*(a + b*x)^9)/(9*b^6) + (e^4*(b*d - a*e)*(a + b*x)^10)/(2*b^6) + (e^5*(a + b*x)^11)/(11*b^6)))/(a + b*x)
3.16.69.3.1 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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(505\) vs. \(2(188)=376\).
Time = 2.82 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.90
| method | result | size |
| gosper | \(\frac {x \left (252 b^{5} e^{5} x^{10}+1386 x^{9} a \,b^{4} e^{5}+1386 x^{9} b^{5} d \,e^{4}+3080 x^{8} a^{2} b^{3} e^{5}+7700 x^{8} a \,b^{4} d \,e^{4}+3080 x^{8} b^{5} d^{2} e^{3}+3465 x^{7} a^{3} b^{2} e^{5}+17325 x^{7} a^{2} b^{3} d \,e^{4}+17325 x^{7} a \,b^{4} d^{2} e^{3}+3465 x^{7} b^{5} d^{3} e^{2}+1980 x^{6} a^{4} b \,e^{5}+19800 x^{6} a^{3} b^{2} d \,e^{4}+39600 x^{6} a^{2} b^{3} d^{2} e^{3}+19800 x^{6} a \,b^{4} d^{3} e^{2}+1980 x^{6} b^{5} d^{4} e +462 x^{5} a^{5} e^{5}+11550 x^{5} a^{4} b d \,e^{4}+46200 x^{5} a^{3} b^{2} d^{2} e^{3}+46200 x^{5} a^{2} b^{3} d^{3} e^{2}+11550 x^{5} a \,b^{4} d^{4} e +462 x^{5} b^{5} d^{5}+2772 a^{5} d \,e^{4} x^{4}+27720 a^{4} b \,d^{2} e^{3} x^{4}+55440 a^{3} b^{2} d^{3} e^{2} x^{4}+27720 a^{2} b^{3} d^{4} e \,x^{4}+2772 a \,b^{4} d^{5} x^{4}+6930 x^{3} a^{5} d^{2} e^{3}+34650 x^{3} a^{4} b \,d^{3} e^{2}+34650 x^{3} a^{3} b^{2} d^{4} e +6930 x^{3} a^{2} b^{3} d^{5}+9240 x^{2} a^{5} d^{3} e^{2}+23100 x^{2} a^{4} b \,d^{4} e +9240 x^{2} a^{3} b^{2} d^{5}+6930 x \,a^{5} d^{4} e +6930 x \,a^{4} b \,d^{5}+2772 a^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2772 \left (b x +a \right )^{5}}\) | \(506\) |
| default | \(\frac {x \left (252 b^{5} e^{5} x^{10}+1386 x^{9} a \,b^{4} e^{5}+1386 x^{9} b^{5} d \,e^{4}+3080 x^{8} a^{2} b^{3} e^{5}+7700 x^{8} a \,b^{4} d \,e^{4}+3080 x^{8} b^{5} d^{2} e^{3}+3465 x^{7} a^{3} b^{2} e^{5}+17325 x^{7} a^{2} b^{3} d \,e^{4}+17325 x^{7} a \,b^{4} d^{2} e^{3}+3465 x^{7} b^{5} d^{3} e^{2}+1980 x^{6} a^{4} b \,e^{5}+19800 x^{6} a^{3} b^{2} d \,e^{4}+39600 x^{6} a^{2} b^{3} d^{2} e^{3}+19800 x^{6} a \,b^{4} d^{3} e^{2}+1980 x^{6} b^{5} d^{4} e +462 x^{5} a^{5} e^{5}+11550 x^{5} a^{4} b d \,e^{4}+46200 x^{5} a^{3} b^{2} d^{2} e^{3}+46200 x^{5} a^{2} b^{3} d^{3} e^{2}+11550 x^{5} a \,b^{4} d^{4} e +462 x^{5} b^{5} d^{5}+2772 a^{5} d \,e^{4} x^{4}+27720 a^{4} b \,d^{2} e^{3} x^{4}+55440 a^{3} b^{2} d^{3} e^{2} x^{4}+27720 a^{2} b^{3} d^{4} e \,x^{4}+2772 a \,b^{4} d^{5} x^{4}+6930 x^{3} a^{5} d^{2} e^{3}+34650 x^{3} a^{4} b \,d^{3} e^{2}+34650 x^{3} a^{3} b^{2} d^{4} e +6930 x^{3} a^{2} b^{3} d^{5}+9240 x^{2} a^{5} d^{3} e^{2}+23100 x^{2} a^{4} b \,d^{4} e +9240 x^{2} a^{3} b^{2} d^{5}+6930 x \,a^{5} d^{4} e +6930 x \,a^{4} b \,d^{5}+2772 a^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2772 \left (b x +a \right )^{5}}\) | \(506\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{5} e^{5} x^{11}}{11 b x +11 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a \,b^{4} e^{5}+5 b^{5} d \,e^{4}\right ) x^{10}}{10 b x +10 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{2} b^{3} e^{5}+25 a \,b^{4} d \,e^{4}+10 b^{5} d^{2} e^{3}\right ) x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{3} b^{2} e^{5}+50 a^{2} b^{3} d \,e^{4}+50 a \,b^{4} d^{2} e^{3}+10 b^{5} d^{3} e^{2}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{4} b \,e^{5}+50 a^{3} b^{2} d \,e^{4}+100 a^{2} b^{3} d^{2} e^{3}+50 a \,b^{4} d^{3} e^{2}+5 b^{5} d^{4} e \right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{5} e^{5}+25 a^{4} b d \,e^{4}+100 a^{3} b^{2} d^{2} e^{3}+100 a^{2} b^{3} d^{3} e^{2}+25 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{5} d \,e^{4}+50 a^{4} b \,d^{2} e^{3}+100 a^{3} b^{2} d^{3} e^{2}+50 a^{2} b^{3} d^{4} e +5 a \,b^{4} d^{5}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{5} d^{2} e^{3}+50 a^{4} b \,d^{3} e^{2}+50 a^{3} b^{2} d^{4} e +10 a^{2} b^{3} d^{5}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{5} d^{3} e^{2}+25 a^{4} b \,d^{4} e +10 a^{3} b^{2} d^{5}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{5} d^{4} e +5 a^{4} b \,d^{5}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{5} d^{5} x}{b x +a}\) | \(617\) |
1/2772*x*(252*b^5*e^5*x^10+1386*a*b^4*e^5*x^9+1386*b^5*d*e^4*x^9+3080*a^2* b^3*e^5*x^8+7700*a*b^4*d*e^4*x^8+3080*b^5*d^2*e^3*x^8+3465*a^3*b^2*e^5*x^7 +17325*a^2*b^3*d*e^4*x^7+17325*a*b^4*d^2*e^3*x^7+3465*b^5*d^3*e^2*x^7+1980 *a^4*b*e^5*x^6+19800*a^3*b^2*d*e^4*x^6+39600*a^2*b^3*d^2*e^3*x^6+19800*a*b ^4*d^3*e^2*x^6+1980*b^5*d^4*e*x^6+462*a^5*e^5*x^5+11550*a^4*b*d*e^4*x^5+46 200*a^3*b^2*d^2*e^3*x^5+46200*a^2*b^3*d^3*e^2*x^5+11550*a*b^4*d^4*e*x^5+46 2*b^5*d^5*x^5+2772*a^5*d*e^4*x^4+27720*a^4*b*d^2*e^3*x^4+55440*a^3*b^2*d^3 *e^2*x^4+27720*a^2*b^3*d^4*e*x^4+2772*a*b^4*d^5*x^4+6930*a^5*d^2*e^3*x^3+3 4650*a^4*b*d^3*e^2*x^3+34650*a^3*b^2*d^4*e*x^3+6930*a^2*b^3*d^5*x^3+9240*a ^5*d^3*e^2*x^2+23100*a^4*b*d^4*e*x^2+9240*a^3*b^2*d^5*x^2+6930*a^5*d^4*e*x +6930*a^4*b*d^5*x+2772*a^5*d^5)*((b*x+a)^2)^(5/2)/(b*x+a)^5
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (188) = 376\).
Time = 0.29 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.61 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{11} \, b^{5} e^{5} x^{11} + a^{5} d^{5} x + \frac {1}{2} \, {\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{5} d^{2} e^{3} + 5 \, a b^{4} d e^{4} + 2 \, a^{2} b^{3} e^{5}\right )} x^{9} + \frac {5}{4} \, {\left (b^{5} d^{3} e^{2} + 5 \, a b^{4} d^{2} e^{3} + 5 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{8} + \frac {5}{7} \, {\left (b^{5} d^{4} e + 10 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{5} + 25 \, a b^{4} d^{4} e + 100 \, a^{2} b^{3} d^{3} e^{2} + 100 \, a^{3} b^{2} d^{2} e^{3} + 25 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} x^{6} + {\left (a b^{4} d^{5} + 10 \, a^{2} b^{3} d^{4} e + 20 \, a^{3} b^{2} d^{3} e^{2} + 10 \, a^{4} b d^{2} e^{3} + a^{5} d e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (a^{2} b^{3} d^{5} + 5 \, a^{3} b^{2} d^{4} e + 5 \, a^{4} b d^{3} e^{2} + a^{5} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d^{5} + 5 \, a^{4} b d^{4} e + 2 \, a^{5} d^{3} e^{2}\right )} x^{3} + \frac {5}{2} \, {\left (a^{4} b d^{5} + a^{5} d^{4} e\right )} x^{2} \]
1/11*b^5*e^5*x^11 + a^5*d^5*x + 1/2*(b^5*d*e^4 + a*b^4*e^5)*x^10 + 5/9*(2* b^5*d^2*e^3 + 5*a*b^4*d*e^4 + 2*a^2*b^3*e^5)*x^9 + 5/4*(b^5*d^3*e^2 + 5*a* b^4*d^2*e^3 + 5*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^8 + 5/7*(b^5*d^4*e + 10*a*b ^4*d^3*e^2 + 20*a^2*b^3*d^2*e^3 + 10*a^3*b^2*d*e^4 + a^4*b*e^5)*x^7 + 1/6* (b^5*d^5 + 25*a*b^4*d^4*e + 100*a^2*b^3*d^3*e^2 + 100*a^3*b^2*d^2*e^3 + 25 *a^4*b*d*e^4 + a^5*e^5)*x^6 + (a*b^4*d^5 + 10*a^2*b^3*d^4*e + 20*a^3*b^2*d ^3*e^2 + 10*a^4*b*d^2*e^3 + a^5*d*e^4)*x^5 + 5/2*(a^2*b^3*d^5 + 5*a^3*b^2* d^4*e + 5*a^4*b*d^3*e^2 + a^5*d^2*e^3)*x^4 + 5/3*(2*a^3*b^2*d^5 + 5*a^4*b* d^4*e + 2*a^5*d^3*e^2)*x^3 + 5/2*(a^4*b*d^5 + a^5*d^4*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 28886 vs. \(2 (192) = 384\).
Time = 1.31 (sec) , antiderivative size = 28886, normalized size of antiderivative = 108.59 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**4*e**5*x**10/11 + x**9*(45 *a*b**5*e**5/11 + 5*b**6*d*e**4)/(10*b**2) + x**8*(155*a**2*b**4*e**5/11 + 30*a*b**5*d*e**4 - 19*a*(45*a*b**5*e**5/11 + 5*b**6*d*e**4)/(10*b) + 10*b **6*d**2*e**3)/(9*b**2) + x**7*(20*a**3*b**3*e**5 + 75*a**2*b**4*d*e**4 - 9*a**2*(45*a*b**5*e**5/11 + 5*b**6*d*e**4)/(10*b**2) + 60*a*b**5*d**2*e**3 - 17*a*(155*a**2*b**4*e**5/11 + 30*a*b**5*d*e**4 - 19*a*(45*a*b**5*e**5/1 1 + 5*b**6*d*e**4)/(10*b) + 10*b**6*d**2*e**3)/(9*b) + 10*b**6*d**3*e**2)/ (8*b**2) + x**6*(15*a**4*b**2*e**5 + 100*a**3*b**3*d*e**4 + 150*a**2*b**4* d**2*e**3 - 8*a**2*(155*a**2*b**4*e**5/11 + 30*a*b**5*d*e**4 - 19*a*(45*a* b**5*e**5/11 + 5*b**6*d*e**4)/(10*b) + 10*b**6*d**2*e**3)/(9*b**2) + 60*a* b**5*d**3*e**2 - 15*a*(20*a**3*b**3*e**5 + 75*a**2*b**4*d*e**4 - 9*a**2*(4 5*a*b**5*e**5/11 + 5*b**6*d*e**4)/(10*b**2) + 60*a*b**5*d**2*e**3 - 17*a*( 155*a**2*b**4*e**5/11 + 30*a*b**5*d*e**4 - 19*a*(45*a*b**5*e**5/11 + 5*b** 6*d*e**4)/(10*b) + 10*b**6*d**2*e**3)/(9*b) + 10*b**6*d**3*e**2)/(8*b) + 5 *b**6*d**4*e)/(7*b**2) + x**5*(6*a**5*b*e**5 + 75*a**4*b**2*d*e**4 + 200*a **3*b**3*d**2*e**3 + 150*a**2*b**4*d**3*e**2 - 7*a**2*(20*a**3*b**3*e**5 + 75*a**2*b**4*d*e**4 - 9*a**2*(45*a*b**5*e**5/11 + 5*b**6*d*e**4)/(10*b**2 ) + 60*a*b**5*d**2*e**3 - 17*a*(155*a**2*b**4*e**5/11 + 30*a*b**5*d*e**4 - 19*a*(45*a*b**5*e**5/11 + 5*b**6*d*e**4)/(10*b) + 10*b**6*d**2*e**3)/(9*b ) + 10*b**6*d**3*e**2)/(8*b**2) + 30*a*b**5*d**4*e - 13*a*(15*a**4*b**2...
Leaf count of result is larger than twice the leaf count of optimal. 815 vs. \(2 (188) = 376\).
Time = 0.24 (sec) , antiderivative size = 815, normalized size of antiderivative = 3.06 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e^{5} x^{4}}{11 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d e^{4} x^{3}}{2 \, b^{2}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e^{5} x^{3}}{22 \, b^{3}} + \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{5} x - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{4} e x}{6 \, b} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{3} e^{2} x}{3 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d^{2} e^{3} x}{3 \, b^{3}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} d e^{4} x}{6 \, b^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5} e^{5} x}{6 \, b^{5}} + \frac {10 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{2} e^{3} x^{2}}{9 \, b^{2}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d e^{4} x^{2}}{18 \, b^{3}} + \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} e^{5} x^{2}}{198 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{5}}{6 \, b} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{4} e}{6 \, b^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d^{3} e^{2}}{3 \, b^{3}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} d^{2} e^{3}}{3 \, b^{4}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5} d e^{4}}{6 \, b^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{6} e^{5}}{6 \, b^{6}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{3} e^{2} x}{4 \, b^{2}} - \frac {55 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d^{2} e^{3} x}{36 \, b^{3}} + \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} d e^{4} x}{36 \, b^{4}} - \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{3} e^{5} x}{396 \, b^{5}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{4} e}{7 \, b^{2}} - \frac {45 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d^{3} e^{2}}{28 \, b^{3}} + \frac {415 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} d^{2} e^{3}}{252 \, b^{4}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{3} d e^{4}}{252 \, b^{5}} + \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{4} e^{5}}{2772 \, b^{6}} \]
1/11*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^5*x^4/b^2 + 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d*e^4*x^3/b^2 - 3/22*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^5*x^ 3/b^3 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d^5*x - 5/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^4*e*x/b + 5/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^3*e^ 2*x/b^2 - 5/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*d^2*e^3*x/b^3 + 5/6*(b^2 *x^2 + 2*a*b*x + a^2)^(5/2)*a^4*d*e^4*x/b^4 - 1/6*(b^2*x^2 + 2*a*b*x + a^2 )^(5/2)*a^5*e^5*x/b^5 + 10/9*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d^2*e^3*x^2/b ^2 - 13/18*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*d*e^4*x^2/b^3 + 31/198*(b^2*x ^2 + 2*a*b*x + a^2)^(7/2)*a^2*e^5*x^2/b^4 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^ (5/2)*a*d^5/b - 5/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^4*e/b^2 + 5/3*(b ^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*d^3*e^2/b^3 - 5/3*(b^2*x^2 + 2*a*b*x + a ^2)^(5/2)*a^4*d^2*e^3/b^4 + 5/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*d*e^4/ b^5 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^6*e^5/b^6 + 5/4*(b^2*x^2 + 2*a *b*x + a^2)^(7/2)*d^3*e^2*x/b^2 - 55/36*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a* d^2*e^3*x/b^3 + 29/36*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*d*e^4*x/b^4 - 65 /396*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*e^5*x/b^5 + 5/7*(b^2*x^2 + 2*a*b* x + a^2)^(7/2)*d^4*e/b^2 - 45/28*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*d^3*e^2 /b^3 + 415/252*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*d^2*e^3/b^4 - 209/252*( b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*d*e^4/b^5 + 461/2772*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*e^5/b^6
Leaf count of result is larger than twice the leaf count of optimal. 785 vs. \(2 (188) = 376\).
Time = 0.29 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.95 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{11} \, b^{5} e^{5} x^{11} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b^{5} d e^{4} x^{10} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a b^{4} e^{5} x^{10} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, b^{5} d^{2} e^{3} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{9} \, a b^{4} d e^{4} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, a^{2} b^{3} e^{5} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, b^{5} d^{3} e^{2} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{4} \, a b^{4} d^{2} e^{3} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{4} \, a^{2} b^{3} d e^{4} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, a^{3} b^{2} e^{5} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, b^{5} d^{4} e x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {50}{7} \, a b^{4} d^{3} e^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {100}{7} \, a^{2} b^{3} d^{2} e^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {50}{7} \, a^{3} b^{2} d e^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, a^{4} b e^{5} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, b^{5} d^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{6} \, a b^{4} d^{4} e x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {50}{3} \, a^{2} b^{3} d^{3} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {50}{3} \, a^{3} b^{2} d^{2} e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{6} \, a^{4} b d e^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{5} e^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + a b^{4} d^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{4} e x^{5} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{2} d^{3} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{4} b d^{2} e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + a^{5} d e^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b^{3} d^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{2} \, a^{3} b^{2} d^{4} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{2} \, a^{4} b d^{3} e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{5} d^{2} e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b^{2} d^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{3} \, a^{4} b d^{4} e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{5} d^{3} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} b d^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{5} d^{4} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{5} d^{5} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (462 \, a^{6} b^{5} d^{5} - 330 \, a^{7} b^{4} d^{4} e + 165 \, a^{8} b^{3} d^{3} e^{2} - 55 \, a^{9} b^{2} d^{2} e^{3} + 11 \, a^{10} b d e^{4} - a^{11} e^{5}\right )} \mathrm {sgn}\left (b x + a\right )}{2772 \, b^{6}} \]
1/11*b^5*e^5*x^11*sgn(b*x + a) + 1/2*b^5*d*e^4*x^10*sgn(b*x + a) + 1/2*a*b ^4*e^5*x^10*sgn(b*x + a) + 10/9*b^5*d^2*e^3*x^9*sgn(b*x + a) + 25/9*a*b^4* d*e^4*x^9*sgn(b*x + a) + 10/9*a^2*b^3*e^5*x^9*sgn(b*x + a) + 5/4*b^5*d^3*e ^2*x^8*sgn(b*x + a) + 25/4*a*b^4*d^2*e^3*x^8*sgn(b*x + a) + 25/4*a^2*b^3*d *e^4*x^8*sgn(b*x + a) + 5/4*a^3*b^2*e^5*x^8*sgn(b*x + a) + 5/7*b^5*d^4*e*x ^7*sgn(b*x + a) + 50/7*a*b^4*d^3*e^2*x^7*sgn(b*x + a) + 100/7*a^2*b^3*d^2* e^3*x^7*sgn(b*x + a) + 50/7*a^3*b^2*d*e^4*x^7*sgn(b*x + a) + 5/7*a^4*b*e^5 *x^7*sgn(b*x + a) + 1/6*b^5*d^5*x^6*sgn(b*x + a) + 25/6*a*b^4*d^4*e*x^6*sg n(b*x + a) + 50/3*a^2*b^3*d^3*e^2*x^6*sgn(b*x + a) + 50/3*a^3*b^2*d^2*e^3* x^6*sgn(b*x + a) + 25/6*a^4*b*d*e^4*x^6*sgn(b*x + a) + 1/6*a^5*e^5*x^6*sgn (b*x + a) + a*b^4*d^5*x^5*sgn(b*x + a) + 10*a^2*b^3*d^4*e*x^5*sgn(b*x + a) + 20*a^3*b^2*d^3*e^2*x^5*sgn(b*x + a) + 10*a^4*b*d^2*e^3*x^5*sgn(b*x + a) + a^5*d*e^4*x^5*sgn(b*x + a) + 5/2*a^2*b^3*d^5*x^4*sgn(b*x + a) + 25/2*a^ 3*b^2*d^4*e*x^4*sgn(b*x + a) + 25/2*a^4*b*d^3*e^2*x^4*sgn(b*x + a) + 5/2*a ^5*d^2*e^3*x^4*sgn(b*x + a) + 10/3*a^3*b^2*d^5*x^3*sgn(b*x + a) + 25/3*a^4 *b*d^4*e*x^3*sgn(b*x + a) + 10/3*a^5*d^3*e^2*x^3*sgn(b*x + a) + 5/2*a^4*b* d^5*x^2*sgn(b*x + a) + 5/2*a^5*d^4*e*x^2*sgn(b*x + a) + a^5*d^5*x*sgn(b*x + a) + 1/2772*(462*a^6*b^5*d^5 - 330*a^7*b^4*d^4*e + 165*a^8*b^3*d^3*e^2 - 55*a^9*b^2*d^2*e^3 + 11*a^10*b*d*e^4 - a^11*e^5)*sgn(b*x + a)/b^6
Timed out. \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^5\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]