Integrand size = 33, antiderivative size = 125 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e)^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^3}+\frac {e (b d-a e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^3}+\frac {e^2 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^3} \]
1/7*(-a*e+b*d)^2*(b*x+a)^6*((b*x+a)^2)^(1/2)/b^3+1/4*e*(-a*e+b*d)*(b*x+a)^ 7*((b*x+a)^2)^(1/2)/b^3+1/9*e^2*(b*x+a)^8*((b*x+a)^2)^(1/2)/b^3
Time = 1.05 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.74 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (84 a^6 \left (3 d^2+3 d e x+e^2 x^2\right )+126 a^5 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+126 a^4 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+84 a^3 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+36 a^2 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+9 a b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )}{252 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(84*a^6*(3*d^2 + 3*d*e*x + e^2*x^2) + 126*a^5*b*x*(6* d^2 + 8*d*e*x + 3*e^2*x^2) + 126*a^4*b^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^ 2) + 84*a^3*b^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 36*a^2*b^4*x^4*(21* d^2 + 35*d*e*x + 15*e^2*x^2) + 9*a*b^5*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2 ) + b^6*x^6*(36*d^2 + 63*d*e*x + 28*e^2*x^2)))/(252*(a + b*x))
Time = 0.34 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.74, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^2 \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^6 (d+e x)^2dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^6 (d+e x)^2dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {e^2 (a+b x)^8}{b^2}+\frac {2 e (b d-a e) (a+b x)^7}{b^2}+\frac {(b d-a e)^2 (a+b x)^6}{b^2}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {e (a+b x)^8 (b d-a e)}{4 b^3}+\frac {(a+b x)^7 (b d-a e)^2}{7 b^3}+\frac {e^2 (a+b x)^9}{9 b^3}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^2*(a + b*x)^7)/(7*b^3) + (e*( b*d - a*e)*(a + b*x)^8)/(4*b^3) + (e^2*(a + b*x)^9)/(9*b^3)))/(a + b*x)
3.20.95.3.1 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_.)*((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. \(270\) vs. \(2(86)=172\).
Time = 0.37 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.17
| method | result | size |
| gosper | \(\frac {x \left (28 b^{6} e^{2} x^{8}+189 x^{7} e^{2} b^{5} a +63 x^{7} d e \,b^{6}+540 x^{6} e^{2} b^{4} a^{2}+432 x^{6} d e \,b^{5} a +36 x^{6} d^{2} b^{6}+840 x^{5} e^{2} a^{3} b^{3}+1260 x^{5} d e \,b^{4} a^{2}+252 x^{5} d^{2} b^{5} a +756 a^{4} b^{2} e^{2} x^{4}+2016 a^{3} b^{3} d e \,x^{4}+756 a^{2} b^{4} d^{2} x^{4}+378 x^{3} e^{2} a^{5} b +1890 x^{3} d e \,a^{4} b^{2}+1260 x^{3} d^{2} a^{3} b^{3}+84 x^{2} e^{2} a^{6}+1008 x^{2} d e \,a^{5} b +1260 x^{2} d^{2} a^{4} b^{2}+252 a^{6} d e x +756 a^{5} b \,d^{2} x +252 d^{2} a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{252 \left (b x +a \right )^{5}}\) | \(271\) |
| default | \(\frac {x \left (28 b^{6} e^{2} x^{8}+189 x^{7} e^{2} b^{5} a +63 x^{7} d e \,b^{6}+540 x^{6} e^{2} b^{4} a^{2}+432 x^{6} d e \,b^{5} a +36 x^{6} d^{2} b^{6}+840 x^{5} e^{2} a^{3} b^{3}+1260 x^{5} d e \,b^{4} a^{2}+252 x^{5} d^{2} b^{5} a +756 a^{4} b^{2} e^{2} x^{4}+2016 a^{3} b^{3} d e \,x^{4}+756 a^{2} b^{4} d^{2} x^{4}+378 x^{3} e^{2} a^{5} b +1890 x^{3} d e \,a^{4} b^{2}+1260 x^{3} d^{2} a^{3} b^{3}+84 x^{2} e^{2} a^{6}+1008 x^{2} d e \,a^{5} b +1260 x^{2} d^{2} a^{4} b^{2}+252 a^{6} d e x +756 a^{5} b \,d^{2} x +252 d^{2} a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{252 \left (b x +a \right )^{5}}\) | \(271\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{6} e^{2} x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 e^{2} b^{5} a +2 d e \,b^{6}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (15 e^{2} b^{4} a^{2}+12 d e \,b^{5} a +d^{2} b^{6}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (20 e^{2} a^{3} b^{3}+30 d e \,b^{4} a^{2}+6 d^{2} b^{5} a \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (15 e^{2} a^{4} b^{2}+40 d e \,a^{3} b^{3}+15 d^{2} b^{4} a^{2}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 e^{2} a^{5} b +30 d e \,a^{4} b^{2}+20 d^{2} a^{3} b^{3}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{2} a^{6}+12 d e \,a^{5} b +15 d^{2} a^{4} b^{2}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 d e \,a^{6}+6 d^{2} a^{5} b \right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, d^{2} a^{6} x}{b x +a}\) | \(383\) |
1/252*x*(28*b^6*e^2*x^8+189*a*b^5*e^2*x^7+63*b^6*d*e*x^7+540*a^2*b^4*e^2*x ^6+432*a*b^5*d*e*x^6+36*b^6*d^2*x^6+840*a^3*b^3*e^2*x^5+1260*a^2*b^4*d*e*x ^5+252*a*b^5*d^2*x^5+756*a^4*b^2*e^2*x^4+2016*a^3*b^3*d*e*x^4+756*a^2*b^4* d^2*x^4+378*a^5*b*e^2*x^3+1890*a^4*b^2*d*e*x^3+1260*a^3*b^3*d^2*x^3+84*a^6 *e^2*x^2+1008*a^5*b*d*e*x^2+1260*a^4*b^2*d^2*x^2+252*a^6*d*e*x+756*a^5*b*d ^2*x+252*a^6*d^2)*((b*x+a)^2)^(5/2)/(b*x+a)^5
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (86) = 172\).
Time = 0.28 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.87 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{9} \, b^{6} e^{2} x^{9} + a^{6} d^{2} x + \frac {1}{4} \, {\left (b^{6} d e + 3 \, a b^{5} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{2} + 12 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x^{7} + \frac {1}{3} \, {\left (3 \, a b^{5} d^{2} + 15 \, a^{2} b^{4} d e + 10 \, a^{3} b^{3} e^{2}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{2} + 8 \, a^{3} b^{3} d e + 3 \, a^{4} b^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (10 \, a^{3} b^{3} d^{2} + 15 \, a^{4} b^{2} d e + 3 \, a^{5} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (15 \, a^{4} b^{2} d^{2} + 12 \, a^{5} b d e + a^{6} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{2} + a^{6} d e\right )} x^{2} \]
1/9*b^6*e^2*x^9 + a^6*d^2*x + 1/4*(b^6*d*e + 3*a*b^5*e^2)*x^8 + 1/7*(b^6*d ^2 + 12*a*b^5*d*e + 15*a^2*b^4*e^2)*x^7 + 1/3*(3*a*b^5*d^2 + 15*a^2*b^4*d* e + 10*a^3*b^3*e^2)*x^6 + (3*a^2*b^4*d^2 + 8*a^3*b^3*d*e + 3*a^4*b^2*e^2)* x^5 + 1/2*(10*a^3*b^3*d^2 + 15*a^4*b^2*d*e + 3*a^5*b*e^2)*x^4 + 1/3*(15*a^ 4*b^2*d^2 + 12*a^5*b*d*e + a^6*e^2)*x^3 + (3*a^5*b*d^2 + a^6*d*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 8136 vs. \(2 (85) = 170\).
Time = 1.34 (sec) , antiderivative size = 8136, normalized size of antiderivative = 65.09 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**5*e**2*x**8/9 + x**7*(46*a *b**6*e**2/9 + 2*b**7*d*e)/(8*b**2) + x**6*(181*a**2*b**5*e**2/9 + 14*a*b* *6*d*e - 15*a*(46*a*b**6*e**2/9 + 2*b**7*d*e)/(8*b) + b**7*d**2)/(7*b**2) + x**5*(35*a**3*b**4*e**2 + 42*a**2*b**5*d*e - 7*a**2*(46*a*b**6*e**2/9 + 2*b**7*d*e)/(8*b**2) + 7*a*b**6*d**2 - 13*a*(181*a**2*b**5*e**2/9 + 14*a*b **6*d*e - 15*a*(46*a*b**6*e**2/9 + 2*b**7*d*e)/(8*b) + b**7*d**2)/(7*b))/( 6*b**2) + x**4*(35*a**4*b**3*e**2 + 70*a**3*b**4*d*e + 21*a**2*b**5*d**2 - 6*a**2*(181*a**2*b**5*e**2/9 + 14*a*b**6*d*e - 15*a*(46*a*b**6*e**2/9 + 2 *b**7*d*e)/(8*b) + b**7*d**2)/(7*b**2) - 11*a*(35*a**3*b**4*e**2 + 42*a**2 *b**5*d*e - 7*a**2*(46*a*b**6*e**2/9 + 2*b**7*d*e)/(8*b**2) + 7*a*b**6*d** 2 - 13*a*(181*a**2*b**5*e**2/9 + 14*a*b**6*d*e - 15*a*(46*a*b**6*e**2/9 + 2*b**7*d*e)/(8*b) + b**7*d**2)/(7*b))/(6*b))/(5*b**2) + x**3*(21*a**5*b**2 *e**2 + 70*a**4*b**3*d*e + 35*a**3*b**4*d**2 - 5*a**2*(35*a**3*b**4*e**2 + 42*a**2*b**5*d*e - 7*a**2*(46*a*b**6*e**2/9 + 2*b**7*d*e)/(8*b**2) + 7*a* b**6*d**2 - 13*a*(181*a**2*b**5*e**2/9 + 14*a*b**6*d*e - 15*a*(46*a*b**6*e **2/9 + 2*b**7*d*e)/(8*b) + b**7*d**2)/(7*b))/(6*b**2) - 9*a*(35*a**4*b**3 *e**2 + 70*a**3*b**4*d*e + 21*a**2*b**5*d**2 - 6*a**2*(181*a**2*b**5*e**2/ 9 + 14*a*b**6*d*e - 15*a*(46*a*b**6*e**2/9 + 2*b**7*d*e)/(8*b) + b**7*d**2 )/(7*b**2) - 11*a*(35*a**3*b**4*e**2 + 42*a**2*b**5*d*e - 7*a**2*(46*a*b** 6*e**2/9 + 2*b**7*d*e)/(8*b**2) + 7*a*b**6*d**2 - 13*a*(181*a**2*b**5*e...
Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (86) = 172\).
Time = 0.20 (sec) , antiderivative size = 452, normalized size of antiderivative = 3.62 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{2} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e^{2} x}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e^{2} x^{2}}{9 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{2}}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} e^{2}}{6 \, b^{3}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e^{2} x}{72 \, b^{2}} + \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} e^{2}}{504 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, b d e + a e^{2}\right )} a^{2} x}{6 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (b d^{2} + 2 \, a d e\right )} a x}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, b d e + a e^{2}\right )} a^{3}}{6 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (b d^{2} + 2 \, a d e\right )} a^{2}}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, b d e + a e^{2}\right )} x}{8 \, b^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, b d e + a e^{2}\right )} a}{56 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (b d^{2} + 2 \, a d e\right )}}{7 \, b^{2}} \]
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^2*x - 1/6*(b^2*x^2 + 2*a*b*x + a^2 )^(5/2)*a^3*e^2*x/b^2 + 1/9*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^2*x^2/b + 1/ 6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^2/b - 1/6*(b^2*x^2 + 2*a*b*x + a^2 )^(5/2)*a^4*e^2/b^3 - 11/72*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^2*x/b^2 + 83/504*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*e^2/b^3 + 1/6*(b^2*x^2 + 2*a*b* x + a^2)^(5/2)*(2*b*d*e + a*e^2)*a^2*x/b^2 - 1/6*(b^2*x^2 + 2*a*b*x + a^2) ^(5/2)*(b*d^2 + 2*a*d*e)*a*x/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(2*b* d*e + a*e^2)*a^3/b^3 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*d^2 + 2*a*d* e)*a^2/b^2 + 1/8*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(2*b*d*e + a*e^2)*x/b^2 - 9/56*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(2*b*d*e + a*e^2)*a/b^3 + 1/7*(b^2*x ^2 + 2*a*b*x + a^2)^(7/2)*(b*d^2 + 2*a*d*e)/b^2
Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (86) = 172\).
Time = 0.27 (sec) , antiderivative size = 417, normalized size of antiderivative = 3.34 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{9} \, b^{6} e^{2} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, b^{6} d e x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, a b^{5} e^{2} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, b^{6} d^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {12}{7} \, a b^{5} d e x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{7} \, a^{2} b^{4} e^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + a b^{5} d^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{2} b^{4} d e x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b^{3} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + 8 \, a^{3} b^{3} d e x^{5} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{4} b^{2} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{4} b^{2} d e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{5} b e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{5} b d e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a^{6} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{6} d e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{6} d^{2} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (36 \, a^{7} b^{2} d^{2} - 9 \, a^{8} b d e + a^{9} e^{2}\right )} \mathrm {sgn}\left (b x + a\right )}{252 \, b^{3}} \]
1/9*b^6*e^2*x^9*sgn(b*x + a) + 1/4*b^6*d*e*x^8*sgn(b*x + a) + 3/4*a*b^5*e^ 2*x^8*sgn(b*x + a) + 1/7*b^6*d^2*x^7*sgn(b*x + a) + 12/7*a*b^5*d*e*x^7*sgn (b*x + a) + 15/7*a^2*b^4*e^2*x^7*sgn(b*x + a) + a*b^5*d^2*x^6*sgn(b*x + a) + 5*a^2*b^4*d*e*x^6*sgn(b*x + a) + 10/3*a^3*b^3*e^2*x^6*sgn(b*x + a) + 3* a^2*b^4*d^2*x^5*sgn(b*x + a) + 8*a^3*b^3*d*e*x^5*sgn(b*x + a) + 3*a^4*b^2* e^2*x^5*sgn(b*x + a) + 5*a^3*b^3*d^2*x^4*sgn(b*x + a) + 15/2*a^4*b^2*d*e*x ^4*sgn(b*x + a) + 3/2*a^5*b*e^2*x^4*sgn(b*x + a) + 5*a^4*b^2*d^2*x^3*sgn(b *x + a) + 4*a^5*b*d*e*x^3*sgn(b*x + a) + 1/3*a^6*e^2*x^3*sgn(b*x + a) + 3* a^5*b*d^2*x^2*sgn(b*x + a) + a^6*d*e*x^2*sgn(b*x + a) + a^6*d^2*x*sgn(b*x + a) + 1/252*(36*a^7*b^2*d^2 - 9*a^8*b*d*e + a^9*e^2)*sgn(b*x + a)/b^3
Timed out. \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]