Integrand size = 28, antiderivative size = 172 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e)^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^4}+\frac {3 e (b d-a e)^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}+\frac {3 e^2 (b d-a e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^4}+\frac {e^3 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^4} \]
1/6*(-a*e+b*d)^3*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^4+3/7*e*(-a*e+b*d)^2*(b*x+a )^6*((b*x+a)^2)^(1/2)/b^4+3/8*e^2*(-a*e+b*d)*(b*x+a)^7*((b*x+a)^2)^(1/2)/b ^4+1/9*e^3*(b*x+a)^8*((b*x+a)^2)^(1/2)/b^4
Time = 1.06 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.47 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (126 a^5 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+126 a^4 b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+84 a^3 b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+36 a^2 b^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+9 a b^4 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+b^5 x^5 \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )\right )}{504 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(126*a^5*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 126*a^4*b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + 84*a^3*b^ 2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 36*a^2*b^3*x^3*( 35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 9*a*b^4*x^4*(56*d^3 + 1 40*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + b^5*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3)))/(504*(a + b*x))
Time = 0.36 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.70, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^5 (d+e x)^3dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^5 (d+e x)^3dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {e^3 (a+b x)^8}{b^3}+\frac {3 e^2 (b d-a e) (a+b x)^7}{b^3}+\frac {3 e (b d-a e)^2 (a+b x)^6}{b^3}+\frac {(b d-a e)^3 (a+b x)^5}{b^3}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {3 e^2 (a+b x)^8 (b d-a e)}{8 b^4}+\frac {3 e (a+b x)^7 (b d-a e)^2}{7 b^4}+\frac {(a+b x)^6 (b d-a e)^3}{6 b^4}+\frac {e^3 (a+b x)^9}{9 b^4}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^3*(a + b*x)^6)/(6*b^4) + (3*e *(b*d - a*e)^2*(a + b*x)^7)/(7*b^4) + (3*e^2*(b*d - a*e)*(a + b*x)^8)/(8*b ^4) + (e^3*(a + b*x)^9)/(9*b^4)))/(a + b*x)
3.16.71.3.1 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. \(321\) vs. \(2(120)=240\).
Time = 2.80 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.87
| method | result | size |
| gosper | \(\frac {x \left (56 e^{3} b^{5} x^{8}+315 x^{7} a \,b^{4} e^{3}+189 x^{7} b^{5} d \,e^{2}+720 x^{6} a^{2} b^{3} e^{3}+1080 x^{6} d \,e^{2} a \,b^{4}+216 x^{6} d^{2} e \,b^{5}+840 x^{5} e^{3} a^{3} b^{2}+2520 x^{5} a^{2} b^{3} d \,e^{2}+1260 x^{5} d^{2} e a \,b^{4}+84 x^{5} d^{3} b^{5}+504 a^{4} b \,e^{3} x^{4}+3024 a^{3} b^{2} d \,e^{2} x^{4}+3024 a^{2} b^{3} d^{2} e \,x^{4}+504 a \,b^{4} d^{3} x^{4}+126 x^{3} e^{3} a^{5}+1890 x^{3} d \,e^{2} a^{4} b +3780 x^{3} d^{2} e \,a^{3} b^{2}+1260 x^{3} a^{2} b^{3} d^{3}+504 x^{2} d \,e^{2} a^{5}+2520 x^{2} d^{2} e \,a^{4} b +1680 x^{2} a^{3} b^{2} d^{3}+756 x \,d^{2} e \,a^{5}+1260 x \,d^{3} a^{4} b +504 a^{5} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5}}\) | \(322\) |
| default | \(\frac {x \left (56 e^{3} b^{5} x^{8}+315 x^{7} a \,b^{4} e^{3}+189 x^{7} b^{5} d \,e^{2}+720 x^{6} a^{2} b^{3} e^{3}+1080 x^{6} d \,e^{2} a \,b^{4}+216 x^{6} d^{2} e \,b^{5}+840 x^{5} e^{3} a^{3} b^{2}+2520 x^{5} a^{2} b^{3} d \,e^{2}+1260 x^{5} d^{2} e a \,b^{4}+84 x^{5} d^{3} b^{5}+504 a^{4} b \,e^{3} x^{4}+3024 a^{3} b^{2} d \,e^{2} x^{4}+3024 a^{2} b^{3} d^{2} e \,x^{4}+504 a \,b^{4} d^{3} x^{4}+126 x^{3} e^{3} a^{5}+1890 x^{3} d \,e^{2} a^{4} b +3780 x^{3} d^{2} e \,a^{3} b^{2}+1260 x^{3} a^{2} b^{3} d^{3}+504 x^{2} d \,e^{2} a^{5}+2520 x^{2} d^{2} e \,a^{4} b +1680 x^{2} a^{3} b^{2} d^{3}+756 x \,d^{2} e \,a^{5}+1260 x \,d^{3} a^{4} b +504 a^{5} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5}}\) | \(322\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{3} b^{5} x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a \,b^{4} e^{3}+3 b^{5} d \,e^{2}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{2} b^{3} e^{3}+15 d \,e^{2} a \,b^{4}+3 d^{2} e \,b^{5}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 e^{3} a^{3} b^{2}+30 a^{2} b^{3} d \,e^{2}+15 d^{2} e a \,b^{4}+d^{3} b^{5}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 e^{3} a^{4} b +30 a^{3} b^{2} d \,e^{2}+30 d^{2} e \,a^{2} b^{3}+5 a \,b^{4} d^{3}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{3} a^{5}+15 d \,e^{2} a^{4} b +30 d^{2} e \,a^{3} b^{2}+10 a^{2} b^{3} d^{3}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 d \,e^{2} a^{5}+15 d^{2} e \,a^{4} b +10 a^{3} b^{2} d^{3}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 d^{2} e \,a^{5}+5 d^{3} a^{4} b \right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{5} d^{3} x}{b x +a}\) | \(425\) |
1/504*x*(56*b^5*e^3*x^8+315*a*b^4*e^3*x^7+189*b^5*d*e^2*x^7+720*a^2*b^3*e^ 3*x^6+1080*a*b^4*d*e^2*x^6+216*b^5*d^2*e*x^6+840*a^3*b^2*e^3*x^5+2520*a^2* b^3*d*e^2*x^5+1260*a*b^4*d^2*e*x^5+84*b^5*d^3*x^5+504*a^4*b*e^3*x^4+3024*a ^3*b^2*d*e^2*x^4+3024*a^2*b^3*d^2*e*x^4+504*a*b^4*d^3*x^4+126*a^5*e^3*x^3+ 1890*a^4*b*d*e^2*x^3+3780*a^3*b^2*d^2*e*x^3+1260*a^2*b^3*d^3*x^3+504*a^5*d *e^2*x^2+2520*a^4*b*d^2*e*x^2+1680*a^3*b^2*d^3*x^2+756*a^5*d^2*e*x+1260*a^ 4*b*d^3*x+504*a^5*d^3)*((b*x+a)^2)^(5/2)/(b*x+a)^5
Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (120) = 240\).
Time = 0.28 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.61 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{9} \, b^{5} e^{3} x^{9} + a^{5} d^{3} x + \frac {1}{8} \, {\left (3 \, b^{5} d e^{2} + 5 \, a b^{4} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, b^{5} d^{2} e + 15 \, a b^{4} d e^{2} + 10 \, a^{2} b^{3} e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{3} + 15 \, a b^{4} d^{2} e + 30 \, a^{2} b^{3} d e^{2} + 10 \, a^{3} b^{2} e^{3}\right )} x^{6} + {\left (a b^{4} d^{3} + 6 \, a^{2} b^{3} d^{2} e + 6 \, a^{3} b^{2} d e^{2} + a^{4} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, a^{2} b^{3} d^{3} + 30 \, a^{3} b^{2} d^{2} e + 15 \, a^{4} b d e^{2} + a^{5} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} d^{3} + 15 \, a^{4} b d^{2} e + 3 \, a^{5} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{3} + 3 \, a^{5} d^{2} e\right )} x^{2} \]
1/9*b^5*e^3*x^9 + a^5*d^3*x + 1/8*(3*b^5*d*e^2 + 5*a*b^4*e^3)*x^8 + 1/7*(3 *b^5*d^2*e + 15*a*b^4*d*e^2 + 10*a^2*b^3*e^3)*x^7 + 1/6*(b^5*d^3 + 15*a*b^ 4*d^2*e + 30*a^2*b^3*d*e^2 + 10*a^3*b^2*e^3)*x^6 + (a*b^4*d^3 + 6*a^2*b^3* d^2*e + 6*a^3*b^2*d*e^2 + a^4*b*e^3)*x^5 + 1/4*(10*a^2*b^3*d^3 + 30*a^3*b^ 2*d^2*e + 15*a^4*b*d*e^2 + a^5*e^3)*x^4 + 1/3*(10*a^3*b^2*d^3 + 15*a^4*b*d ^2*e + 3*a^5*d*e^2)*x^3 + 1/2*(5*a^4*b*d^3 + 3*a^5*d^2*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 9421 vs. \(2 (122) = 244\).
Time = 1.07 (sec) , antiderivative size = 9421, normalized size of antiderivative = 54.77 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**4*e**3*x**8/9 + x**7*(37*a *b**5*e**3/9 + 3*b**6*d*e**2)/(8*b**2) + x**6*(127*a**2*b**4*e**3/9 + 18*a *b**5*d*e**2 - 15*a*(37*a*b**5*e**3/9 + 3*b**6*d*e**2)/(8*b) + 3*b**6*d**2 *e)/(7*b**2) + x**5*(20*a**3*b**3*e**3 + 45*a**2*b**4*d*e**2 - 7*a**2*(37* a*b**5*e**3/9 + 3*b**6*d*e**2)/(8*b**2) + 18*a*b**5*d**2*e - 13*a*(127*a** 2*b**4*e**3/9 + 18*a*b**5*d*e**2 - 15*a*(37*a*b**5*e**3/9 + 3*b**6*d*e**2) /(8*b) + 3*b**6*d**2*e)/(7*b) + b**6*d**3)/(6*b**2) + x**4*(15*a**4*b**2*e **3 + 60*a**3*b**3*d*e**2 + 45*a**2*b**4*d**2*e - 6*a**2*(127*a**2*b**4*e* *3/9 + 18*a*b**5*d*e**2 - 15*a*(37*a*b**5*e**3/9 + 3*b**6*d*e**2)/(8*b) + 3*b**6*d**2*e)/(7*b**2) + 6*a*b**5*d**3 - 11*a*(20*a**3*b**3*e**3 + 45*a** 2*b**4*d*e**2 - 7*a**2*(37*a*b**5*e**3/9 + 3*b**6*d*e**2)/(8*b**2) + 18*a* b**5*d**2*e - 13*a*(127*a**2*b**4*e**3/9 + 18*a*b**5*d*e**2 - 15*a*(37*a*b **5*e**3/9 + 3*b**6*d*e**2)/(8*b) + 3*b**6*d**2*e)/(7*b) + b**6*d**3)/(6*b ))/(5*b**2) + x**3*(6*a**5*b*e**3 + 45*a**4*b**2*d*e**2 + 60*a**3*b**3*d** 2*e + 15*a**2*b**4*d**3 - 5*a**2*(20*a**3*b**3*e**3 + 45*a**2*b**4*d*e**2 - 7*a**2*(37*a*b**5*e**3/9 + 3*b**6*d*e**2)/(8*b**2) + 18*a*b**5*d**2*e - 13*a*(127*a**2*b**4*e**3/9 + 18*a*b**5*d*e**2 - 15*a*(37*a*b**5*e**3/9 + 3 *b**6*d*e**2)/(8*b) + 3*b**6*d**2*e)/(7*b) + b**6*d**3)/(6*b**2) - 9*a*(15 *a**4*b**2*e**3 + 60*a**3*b**3*d*e**2 + 45*a**2*b**4*d**2*e - 6*a**2*(127* a**2*b**4*e**3/9 + 18*a*b**5*d*e**2 - 15*a*(37*a*b**5*e**3/9 + 3*b**6*d...
Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (120) = 240\).
Time = 0.22 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.33 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{3} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{2} e x}{2 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d e^{2} x}{2 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e^{3} x}{6 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e^{3} x^{2}}{9 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{3}}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{2} e}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d e^{2}}{2 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} e^{3}}{6 \, b^{4}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d e^{2} x}{8 \, b^{2}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e^{3} x}{72 \, b^{3}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{2} e}{7 \, b^{2}} - \frac {27 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d e^{2}}{56 \, b^{3}} + \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} e^{3}}{504 \, b^{4}} \]
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d^3*x - 1/2*(b^2*x^2 + 2*a*b*x + a^2)^ (5/2)*a*d^2*e*x/b + 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d*e^2*x/b^2 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*e^3*x/b^3 + 1/9*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^3*x^2/b^2 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^3/b - 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^2*e/b^2 + 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*d*e^2/b^3 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*e^3/ b^4 + 3/8*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d*e^2*x/b^2 - 11/72*(b^2*x^2 + 2 *a*b*x + a^2)^(7/2)*a*e^3*x/b^3 + 3/7*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d^2* e/b^2 - 27/56*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*d*e^2/b^3 + 83/504*(b^2*x^ 2 + 2*a*b*x + a^2)^(7/2)*a^2*e^3/b^4
Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (120) = 240\).
Time = 0.27 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.91 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{9} \, b^{5} e^{3} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{8} \, b^{5} d e^{2} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, a b^{4} e^{3} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, b^{5} d^{2} e x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{7} \, a b^{4} d e^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, a^{2} b^{3} e^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, b^{5} d^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a b^{4} d^{2} e x^{6} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{2} b^{3} d e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a^{3} b^{2} e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + a b^{4} d^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e x^{5} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{3} b^{2} d e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + a^{4} b e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b^{3} d^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{3} b^{2} d^{2} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{4} \, a^{4} b d e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, a^{5} e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b^{2} d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d^{2} e x^{3} \mathrm {sgn}\left (b x + a\right ) + a^{5} d e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} b d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{5} d^{2} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{5} d^{3} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (84 \, a^{6} b^{3} d^{3} - 36 \, a^{7} b^{2} d^{2} e + 9 \, a^{8} b d e^{2} - a^{9} e^{3}\right )} \mathrm {sgn}\left (b x + a\right )}{504 \, b^{4}} \]
1/9*b^5*e^3*x^9*sgn(b*x + a) + 3/8*b^5*d*e^2*x^8*sgn(b*x + a) + 5/8*a*b^4* e^3*x^8*sgn(b*x + a) + 3/7*b^5*d^2*e*x^7*sgn(b*x + a) + 15/7*a*b^4*d*e^2*x ^7*sgn(b*x + a) + 10/7*a^2*b^3*e^3*x^7*sgn(b*x + a) + 1/6*b^5*d^3*x^6*sgn( b*x + a) + 5/2*a*b^4*d^2*e*x^6*sgn(b*x + a) + 5*a^2*b^3*d*e^2*x^6*sgn(b*x + a) + 5/3*a^3*b^2*e^3*x^6*sgn(b*x + a) + a*b^4*d^3*x^5*sgn(b*x + a) + 6*a ^2*b^3*d^2*e*x^5*sgn(b*x + a) + 6*a^3*b^2*d*e^2*x^5*sgn(b*x + a) + a^4*b*e ^3*x^5*sgn(b*x + a) + 5/2*a^2*b^3*d^3*x^4*sgn(b*x + a) + 15/2*a^3*b^2*d^2* e*x^4*sgn(b*x + a) + 15/4*a^4*b*d*e^2*x^4*sgn(b*x + a) + 1/4*a^5*e^3*x^4*s gn(b*x + a) + 10/3*a^3*b^2*d^3*x^3*sgn(b*x + a) + 5*a^4*b*d^2*e*x^3*sgn(b* x + a) + a^5*d*e^2*x^3*sgn(b*x + a) + 5/2*a^4*b*d^3*x^2*sgn(b*x + a) + 3/2 *a^5*d^2*e*x^2*sgn(b*x + a) + a^5*d^3*x*sgn(b*x + a) + 1/504*(84*a^6*b^3*d ^3 - 36*a^7*b^2*d^2*e + 9*a^8*b*d*e^2 - a^9*e^3)*sgn(b*x + a)/b^4
Timed out. \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]