Integrand size = 33, antiderivative size = 172 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(b d-a e)^3 (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^4}+\frac {e (b d-a e)^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^4}+\frac {3 e^2 (b d-a e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}+\frac {e^3 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^4} \]
1/5*(-a*e+b*d)^3*(b*x+a)^4*((b*x+a)^2)^(1/2)/b^4+1/2*e*(-a*e+b*d)^2*(b*x+a )^5*((b*x+a)^2)^(1/2)/b^4+3/7*e^2*(-a*e+b*d)*(b*x+a)^6*((b*x+a)^2)^(1/2)/b ^4+1/8*e^3*(b*x+a)^7*((b*x+a)^2)^(1/2)/b^4
Time = 1.05 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.23 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (70 a^4 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+56 a^3 b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+28 a^2 b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+8 a b^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+b^4 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )\right )}{280 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(70*a^4*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 56*a^3*b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + 28*a^2*b^2* x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 8*a*b^3*x^3*(35*d^ 3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + b^4*x^4*(56*d^3 + 140*d^2*e* x + 120*d*e^2*x^2 + 35*e^3*x^3)))/(280*(a + b*x))
Time = 0.35 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.70, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^4 (d+e x)^3dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^4 (d+e x)^3dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {e^3 (a+b x)^7}{b^3}+\frac {3 e^2 (b d-a e) (a+b x)^6}{b^3}+\frac {3 e (b d-a e)^2 (a+b x)^5}{b^3}+\frac {(b d-a e)^3 (a+b x)^4}{b^3}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {3 e^2 (a+b x)^7 (b d-a e)}{7 b^4}+\frac {e (a+b x)^6 (b d-a e)^2}{2 b^4}+\frac {(a+b x)^5 (b d-a e)^3}{5 b^4}+\frac {e^3 (a+b x)^8}{8 b^4}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^3*(a + b*x)^5)/(5*b^4) + (e*( b*d - a*e)^2*(a + b*x)^6)/(2*b^4) + (3*e^2*(b*d - a*e)*(a + b*x)^7)/(7*b^4 ) + (e^3*(a + b*x)^8)/(8*b^4)))/(a + b*x)
3.20.72.3.1 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. \(263\) vs. \(2(120)=240\).
Time = 0.40 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.53
| method | result | size |
| gosper | \(\frac {x \left (35 x^{7} b^{4} e^{3}+160 x^{6} a \,b^{3} e^{3}+120 x^{6} b^{4} d \,e^{2}+280 x^{5} a^{2} b^{2} e^{3}+560 x^{5} a \,b^{3} d \,e^{2}+140 x^{5} b^{4} d^{2} e +224 x^{4} a^{3} b \,e^{3}+1008 x^{4} a^{2} b^{2} d \,e^{2}+672 x^{4} a \,b^{3} d^{2} e +56 x^{4} b^{4} d^{3}+70 x^{3} a^{4} e^{3}+840 a^{3} b d \,e^{2} x^{3}+1260 a^{2} b^{2} d^{2} e \,x^{3}+280 a \,b^{3} d^{3} x^{3}+280 a^{4} d \,e^{2} x^{2}+1120 x^{2} a^{3} b \,d^{2} e +560 x^{2} b^{2} d^{3} a^{2}+420 x \,d^{2} e \,a^{4}+560 b \,d^{3} a^{3} x +280 d^{3} a^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 \left (b x +a \right )^{3}}\) | \(264\) |
| default | \(\frac {x \left (35 x^{7} b^{4} e^{3}+160 x^{6} a \,b^{3} e^{3}+120 x^{6} b^{4} d \,e^{2}+280 x^{5} a^{2} b^{2} e^{3}+560 x^{5} a \,b^{3} d \,e^{2}+140 x^{5} b^{4} d^{2} e +224 x^{4} a^{3} b \,e^{3}+1008 x^{4} a^{2} b^{2} d \,e^{2}+672 x^{4} a \,b^{3} d^{2} e +56 x^{4} b^{4} d^{3}+70 x^{3} a^{4} e^{3}+840 a^{3} b d \,e^{2} x^{3}+1260 a^{2} b^{2} d^{2} e \,x^{3}+280 a \,b^{3} d^{3} x^{3}+280 a^{4} d \,e^{2} x^{2}+1120 x^{2} a^{3} b \,d^{2} e +560 x^{2} b^{2} d^{3} a^{2}+420 x \,d^{2} e \,a^{4}+560 b \,d^{3} a^{3} x +280 d^{3} a^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 \left (b x +a \right )^{3}}\) | \(264\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{4} e^{3} x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 e^{3} b^{3} a +3 d \,e^{2} b^{4}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 e^{3} b^{2} a^{2}+12 d \,e^{2} b^{3} a +3 d^{2} e \,b^{4}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 e^{3} b \,a^{3}+18 a^{2} b^{2} d \,e^{2}+12 d^{2} e \,b^{3} a +b^{4} d^{3}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{4} e^{3}+12 d \,e^{2} b \,a^{3}+18 d^{2} e \,b^{2} a^{2}+4 d^{3} b^{3} a \right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 d \,e^{2} a^{4}+12 d^{2} e b \,a^{3}+6 b^{2} d^{3} a^{2}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 d^{2} e \,a^{4}+4 d^{3} b \,a^{3}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, d^{3} a^{4} x}{b x +a}\) | \(357\) |
1/280*x*(35*b^4*e^3*x^7+160*a*b^3*e^3*x^6+120*b^4*d*e^2*x^6+280*a^2*b^2*e^ 3*x^5+560*a*b^3*d*e^2*x^5+140*b^4*d^2*e*x^5+224*a^3*b*e^3*x^4+1008*a^2*b^2 *d*e^2*x^4+672*a*b^3*d^2*e*x^4+56*b^4*d^3*x^4+70*a^4*e^3*x^3+840*a^3*b*d*e ^2*x^3+1260*a^2*b^2*d^2*e*x^3+280*a*b^3*d^3*x^3+280*a^4*d*e^2*x^2+1120*a^3 *b*d^2*e*x^2+560*a^2*b^2*d^3*x^2+420*a^4*d^2*e*x+560*a^3*b*d^3*x+280*a^4*d ^3)*((b*x+a)^2)^(3/2)/(b*x+a)^3
Time = 0.71 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.31 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{8} \, b^{4} e^{3} x^{8} + a^{4} d^{3} x + \frac {1}{7} \, {\left (3 \, b^{4} d e^{2} + 4 \, a b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b^{4} d^{2} e + 4 \, a b^{3} d e^{2} + 2 \, a^{2} b^{2} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{3} + 12 \, a b^{3} d^{2} e + 18 \, a^{2} b^{2} d e^{2} + 4 \, a^{3} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} d^{3} + 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} + a^{4} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{3} + 4 \, a^{3} b d^{2} e + a^{4} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \]
1/8*b^4*e^3*x^8 + a^4*d^3*x + 1/7*(3*b^4*d*e^2 + 4*a*b^3*e^3)*x^7 + 1/2*(b ^4*d^2*e + 4*a*b^3*d*e^2 + 2*a^2*b^2*e^3)*x^6 + 1/5*(b^4*d^3 + 12*a*b^3*d^ 2*e + 18*a^2*b^2*d*e^2 + 4*a^3*b*e^3)*x^5 + 1/4*(4*a*b^3*d^3 + 18*a^2*b^2* d^2*e + 12*a^3*b*d*e^2 + a^4*e^3)*x^4 + (2*a^2*b^2*d^3 + 4*a^3*b*d^2*e + a ^4*d*e^2)*x^3 + 1/2*(4*a^3*b*d^3 + 3*a^4*d^2*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 5727 vs. \(2 (121) = 242\).
Time = 1.33 (sec) , antiderivative size = 5727, normalized size of antiderivative = 33.30 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**3*e**3*x**7/8 + x**6*(25*a *b**4*e**3/8 + 3*b**5*d*e**2)/(7*b**2) + x**5*(73*a**2*b**3*e**3/8 + 15*a* b**4*d*e**2 - 13*a*(25*a*b**4*e**3/8 + 3*b**5*d*e**2)/(7*b) + 3*b**5*d**2* e)/(6*b**2) + x**4*(10*a**3*b**2*e**3 + 30*a**2*b**3*d*e**2 - 6*a**2*(25*a *b**4*e**3/8 + 3*b**5*d*e**2)/(7*b**2) + 15*a*b**4*d**2*e - 11*a*(73*a**2* b**3*e**3/8 + 15*a*b**4*d*e**2 - 13*a*(25*a*b**4*e**3/8 + 3*b**5*d*e**2)/( 7*b) + 3*b**5*d**2*e)/(6*b) + b**5*d**3)/(5*b**2) + x**3*(5*a**4*b*e**3 + 30*a**3*b**2*d*e**2 + 30*a**2*b**3*d**2*e - 5*a**2*(73*a**2*b**3*e**3/8 + 15*a*b**4*d*e**2 - 13*a*(25*a*b**4*e**3/8 + 3*b**5*d*e**2)/(7*b) + 3*b**5* d**2*e)/(6*b**2) + 5*a*b**4*d**3 - 9*a*(10*a**3*b**2*e**3 + 30*a**2*b**3*d *e**2 - 6*a**2*(25*a*b**4*e**3/8 + 3*b**5*d*e**2)/(7*b**2) + 15*a*b**4*d** 2*e - 11*a*(73*a**2*b**3*e**3/8 + 15*a*b**4*d*e**2 - 13*a*(25*a*b**4*e**3/ 8 + 3*b**5*d*e**2)/(7*b) + 3*b**5*d**2*e)/(6*b) + b**5*d**3)/(5*b))/(4*b** 2) + x**2*(a**5*e**3 + 15*a**4*b*d*e**2 + 30*a**3*b**2*d**2*e + 10*a**2*b* *3*d**3 - 4*a**2*(10*a**3*b**2*e**3 + 30*a**2*b**3*d*e**2 - 6*a**2*(25*a*b **4*e**3/8 + 3*b**5*d*e**2)/(7*b**2) + 15*a*b**4*d**2*e - 11*a*(73*a**2*b* *3*e**3/8 + 15*a*b**4*d*e**2 - 13*a*(25*a*b**4*e**3/8 + 3*b**5*d*e**2)/(7* b) + 3*b**5*d**2*e)/(6*b) + b**5*d**3)/(5*b**2) - 7*a*(5*a**4*b*e**3 + 30* a**3*b**2*d*e**2 + 30*a**2*b**3*d**2*e - 5*a**2*(73*a**2*b**3*e**3/8 + 15* a*b**4*d*e**2 - 13*a*(25*a*b**4*e**3/8 + 3*b**5*d*e**2)/(7*b) + 3*b**5*...
Leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (120) = 240\).
Time = 0.21 (sec) , antiderivative size = 693, normalized size of antiderivative = 4.03 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} e^{3} x^{3}}{8 \, b} + \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{3} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4} e^{3} x}{4 \, b^{3}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e^{3} x^{2}}{56 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{3}}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{5} e^{3}}{4 \, b^{4}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e^{3} x}{56 \, b^{3}} - \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e^{3}}{280 \, b^{4}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} x}{4 \, b^{3}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} x}{4 \, b^{2}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a x}{4 \, b} + \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x^{2}}{7 \, b^{2}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4}}{4 \, b^{4}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3}}{4 \, b^{3}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2}}{4 \, b^{2}} - \frac {3 \, {\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x}{14 \, b^{3}} + \frac {{\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x}{2 \, b^{2}} + \frac {17 \, {\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2}}{70 \, b^{4}} - \frac {7 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a}{10 \, b^{3}} + \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{5 \, b^{2}} \]
1/8*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^3*x^3/b + 1/4*(b^2*x^2 + 2*a*b*x + a ^2)^(3/2)*a*d^3*x + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*e^3*x/b^3 - 11 /56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*e^3*x^2/b^2 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^3/b + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5*e^3/b^4 + 13/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*e^3*x/b^3 - 69/280*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*e^3/b^4 - 1/4*(3*b*d*e^2 + a*e^3)*(b^2*x^2 + 2*a* b*x + a^2)^(3/2)*a^3*x/b^3 + 3/4*(b*d^2*e + a*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*x/b^2 - 1/4*(b*d^3 + 3*a*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^( 3/2)*a*x/b + 1/7*(3*b*d*e^2 + a*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^2/b ^2 - 1/4*(3*b*d*e^2 + a*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4/b^4 + 3/4 *(b*d^2*e + a*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3/b^3 - 1/4*(b*d^3 + 3*a*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2/b^2 - 3/14*(3*b*d*e^2 + a *e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x/b^3 + 1/2*(b*d^2*e + a*d*e^2)*(b ^2*x^2 + 2*a*b*x + a^2)^(5/2)*x/b^2 + 17/70*(3*b*d*e^2 + a*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^4 - 7/10*(b*d^2*e + a*d*e^2)*(b^2*x^2 + 2*a*b* x + a^2)^(5/2)*a/b^3 + 1/5*(b*d^3 + 3*a*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^( 5/2)/b^2
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (120) = 240\).
Time = 0.28 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.43 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{8} \, b^{4} e^{3} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, b^{4} d e^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{7} \, a b^{3} e^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b^{4} d^{2} e x^{6} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b^{3} d e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{2} e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, b^{4} d^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {12}{5} \, a b^{3} d^{2} e x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {18}{5} \, a^{2} b^{2} d e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, a^{3} b e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + a b^{3} d^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{2} \, a^{2} b^{2} d^{2} e x^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{3} b d e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, a^{4} e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} b d^{2} e x^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} d e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{4} d^{2} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{4} d^{3} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (56 \, a^{5} b^{3} d^{3} - 28 \, a^{6} b^{2} d^{2} e + 8 \, a^{7} b d e^{2} - a^{8} e^{3}\right )} \mathrm {sgn}\left (b x + a\right )}{280 \, b^{4}} \]
1/8*b^4*e^3*x^8*sgn(b*x + a) + 3/7*b^4*d*e^2*x^7*sgn(b*x + a) + 4/7*a*b^3* e^3*x^7*sgn(b*x + a) + 1/2*b^4*d^2*e*x^6*sgn(b*x + a) + 2*a*b^3*d*e^2*x^6* sgn(b*x + a) + a^2*b^2*e^3*x^6*sgn(b*x + a) + 1/5*b^4*d^3*x^5*sgn(b*x + a) + 12/5*a*b^3*d^2*e*x^5*sgn(b*x + a) + 18/5*a^2*b^2*d*e^2*x^5*sgn(b*x + a) + 4/5*a^3*b*e^3*x^5*sgn(b*x + a) + a*b^3*d^3*x^4*sgn(b*x + a) + 9/2*a^2*b ^2*d^2*e*x^4*sgn(b*x + a) + 3*a^3*b*d*e^2*x^4*sgn(b*x + a) + 1/4*a^4*e^3*x ^4*sgn(b*x + a) + 2*a^2*b^2*d^3*x^3*sgn(b*x + a) + 4*a^3*b*d^2*e*x^3*sgn(b *x + a) + a^4*d*e^2*x^3*sgn(b*x + a) + 2*a^3*b*d^3*x^2*sgn(b*x + a) + 3/2* a^4*d^2*e*x^2*sgn(b*x + a) + a^4*d^3*x*sgn(b*x + a) + 1/280*(56*a^5*b^3*d^ 3 - 28*a^6*b^2*d^2*e + 8*a^7*b*d*e^2 - a^8*e^3)*sgn(b*x + a)/b^4
Timed out. \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]