Integrand size = 28, antiderivative size = 200 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=-\frac {(b d-a e)^3 (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x)}+\frac {b (b d-a e)^2 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x)}-\frac {3 b^2 (b d-a e) (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x)}+\frac {b^3 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^4 (a+b x)} \]
-1/5*(-a*e+b*d)^3*(e*x+d)^5*((b*x+a)^2)^(1/2)/e^4/(b*x+a)+1/2*b*(-a*e+b*d) ^2*(e*x+d)^6*((b*x+a)^2)^(1/2)/e^4/(b*x+a)-3/7*b^2*(-a*e+b*d)*(e*x+d)^7*(( b*x+a)^2)^(1/2)/e^4/(b*x+a)+1/8*b^3*(e*x+d)^8*((b*x+a)^2)^(1/2)/e^4/(b*x+a )
Time = 1.05 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.08 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (56 a^3 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+28 a^2 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+8 a b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )\right )}{280 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(56*a^3*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^ 3*x^3 + e^4*x^4) + 28*a^2*b*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d *e^3*x^3 + 5*e^4*x^4) + 8*a*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^ 2 + 70*d*e^3*x^3 + 15*e^4*x^4) + b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e ^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4)))/(280*(a + b*x))
Time = 0.34 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.60, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^4 \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^3 (d+e x)^4dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^3 (d+e x)^4dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^3 (d+e x)^7}{e^3}-\frac {3 b^2 (b d-a e) (d+e x)^6}{e^3}+\frac {3 b (b d-a e)^2 (d+e x)^5}{e^3}+\frac {(a e-b d)^3 (d+e x)^4}{e^3}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {3 b^2 (d+e x)^7 (b d-a e)}{7 e^4}+\frac {b (d+e x)^6 (b d-a e)^2}{2 e^4}-\frac {(d+e x)^5 (b d-a e)^3}{5 e^4}+\frac {b^3 (d+e x)^8}{8 e^4}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/5*((b*d - a*e)^3*(d + e*x)^5)/e^4 + (b* (b*d - a*e)^2*(d + e*x)^6)/(2*e^4) - (3*b^2*(b*d - a*e)*(d + e*x)^7)/(7*e^ 4) + (b^3*(d + e*x)^8)/(8*e^4)))/(a + b*x)
3.16.55.3.1 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_))^(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]
Time = 2.54 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.32
| method | result | size |
| gosper | \(\frac {x \left (35 b^{3} e^{4} x^{7}+120 x^{6} a \,b^{2} e^{4}+160 x^{6} b^{3} d \,e^{3}+140 x^{5} a^{2} b \,e^{4}+560 x^{5} a \,b^{2} d \,e^{3}+280 x^{5} b^{3} d^{2} e^{2}+56 x^{4} e^{4} a^{3}+672 x^{4} a^{2} b d \,e^{3}+1008 x^{4} a \,b^{2} d^{2} e^{2}+224 x^{4} b^{3} d^{3} e +280 x^{3} a^{3} d \,e^{3}+1260 x^{3} a^{2} b \,d^{2} e^{2}+840 x^{3} a \,b^{2} d^{3} e +70 x^{3} b^{3} d^{4}+560 a^{3} d^{2} e^{2} x^{2}+1120 a^{2} b \,d^{3} e \,x^{2}+280 a \,b^{2} d^{4} x^{2}+560 x \,a^{3} d^{3} e +420 a^{2} b \,d^{4} x +280 d^{4} a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 \left (b x +a \right )^{3}}\) | \(264\) |
| default | \(\frac {x \left (35 b^{3} e^{4} x^{7}+120 x^{6} a \,b^{2} e^{4}+160 x^{6} b^{3} d \,e^{3}+140 x^{5} a^{2} b \,e^{4}+560 x^{5} a \,b^{2} d \,e^{3}+280 x^{5} b^{3} d^{2} e^{2}+56 x^{4} e^{4} a^{3}+672 x^{4} a^{2} b d \,e^{3}+1008 x^{4} a \,b^{2} d^{2} e^{2}+224 x^{4} b^{3} d^{3} e +280 x^{3} a^{3} d \,e^{3}+1260 x^{3} a^{2} b \,d^{2} e^{2}+840 x^{3} a \,b^{2} d^{3} e +70 x^{3} b^{3} d^{4}+560 a^{3} d^{2} e^{2} x^{2}+1120 a^{2} b \,d^{3} e \,x^{2}+280 a \,b^{2} d^{4} x^{2}+560 x \,a^{3} d^{3} e +420 a^{2} b \,d^{4} x +280 d^{4} a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 \left (b x +a \right )^{3}}\) | \(264\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{3} e^{4} x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 a \,b^{2} e^{4}+4 b^{3} d \,e^{3}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 a^{2} b \,e^{4}+12 a \,b^{2} d \,e^{3}+6 b^{3} d^{2} e^{2}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{4} a^{3}+12 a^{2} b d \,e^{3}+18 a \,b^{2} d^{2} e^{2}+4 b^{3} d^{3} e \right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a^{3} d \,e^{3}+18 a^{2} b \,d^{2} e^{2}+12 a \,b^{2} d^{3} e +b^{3} d^{4}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a^{3} d^{2} e^{2}+12 a^{2} b \,d^{3} e +3 a \,b^{2} d^{4}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a^{3} d^{3} e +3 a^{2} b \,d^{4}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, d^{4} x \,a^{3}}{b x +a}\) | \(357\) |
1/280*x*(35*b^3*e^4*x^7+120*a*b^2*e^4*x^6+160*b^3*d*e^3*x^6+140*a^2*b*e^4* x^5+560*a*b^2*d*e^3*x^5+280*b^3*d^2*e^2*x^5+56*a^3*e^4*x^4+672*a^2*b*d*e^3 *x^4+1008*a*b^2*d^2*e^2*x^4+224*b^3*d^3*e*x^4+280*a^3*d*e^3*x^3+1260*a^2*b *d^2*e^2*x^3+840*a*b^2*d^3*e*x^3+70*b^3*d^4*x^3+560*a^3*d^2*e^2*x^2+1120*a ^2*b*d^3*e*x^2+280*a*b^2*d^4*x^2+560*a^3*d^3*e*x+420*a^2*b*d^4*x+280*a^3*d ^4)*((b*x+a)^2)^(3/2)/(b*x+a)^3
Time = 0.29 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.12 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{8} \, b^{3} e^{4} x^{8} + a^{3} d^{4} x + \frac {1}{7} \, {\left (4 \, b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, b^{3} d^{2} e^{2} + 4 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, b^{3} d^{3} e + 18 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{4} + 12 \, a b^{2} d^{3} e + 18 \, a^{2} b d^{2} e^{2} + 4 \, a^{3} d e^{3}\right )} x^{4} + {\left (a b^{2} d^{4} + 4 \, a^{2} b d^{3} e + 2 \, a^{3} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{4} + 4 \, a^{3} d^{3} e\right )} x^{2} \]
1/8*b^3*e^4*x^8 + a^3*d^4*x + 1/7*(4*b^3*d*e^3 + 3*a*b^2*e^4)*x^7 + 1/2*(2 *b^3*d^2*e^2 + 4*a*b^2*d*e^3 + a^2*b*e^4)*x^6 + 1/5*(4*b^3*d^3*e + 18*a*b^ 2*d^2*e^2 + 12*a^2*b*d*e^3 + a^3*e^4)*x^5 + 1/4*(b^3*d^4 + 12*a*b^2*d^3*e + 18*a^2*b*d^2*e^2 + 4*a^3*d*e^3)*x^4 + (a*b^2*d^4 + 4*a^2*b*d^3*e + 2*a^3 *d^2*e^2)*x^3 + 1/2*(3*a^2*b*d^4 + 4*a^3*d^3*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 5923 vs. \(2 (141) = 282\).
Time = 1.05 (sec) , antiderivative size = 5923, normalized size of antiderivative = 29.62 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**2*e**4*x**7/8 + x**6*(17*a *b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + x**5*(41*a**2*b**2*e**4/8 + 16*a* b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2* e**2)/(6*b**2) + x**4*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a* b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a** 2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3) /(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b**2) + x**3*(a**4*e* *4 + 16*a**3*b*d*e**3 + 36*a**2*b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e**4 /8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6* b**4*d**2*e**2)/(6*b**2) + 16*a*b**3*d**3*e - 9*a*(4*a**3*b*e**4 + 24*a**2 *b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b **3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a* b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3 *e)/(5*b) + b**4*d**4)/(4*b**2) + x**2*(4*a**4*d*e**3 + 24*a**3*b*d**2*e** 2 + 24*a**2*b**2*d**3*e - 4*a**2*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6* a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 1 1*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b **4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b**2) + 4* a*b**3*d**4 - 7*a*(a**4*e**4 + 16*a**3*b*d*e**3 + 36*a**2*b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/...
Leaf count of result is larger than twice the leaf count of optimal. 589 vs. \(2 (148) = 296\).
Time = 0.20 (sec) , antiderivative size = 589, normalized size of antiderivative = 2.94 \[ \int (d+e x)^4 \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^{4} x^{3}}{8 \, b^{2}} + \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} d^{4} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{3} e x}{b} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{2} e^{2} x}{2 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} d e^{3} x}{b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4} e^{4} x}{4 \, b^{4}} + \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d e^{3} x^{2}}{7 \, b^{2}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e^{4} x^{2}}{56 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{4}}{4 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{3} e}{b^{2}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} d^{2} e^{2}}{2 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4} d e^{3}}{b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{5} e^{4}}{4 \, b^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{2} e^{2} x}{b^{2}} - \frac {6 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d e^{3} x}{7 \, b^{3}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e^{4} x}{56 \, b^{4}} + \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{3} e}{5 \, b^{2}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{2} e^{2}}{5 \, b^{3}} + \frac {34 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d e^{3}}{35 \, b^{4}} - \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e^{4}}{280 \, b^{5}} \]
1/8*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^4*x^3/b^2 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*d^4*x - (b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*d^3*e*x/b + 3/2*(b^2 *x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^2*e^2*x/b^2 - (b^2*x^2 + 2*a*b*x + a^2)^ (3/2)*a^3*d*e^3*x/b^3 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*e^4*x/b^4 + 4/7*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d*e^3*x^2/b^2 - 11/56*(b^2*x^2 + 2*a *b*x + a^2)^(5/2)*a*e^4*x^2/b^3 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*d^ 4/b - (b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^3*e/b^2 + 3/2*(b^2*x^2 + 2*a*b *x + a^2)^(3/2)*a^3*d^2*e^2/b^3 - (b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*d*e^ 3/b^4 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5*e^4/b^5 + (b^2*x^2 + 2*a*b *x + a^2)^(5/2)*d^2*e^2*x/b^2 - 6/7*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d*e^ 3*x/b^3 + 13/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*e^4*x/b^4 + 4/5*(b^2*x ^2 + 2*a*b*x + a^2)^(5/2)*d^3*e/b^2 - 7/5*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)* a*d^2*e^2/b^3 + 34/35*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d*e^3/b^4 - 69/2 80*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*e^4/b^5
Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (148) = 296\).
Time = 0.29 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.16 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{8} \, b^{3} e^{4} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{7} \, b^{3} d e^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, a b^{2} e^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + b^{3} d^{2} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b^{2} d e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a^{2} b e^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, b^{3} d^{3} e x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {18}{5} \, a b^{2} d^{2} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {12}{5} \, a^{2} b d e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a^{3} e^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, b^{3} d^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a b^{2} d^{3} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{2} \, a^{2} b d^{2} e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + a^{3} d e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + a b^{2} d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{2} b d^{3} e x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} d^{2} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{2} b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} d^{3} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} d^{4} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (70 \, a^{4} b^{4} d^{4} - 56 \, a^{5} b^{3} d^{3} e + 28 \, a^{6} b^{2} d^{2} e^{2} - 8 \, a^{7} b d e^{3} + a^{8} e^{4}\right )} \mathrm {sgn}\left (b x + a\right )}{280 \, b^{5}} \]
1/8*b^3*e^4*x^8*sgn(b*x + a) + 4/7*b^3*d*e^3*x^7*sgn(b*x + a) + 3/7*a*b^2* e^4*x^7*sgn(b*x + a) + b^3*d^2*e^2*x^6*sgn(b*x + a) + 2*a*b^2*d*e^3*x^6*sg n(b*x + a) + 1/2*a^2*b*e^4*x^6*sgn(b*x + a) + 4/5*b^3*d^3*e*x^5*sgn(b*x + a) + 18/5*a*b^2*d^2*e^2*x^5*sgn(b*x + a) + 12/5*a^2*b*d*e^3*x^5*sgn(b*x + a) + 1/5*a^3*e^4*x^5*sgn(b*x + a) + 1/4*b^3*d^4*x^4*sgn(b*x + a) + 3*a*b^2 *d^3*e*x^4*sgn(b*x + a) + 9/2*a^2*b*d^2*e^2*x^4*sgn(b*x + a) + a^3*d*e^3*x ^4*sgn(b*x + a) + a*b^2*d^4*x^3*sgn(b*x + a) + 4*a^2*b*d^3*e*x^3*sgn(b*x + a) + 2*a^3*d^2*e^2*x^3*sgn(b*x + a) + 3/2*a^2*b*d^4*x^2*sgn(b*x + a) + 2* a^3*d^3*e*x^2*sgn(b*x + a) + a^3*d^4*x*sgn(b*x + a) + 1/280*(70*a^4*b^4*d^ 4 - 56*a^5*b^3*d^3*e + 28*a^6*b^2*d^2*e^2 - 8*a^7*b*d*e^3 + a^8*e^4)*sgn(b *x + a)/b^5
Timed out. \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int {\left (d+e\,x\right )}^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]