Integrand size = 33, antiderivative size = 198 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(A b-a B) (b d-a e)^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^4}+\frac {B e^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4} \]
1/4*(A*b-B*a)*(-a*e+b*d)^2*(b*x+a)^3*((b*x+a)^2)^(1/2)/b^4+1/5*(-a*e+b*d)* (2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^4*((b*x+a)^2)^(1/2)/b^4+1/6*e*(A*b*e-3*B*a *e+2*B*b*d)*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^4+1/7*B*e^2*(b*x+a)^6*((b*x+a)^2 )^(1/2)/b^4
Time = 1.07 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.18 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (35 a^3 \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+21 a^2 b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+21 a b^2 x^2 \left (2 A \left (10 d^2+15 d e x+6 e^2 x^2\right )+B x \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+b^3 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )\right )}{420 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(35*a^3*(4*A*(3*d^2 + 3*d*e*x + e^2*x^2) + B*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)) + 21*a^2*b*x*(5*A*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 2*B*x*(10*d^2 + 15*d*e*x + 6*e^2*x^2)) + 21*a*b^2*x^2*(2*A*(10*d^2 + 15*d *e*x + 6*e^2*x^2) + B*x*(15*d^2 + 24*d*e*x + 10*e^2*x^2)) + b^3*x^3*(7*A*( 15*d^2 + 24*d*e*x + 10*e^2*x^2) + 4*B*x*(21*d^2 + 35*d*e*x + 15*e^2*x^2))) )/(420*(a + b*x))
Time = 0.41 (sec) , antiderivative size = 146, 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, 86, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (A+B x) (d+e x)^2 \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^3 (A+B x) (d+e x)^2dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^3 (A+B x) (d+e x)^2dx}{a+b x}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B e^2 (a+b x)^6}{b^3}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^5}{b^3}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^4}{b^3}+\frac {(A b-a B) (b d-a e)^2 (a+b x)^3}{b^3}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {e (a+b x)^6 (-3 a B e+A b e+2 b B d)}{6 b^4}+\frac {(a+b x)^5 (b d-a e) (-3 a B e+2 A b e+b B d)}{5 b^4}+\frac {(a+b x)^4 (A b-a B) (b d-a e)^2}{4 b^4}+\frac {B e^2 (a+b x)^7}{7 b^4}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((A*b - a*B)*(b*d - a*e)^2*(a + b*x)^4)/(4 *b^4) + ((b*d - a*e)*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^5)/(5*b^4) + (e *(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^6)/(6*b^4) + (B*e^2*(a + b*x)^7)/(7 *b^4)))/(a + b*x)
3.18.23.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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. \(303\) vs. \(2(146)=292\).
Time = 0.34 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.54
| method | result | size |
| gosper | \(\frac {x \left (60 b^{3} B \,e^{2} x^{6}+70 x^{5} A \,b^{3} e^{2}+210 x^{5} B \,e^{2} b^{2} a +140 x^{5} B \,b^{3} d e +252 x^{4} A a \,b^{2} e^{2}+168 x^{4} A \,b^{3} d e +252 x^{4} B \,e^{2} b \,a^{2}+504 x^{4} B a \,b^{2} d e +84 x^{4} B \,b^{3} d^{2}+315 x^{3} A \,a^{2} b \,e^{2}+630 x^{3} A \,b^{2} d e a +105 x^{3} A \,d^{2} b^{3}+105 x^{3} B \,e^{2} a^{3}+630 x^{3} B \,a^{2} b d e +315 x^{3} B \,b^{2} d^{2} a +140 x^{2} A \,a^{3} e^{2}+840 x^{2} A \,a^{2} b d e +420 x^{2} A \,d^{2} b^{2} a +280 x^{2} B \,a^{3} d e +420 x^{2} B \,a^{2} b \,d^{2}+420 x A \,a^{3} d e +630 x A \,d^{2} b \,a^{2}+210 x B \,a^{3} d^{2}+420 A \,d^{2} a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{420 \left (b x +a \right )^{3}}\) | \(304\) |
| default | \(\frac {x \left (60 b^{3} B \,e^{2} x^{6}+70 x^{5} A \,b^{3} e^{2}+210 x^{5} B \,e^{2} b^{2} a +140 x^{5} B \,b^{3} d e +252 x^{4} A a \,b^{2} e^{2}+168 x^{4} A \,b^{3} d e +252 x^{4} B \,e^{2} b \,a^{2}+504 x^{4} B a \,b^{2} d e +84 x^{4} B \,b^{3} d^{2}+315 x^{3} A \,a^{2} b \,e^{2}+630 x^{3} A \,b^{2} d e a +105 x^{3} A \,d^{2} b^{3}+105 x^{3} B \,e^{2} a^{3}+630 x^{3} B \,a^{2} b d e +315 x^{3} B \,b^{2} d^{2} a +140 x^{2} A \,a^{3} e^{2}+840 x^{2} A \,a^{2} b d e +420 x^{2} A \,d^{2} b^{2} a +280 x^{2} B \,a^{3} d e +420 x^{2} B \,a^{2} b \,d^{2}+420 x A \,a^{3} d e +630 x A \,d^{2} b \,a^{2}+210 x B \,a^{3} d^{2}+420 A \,d^{2} a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{420 \left (b x +a \right )^{3}}\) | \(304\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{3} B \,e^{2} x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (A \,e^{2}+2 B d e \right ) b^{3}+3 B \,e^{2} b^{2} a \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (2 A d e +B \,d^{2}\right ) b^{3}+3 \left (A \,e^{2}+2 B d e \right ) b^{2} a +3 B \,e^{2} b \,a^{2}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A \,d^{2} b^{3}+3 \left (2 A d e +B \,d^{2}\right ) b^{2} a +3 \left (A \,e^{2}+2 B d e \right ) b \,a^{2}+B \,e^{2} a^{3}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 A \,d^{2} b^{2} a +3 \left (2 A d e +B \,d^{2}\right ) b \,a^{2}+\left (A \,e^{2}+2 B d e \right ) a^{3}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 A \,d^{2} b \,a^{2}+\left (2 A d e +B \,d^{2}\right ) a^{3}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, A \,d^{2} a^{3} x}{b x +a}\) | \(349\) |
1/420*x*(60*B*b^3*e^2*x^6+70*A*b^3*e^2*x^5+210*B*a*b^2*e^2*x^5+140*B*b^3*d *e*x^5+252*A*a*b^2*e^2*x^4+168*A*b^3*d*e*x^4+252*B*a^2*b*e^2*x^4+504*B*a*b ^2*d*e*x^4+84*B*b^3*d^2*x^4+315*A*a^2*b*e^2*x^3+630*A*a*b^2*d*e*x^3+105*A* b^3*d^2*x^3+105*B*a^3*e^2*x^3+630*B*a^2*b*d*e*x^3+315*B*a*b^2*d^2*x^3+140* A*a^3*e^2*x^2+840*A*a^2*b*d*e*x^2+420*A*a*b^2*d^2*x^2+280*B*a^3*d*e*x^2+42 0*B*a^2*b*d^2*x^2+420*A*a^3*d*e*x+630*A*a^2*b*d^2*x+210*B*a^3*d^2*x+420*A* a^3*d^2)*((b*x+a)^2)^(3/2)/(b*x+a)^3
Time = 0.39 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.21 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{7} \, B b^{3} e^{2} x^{7} + A a^{3} d^{2} x + \frac {1}{6} \, {\left (2 \, B b^{3} d e + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{2} + 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{3} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{3} d e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2}\right )} x^{2} \]
1/7*B*b^3*e^2*x^7 + A*a^3*d^2*x + 1/6*(2*B*b^3*d*e + (3*B*a*b^2 + A*b^3)*e ^2)*x^6 + 1/5*(B*b^3*d^2 + 2*(3*B*a*b^2 + A*b^3)*d*e + 3*(B*a^2*b + A*a*b^ 2)*e^2)*x^5 + 1/4*((3*B*a*b^2 + A*b^3)*d^2 + 6*(B*a^2*b + A*a*b^2)*d*e + ( B*a^3 + 3*A*a^2*b)*e^2)*x^4 + 1/3*(A*a^3*e^2 + 3*(B*a^2*b + A*a*b^2)*d^2 + 2*(B*a^3 + 3*A*a^2*b)*d*e)*x^3 + 1/2*(2*A*a^3*d*e + (B*a^3 + 3*A*a^2*b)*d ^2)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 4842 vs. \(2 (156) = 312\).
Time = 1.28 (sec) , antiderivative size = 4842, normalized size of antiderivative = 24.45 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(B*b**2*e**2*x**6/7 + x**5*(A* b**4*e**2 + 15*B*a*b**3*e**2/7 + 2*B*b**4*d*e)/(6*b**2) + x**4*(4*A*a*b**3 *e**2 + 2*A*b**4*d*e + 36*B*a**2*b**2*e**2/7 + 8*B*a*b**3*d*e + B*b**4*d** 2 - 11*a*(A*b**4*e**2 + 15*B*a*b**3*e**2/7 + 2*B*b**4*d*e)/(6*b))/(5*b**2) + x**3*(6*A*a**2*b**2*e**2 + 8*A*a*b**3*d*e + A*b**4*d**2 + 4*B*a**3*b*e* *2 + 12*B*a**2*b**2*d*e + 4*B*a*b**3*d**2 - 5*a**2*(A*b**4*e**2 + 15*B*a*b **3*e**2/7 + 2*B*b**4*d*e)/(6*b**2) - 9*a*(4*A*a*b**3*e**2 + 2*A*b**4*d*e + 36*B*a**2*b**2*e**2/7 + 8*B*a*b**3*d*e + B*b**4*d**2 - 11*a*(A*b**4*e**2 + 15*B*a*b**3*e**2/7 + 2*B*b**4*d*e)/(6*b))/(5*b))/(4*b**2) + x**2*(4*A*a **3*b*e**2 + 12*A*a**2*b**2*d*e + 4*A*a*b**3*d**2 + B*a**4*e**2 + 8*B*a**3 *b*d*e + 6*B*a**2*b**2*d**2 - 4*a**2*(4*A*a*b**3*e**2 + 2*A*b**4*d*e + 36* B*a**2*b**2*e**2/7 + 8*B*a*b**3*d*e + B*b**4*d**2 - 11*a*(A*b**4*e**2 + 15 *B*a*b**3*e**2/7 + 2*B*b**4*d*e)/(6*b))/(5*b**2) - 7*a*(6*A*a**2*b**2*e**2 + 8*A*a*b**3*d*e + A*b**4*d**2 + 4*B*a**3*b*e**2 + 12*B*a**2*b**2*d*e + 4 *B*a*b**3*d**2 - 5*a**2*(A*b**4*e**2 + 15*B*a*b**3*e**2/7 + 2*B*b**4*d*e)/ (6*b**2) - 9*a*(4*A*a*b**3*e**2 + 2*A*b**4*d*e + 36*B*a**2*b**2*e**2/7 + 8 *B*a*b**3*d*e + B*b**4*d**2 - 11*a*(A*b**4*e**2 + 15*B*a*b**3*e**2/7 + 2*B *b**4*d*e)/(6*b))/(5*b))/(4*b))/(3*b**2) + x*(A*a**4*e**2 + 8*A*a**3*b*d*e + 6*A*a**2*b**2*d**2 + 2*B*a**4*d*e + 4*B*a**3*b*d**2 - 3*a**2*(6*A*a**2* b**2*e**2 + 8*A*a*b**3*d*e + A*b**4*d**2 + 4*B*a**3*b*e**2 + 12*B*a**2*...
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (146) = 292\).
Time = 0.20 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.30 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A d^{2} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{3} e^{2} x}{4 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B e^{2} x^{2}}{7 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A a d^{2}}{4 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{4} e^{2}}{4 \, b^{4}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a e^{2} x}{14 \, b^{3}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{2} e^{2}}{70 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (2 \, B d e + A e^{2}\right )} a^{2} x}{4 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (B d^{2} + 2 \, A d e\right )} a x}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (2 \, B d e + A e^{2}\right )} a^{3}}{4 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (B d^{2} + 2 \, A d e\right )} a^{2}}{4 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, B d e + A e^{2}\right )} x}{6 \, b^{2}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, B d e + A e^{2}\right )} a}{30 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (B d^{2} + 2 \, A d e\right )}}{5 \, b^{2}} \]
1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*d^2*x - 1/4*(b^2*x^2 + 2*a*b*x + a^2 )^(3/2)*B*a^3*e^2*x/b^3 + 1/7*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*e^2*x^2/b^ 2 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*a*d^2/b - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a^4*e^2/b^4 - 3/14*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a*e^2 *x/b^3 + 17/70*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^2*e^2/b^4 + 1/4*(b^2*x^ 2 + 2*a*b*x + a^2)^(3/2)*(2*B*d*e + A*e^2)*a^2*x/b^2 - 1/4*(b^2*x^2 + 2*a* b*x + a^2)^(3/2)*(B*d^2 + 2*A*d*e)*a*x/b + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^( 3/2)*(2*B*d*e + A*e^2)*a^3/b^3 - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*d^ 2 + 2*A*d*e)*a^2/b^2 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(2*B*d*e + A*e^ 2)*x/b^2 - 7/30*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(2*B*d*e + A*e^2)*a/b^3 + 1/5*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*d^2 + 2*A*d*e)/b^2
Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (146) = 292\).
Time = 0.30 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.56 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{7} \, B b^{3} e^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, B b^{3} d e x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a b^{2} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, A b^{3} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, B b^{3} d^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{5} \, B a b^{2} d e x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, A b^{3} d e x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, B a^{2} b e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, A a b^{2} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, B a b^{2} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, A b^{3} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, B a^{2} b d e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, A a b^{2} d e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, B a^{3} e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, A a^{2} b e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + B a^{2} b d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + A a b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B a^{3} d e x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{2} b d e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, A a^{3} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{3} d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, A a^{2} b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + A a^{3} d e x^{2} \mathrm {sgn}\left (b x + a\right ) + A a^{3} d^{2} x \mathrm {sgn}\left (b x + a\right ) - \frac {{\left (21 \, B a^{5} b^{2} d^{2} - 105 \, A a^{4} b^{3} d^{2} - 14 \, B a^{6} b d e + 42 \, A a^{5} b^{2} d e + 3 \, B a^{7} e^{2} - 7 \, A a^{6} b e^{2}\right )} \mathrm {sgn}\left (b x + a\right )}{420 \, b^{4}} \]
1/7*B*b^3*e^2*x^7*sgn(b*x + a) + 1/3*B*b^3*d*e*x^6*sgn(b*x + a) + 1/2*B*a* b^2*e^2*x^6*sgn(b*x + a) + 1/6*A*b^3*e^2*x^6*sgn(b*x + a) + 1/5*B*b^3*d^2* x^5*sgn(b*x + a) + 6/5*B*a*b^2*d*e*x^5*sgn(b*x + a) + 2/5*A*b^3*d*e*x^5*sg n(b*x + a) + 3/5*B*a^2*b*e^2*x^5*sgn(b*x + a) + 3/5*A*a*b^2*e^2*x^5*sgn(b* x + a) + 3/4*B*a*b^2*d^2*x^4*sgn(b*x + a) + 1/4*A*b^3*d^2*x^4*sgn(b*x + a) + 3/2*B*a^2*b*d*e*x^4*sgn(b*x + a) + 3/2*A*a*b^2*d*e*x^4*sgn(b*x + a) + 1 /4*B*a^3*e^2*x^4*sgn(b*x + a) + 3/4*A*a^2*b*e^2*x^4*sgn(b*x + a) + B*a^2*b *d^2*x^3*sgn(b*x + a) + A*a*b^2*d^2*x^3*sgn(b*x + a) + 2/3*B*a^3*d*e*x^3*s gn(b*x + a) + 2*A*a^2*b*d*e*x^3*sgn(b*x + a) + 1/3*A*a^3*e^2*x^3*sgn(b*x + a) + 1/2*B*a^3*d^2*x^2*sgn(b*x + a) + 3/2*A*a^2*b*d^2*x^2*sgn(b*x + a) + A*a^3*d*e*x^2*sgn(b*x + a) + A*a^3*d^2*x*sgn(b*x + a) - 1/420*(21*B*a^5*b^ 2*d^2 - 105*A*a^4*b^3*d^2 - 14*B*a^6*b*d*e + 42*A*a^5*b^2*d*e + 3*B*a^7*e^ 2 - 7*A*a^6*b*e^2)*sgn(b*x + a)/b^4
Timed out. \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]