Integrand size = 33, antiderivative size = 158 \[ \int (A+B x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {(b d-a e) (B d-A e) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}-\frac {(2 b B d-A b e-a B e) (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x)}+\frac {b B (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)} \]
1/5*(-a*e+b*d)*(-A*e+B*d)*(e*x+d)^5*((b*x+a)^2)^(1/2)/e^3/(b*x+a)-1/6*(-A* b*e-B*a*e+2*B*b*d)*(e*x+d)^6*((b*x+a)^2)^(1/2)/e^3/(b*x+a)+1/7*b*B*(e*x+d) ^7*((b*x+a)^2)^(1/2)/e^3/(b*x+a)
Time = 1.06 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.32 \[ \int (A+B x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (7 a \left (6 A \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+B x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )\right )+b x \left (7 A \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+2 B x \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )\right )\right )}{210 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(7*a*(6*A*(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) + 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)) + b*x*(7*A*(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) + 2*B*x*(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))))/(210*(a + b*x))
Time = 0.35 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.66, 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.
| \(\displaystyle \int \sqrt {a^2+2 a b x+b^2 x^2} (A+B x) (d+e x)^4 \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b (a+b x) (A+B x) (d+e x)^4dx}{b (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) (A+B x) (d+e x)^4dx}{a+b x}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b B (d+e x)^6}{e^2}+\frac {(-2 b B d+A b e+a B e) (d+e x)^5}{e^2}+\frac {(a e-b d) (A e-B d) (d+e x)^4}{e^2}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {(d+e x)^6 (-a B e-A b e+2 b B d)}{6 e^3}+\frac {(d+e x)^5 (b d-a e) (B d-A e)}{5 e^3}+\frac {b B (d+e x)^7}{7 e^3}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)*(B*d - A*e)*(d + e*x)^5)/(5*e ^3) - ((2*b*B*d - A*b*e - a*B*e)*(d + e*x)^6)/(6*e^3) + (b*B*(d + e*x)^7)/ (7*e^3)))/(a + b*x)
3.18.8.3.1 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]
Time = 0.52 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.47
| method | result | size |
| gosper | \(\frac {x \left (30 x^{6} B \,e^{4} b +35 x^{5} A b \,e^{4}+35 x^{5} B \,e^{4} a +140 x^{5} B b d \,e^{3}+42 x^{4} A a \,e^{4}+168 x^{4} A b d \,e^{3}+168 x^{4} B a d \,e^{3}+252 x^{4} B b \,d^{2} e^{2}+210 x^{3} A a d \,e^{3}+315 x^{3} A b \,d^{2} e^{2}+315 x^{3} B a \,d^{2} e^{2}+210 x^{3} B b \,d^{3} e +420 x^{2} A a \,d^{2} e^{2}+280 x^{2} A b \,d^{3} e +280 x^{2} B a \,d^{3} e +70 x^{2} B b \,d^{4}+420 x A a \,d^{3} e +105 x A \,d^{4} b +105 x B a \,d^{4}+210 a A \,d^{4}\right ) \sqrt {\left (b x +a \right )^{2}}}{210 b x +210 a}\) | \(232\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, x^{7} B \,e^{4} b}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (A b +B a \right ) e^{4}+4 B b d \,e^{3}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A a \,e^{4}+4 \left (A b +B a \right ) e^{3} d +6 B b \,d^{2} e^{2}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 A a d \,e^{3}+6 \left (A b +B a \right ) d^{2} e^{2}+4 B b \,d^{3} e \right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 A a \,d^{2} e^{2}+4 \left (A b +B a \right ) d^{3} e +B b \,d^{4}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 A a \,d^{3} e +\left (A b +B a \right ) d^{4}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, d^{4} A a x}{b x +a}\) | \(288\) |
| default | \(\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{2} \left (30 B \,x^{5} e^{4} b^{5}+35 A \,x^{4} b^{5} e^{4}-25 B \,x^{4} a \,b^{4} e^{4}+140 B \,x^{4} b^{5} d \,e^{3}-28 A \,x^{3} a \,b^{4} e^{4}+168 A \,x^{3} b^{5} d \,e^{3}+20 B \,x^{3} a^{2} b^{3} e^{4}-112 B \,x^{3} a \,b^{4} d \,e^{3}+252 B \,x^{3} b^{5} d^{2} e^{2}+21 A \,x^{2} a^{2} b^{3} e^{4}-126 A \,x^{2} a \,b^{4} d \,e^{3}+315 A \,x^{2} b^{5} d^{2} e^{2}-15 B \,x^{2} a^{3} b^{2} e^{4}+84 B \,x^{2} a^{2} b^{3} d \,e^{3}-189 B \,x^{2} a \,b^{4} d^{2} e^{2}+210 B \,x^{2} b^{5} d^{3} e -14 A \,a^{3} b^{2} e^{4} x +84 A x \,a^{2} b^{3} d \,e^{3}-210 A x a \,b^{4} d^{2} e^{2}+280 A \,b^{5} d^{3} e x +10 B \,a^{4} b \,e^{4} x -56 B \,a^{3} b^{2} d \,e^{3} x +126 B x \,a^{2} b^{3} d^{2} e^{2}-140 B x a \,b^{4} d^{3} e +70 B \,b^{5} d^{4} x +7 A \,a^{4} b \,e^{4}-42 A \,a^{3} b^{2} d \,e^{3}+105 A \,a^{2} b^{3} d^{2} e^{2}-140 A \,b^{4} d^{3} e a +105 A \,b^{5} d^{4}-5 B \,a^{5} e^{4}+28 B \,a^{4} b d \,e^{3}-63 B \,a^{3} b^{2} d^{2} e^{2}+70 B \,a^{2} b^{3} d^{3} e -35 B \,b^{4} d^{4} a \right )}{210 b^{6}}\) | \(475\) |
1/210*x*(30*B*b*e^4*x^6+35*A*b*e^4*x^5+35*B*a*e^4*x^5+140*B*b*d*e^3*x^5+42 *A*a*e^4*x^4+168*A*b*d*e^3*x^4+168*B*a*d*e^3*x^4+252*B*b*d^2*e^2*x^4+210*A *a*d*e^3*x^3+315*A*b*d^2*e^2*x^3+315*B*a*d^2*e^2*x^3+210*B*b*d^3*e*x^3+420 *A*a*d^2*e^2*x^2+280*A*b*d^3*e*x^2+280*B*a*d^3*e*x^2+70*B*b*d^4*x^2+420*A* a*d^3*e*x+105*A*b*d^4*x+105*B*a*d^4*x+210*A*a*d^4)*((b*x+a)^2)^(1/2)/(b*x+ a)
Time = 0.29 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.11 \[ \int (A+B x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{7} \, B b e^{4} x^{7} + A a d^{4} x + \frac {1}{6} \, {\left (4 \, B b d e^{3} + {\left (B a + A b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, B b d^{2} e^{2} + A a e^{4} + 4 \, {\left (B a + A b\right )} d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, B b d^{3} e + 2 \, A a d e^{3} + 3 \, {\left (B a + A b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b d^{4} + 6 \, A a d^{2} e^{2} + 4 \, {\left (B a + A b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a d^{3} e + {\left (B a + A b\right )} d^{4}\right )} x^{2} \]
1/7*B*b*e^4*x^7 + A*a*d^4*x + 1/6*(4*B*b*d*e^3 + (B*a + A*b)*e^4)*x^6 + 1/ 5*(6*B*b*d^2*e^2 + A*a*e^4 + 4*(B*a + A*b)*d*e^3)*x^5 + 1/2*(2*B*b*d^3*e + 2*A*a*d*e^3 + 3*(B*a + A*b)*d^2*e^2)*x^4 + 1/3*(B*b*d^4 + 6*A*a*d^2*e^2 + 4*(B*a + A*b)*d^3*e)*x^3 + 1/2*(4*A*a*d^3*e + (B*a + A*b)*d^4)*x^2
Time = 4.09 (sec) , antiderivative size = 1516, normalized size of antiderivative = 9.59 \[ \int (A+B x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]
A*d**4*Piecewise(((a/(2*b) + x/2)*sqrt(a**2 + 2*a*b*x + b**2*x**2), Ne(b** 2, 0)), ((a**2 + 2*a*b*x)**(3/2)/(3*a*b), Ne(a*b, 0)), (x*sqrt(a**2), True )) + 4*A*d**3*e*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(-a**2/(6*b**2 ) + a*x/(6*b) + x**2/3), Ne(b**2, 0)), ((-a**2*(a**2 + 2*a*b*x)**(3/2)/3 + (a**2 + 2*a*b*x)**(5/2)/5)/(2*a**2*b**2), Ne(a*b, 0)), (x**2*sqrt(a**2)/2 , True)) + 6*A*d**2*e**2*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(a**3 /(12*b**3) - a**2*x/(12*b**2) + a*x**2/(12*b) + x**3/4), Ne(b**2, 0)), ((a **4*(a**2 + 2*a*b*x)**(3/2)/3 - 2*a**2*(a**2 + 2*a*b*x)**(5/2)/5 + (a**2 + 2*a*b*x)**(7/2)/7)/(4*a**3*b**3), Ne(a*b, 0)), (x**3*sqrt(a**2)/3, True)) + 4*A*d*e**3*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(-a**4/(20*b**4) + a**3*x/(20*b**3) - a**2*x**2/(20*b**2) + a*x**3/(20*b) + x**4/5), Ne(b* *2, 0)), ((-a**6*(a**2 + 2*a*b*x)**(3/2)/3 + 3*a**4*(a**2 + 2*a*b*x)**(5/2 )/5 - 3*a**2*(a**2 + 2*a*b*x)**(7/2)/7 + (a**2 + 2*a*b*x)**(9/2)/9)/(8*a** 4*b**4), Ne(a*b, 0)), (x**4*sqrt(a**2)/4, True)) + A*e**4*Piecewise((sqrt( a**2 + 2*a*b*x + b**2*x**2)*(a**5/(30*b**5) - a**4*x/(30*b**4) + a**3*x**2 /(30*b**3) - a**2*x**3/(30*b**2) + a*x**4/(30*b) + x**5/6), Ne(b**2, 0)), ((a**8*(a**2 + 2*a*b*x)**(3/2)/3 - 4*a**6*(a**2 + 2*a*b*x)**(5/2)/5 + 6*a* *4*(a**2 + 2*a*b*x)**(7/2)/7 - 4*a**2*(a**2 + 2*a*b*x)**(9/2)/9 + (a**2 + 2*a*b*x)**(11/2)/11)/(16*a**5*b**5), Ne(a*b, 0)), (x**5*sqrt(a**2)/5, True )) + B*d**4*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(-a**2/(6*b**2)...
Leaf count of result is larger than twice the leaf count of optimal. 1002 vs. \(2 (119) = 238\).
Time = 0.21 (sec) , antiderivative size = 1002, normalized size of antiderivative = 6.34 \[ \int (A+B x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]
1/7*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*e^4*x^4/b^2 - 11/42*(b^2*x^2 + 2*a*b *x + a^2)^(3/2)*B*a*e^4*x^3/b^3 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*d^4* x - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a^5*e^4*x/b^5 + 5/14*(b^2*x^2 + 2* a*b*x + a^2)^(3/2)*B*a^2*e^4*x^2/b^4 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A *a*d^4/b - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a^6*e^4/b^6 - 3/7*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a^3*e^4*x/b^5 + 10/21*(b^2*x^2 + 2*a*b*x + a^2)^( 3/2)*B*a^4*e^4/b^6 + 1/6*(4*B*d*e^3 + A*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/ 2)*x^3/b^2 + 1/2*(4*B*d*e^3 + A*e^4)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^4*x/b ^4 - (3*B*d^2*e^2 + 2*A*d*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^3*x/b^3 + ( 2*B*d^3*e + 3*A*d^2*e^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*x/b^2 - 1/2*(B* d^4 + 4*A*d^3*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*x/b - 3/10*(4*B*d*e^3 + A *e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x^2/b^3 + 2/5*(3*B*d^2*e^2 + 2*A*d *e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^2/b^2 + 1/2*(4*B*d*e^3 + A*e^4)*sq rt(b^2*x^2 + 2*a*b*x + a^2)*a^5/b^5 - (3*B*d^2*e^2 + 2*A*d*e^3)*sqrt(b^2*x ^2 + 2*a*b*x + a^2)*a^4/b^4 + (2*B*d^3*e + 3*A*d^2*e^2)*sqrt(b^2*x^2 + 2*a *b*x + a^2)*a^3/b^3 - 1/2*(B*d^4 + 4*A*d^3*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2 )*a^2/b^2 + 2/5*(4*B*d*e^3 + A*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*x/ b^4 - 7/10*(3*B*d^2*e^2 + 2*A*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x/b ^3 + 1/2*(2*B*d^3*e + 3*A*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x/b^2 - 7/15*(4*B*d*e^3 + A*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3/b^5 + 9/1...
Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (119) = 238\).
Time = 0.29 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.98 \[ \int (A+B x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{7} \, B b e^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B b d e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, B a e^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, A b e^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{5} \, B b d^{2} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, B a d e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, A b d e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, A a e^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + B b d^{3} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, B a d^{2} e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, A b d^{2} e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + A a d e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, B b d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, B a d^{3} e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, A b d^{3} e x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a d^{2} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a d^{3} e x^{2} \mathrm {sgn}\left (b x + a\right ) + A a d^{4} x \mathrm {sgn}\left (b x + a\right ) - \frac {{\left (35 \, B a^{3} b^{4} d^{4} - 105 \, A a^{2} b^{5} d^{4} - 70 \, B a^{4} b^{3} d^{3} e + 140 \, A a^{3} b^{4} d^{3} e + 63 \, B a^{5} b^{2} d^{2} e^{2} - 105 \, A a^{4} b^{3} d^{2} e^{2} - 28 \, B a^{6} b d e^{3} + 42 \, A a^{5} b^{2} d e^{3} + 5 \, B a^{7} e^{4} - 7 \, A a^{6} b e^{4}\right )} \mathrm {sgn}\left (b x + a\right )}{210 \, b^{6}} \]
1/7*B*b*e^4*x^7*sgn(b*x + a) + 2/3*B*b*d*e^3*x^6*sgn(b*x + a) + 1/6*B*a*e^ 4*x^6*sgn(b*x + a) + 1/6*A*b*e^4*x^6*sgn(b*x + a) + 6/5*B*b*d^2*e^2*x^5*sg n(b*x + a) + 4/5*B*a*d*e^3*x^5*sgn(b*x + a) + 4/5*A*b*d*e^3*x^5*sgn(b*x + a) + 1/5*A*a*e^4*x^5*sgn(b*x + a) + B*b*d^3*e*x^4*sgn(b*x + a) + 3/2*B*a*d ^2*e^2*x^4*sgn(b*x + a) + 3/2*A*b*d^2*e^2*x^4*sgn(b*x + a) + A*a*d*e^3*x^4 *sgn(b*x + a) + 1/3*B*b*d^4*x^3*sgn(b*x + a) + 4/3*B*a*d^3*e*x^3*sgn(b*x + a) + 4/3*A*b*d^3*e*x^3*sgn(b*x + a) + 2*A*a*d^2*e^2*x^3*sgn(b*x + a) + 1/ 2*B*a*d^4*x^2*sgn(b*x + a) + 1/2*A*b*d^4*x^2*sgn(b*x + a) + 2*A*a*d^3*e*x^ 2*sgn(b*x + a) + A*a*d^4*x*sgn(b*x + a) - 1/210*(35*B*a^3*b^4*d^4 - 105*A* a^2*b^5*d^4 - 70*B*a^4*b^3*d^3*e + 140*A*a^3*b^4*d^3*e + 63*B*a^5*b^2*d^2* e^2 - 105*A*a^4*b^3*d^2*e^2 - 28*B*a^6*b*d*e^3 + 42*A*a^5*b^2*d*e^3 + 5*B* a^7*e^4 - 7*A*a^6*b*e^4)*sgn(b*x + a)/b^6
Time = 12.87 (sec) , antiderivative size = 1400, normalized size of antiderivative = 8.86 \[ \int (A+B x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]
A*d^4*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2) + (A*e^4*x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(6*b^2) + (B*e^4*x^4*(a^2 + b^2*x^2 + 2*a*b*x)^( 3/2))/(7*b^2) + (B*d^4*(8*b^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*(a ^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(24*b^4) + (A*d^3*e*(8*b^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(6*b^4) + (B*d^ 3*e*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/b^2 - (B*a^2*e^4*(a^2 + b^2*x^2 + 2 *a*b*x)^(1/2)*(4*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) - a^4 + 9*a^2*b^2*x^2 + 8*a^3*b*x - 7*a*b*x*(a^2 + b^2*x^2 + 2*a*b*x)))/(35*b^6) - (A*a^2*e^4*(a^ 2 + b^2*x^2 + 2*a*b*x)^(1/2)*(a^3 - 5*a*b^2*x^2 + 3*b*x*(a^2 + b^2*x^2 + 2 *a*b*x) - 4*a^2*b*x))/(24*b^5) + (3*A*d^2*e^2*x*(a^2 + b^2*x^2 + 2*a*b*x)^ (3/2))/(2*b^2) + (4*A*d*e^3*x^2*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(5*b^2) + (2*B*d*e^3*x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(3*b^2) - (3*A*a*e^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(4*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) - a^4 + 9 *a^2*b^2*x^2 + 8*a^3*b*x - 7*a*b*x*(a^2 + b^2*x^2 + 2*a*b*x)))/(40*b^5) - (11*B*a*e^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(a^5 + 5*b^3*x^3*(a^2 + b^2*x^ 2 + 2*a*b*x) - 14*a^3*b^2*x^2 - 13*a^4*b*x - 9*a*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) + 12*a^2*b*x*(a^2 + b^2*x^2 + 2*a*b*x)))/(210*b^6) + (6*B*d^2*e^2 *x^2*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(5*b^2) - (B*a^2*d^3*e*(x/2 + a/(2*b ))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/b^2 - (3*B*a*d*e^3*(a^2 + b^2*x^2 + 2* a*b*x)^(1/2)*(4*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) - a^4 + 9*a^2*b^2*x^2...