Integrand size = 33, antiderivative size = 198 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(A b-a B) (b d-a e)^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^4}+\frac {B e^2 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^4} \]
1/6*(A*b-B*a)*(-a*e+b*d)^2*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^4+1/7*(-a*e+b*d)* (2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^6*((b*x+a)^2)^(1/2)/b^4+1/8*e*(A*b*e-3*B*a *e+2*B*b*d)*(b*x+a)^7*((b*x+a)^2)^(1/2)/b^4+1/9*B*e^2*(b*x+a)^8*((b*x+a)^2 )^(1/2)/b^4
Time = 1.10 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.75 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (42 a^5 \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+42 a^4 b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+84 a^3 b^2 x^2 \left (2 A \left (10 d^2+15 d e x+6 e^2 x^2\right )+B x \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+12 a^2 b^3 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )+3 a b^4 x^4 \left (8 A \left (21 d^2+35 d e x+15 e^2 x^2\right )+5 B x \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )+b^5 x^5 \left (3 A \left (28 d^2+48 d e x+21 e^2 x^2\right )+2 B x \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )\right )}{504 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(42*a^5*(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)) + 42*a^4*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)) + 84*a^3*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)) + 12*a^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)) + 3*a*b^4*x^4*(8*A*(21*d^2 + 35*d*e*x + 15*e^2*x^2) + 5*B*x*(28*d ^2 + 48*d*e*x + 21*e^2*x^2)) + b^5*x^5*(3*A*(28*d^2 + 48*d*e*x + 21*e^2*x^ 2) + 2*B*x*(36*d^2 + 63*d*e*x + 28*e^2*x^2))))/(504*(a + b*x))
Time = 0.49 (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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x) (d+e x)^2 \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^5 (A+B x) (d+e x)^2dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^5 (A+B x) (d+e x)^2dx}{a+b x}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B e^2 (a+b x)^8}{b^3}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^7}{b^3}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^6}{b^3}+\frac {(A b-a B) (b d-a e)^2 (a+b x)^5}{b^3}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {e (a+b x)^8 (-3 a B e+A b e+2 b B d)}{8 b^4}+\frac {(a+b x)^7 (b d-a e) (-3 a B e+2 A b e+b B d)}{7 b^4}+\frac {(a+b x)^6 (A b-a B) (b d-a e)^2}{6 b^4}+\frac {B e^2 (a+b x)^9}{9 b^4}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((A*b - a*B)*(b*d - a*e)^2*(a + b*x)^6)/(6 *b^4) + ((b*d - a*e)*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^7)/(7*b^4) + (e *(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^8)/(8*b^4) + (B*e^2*(a + b*x)^9)/(9 *b^4)))/(a + b*x)
3.18.42.3.1 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. \(479\) vs. \(2(146)=292\).
Time = 0.36 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.42
| method | result | size |
| gosper | \(\frac {x \left (56 B \,e^{2} b^{5} x^{8}+63 x^{7} A \,b^{5} e^{2}+315 x^{7} B \,e^{2} b^{4} a +126 x^{7} B \,b^{5} d e +360 x^{6} A a \,b^{4} e^{2}+144 x^{6} A \,b^{5} d e +720 x^{6} B \,e^{2} a^{2} b^{3}+720 x^{6} B a \,b^{4} d e +72 x^{6} B \,b^{5} d^{2}+840 x^{5} A \,a^{2} b^{3} e^{2}+840 x^{5} A a \,b^{4} d e +84 x^{5} A \,d^{2} b^{5}+840 x^{5} B \,e^{2} a^{3} b^{2}+1680 x^{5} B \,a^{2} b^{3} d e +420 x^{5} B a \,b^{4} d^{2}+1008 A \,a^{3} b^{2} e^{2} x^{4}+2016 A \,a^{2} b^{3} d e \,x^{4}+504 A a \,b^{4} d^{2} x^{4}+504 B \,a^{4} b \,e^{2} x^{4}+2016 B \,a^{3} b^{2} d e \,x^{4}+1008 B \,a^{2} b^{3} d^{2} x^{4}+630 x^{3} A \,a^{4} b \,e^{2}+2520 x^{3} A \,a^{3} b^{2} d e +1260 x^{3} A \,d^{2} a^{2} b^{3}+126 x^{3} B \,e^{2} a^{5}+1260 x^{3} B \,a^{4} b d e +1260 x^{3} B \,a^{3} b^{2} d^{2}+168 x^{2} A \,a^{5} e^{2}+1680 x^{2} A \,a^{4} b d e +1680 x^{2} A \,d^{2} a^{3} b^{2}+336 x^{2} B \,a^{5} d e +840 x^{2} B \,a^{4} b \,d^{2}+504 x A \,a^{5} d e +1260 x A \,d^{2} a^{4} b +252 x B \,a^{5} d^{2}+504 A \,d^{2} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5}}\) | \(480\) |
| default | \(\frac {x \left (56 B \,e^{2} b^{5} x^{8}+63 x^{7} A \,b^{5} e^{2}+315 x^{7} B \,e^{2} b^{4} a +126 x^{7} B \,b^{5} d e +360 x^{6} A a \,b^{4} e^{2}+144 x^{6} A \,b^{5} d e +720 x^{6} B \,e^{2} a^{2} b^{3}+720 x^{6} B a \,b^{4} d e +72 x^{6} B \,b^{5} d^{2}+840 x^{5} A \,a^{2} b^{3} e^{2}+840 x^{5} A a \,b^{4} d e +84 x^{5} A \,d^{2} b^{5}+840 x^{5} B \,e^{2} a^{3} b^{2}+1680 x^{5} B \,a^{2} b^{3} d e +420 x^{5} B a \,b^{4} d^{2}+1008 A \,a^{3} b^{2} e^{2} x^{4}+2016 A \,a^{2} b^{3} d e \,x^{4}+504 A a \,b^{4} d^{2} x^{4}+504 B \,a^{4} b \,e^{2} x^{4}+2016 B \,a^{3} b^{2} d e \,x^{4}+1008 B \,a^{2} b^{3} d^{2} x^{4}+630 x^{3} A \,a^{4} b \,e^{2}+2520 x^{3} A \,a^{3} b^{2} d e +1260 x^{3} A \,d^{2} a^{2} b^{3}+126 x^{3} B \,e^{2} a^{5}+1260 x^{3} B \,a^{4} b d e +1260 x^{3} B \,a^{3} b^{2} d^{2}+168 x^{2} A \,a^{5} e^{2}+1680 x^{2} A \,a^{4} b d e +1680 x^{2} A \,d^{2} a^{3} b^{2}+336 x^{2} B \,a^{5} d e +840 x^{2} B \,a^{4} b \,d^{2}+504 x A \,a^{5} d e +1260 x A \,d^{2} a^{4} b +252 x B \,a^{5} d^{2}+504 A \,d^{2} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5}}\) | \(480\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, B \,e^{2} b^{5} x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (A \,e^{2}+2 B d e \right ) b^{5}+5 B \,e^{2} b^{4} a \right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (2 A d e +B \,d^{2}\right ) b^{5}+5 \left (A \,e^{2}+2 B d e \right ) b^{4} a +10 B \,e^{2} a^{2} b^{3}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A \,d^{2} b^{5}+5 \left (2 A d e +B \,d^{2}\right ) b^{4} a +10 \left (A \,e^{2}+2 B d e \right ) a^{2} b^{3}+10 B \,e^{2} a^{3} b^{2}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,d^{2} b^{4} a +10 \left (2 A d e +B \,d^{2}\right ) a^{2} b^{3}+10 \left (A \,e^{2}+2 B d e \right ) a^{3} b^{2}+5 B \,e^{2} a^{4} b \right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A \,d^{2} a^{2} b^{3}+10 \left (2 A d e +B \,d^{2}\right ) a^{3} b^{2}+5 \left (A \,e^{2}+2 B d e \right ) a^{4} b +B \,e^{2} a^{5}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A \,d^{2} a^{3} b^{2}+5 \left (2 A d e +B \,d^{2}\right ) a^{4} b +\left (A \,e^{2}+2 B d e \right ) a^{5}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,d^{2} a^{4} b +\left (2 A d e +B \,d^{2}\right ) a^{5}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, A \,d^{2} a^{5} x}{b x +a}\) | \(517\) |
1/504*x*(56*B*b^5*e^2*x^8+63*A*b^5*e^2*x^7+315*B*a*b^4*e^2*x^7+126*B*b^5*d *e*x^7+360*A*a*b^4*e^2*x^6+144*A*b^5*d*e*x^6+720*B*a^2*b^3*e^2*x^6+720*B*a *b^4*d*e*x^6+72*B*b^5*d^2*x^6+840*A*a^2*b^3*e^2*x^5+840*A*a*b^4*d*e*x^5+84 *A*b^5*d^2*x^5+840*B*a^3*b^2*e^2*x^5+1680*B*a^2*b^3*d*e*x^5+420*B*a*b^4*d^ 2*x^5+1008*A*a^3*b^2*e^2*x^4+2016*A*a^2*b^3*d*e*x^4+504*A*a*b^4*d^2*x^4+50 4*B*a^4*b*e^2*x^4+2016*B*a^3*b^2*d*e*x^4+1008*B*a^2*b^3*d^2*x^4+630*A*a^4* b*e^2*x^3+2520*A*a^3*b^2*d*e*x^3+1260*A*a^2*b^3*d^2*x^3+126*B*a^5*e^2*x^3+ 1260*B*a^4*b*d*e*x^3+1260*B*a^3*b^2*d^2*x^3+168*A*a^5*e^2*x^2+1680*A*a^4*b *d*e*x^2+1680*A*a^3*b^2*d^2*x^2+336*B*a^5*d*e*x^2+840*B*a^4*b*d^2*x^2+504* A*a^5*d*e*x+1260*A*a^4*b*d^2*x+252*B*a^5*d^2*x+504*A*a^5*d^2)*((b*x+a)^2)^ (5/2)/(b*x+a)^5
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (146) = 292\).
Time = 0.39 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.94 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{9} \, B b^{5} e^{2} x^{9} + A a^{5} d^{2} x + \frac {1}{8} \, {\left (2 \, B b^{5} d e + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{5} d^{2} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left ({\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{2}\right )} x^{6} + {\left ({\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} + 4 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{5} e^{2} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} + 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{5} d e + {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2}\right )} x^{2} \]
1/9*B*b^5*e^2*x^9 + A*a^5*d^2*x + 1/8*(2*B*b^5*d*e + (5*B*a*b^4 + A*b^5)*e ^2)*x^8 + 1/7*(B*b^5*d^2 + 2*(5*B*a*b^4 + A*b^5)*d*e + 5*(2*B*a^2*b^3 + A* a*b^4)*e^2)*x^7 + 1/6*((5*B*a*b^4 + A*b^5)*d^2 + 10*(2*B*a^2*b^3 + A*a*b^4 )*d*e + 10*(B*a^3*b^2 + A*a^2*b^3)*e^2)*x^6 + ((2*B*a^2*b^3 + A*a*b^4)*d^2 + 4*(B*a^3*b^2 + A*a^2*b^3)*d*e + (B*a^4*b + 2*A*a^3*b^2)*e^2)*x^5 + 1/4* (10*(B*a^3*b^2 + A*a^2*b^3)*d^2 + 10*(B*a^4*b + 2*A*a^3*b^2)*d*e + (B*a^5 + 5*A*a^4*b)*e^2)*x^4 + 1/3*(A*a^5*e^2 + 5*(B*a^4*b + 2*A*a^3*b^2)*d^2 + 2 *(B*a^5 + 5*A*a^4*b)*d*e)*x^3 + 1/2*(2*A*a^5*d*e + (B*a^5 + 5*A*a^4*b)*d^2 )*x^2
Leaf count of result is larger than twice the leaf count of optimal. 13857 vs. \(2 (156) = 312\).
Time = 1.43 (sec) , antiderivative size = 13857, normalized size of antiderivative = 69.98 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(B*b**4*e**2*x**8/9 + x**7*(A* b**6*e**2 + 37*B*a*b**5*e**2/9 + 2*B*b**6*d*e)/(8*b**2) + x**6*(6*A*a*b**5 *e**2 + 2*A*b**6*d*e + 127*B*a**2*b**4*e**2/9 + 12*B*a*b**5*d*e + B*b**6*d **2 - 15*a*(A*b**6*e**2 + 37*B*a*b**5*e**2/9 + 2*B*b**6*d*e)/(8*b))/(7*b** 2) + x**5*(15*A*a**2*b**4*e**2 + 12*A*a*b**5*d*e + A*b**6*d**2 + 20*B*a**3 *b**3*e**2 + 30*B*a**2*b**4*d*e + 6*B*a*b**5*d**2 - 7*a**2*(A*b**6*e**2 + 37*B*a*b**5*e**2/9 + 2*B*b**6*d*e)/(8*b**2) - 13*a*(6*A*a*b**5*e**2 + 2*A* b**6*d*e + 127*B*a**2*b**4*e**2/9 + 12*B*a*b**5*d*e + B*b**6*d**2 - 15*a*( A*b**6*e**2 + 37*B*a*b**5*e**2/9 + 2*B*b**6*d*e)/(8*b))/(7*b))/(6*b**2) + x**4*(20*A*a**3*b**3*e**2 + 30*A*a**2*b**4*d*e + 6*A*a*b**5*d**2 + 15*B*a* *4*b**2*e**2 + 40*B*a**3*b**3*d*e + 15*B*a**2*b**4*d**2 - 6*a**2*(6*A*a*b* *5*e**2 + 2*A*b**6*d*e + 127*B*a**2*b**4*e**2/9 + 12*B*a*b**5*d*e + B*b**6 *d**2 - 15*a*(A*b**6*e**2 + 37*B*a*b**5*e**2/9 + 2*B*b**6*d*e)/(8*b))/(7*b **2) - 11*a*(15*A*a**2*b**4*e**2 + 12*A*a*b**5*d*e + A*b**6*d**2 + 20*B*a* *3*b**3*e**2 + 30*B*a**2*b**4*d*e + 6*B*a*b**5*d**2 - 7*a**2*(A*b**6*e**2 + 37*B*a*b**5*e**2/9 + 2*B*b**6*d*e)/(8*b**2) - 13*a*(6*A*a*b**5*e**2 + 2* A*b**6*d*e + 127*B*a**2*b**4*e**2/9 + 12*B*a*b**5*d*e + B*b**6*d**2 - 15*a *(A*b**6*e**2 + 37*B*a*b**5*e**2/9 + 2*B*b**6*d*e)/(8*b))/(7*b))/(6*b))/(5 *b**2) + x**3*(15*A*a**4*b**2*e**2 + 40*A*a**3*b**3*d*e + 15*A*a**2*b**4*d **2 + 6*B*a**5*b*e**2 + 30*B*a**4*b**2*d*e + 20*B*a**3*b**3*d**2 - 5*a*...
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 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A d^{2} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{3} e^{2} x}{6 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B e^{2} x^{2}}{9 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a d^{2}}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{4} e^{2}}{6 \, b^{4}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a e^{2} x}{72 \, b^{3}} + \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{2} e^{2}}{504 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, B d e + A e^{2}\right )} a^{2} x}{6 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (B d^{2} + 2 \, A d e\right )} a x}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, B d e + A e^{2}\right )} a^{3}}{6 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (B d^{2} + 2 \, A d e\right )} a^{2}}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, B d e + A e^{2}\right )} x}{8 \, b^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, B d e + A e^{2}\right )} a}{56 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (B d^{2} + 2 \, A d e\right )}}{7 \, b^{2}} \]
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*d^2*x - 1/6*(b^2*x^2 + 2*a*b*x + a^2 )^(5/2)*B*a^3*e^2*x/b^3 + 1/9*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*e^2*x^2/b^ 2 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*a*d^2/b - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^4*e^2/b^4 - 11/72*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a*e^ 2*x/b^3 + 83/504*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^2*e^2/b^4 + 1/6*(b^2* x^2 + 2*a*b*x + a^2)^(5/2)*(2*B*d*e + A*e^2)*a^2*x/b^2 - 1/6*(b^2*x^2 + 2* a*b*x + a^2)^(5/2)*(B*d^2 + 2*A*d*e)*a*x/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2) ^(5/2)*(2*B*d*e + A*e^2)*a^3/b^3 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B* d^2 + 2*A*d*e)*a^2/b^2 + 1/8*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(2*B*d*e + A* e^2)*x/b^2 - 9/56*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(2*B*d*e + A*e^2)*a/b^3 + 1/7*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(B*d^2 + 2*A*d*e)/b^2
Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (146) = 292\).
Time = 0.28 (sec) , antiderivative size = 753, normalized size of antiderivative = 3.80 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{9} \, B b^{5} e^{2} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, B b^{5} d e x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, B a b^{4} e^{2} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{8} \, A b^{5} e^{2} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, B b^{5} d^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, B a b^{4} d e x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, A b^{5} d e x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, B a^{2} b^{3} e^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, A a b^{4} e^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, B a b^{4} d^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, A b^{5} d^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, B a^{2} b^{3} d e x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, A a b^{4} d e x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a^{3} b^{2} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, A a^{2} b^{3} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{2} b^{3} d^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + A a b^{4} d^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + 4 \, B a^{3} b^{2} d e x^{5} \mathrm {sgn}\left (b x + a\right ) + 4 \, A a^{2} b^{3} d e x^{5} \mathrm {sgn}\left (b x + a\right ) + B a^{4} b e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} b^{2} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, B a^{3} b^{2} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{2} b^{3} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, B a^{4} b d e x^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, A a^{3} b^{2} d e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, B a^{5} e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, A a^{4} b e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a^{4} b d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, A a^{3} b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B a^{5} d e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, A a^{4} b d e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, A a^{5} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{5} d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{4} b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + A a^{5} d e x^{2} \mathrm {sgn}\left (b x + a\right ) + A a^{5} d^{2} x \mathrm {sgn}\left (b x + a\right ) - \frac {{\left (12 \, B a^{7} b^{2} d^{2} - 84 \, A a^{6} b^{3} d^{2} - 6 \, B a^{8} b d e + 24 \, A a^{7} b^{2} d e + B a^{9} e^{2} - 3 \, A a^{8} b e^{2}\right )} \mathrm {sgn}\left (b x + a\right )}{504 \, b^{4}} \]
1/9*B*b^5*e^2*x^9*sgn(b*x + a) + 1/4*B*b^5*d*e*x^8*sgn(b*x + a) + 5/8*B*a* b^4*e^2*x^8*sgn(b*x + a) + 1/8*A*b^5*e^2*x^8*sgn(b*x + a) + 1/7*B*b^5*d^2* x^7*sgn(b*x + a) + 10/7*B*a*b^4*d*e*x^7*sgn(b*x + a) + 2/7*A*b^5*d*e*x^7*s gn(b*x + a) + 10/7*B*a^2*b^3*e^2*x^7*sgn(b*x + a) + 5/7*A*a*b^4*e^2*x^7*sg n(b*x + a) + 5/6*B*a*b^4*d^2*x^6*sgn(b*x + a) + 1/6*A*b^5*d^2*x^6*sgn(b*x + a) + 10/3*B*a^2*b^3*d*e*x^6*sgn(b*x + a) + 5/3*A*a*b^4*d*e*x^6*sgn(b*x + a) + 5/3*B*a^3*b^2*e^2*x^6*sgn(b*x + a) + 5/3*A*a^2*b^3*e^2*x^6*sgn(b*x + a) + 2*B*a^2*b^3*d^2*x^5*sgn(b*x + a) + A*a*b^4*d^2*x^5*sgn(b*x + a) + 4* B*a^3*b^2*d*e*x^5*sgn(b*x + a) + 4*A*a^2*b^3*d*e*x^5*sgn(b*x + a) + B*a^4* b*e^2*x^5*sgn(b*x + a) + 2*A*a^3*b^2*e^2*x^5*sgn(b*x + a) + 5/2*B*a^3*b^2* d^2*x^4*sgn(b*x + a) + 5/2*A*a^2*b^3*d^2*x^4*sgn(b*x + a) + 5/2*B*a^4*b*d* e*x^4*sgn(b*x + a) + 5*A*a^3*b^2*d*e*x^4*sgn(b*x + a) + 1/4*B*a^5*e^2*x^4* sgn(b*x + a) + 5/4*A*a^4*b*e^2*x^4*sgn(b*x + a) + 5/3*B*a^4*b*d^2*x^3*sgn( b*x + a) + 10/3*A*a^3*b^2*d^2*x^3*sgn(b*x + a) + 2/3*B*a^5*d*e*x^3*sgn(b*x + a) + 10/3*A*a^4*b*d*e*x^3*sgn(b*x + a) + 1/3*A*a^5*e^2*x^3*sgn(b*x + a) + 1/2*B*a^5*d^2*x^2*sgn(b*x + a) + 5/2*A*a^4*b*d^2*x^2*sgn(b*x + a) + A*a ^5*d*e*x^2*sgn(b*x + a) + A*a^5*d^2*x*sgn(b*x + a) - 1/504*(12*B*a^7*b^2*d ^2 - 84*A*a^6*b^3*d^2 - 6*B*a^8*b*d*e + 24*A*a^7*b^2*d*e + B*a^9*e^2 - 3*A *a^8*b*e^2)*sgn(b*x + a)/b^4
Timed out. \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]