\(\int (A+B x) (d+e x) (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [413]

Optimal result

Integrand size = 31, antiderivative size = 135 \[ \int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(A b-a B) (b d-a e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^3}+\frac {B e (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^3} \] Output:

1/6*(A*b-B*a)*(-a*e+b*d)*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^3+1/7*(A*b*e-2*B*a* 
e+B*b*d)*(b*x+a)^6*((b*x+a)^2)^(1/2)/b^3+1/8*B*e*(b*x+a)^7*((b*x+a)^2)^(1/ 
2)/b^3
 

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.59 \[ \int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (28 a^5 (3 A (2 d+e x)+B x (3 d+2 e x))+28 a^3 b^2 x^2 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+70 a^4 b x (B x (4 d+3 e x)+A (6 d+4 e x))+28 a^2 b^3 x^3 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))+4 a b^4 x^4 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+b^5 x^5 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))\right )}{168 (a+b x)} \] Input:

Integrate[(A + B*x)*(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
 

Output:

(x*Sqrt[(a + b*x)^2]*(28*a^5*(3*A*(2*d + e*x) + B*x*(3*d + 2*e*x)) + 28*a^ 
3*b^2*x^2*(5*A*(4*d + 3*e*x) + 3*B*x*(5*d + 4*e*x)) + 70*a^4*b*x*(B*x*(4*d 
 + 3*e*x) + A*(6*d + 4*e*x)) + 28*a^2*b^3*x^3*(3*A*(5*d + 4*e*x) + 2*B*x*( 
6*d + 5*e*x)) + 4*a*b^4*x^4*(7*A*(6*d + 5*e*x) + 5*B*x*(7*d + 6*e*x)) + b^ 
5*x^5*(4*A*(7*d + 6*e*x) + 3*B*x*(8*d + 7*e*x))))/(168*(a + b*x))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, 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.

\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x) (d+e x) \, dx\)

\(\Big \downarrow \) 1187

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^5 (A+B x) (d+e x)dx}{b^5 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^5 (A+B x) (d+e x)dx}{a+b x}\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B e (a+b x)^7}{b^2}+\frac {(b B d+A b e-2 a B e) (a+b x)^6}{b^2}+\frac {(A b-a B) (b d-a e) (a+b x)^5}{b^2}\right )dx}{a+b x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(a+b x)^7 (-2 a B e+A b e+b B d)}{7 b^3}+\frac {(a+b x)^6 (A b-a B) (b d-a e)}{6 b^3}+\frac {B e (a+b x)^8}{8 b^3}\right )}{a+b x}\)

Input:

Int[(A + B*x)*(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
 

Output:

(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((A*b - a*B)*(b*d - a*e)*(a + b*x)^6)/(6*b 
^3) + ((b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^7)/(7*b^3) + (B*e*(a + b*x)^8)/ 
(8*b^3)))/(a + b*x)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 86

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]))))
 

rule 1187

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(283\) vs. \(2(96)=192\).

Time = 1.22 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.10

method	result	size
gosper	\(\frac {x \left (21 B e \,b^{5} x^{7}+24 x^{6} A \,b^{5} e +120 x^{6} B a \,b^{4} e +24 x^{6} B \,b^{5} d +140 x^{5} A a \,b^{4} e +28 x^{5} A d \,b^{5}+280 x^{5} B e \,a^{2} b^{3}+140 x^{5} B a \,b^{4} d +336 A \,a^{2} b^{3} e \,x^{4}+168 A a \,b^{4} d \,x^{4}+336 B \,a^{3} b^{2} e \,x^{4}+336 B \,a^{2} b^{3} d \,x^{4}+420 x^{3} A \,a^{3} b^{2} e +420 x^{3} A d \,a^{2} b^{3}+210 x^{3} B e \,a^{4} b +420 x^{3} B \,a^{3} b^{2} d +280 x^{2} A \,a^{4} b e +560 x^{2} A d \,a^{3} b^{2}+56 x^{2} B e \,a^{5}+280 x^{2} B \,a^{4} b d +84 x A \,a^{5} e +420 x A d \,a^{4} b +84 x B \,a^{5} d +168 A d \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 \left (b x +a \right )^{5}}\)	\(284\)
default	\(\frac {x \left (21 B e \,b^{5} x^{7}+24 x^{6} A \,b^{5} e +120 x^{6} B a \,b^{4} e +24 x^{6} B \,b^{5} d +140 x^{5} A a \,b^{4} e +28 x^{5} A d \,b^{5}+280 x^{5} B e \,a^{2} b^{3}+140 x^{5} B a \,b^{4} d +336 A \,a^{2} b^{3} e \,x^{4}+168 A a \,b^{4} d \,x^{4}+336 B \,a^{3} b^{2} e \,x^{4}+336 B \,a^{2} b^{3} d \,x^{4}+420 x^{3} A \,a^{3} b^{2} e +420 x^{3} A d \,a^{2} b^{3}+210 x^{3} B e \,a^{4} b +420 x^{3} B \,a^{3} b^{2} d +280 x^{2} A \,a^{4} b e +560 x^{2} A d \,a^{3} b^{2}+56 x^{2} B e \,a^{5}+280 x^{2} B \,a^{4} b d +84 x A \,a^{5} e +420 x A d \,a^{4} b +84 x B \,a^{5} d +168 A d \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 \left (b x +a \right )^{5}}\)	\(284\)
orering	\(\frac {x \left (21 B e \,b^{5} x^{7}+24 x^{6} A \,b^{5} e +120 x^{6} B a \,b^{4} e +24 x^{6} B \,b^{5} d +140 x^{5} A a \,b^{4} e +28 x^{5} A d \,b^{5}+280 x^{5} B e \,a^{2} b^{3}+140 x^{5} B a \,b^{4} d +336 A \,a^{2} b^{3} e \,x^{4}+168 A a \,b^{4} d \,x^{4}+336 B \,a^{3} b^{2} e \,x^{4}+336 B \,a^{2} b^{3} d \,x^{4}+420 x^{3} A \,a^{3} b^{2} e +420 x^{3} A d \,a^{2} b^{3}+210 x^{3} B e \,a^{4} b +420 x^{3} B \,a^{3} b^{2} d +280 x^{2} A \,a^{4} b e +560 x^{2} A d \,a^{3} b^{2}+56 x^{2} B e \,a^{5}+280 x^{2} B \,a^{4} b d +84 x A \,a^{5} e +420 x A d \,a^{4} b +84 x B \,a^{5} d +168 A d \,a^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{168 \left (b x +a \right )^{5}}\)	\(293\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, B e \,b^{5} x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (A e +B d \right ) b^{5}+5 B a \,b^{4} e \right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A d \,b^{5}+5 \left (A e +B d \right ) a \,b^{4}+10 B e \,a^{2} b^{3}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A d a \,b^{4}+10 \left (A e +B d \right ) a^{2} b^{3}+10 B e \,a^{3} b^{2}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A d \,a^{2} b^{3}+10 \left (A e +B d \right ) a^{3} b^{2}+5 B e \,a^{4} b \right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A d \,a^{3} b^{2}+5 \left (A e +B d \right ) a^{4} b +B e \,a^{5}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A d \,a^{4} b +\left (A e +B d \right ) a^{5}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, A d \,a^{5} x}{b x +a}\)	\(345\)

Input:

int((B*x+A)*(e*x+d)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/168*x*(21*B*b^5*e*x^7+24*A*b^5*e*x^6+120*B*a*b^4*e*x^6+24*B*b^5*d*x^6+14 
0*A*a*b^4*e*x^5+28*A*b^5*d*x^5+280*B*a^2*b^3*e*x^5+140*B*a*b^4*d*x^5+336*A 
*a^2*b^3*e*x^4+168*A*a*b^4*d*x^4+336*B*a^3*b^2*e*x^4+336*B*a^2*b^3*d*x^4+4 
20*A*a^3*b^2*e*x^3+420*A*a^2*b^3*d*x^3+210*B*a^4*b*e*x^3+420*B*a^3*b^2*d*x 
^3+280*A*a^4*b*e*x^2+560*A*a^3*b^2*d*x^2+56*B*a^5*e*x^2+280*B*a^4*b*d*x^2+ 
84*A*a^5*e*x+420*A*a^4*b*d*x+84*B*a^5*d*x+168*A*a^5*d)*((b*x+a)^2)^(5/2)/( 
b*x+a)^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (96) = 192\).

Time = 0.07 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.77 \[ \int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{8} \, B b^{5} e x^{8} + A a^{5} d x + \frac {1}{7} \, {\left (B b^{5} d + {\left (5 \, B a b^{4} + A b^{5}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left ({\left (5 \, B a b^{4} + A b^{5}\right )} d + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e\right )} x^{6} + {\left ({\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e\right )} x^{5} + \frac {5}{4} \, {\left (2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d + {\left (B a^{5} + 5 \, A a^{4} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{5} e + {\left (B a^{5} + 5 \, A a^{4} b\right )} d\right )} x^{2} \] Input:

integrate((B*x+A)*(e*x+d)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas 
")
 

Output:

1/8*B*b^5*e*x^8 + A*a^5*d*x + 1/7*(B*b^5*d + (5*B*a*b^4 + A*b^5)*e)*x^7 + 
1/6*((5*B*a*b^4 + A*b^5)*d + 5*(2*B*a^2*b^3 + A*a*b^4)*e)*x^6 + ((2*B*a^2* 
b^3 + A*a*b^4)*d + 2*(B*a^3*b^2 + A*a^2*b^3)*e)*x^5 + 5/4*(2*(B*a^3*b^2 + 
A*a^2*b^3)*d + (B*a^4*b + 2*A*a^3*b^2)*e)*x^4 + 1/3*(5*(B*a^4*b + 2*A*a^3* 
b^2)*d + (B*a^5 + 5*A*a^4*b)*e)*x^3 + 1/2*(A*a^5*e + (B*a^5 + 5*A*a^4*b)*d 
)*x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6004 vs. \(2 (104) = 208\).

Time = 1.73 (sec) , antiderivative size = 6004, normalized size of antiderivative = 44.47 \[ \int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
 

Output:

Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(B*b**4*e*x**7/8 + x**6*(A*b** 
6*e + 33*B*a*b**5*e/8 + B*b**6*d)/(7*b**2) + x**5*(6*A*a*b**5*e + A*b**6*d 
 + 113*B*a**2*b**4*e/8 + 6*B*a*b**5*d - 13*a*(A*b**6*e + 33*B*a*b**5*e/8 + 
 B*b**6*d)/(7*b))/(6*b**2) + x**4*(15*A*a**2*b**4*e + 6*A*a*b**5*d + 20*B* 
a**3*b**3*e + 15*B*a**2*b**4*d - 6*a**2*(A*b**6*e + 33*B*a*b**5*e/8 + B*b* 
*6*d)/(7*b**2) - 11*a*(6*A*a*b**5*e + A*b**6*d + 113*B*a**2*b**4*e/8 + 6*B 
*a*b**5*d - 13*a*(A*b**6*e + 33*B*a*b**5*e/8 + B*b**6*d)/(7*b))/(6*b))/(5* 
b**2) + x**3*(20*A*a**3*b**3*e + 15*A*a**2*b**4*d + 15*B*a**4*b**2*e + 20* 
B*a**3*b**3*d - 5*a**2*(6*A*a*b**5*e + A*b**6*d + 113*B*a**2*b**4*e/8 + 6* 
B*a*b**5*d - 13*a*(A*b**6*e + 33*B*a*b**5*e/8 + B*b**6*d)/(7*b))/(6*b**2) 
- 9*a*(15*A*a**2*b**4*e + 6*A*a*b**5*d + 20*B*a**3*b**3*e + 15*B*a**2*b**4 
*d - 6*a**2*(A*b**6*e + 33*B*a*b**5*e/8 + B*b**6*d)/(7*b**2) - 11*a*(6*A*a 
*b**5*e + A*b**6*d + 113*B*a**2*b**4*e/8 + 6*B*a*b**5*d - 13*a*(A*b**6*e + 
 33*B*a*b**5*e/8 + B*b**6*d)/(7*b))/(6*b))/(5*b))/(4*b**2) + x**2*(15*A*a* 
*4*b**2*e + 20*A*a**3*b**3*d + 6*B*a**5*b*e + 15*B*a**4*b**2*d - 4*a**2*(1 
5*A*a**2*b**4*e + 6*A*a*b**5*d + 20*B*a**3*b**3*e + 15*B*a**2*b**4*d - 6*a 
**2*(A*b**6*e + 33*B*a*b**5*e/8 + B*b**6*d)/(7*b**2) - 11*a*(6*A*a*b**5*e 
+ A*b**6*d + 113*B*a**2*b**4*e/8 + 6*B*a*b**5*d - 13*a*(A*b**6*e + 33*B*a* 
b**5*e/8 + B*b**6*d)/(7*b))/(6*b))/(5*b**2) - 7*a*(20*A*a**3*b**3*e + 15*A 
*a**2*b**4*d + 15*B*a**4*b**2*e + 20*B*a**3*b**3*d - 5*a**2*(6*A*a*b**5...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (96) = 192\).

Time = 0.04 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.88 \[ \int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A d x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{2} e x}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a d}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{3} e}{6 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (B d + A e\right )} a x}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B e x}{8 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (B d + A e\right )} a^{2}}{6 \, b^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a e}{56 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (B d + A e\right )}}{7 \, b^{2}} \] Input:

integrate((B*x+A)*(e*x+d)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima 
")
 

Output:

1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*d*x + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^ 
(5/2)*B*a^2*e*x/b^2 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*a*d/b + 1/6*(b 
^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^3*e/b^3 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^ 
(5/2)*(B*d + A*e)*a*x/b + 1/8*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*e*x/b^2 - 
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*d + A*e)*a^2/b^2 - 9/56*(b^2*x^2 + 
2*a*b*x + a^2)^(7/2)*B*a*e/b^3 + 1/7*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(B*d 
+ A*e)/b^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (96) = 192\).

Time = 0.19 (sec) , antiderivative size = 458, normalized size of antiderivative = 3.39 \[ \int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{8} \, B b^{5} e x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, B b^{5} d x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, B a b^{4} e x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, A b^{5} e x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, B a b^{4} d x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, A b^{5} d x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a^{2} b^{3} e x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, A a b^{4} e x^{6} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{2} b^{3} d x^{5} \mathrm {sgn}\left (b x + a\right ) + A a b^{4} d x^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{3} b^{2} e x^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{2} b^{3} e x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, B a^{3} b^{2} d x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{2} b^{3} d x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, B a^{4} b e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{3} b^{2} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a^{4} b d x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, A a^{3} b^{2} d x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, B a^{5} e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, A a^{4} b e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{5} d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{4} b d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A a^{5} e x^{2} \mathrm {sgn}\left (b x + a\right ) + A a^{5} d x \mathrm {sgn}\left (b x + a\right ) - \frac {{\left (4 \, B a^{7} b d - 28 \, A a^{6} b^{2} d - B a^{8} e + 4 \, A a^{7} b e\right )} \mathrm {sgn}\left (b x + a\right )}{168 \, b^{3}} \] Input:

integrate((B*x+A)*(e*x+d)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
 

Output:

1/8*B*b^5*e*x^8*sgn(b*x + a) + 1/7*B*b^5*d*x^7*sgn(b*x + a) + 5/7*B*a*b^4* 
e*x^7*sgn(b*x + a) + 1/7*A*b^5*e*x^7*sgn(b*x + a) + 5/6*B*a*b^4*d*x^6*sgn( 
b*x + a) + 1/6*A*b^5*d*x^6*sgn(b*x + a) + 5/3*B*a^2*b^3*e*x^6*sgn(b*x + a) 
 + 5/6*A*a*b^4*e*x^6*sgn(b*x + a) + 2*B*a^2*b^3*d*x^5*sgn(b*x + a) + A*a*b 
^4*d*x^5*sgn(b*x + a) + 2*B*a^3*b^2*e*x^5*sgn(b*x + a) + 2*A*a^2*b^3*e*x^5 
*sgn(b*x + a) + 5/2*B*a^3*b^2*d*x^4*sgn(b*x + a) + 5/2*A*a^2*b^3*d*x^4*sgn 
(b*x + a) + 5/4*B*a^4*b*e*x^4*sgn(b*x + a) + 5/2*A*a^3*b^2*e*x^4*sgn(b*x + 
 a) + 5/3*B*a^4*b*d*x^3*sgn(b*x + a) + 10/3*A*a^3*b^2*d*x^3*sgn(b*x + a) + 
 1/3*B*a^5*e*x^3*sgn(b*x + a) + 5/3*A*a^4*b*e*x^3*sgn(b*x + a) + 1/2*B*a^5 
*d*x^2*sgn(b*x + a) + 5/2*A*a^4*b*d*x^2*sgn(b*x + a) + 1/2*A*a^5*e*x^2*sgn 
(b*x + a) + A*a^5*d*x*sgn(b*x + a) - 1/168*(4*B*a^7*b*d - 28*A*a^6*b^2*d - 
 B*a^8*e + 4*A*a^7*b*e)*sgn(b*x + a)/b^3
 

Mupad [F(-1)]

Timed out. \[ \int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (A+B\,x\right )\,\left (d+e\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \] Input:

int((A + B*x)*(d + e*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
 

Output:

int((A + B*x)*(d + e*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.07 \[ \int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \left (7 b^{6} e \,x^{7}+48 a \,b^{5} e \,x^{6}+8 b^{6} d \,x^{6}+140 a^{2} b^{4} e \,x^{5}+56 a \,b^{5} d \,x^{5}+224 a^{3} b^{3} e \,x^{4}+168 a^{2} b^{4} d \,x^{4}+210 a^{4} b^{2} e \,x^{3}+280 a^{3} b^{3} d \,x^{3}+112 a^{5} b e \,x^{2}+280 a^{4} b^{2} d \,x^{2}+28 a^{6} e x +168 a^{5} b d x +56 a^{6} d \right )}{56} \] Input:

int((B*x+A)*(e*x+d)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
 

Output:

(x*(56*a**6*d + 28*a**6*e*x + 168*a**5*b*d*x + 112*a**5*b*e*x**2 + 280*a** 
4*b**2*d*x**2 + 210*a**4*b**2*e*x**3 + 280*a**3*b**3*d*x**3 + 224*a**3*b** 
3*e*x**4 + 168*a**2*b**4*d*x**4 + 140*a**2*b**4*e*x**5 + 56*a*b**5*d*x**5 
+ 48*a*b**5*e*x**6 + 8*b**6*d*x**6 + 7*b**6*e*x**7))/56