\(\int (A+B x) (d+e x)^3 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\) [392]

Optimal result

Integrand size = 33, antiderivative size = 259 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(A b-a B) (b d-a e)^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^5}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^5}+\frac {e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^5}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {B e^3 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^5} \] Output:

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

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.24 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (14 a^3 \left (5 A \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+B x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )+14 a^2 b x \left (3 A \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )+2 a b^2 x^2 \left (7 A \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+3 B x \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )+b^3 x^3 \left (2 A \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+B x \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )\right )\right )}{280 (a+b x)} \] Input:

Integrate[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
 

Output:

(x*Sqrt[(a + b*x)^2]*(14*a^3*(5*A*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x 
^3) + B*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3)) + 14*a^2*b*x*( 
3*A*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + B*x*(20*d^3 + 45*d^ 
2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3)) + 2*a*b^2*x^2*(7*A*(20*d^3 + 45*d^2*e* 
x + 36*d*e^2*x^2 + 10*e^3*x^3) + 3*B*x*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 
 + 20*e^3*x^3)) + b^3*x^3*(2*A*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^ 
3*x^3) + B*x*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3))))/(280*( 
a + b*x))
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.72, 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.

\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (A+B x) (d+e x)^3 \, dx\)

\(\Big \downarrow \) 1187

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^3 (A+B x) (d+e x)^3dx}{b^3 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^3 (A+B x) (d+e x)^3dx}{a+b x}\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B e^3 (a+b x)^7}{b^4}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^6}{b^4}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^5}{b^4}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^4}{b^4}+\frac {(A b-a B) (b d-a e)^3 (a+b x)^3}{b^4}\right )dx}{a+b x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {e^2 (a+b x)^7 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac {e (a+b x)^6 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac {(a+b x)^5 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac {(a+b x)^4 (A b-a B) (b d-a e)^3}{4 b^5}+\frac {B e^3 (a+b x)^8}{8 b^5}\right )}{a+b x}\)

Input:

Int[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
 

Output:

(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((A*b - a*B)*(b*d - a*e)^3*(a + b*x)^4)/(4 
*b^5) + ((b*d - a*e)^2*(b*B*d + 3*A*b*e - 4*a*B*e)*(a + b*x)^5)/(5*b^5) + 
(e*(b*d - a*e)*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^6)/(2*b^5) + (e^2*(3*b* 
B*d + A*b*e - 4*a*B*e)*(a + b*x)^7)/(7*b^5) + (B*e^3*(a + b*x)^8)/(8*b^5)) 
)/(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. \(427\) vs. \(2(194)=388\).

Time = 1.39 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.65

method	result	size
gosper	\(\frac {x \left (35 B \,e^{3} b^{3} x^{7}+40 x^{6} A \,b^{3} e^{3}+120 x^{6} B \,e^{3} a \,b^{2}+120 x^{6} B \,b^{3} d \,e^{2}+140 x^{5} A a \,b^{2} e^{3}+140 x^{5} A \,b^{3} d \,e^{2}+140 x^{5} B \,e^{3} a^{2} b +420 x^{5} B a \,b^{2} d \,e^{2}+140 x^{5} B \,b^{3} d^{2} e +168 x^{4} A \,a^{2} b \,e^{3}+504 x^{4} A a \,b^{2} d \,e^{2}+168 x^{4} A \,b^{3} d^{2} e +56 x^{4} B \,e^{3} a^{3}+504 x^{4} B \,a^{2} b d \,e^{2}+504 x^{4} B a \,b^{2} d^{2} e +56 x^{4} B \,b^{3} d^{3}+70 x^{3} A \,a^{3} e^{3}+630 x^{3} A \,a^{2} b d \,e^{2}+630 x^{3} A a \,b^{2} d^{2} e +70 x^{3} A \,d^{3} b^{3}+210 x^{3} B \,a^{3} d \,e^{2}+630 x^{3} B \,a^{2} b \,d^{2} e +210 x^{3} B a \,b^{2} d^{3}+280 A \,a^{3} d \,e^{2} x^{2}+840 A \,a^{2} b \,d^{2} e \,x^{2}+280 A a \,b^{2} d^{3} x^{2}+280 B \,a^{3} d^{2} e \,x^{2}+280 B \,a^{2} b \,d^{3} x^{2}+420 x A \,a^{3} d^{2} e +420 x A \,a^{2} b \,d^{3}+140 x \,a^{3} B \,d^{3}+280 A \,d^{3} a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 \left (b x +a \right )^{3}}\)	\(428\)
default	\(\frac {x \left (35 B \,e^{3} b^{3} x^{7}+40 x^{6} A \,b^{3} e^{3}+120 x^{6} B \,e^{3} a \,b^{2}+120 x^{6} B \,b^{3} d \,e^{2}+140 x^{5} A a \,b^{2} e^{3}+140 x^{5} A \,b^{3} d \,e^{2}+140 x^{5} B \,e^{3} a^{2} b +420 x^{5} B a \,b^{2} d \,e^{2}+140 x^{5} B \,b^{3} d^{2} e +168 x^{4} A \,a^{2} b \,e^{3}+504 x^{4} A a \,b^{2} d \,e^{2}+168 x^{4} A \,b^{3} d^{2} e +56 x^{4} B \,e^{3} a^{3}+504 x^{4} B \,a^{2} b d \,e^{2}+504 x^{4} B a \,b^{2} d^{2} e +56 x^{4} B \,b^{3} d^{3}+70 x^{3} A \,a^{3} e^{3}+630 x^{3} A \,a^{2} b d \,e^{2}+630 x^{3} A a \,b^{2} d^{2} e +70 x^{3} A \,d^{3} b^{3}+210 x^{3} B \,a^{3} d \,e^{2}+630 x^{3} B \,a^{2} b \,d^{2} e +210 x^{3} B a \,b^{2} d^{3}+280 A \,a^{3} d \,e^{2} x^{2}+840 A \,a^{2} b \,d^{2} e \,x^{2}+280 A a \,b^{2} d^{3} x^{2}+280 B \,a^{3} d^{2} e \,x^{2}+280 B \,a^{2} b \,d^{3} x^{2}+420 x A \,a^{3} d^{2} e +420 x A \,a^{2} b \,d^{3}+140 x \,a^{3} B \,d^{3}+280 A \,d^{3} a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 \left (b x +a \right )^{3}}\)	\(428\)
orering	\(\frac {x \left (35 B \,e^{3} b^{3} x^{7}+40 x^{6} A \,b^{3} e^{3}+120 x^{6} B \,e^{3} a \,b^{2}+120 x^{6} B \,b^{3} d \,e^{2}+140 x^{5} A a \,b^{2} e^{3}+140 x^{5} A \,b^{3} d \,e^{2}+140 x^{5} B \,e^{3} a^{2} b +420 x^{5} B a \,b^{2} d \,e^{2}+140 x^{5} B \,b^{3} d^{2} e +168 x^{4} A \,a^{2} b \,e^{3}+504 x^{4} A a \,b^{2} d \,e^{2}+168 x^{4} A \,b^{3} d^{2} e +56 x^{4} B \,e^{3} a^{3}+504 x^{4} B \,a^{2} b d \,e^{2}+504 x^{4} B a \,b^{2} d^{2} e +56 x^{4} B \,b^{3} d^{3}+70 x^{3} A \,a^{3} e^{3}+630 x^{3} A \,a^{2} b d \,e^{2}+630 x^{3} A a \,b^{2} d^{2} e +70 x^{3} A \,d^{3} b^{3}+210 x^{3} B \,a^{3} d \,e^{2}+630 x^{3} B \,a^{2} b \,d^{2} e +210 x^{3} B a \,b^{2} d^{3}+280 A \,a^{3} d \,e^{2} x^{2}+840 A \,a^{2} b \,d^{2} e \,x^{2}+280 A a \,b^{2} d^{3} x^{2}+280 B \,a^{3} d^{2} e \,x^{2}+280 B \,a^{2} b \,d^{3} x^{2}+420 x A \,a^{3} d^{2} e +420 x A \,a^{2} b \,d^{3}+140 x \,a^{3} B \,d^{3}+280 A \,d^{3} a^{3}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}{280 \left (b x +a \right )^{3}}\)	\(437\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, B \,e^{3} b^{3} x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (A \,e^{3}+3 B d \,e^{2}\right ) b^{3}+3 B \,e^{3} a \,b^{2}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (3 A d \,e^{2}+3 e B \,d^{2}\right ) b^{3}+3 \left (A \,e^{3}+3 B d \,e^{2}\right ) a \,b^{2}+3 B \,e^{3} a^{2} b \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (3 A \,d^{2} e +B \,d^{3}\right ) b^{3}+3 \left (3 A d \,e^{2}+3 e B \,d^{2}\right ) a \,b^{2}+3 \left (A \,e^{3}+3 B d \,e^{2}\right ) a^{2} b +B \,e^{3} a^{3}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A \,d^{3} b^{3}+3 \left (3 A \,d^{2} e +B \,d^{3}\right ) a \,b^{2}+3 \left (3 A d \,e^{2}+3 e B \,d^{2}\right ) a^{2} b +\left (A \,e^{3}+3 B d \,e^{2}\right ) a^{3}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 A \,d^{3} a \,b^{2}+3 \left (3 A \,d^{2} e +B \,d^{3}\right ) a^{2} b +\left (3 A d \,e^{2}+3 e B \,d^{2}\right ) a^{3}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 A \,a^{2} b \,d^{3}+\left (3 A \,d^{2} e +B \,d^{3}\right ) a^{3}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, A \,d^{3} a^{3} x}{b x +a}\)	\(467\)

Input:

int((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/280*x*(35*B*b^3*e^3*x^7+40*A*b^3*e^3*x^6+120*B*a*b^2*e^3*x^6+120*B*b^3*d 
*e^2*x^6+140*A*a*b^2*e^3*x^5+140*A*b^3*d*e^2*x^5+140*B*a^2*b*e^3*x^5+420*B 
*a*b^2*d*e^2*x^5+140*B*b^3*d^2*e*x^5+168*A*a^2*b*e^3*x^4+504*A*a*b^2*d*e^2 
*x^4+168*A*b^3*d^2*e*x^4+56*B*a^3*e^3*x^4+504*B*a^2*b*d*e^2*x^4+504*B*a*b^ 
2*d^2*e*x^4+56*B*b^3*d^3*x^4+70*A*a^3*e^3*x^3+630*A*a^2*b*d*e^2*x^3+630*A* 
a*b^2*d^2*e*x^3+70*A*b^3*d^3*x^3+210*B*a^3*d*e^2*x^3+630*B*a^2*b*d^2*e*x^3 
+210*B*a*b^2*d^3*x^3+280*A*a^3*d*e^2*x^2+840*A*a^2*b*d^2*e*x^2+280*A*a*b^2 
*d^3*x^2+280*B*a^3*d^2*e*x^2+280*B*a^2*b*d^3*x^2+420*A*a^3*d^2*e*x+420*A*a 
^2*b*d^3*x+140*B*a^3*d^3*x+280*A*a^3*d^3)*((b*x+a)^2)^(3/2)/(b*x+a)^3
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.25 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{8} \, B b^{3} e^{3} x^{8} + A a^{3} d^{3} x + \frac {1}{7} \, {\left (3 \, B b^{3} d e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B b^{3} d^{2} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{3} e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{2}\right )} x^{4} + {\left (A a^{3} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} d^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{3} d^{2} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3}\right )} x^{2} \] Input:

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fric 
as")
 

Output:

1/8*B*b^3*e^3*x^8 + A*a^3*d^3*x + 1/7*(3*B*b^3*d*e^2 + (3*B*a*b^2 + A*b^3) 
*e^3)*x^7 + 1/2*(B*b^3*d^2*e + (3*B*a*b^2 + A*b^3)*d*e^2 + (B*a^2*b + A*a* 
b^2)*e^3)*x^6 + 1/5*(B*b^3*d^3 + 3*(3*B*a*b^2 + A*b^3)*d^2*e + 9*(B*a^2*b 
+ A*a*b^2)*d*e^2 + (B*a^3 + 3*A*a^2*b)*e^3)*x^5 + 1/4*(A*a^3*e^3 + (3*B*a* 
b^2 + A*b^3)*d^3 + 9*(B*a^2*b + A*a*b^2)*d^2*e + 3*(B*a^3 + 3*A*a^2*b)*d*e 
^2)*x^4 + (A*a^3*d*e^2 + (B*a^2*b + A*a*b^2)*d^3 + (B*a^3 + 3*A*a^2*b)*d^2 
*e)*x^3 + 1/2*(3*A*a^3*d^2*e + (B*a^3 + 3*A*a^2*b)*d^3)*x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9440 vs. \(2 (206) = 412\).

Time = 1.52 (sec) , antiderivative size = 9440, normalized size of antiderivative = 36.45 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
 

Output:

Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(B*b**2*e**3*x**7/8 + x**6*(A* 
b**4*e**3 + 17*B*a*b**3*e**3/8 + 3*B*b**4*d*e**2)/(7*b**2) + x**5*(4*A*a*b 
**3*e**3 + 3*A*b**4*d*e**2 + 41*B*a**2*b**2*e**3/8 + 12*B*a*b**3*d*e**2 + 
3*B*b**4*d**2*e - 13*a*(A*b**4*e**3 + 17*B*a*b**3*e**3/8 + 3*B*b**4*d*e**2 
)/(7*b))/(6*b**2) + x**4*(6*A*a**2*b**2*e**3 + 12*A*a*b**3*d*e**2 + 3*A*b* 
*4*d**2*e + 4*B*a**3*b*e**3 + 18*B*a**2*b**2*d*e**2 + 12*B*a*b**3*d**2*e + 
 B*b**4*d**3 - 6*a**2*(A*b**4*e**3 + 17*B*a*b**3*e**3/8 + 3*B*b**4*d*e**2) 
/(7*b**2) - 11*a*(4*A*a*b**3*e**3 + 3*A*b**4*d*e**2 + 41*B*a**2*b**2*e**3/ 
8 + 12*B*a*b**3*d*e**2 + 3*B*b**4*d**2*e - 13*a*(A*b**4*e**3 + 17*B*a*b**3 
*e**3/8 + 3*B*b**4*d*e**2)/(7*b))/(6*b))/(5*b**2) + x**3*(4*A*a**3*b*e**3 
+ 18*A*a**2*b**2*d*e**2 + 12*A*a*b**3*d**2*e + A*b**4*d**3 + B*a**4*e**3 + 
 12*B*a**3*b*d*e**2 + 18*B*a**2*b**2*d**2*e + 4*B*a*b**3*d**3 - 5*a**2*(4* 
A*a*b**3*e**3 + 3*A*b**4*d*e**2 + 41*B*a**2*b**2*e**3/8 + 12*B*a*b**3*d*e* 
*2 + 3*B*b**4*d**2*e - 13*a*(A*b**4*e**3 + 17*B*a*b**3*e**3/8 + 3*B*b**4*d 
*e**2)/(7*b))/(6*b**2) - 9*a*(6*A*a**2*b**2*e**3 + 12*A*a*b**3*d*e**2 + 3* 
A*b**4*d**2*e + 4*B*a**3*b*e**3 + 18*B*a**2*b**2*d*e**2 + 12*B*a*b**3*d**2 
*e + B*b**4*d**3 - 6*a**2*(A*b**4*e**3 + 17*B*a*b**3*e**3/8 + 3*B*b**4*d*e 
**2)/(7*b**2) - 11*a*(4*A*a*b**3*e**3 + 3*A*b**4*d*e**2 + 41*B*a**2*b**2*e 
**3/8 + 12*B*a*b**3*d*e**2 + 3*B*b**4*d**2*e - 13*a*(A*b**4*e**3 + 17*B*a* 
b**3*e**3/8 + 3*B*b**4*d*e**2)/(7*b))/(6*b))/(5*b))/(4*b**2) + x**2*(A*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 698 vs. \(2 (194) = 388\).

Time = 0.06 (sec) , antiderivative size = 698, normalized size of antiderivative = 2.69 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxi 
ma")
 

Output:

1/8*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*e^3*x^3/b^2 + 1/4*(b^2*x^2 + 2*a*b*x 
 + a^2)^(3/2)*A*d^3*x + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a^4*e^3*x/b^ 
4 - 11/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a*e^3*x^2/b^3 + 1/4*(b^2*x^2 + 
 2*a*b*x + a^2)^(3/2)*A*a*d^3/b + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a^ 
5*e^3/b^5 + 13/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^2*e^3*x/b^4 - 69/280 
*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^3*e^3/b^5 - 1/4*(3*B*d*e^2 + A*e^3)*( 
b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*x/b^3 + 3/4*(B*d^2*e + A*d*e^2)*(b^2*x^ 
2 + 2*a*b*x + a^2)^(3/2)*a^2*x/b^2 - 1/4*(B*d^3 + 3*A*d^2*e)*(b^2*x^2 + 2* 
a*b*x + a^2)^(3/2)*a*x/b + 1/7*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^ 
2)^(5/2)*x^2/b^2 - 1/4*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2) 
*a^4/b^4 + 3/4*(B*d^2*e + A*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3/b^3 
 - 1/4*(B*d^3 + 3*A*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2/b^2 - 3/14* 
(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x/b^3 + 1/2*(B*d^2*e 
 + A*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x/b^2 + 17/70*(3*B*d*e^2 + A*e 
^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^4 - 7/10*(B*d^2*e + A*d*e^2)*(b^ 
2*x^2 + 2*a*b*x + a^2)^(5/2)*a/b^3 + 1/5*(B*d^3 + 3*A*d^2*e)*(b^2*x^2 + 2* 
a*b*x + a^2)^(5/2)/b^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 707 vs. \(2 (194) = 388\).

Time = 0.21 (sec) , antiderivative size = 707, normalized size of antiderivative = 2.73 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac 
")
 

Output:

1/8*B*b^3*e^3*x^8*sgn(b*x + a) + 3/7*B*b^3*d*e^2*x^7*sgn(b*x + a) + 3/7*B* 
a*b^2*e^3*x^7*sgn(b*x + a) + 1/7*A*b^3*e^3*x^7*sgn(b*x + a) + 1/2*B*b^3*d^ 
2*e*x^6*sgn(b*x + a) + 3/2*B*a*b^2*d*e^2*x^6*sgn(b*x + a) + 1/2*A*b^3*d*e^ 
2*x^6*sgn(b*x + a) + 1/2*B*a^2*b*e^3*x^6*sgn(b*x + a) + 1/2*A*a*b^2*e^3*x^ 
6*sgn(b*x + a) + 1/5*B*b^3*d^3*x^5*sgn(b*x + a) + 9/5*B*a*b^2*d^2*e*x^5*sg 
n(b*x + a) + 3/5*A*b^3*d^2*e*x^5*sgn(b*x + a) + 9/5*B*a^2*b*d*e^2*x^5*sgn( 
b*x + a) + 9/5*A*a*b^2*d*e^2*x^5*sgn(b*x + a) + 1/5*B*a^3*e^3*x^5*sgn(b*x 
+ a) + 3/5*A*a^2*b*e^3*x^5*sgn(b*x + a) + 3/4*B*a*b^2*d^3*x^4*sgn(b*x + a) 
 + 1/4*A*b^3*d^3*x^4*sgn(b*x + a) + 9/4*B*a^2*b*d^2*e*x^4*sgn(b*x + a) + 9 
/4*A*a*b^2*d^2*e*x^4*sgn(b*x + a) + 3/4*B*a^3*d*e^2*x^4*sgn(b*x + a) + 9/4 
*A*a^2*b*d*e^2*x^4*sgn(b*x + a) + 1/4*A*a^3*e^3*x^4*sgn(b*x + a) + B*a^2*b 
*d^3*x^3*sgn(b*x + a) + A*a*b^2*d^3*x^3*sgn(b*x + a) + B*a^3*d^2*e*x^3*sgn 
(b*x + a) + 3*A*a^2*b*d^2*e*x^3*sgn(b*x + a) + A*a^3*d*e^2*x^3*sgn(b*x + a 
) + 1/2*B*a^3*d^3*x^2*sgn(b*x + a) + 3/2*A*a^2*b*d^3*x^2*sgn(b*x + a) + 3/ 
2*A*a^3*d^2*e*x^2*sgn(b*x + a) + A*a^3*d^3*x*sgn(b*x + a) - 1/280*(14*B*a^ 
5*b^3*d^3 - 70*A*a^4*b^4*d^3 - 14*B*a^6*b^2*d^2*e + 42*A*a^5*b^3*d^2*e + 6 
*B*a^7*b*d*e^2 - 14*A*a^6*b^2*d*e^2 - B*a^8*e^3 + 2*A*a^7*b*e^3)*sgn(b*x + 
 a)/b^5
 

Mupad [F(-1)]

Timed out. \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \] Input:

int((A + B*x)*(d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)
 

Output:

int((A + B*x)*(d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.95 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \left (35 b^{4} e^{3} x^{7}+160 a \,b^{3} e^{3} x^{6}+120 b^{4} d \,e^{2} x^{6}+280 a^{2} b^{2} e^{3} x^{5}+560 a \,b^{3} d \,e^{2} x^{5}+140 b^{4} d^{2} e \,x^{5}+224 a^{3} b \,e^{3} x^{4}+1008 a^{2} b^{2} d \,e^{2} x^{4}+672 a \,b^{3} d^{2} e \,x^{4}+56 b^{4} d^{3} x^{4}+70 a^{4} e^{3} x^{3}+840 a^{3} b d \,e^{2} x^{3}+1260 a^{2} b^{2} d^{2} e \,x^{3}+280 a \,b^{3} d^{3} x^{3}+280 a^{4} d \,e^{2} x^{2}+1120 a^{3} b \,d^{2} e \,x^{2}+560 a^{2} b^{2} d^{3} x^{2}+420 a^{4} d^{2} e x +560 a^{3} b \,d^{3} x +280 a^{4} d^{3}\right )}{280} \] Input:

int((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
 

Output:

(x*(280*a**4*d**3 + 420*a**4*d**2*e*x + 280*a**4*d*e**2*x**2 + 70*a**4*e** 
3*x**3 + 560*a**3*b*d**3*x + 1120*a**3*b*d**2*e*x**2 + 840*a**3*b*d*e**2*x 
**3 + 224*a**3*b*e**3*x**4 + 560*a**2*b**2*d**3*x**2 + 1260*a**2*b**2*d**2 
*e*x**3 + 1008*a**2*b**2*d*e**2*x**4 + 280*a**2*b**2*e**3*x**5 + 280*a*b** 
3*d**3*x**3 + 672*a*b**3*d**2*e*x**4 + 560*a*b**3*d*e**2*x**5 + 160*a*b**3 
*e**3*x**6 + 56*b**4*d**3*x**4 + 140*b**4*d**2*e*x**5 + 120*b**4*d*e**2*x* 
*6 + 35*b**4*e**3*x**7))/280