\(\int (A+B x) (d+e x)^3 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [411]

Optimal result

Integrand size = 33, antiderivative size = 259 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(A b-a B) (b d-a e)^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^5}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^5}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^5}+\frac {B e^3 (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{10 b^5} \] Output:

1/6*(A*b-B*a)*(-a*e+b*d)^3*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^5+1/7*(-a*e+b*d)^ 
2*(3*A*b*e-4*B*a*e+B*b*d)*(b*x+a)^6*((b*x+a)^2)^(1/2)/b^5+3/8*e*(-a*e+b*d) 
*(A*b*e-2*B*a*e+B*b*d)*(b*x+a)^7*((b*x+a)^2)^(1/2)/b^5+1/9*e^2*(A*b*e-4*B* 
a*e+3*B*b*d)*(b*x+a)^8*((b*x+a)^2)^(1/2)/b^5+1/10*B*e^3*(b*x+a)^9*((b*x+a) 
^2)^(1/2)/b^5
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.85 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (126 a^5 \left (5 A \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+B x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )+210 a^4 b x \left (3 A \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )+60 a^3 b^2 x^2 \left (7 A \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+3 B x \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )+90 a^2 b^3 x^3 \left (2 A \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+B x \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )\right )+5 a b^4 x^4 \left (9 A \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+5 B x \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )\right )+b^5 x^5 \left (5 A \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )+3 B x \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )\right )\right )}{2520 (a+b x)} \] Input:

Integrate[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
 

Output:

(x*Sqrt[(a + b*x)^2]*(126*a^5*(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)) + 210*a^4*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)) + 60*a^3*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)) + 90*a^2*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)) + 5*a*b^4*x^4*(9*A*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) 
 + 5*B*x*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3)) + b^5*x^5*(5 
*A*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + 3*B*x*(120*d^3 + 
315*d^2*e*x + 280*d*e^2*x^2 + 84*e^3*x^3))))/(2520*(a + b*x))
 

Rubi [A] (verified)

Time = 0.98 (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.

\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x) (d+e x)^3 \, dx\)

\(\Big \downarrow \) 1187

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^5 (A+B x) (d+e x)^3dx}{b^5 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^5 (A+B x) (d+e x)^3dx}{a+b x}\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B e^3 (a+b x)^9}{b^4}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^8}{b^4}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^7}{b^4}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^6}{b^4}+\frac {(A b-a B) (b d-a e)^3 (a+b x)^5}{b^4}\right )dx}{a+b x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {e^2 (a+b x)^9 (-4 a B e+A b e+3 b B d)}{9 b^5}+\frac {3 e (a+b x)^8 (b d-a e) (-2 a B e+A b e+b B d)}{8 b^5}+\frac {(a+b x)^7 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{7 b^5}+\frac {(a+b x)^6 (A b-a B) (b d-a e)^3}{6 b^5}+\frac {B e^3 (a+b x)^{10}}{10 b^5}\right )}{a+b x}\)

Input:

Int[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),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)^6)/(6 
*b^5) + ((b*d - a*e)^2*(b*B*d + 3*A*b*e - 4*a*B*e)*(a + b*x)^7)/(7*b^5) + 
(3*e*(b*d - a*e)*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^8)/(8*b^5) + (e^2*(3* 
b*B*d + A*b*e - 4*a*B*e)*(a + b*x)^9)/(9*b^5) + (B*e^3*(a + b*x)^10)/(10*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. \(675\) vs. \(2(194)=388\).

Time = 1.45 (sec) , antiderivative size = 676, normalized size of antiderivative = 2.61

method	result	size
gosper	\(\frac {x \left (252 B \,e^{3} b^{5} x^{9}+280 x^{8} A \,b^{5} e^{3}+1400 x^{8} B \,e^{3} a \,b^{4}+840 x^{8} B \,b^{5} d \,e^{2}+1575 x^{7} A a \,b^{4} e^{3}+945 x^{7} A \,b^{5} d \,e^{2}+3150 x^{7} B \,e^{3} a^{2} b^{3}+4725 x^{7} B a \,b^{4} d \,e^{2}+945 x^{7} B \,b^{5} d^{2} e +3600 x^{6} A \,a^{2} b^{3} e^{3}+5400 x^{6} A a \,b^{4} d \,e^{2}+1080 x^{6} A \,b^{5} d^{2} e +3600 x^{6} B \,a^{3} b^{2} e^{3}+10800 x^{6} B \,a^{2} b^{3} d \,e^{2}+5400 x^{6} B a \,b^{4} d^{2} e +360 x^{6} B \,b^{5} d^{3}+4200 x^{5} A \,a^{3} b^{2} e^{3}+12600 x^{5} A \,a^{2} b^{3} d \,e^{2}+6300 x^{5} A a \,b^{4} d^{2} e +420 x^{5} A \,d^{3} b^{5}+2100 x^{5} B \,e^{3} a^{4} b +12600 x^{5} B \,a^{3} b^{2} d \,e^{2}+12600 x^{5} B \,a^{2} b^{3} d^{2} e +2100 x^{5} B a \,b^{4} d^{3}+2520 x^{4} A \,a^{4} b \,e^{3}+15120 x^{4} A \,a^{3} b^{2} d \,e^{2}+15120 x^{4} A \,a^{2} b^{3} d^{2} e +2520 x^{4} A \,d^{3} a \,b^{4}+504 x^{4} B \,e^{3} a^{5}+7560 x^{4} B \,a^{4} b d \,e^{2}+15120 x^{4} B \,a^{3} b^{2} d^{2} e +5040 x^{4} B \,a^{2} b^{3} d^{3}+630 x^{3} A \,a^{5} e^{3}+9450 x^{3} A \,a^{4} b d \,e^{2}+18900 x^{3} A \,a^{3} b^{2} d^{2} e +6300 x^{3} A \,d^{3} a^{2} b^{3}+1890 x^{3} B \,a^{5} d \,e^{2}+9450 x^{3} B \,a^{4} b \,d^{2} e +6300 x^{3} B \,a^{3} b^{2} d^{3}+2520 x^{2} A \,a^{5} d \,e^{2}+12600 x^{2} A \,a^{4} b \,d^{2} e +8400 x^{2} A \,d^{3} a^{3} b^{2}+2520 x^{2} B \,a^{5} d^{2} e +4200 x^{2} B \,a^{4} b \,d^{3}+3780 x A \,a^{5} d^{2} e +6300 x A \,d^{3} a^{4} b +1260 x B \,a^{5} d^{3}+2520 A \,d^{3} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2520 \left (b x +a \right )^{5}}\)	\(676\)
default	\(\frac {x \left (252 B \,e^{3} b^{5} x^{9}+280 x^{8} A \,b^{5} e^{3}+1400 x^{8} B \,e^{3} a \,b^{4}+840 x^{8} B \,b^{5} d \,e^{2}+1575 x^{7} A a \,b^{4} e^{3}+945 x^{7} A \,b^{5} d \,e^{2}+3150 x^{7} B \,e^{3} a^{2} b^{3}+4725 x^{7} B a \,b^{4} d \,e^{2}+945 x^{7} B \,b^{5} d^{2} e +3600 x^{6} A \,a^{2} b^{3} e^{3}+5400 x^{6} A a \,b^{4} d \,e^{2}+1080 x^{6} A \,b^{5} d^{2} e +3600 x^{6} B \,a^{3} b^{2} e^{3}+10800 x^{6} B \,a^{2} b^{3} d \,e^{2}+5400 x^{6} B a \,b^{4} d^{2} e +360 x^{6} B \,b^{5} d^{3}+4200 x^{5} A \,a^{3} b^{2} e^{3}+12600 x^{5} A \,a^{2} b^{3} d \,e^{2}+6300 x^{5} A a \,b^{4} d^{2} e +420 x^{5} A \,d^{3} b^{5}+2100 x^{5} B \,e^{3} a^{4} b +12600 x^{5} B \,a^{3} b^{2} d \,e^{2}+12600 x^{5} B \,a^{2} b^{3} d^{2} e +2100 x^{5} B a \,b^{4} d^{3}+2520 x^{4} A \,a^{4} b \,e^{3}+15120 x^{4} A \,a^{3} b^{2} d \,e^{2}+15120 x^{4} A \,a^{2} b^{3} d^{2} e +2520 x^{4} A \,d^{3} a \,b^{4}+504 x^{4} B \,e^{3} a^{5}+7560 x^{4} B \,a^{4} b d \,e^{2}+15120 x^{4} B \,a^{3} b^{2} d^{2} e +5040 x^{4} B \,a^{2} b^{3} d^{3}+630 x^{3} A \,a^{5} e^{3}+9450 x^{3} A \,a^{4} b d \,e^{2}+18900 x^{3} A \,a^{3} b^{2} d^{2} e +6300 x^{3} A \,d^{3} a^{2} b^{3}+1890 x^{3} B \,a^{5} d \,e^{2}+9450 x^{3} B \,a^{4} b \,d^{2} e +6300 x^{3} B \,a^{3} b^{2} d^{3}+2520 x^{2} A \,a^{5} d \,e^{2}+12600 x^{2} A \,a^{4} b \,d^{2} e +8400 x^{2} A \,d^{3} a^{3} b^{2}+2520 x^{2} B \,a^{5} d^{2} e +4200 x^{2} B \,a^{4} b \,d^{3}+3780 x A \,a^{5} d^{2} e +6300 x A \,d^{3} a^{4} b +1260 x B \,a^{5} d^{3}+2520 A \,d^{3} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2520 \left (b x +a \right )^{5}}\)	\(676\)
orering	\(\frac {x \left (252 B \,e^{3} b^{5} x^{9}+280 x^{8} A \,b^{5} e^{3}+1400 x^{8} B \,e^{3} a \,b^{4}+840 x^{8} B \,b^{5} d \,e^{2}+1575 x^{7} A a \,b^{4} e^{3}+945 x^{7} A \,b^{5} d \,e^{2}+3150 x^{7} B \,e^{3} a^{2} b^{3}+4725 x^{7} B a \,b^{4} d \,e^{2}+945 x^{7} B \,b^{5} d^{2} e +3600 x^{6} A \,a^{2} b^{3} e^{3}+5400 x^{6} A a \,b^{4} d \,e^{2}+1080 x^{6} A \,b^{5} d^{2} e +3600 x^{6} B \,a^{3} b^{2} e^{3}+10800 x^{6} B \,a^{2} b^{3} d \,e^{2}+5400 x^{6} B a \,b^{4} d^{2} e +360 x^{6} B \,b^{5} d^{3}+4200 x^{5} A \,a^{3} b^{2} e^{3}+12600 x^{5} A \,a^{2} b^{3} d \,e^{2}+6300 x^{5} A a \,b^{4} d^{2} e +420 x^{5} A \,d^{3} b^{5}+2100 x^{5} B \,e^{3} a^{4} b +12600 x^{5} B \,a^{3} b^{2} d \,e^{2}+12600 x^{5} B \,a^{2} b^{3} d^{2} e +2100 x^{5} B a \,b^{4} d^{3}+2520 x^{4} A \,a^{4} b \,e^{3}+15120 x^{4} A \,a^{3} b^{2} d \,e^{2}+15120 x^{4} A \,a^{2} b^{3} d^{2} e +2520 x^{4} A \,d^{3} a \,b^{4}+504 x^{4} B \,e^{3} a^{5}+7560 x^{4} B \,a^{4} b d \,e^{2}+15120 x^{4} B \,a^{3} b^{2} d^{2} e +5040 x^{4} B \,a^{2} b^{3} d^{3}+630 x^{3} A \,a^{5} e^{3}+9450 x^{3} A \,a^{4} b d \,e^{2}+18900 x^{3} A \,a^{3} b^{2} d^{2} e +6300 x^{3} A \,d^{3} a^{2} b^{3}+1890 x^{3} B \,a^{5} d \,e^{2}+9450 x^{3} B \,a^{4} b \,d^{2} e +6300 x^{3} B \,a^{3} b^{2} d^{3}+2520 x^{2} A \,a^{5} d \,e^{2}+12600 x^{2} A \,a^{4} b \,d^{2} e +8400 x^{2} A \,d^{3} a^{3} b^{2}+2520 x^{2} B \,a^{5} d^{2} e +4200 x^{2} B \,a^{4} b \,d^{3}+3780 x A \,a^{5} d^{2} e +6300 x A \,d^{3} a^{4} b +1260 x B \,a^{5} d^{3}+2520 A \,d^{3} a^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{2520 \left (b x +a \right )^{5}}\)	\(685\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, B \,e^{3} b^{5} x^{10}}{10 b x +10 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (A \,e^{3}+3 B d \,e^{2}\right ) b^{5}+5 B \,e^{3} a \,b^{4}\right ) x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (3 A d \,e^{2}+3 e B \,d^{2}\right ) b^{5}+5 \left (A \,e^{3}+3 B d \,e^{2}\right ) a \,b^{4}+10 B \,e^{3} a^{2} b^{3}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (3 A \,d^{2} e +B \,d^{3}\right ) b^{5}+5 \left (3 A d \,e^{2}+3 e B \,d^{2}\right ) a \,b^{4}+10 \left (A \,e^{3}+3 B d \,e^{2}\right ) a^{2} b^{3}+10 B \,a^{3} b^{2} e^{3}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A \,d^{3} b^{5}+5 \left (3 A \,d^{2} e +B \,d^{3}\right ) a \,b^{4}+10 \left (3 A d \,e^{2}+3 e B \,d^{2}\right ) a^{2} b^{3}+10 \left (A \,e^{3}+3 B d \,e^{2}\right ) a^{3} b^{2}+5 B \,e^{3} a^{4} b \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,d^{3} a \,b^{4}+10 \left (3 A \,d^{2} e +B \,d^{3}\right ) a^{2} b^{3}+10 \left (3 A d \,e^{2}+3 e B \,d^{2}\right ) a^{3} b^{2}+5 \left (A \,e^{3}+3 B d \,e^{2}\right ) a^{4} b +B \,e^{3} a^{5}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A \,d^{3} a^{2} b^{3}+10 \left (3 A \,d^{2} e +B \,d^{3}\right ) a^{3} b^{2}+5 \left (3 A d \,e^{2}+3 e B \,d^{2}\right ) a^{4} b +\left (A \,e^{3}+3 B d \,e^{2}\right ) a^{5}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A \,d^{3} a^{3} b^{2}+5 \left (3 A \,d^{2} e +B \,d^{3}\right ) a^{4} b +\left (3 A d \,e^{2}+3 e B \,d^{2}\right ) a^{5}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,d^{3} a^{4} b +\left (3 A \,d^{2} e +B \,d^{3}\right ) a^{5}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, A \,d^{3} a^{5} x}{b x +a}\)	\(689\)

Input:

int((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/2520*x*(252*B*b^5*e^3*x^9+280*A*b^5*e^3*x^8+1400*B*a*b^4*e^3*x^8+840*B*b 
^5*d*e^2*x^8+1575*A*a*b^4*e^3*x^7+945*A*b^5*d*e^2*x^7+3150*B*a^2*b^3*e^3*x 
^7+4725*B*a*b^4*d*e^2*x^7+945*B*b^5*d^2*e*x^7+3600*A*a^2*b^3*e^3*x^6+5400* 
A*a*b^4*d*e^2*x^6+1080*A*b^5*d^2*e*x^6+3600*B*a^3*b^2*e^3*x^6+10800*B*a^2* 
b^3*d*e^2*x^6+5400*B*a*b^4*d^2*e*x^6+360*B*b^5*d^3*x^6+4200*A*a^3*b^2*e^3* 
x^5+12600*A*a^2*b^3*d*e^2*x^5+6300*A*a*b^4*d^2*e*x^5+420*A*b^5*d^3*x^5+210 
0*B*a^4*b*e^3*x^5+12600*B*a^3*b^2*d*e^2*x^5+12600*B*a^2*b^3*d^2*e*x^5+2100 
*B*a*b^4*d^3*x^5+2520*A*a^4*b*e^3*x^4+15120*A*a^3*b^2*d*e^2*x^4+15120*A*a^ 
2*b^3*d^2*e*x^4+2520*A*a*b^4*d^3*x^4+504*B*a^5*e^3*x^4+7560*B*a^4*b*d*e^2* 
x^4+15120*B*a^3*b^2*d^2*e*x^4+5040*B*a^2*b^3*d^3*x^4+630*A*a^5*e^3*x^3+945 
0*A*a^4*b*d*e^2*x^3+18900*A*a^3*b^2*d^2*e*x^3+6300*A*a^2*b^3*d^3*x^3+1890* 
B*a^5*d*e^2*x^3+9450*B*a^4*b*d^2*e*x^3+6300*B*a^3*b^2*d^3*x^3+2520*A*a^5*d 
*e^2*x^2+12600*A*a^4*b*d^2*e*x^2+8400*A*a^3*b^2*d^3*x^2+2520*B*a^5*d^2*e*x 
^2+4200*B*a^4*b*d^3*x^2+3780*A*a^5*d^2*e*x+6300*A*a^4*b*d^3*x+1260*B*a^5*d 
^3*x+2520*A*a^5*d^3)*((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. 532 vs. \(2 (194) = 388\).

Time = 0.08 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.05 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{10} \, B b^{5} e^{3} x^{10} + A a^{5} d^{3} x + \frac {1}{9} \, {\left (3 \, B b^{5} d e^{2} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (3 \, B b^{5} d^{2} e + 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{2} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{5} d^{3} + 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e + 15 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{2} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{6} \, {\left ({\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} + 15 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{2} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e + 15 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{2} + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{5} e^{3} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} + 15 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{5} d e^{2} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{5} d^{2} e + {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3}\right )} x^{2} \] Input:

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fric 
as")
 

Output:

1/10*B*b^5*e^3*x^10 + A*a^5*d^3*x + 1/9*(3*B*b^5*d*e^2 + (5*B*a*b^4 + A*b^ 
5)*e^3)*x^9 + 1/8*(3*B*b^5*d^2*e + 3*(5*B*a*b^4 + A*b^5)*d*e^2 + 5*(2*B*a^ 
2*b^3 + A*a*b^4)*e^3)*x^8 + 1/7*(B*b^5*d^3 + 3*(5*B*a*b^4 + A*b^5)*d^2*e + 
 15*(2*B*a^2*b^3 + A*a*b^4)*d*e^2 + 10*(B*a^3*b^2 + A*a^2*b^3)*e^3)*x^7 + 
1/6*((5*B*a*b^4 + A*b^5)*d^3 + 15*(2*B*a^2*b^3 + A*a*b^4)*d^2*e + 30*(B*a^ 
3*b^2 + A*a^2*b^3)*d*e^2 + 5*(B*a^4*b + 2*A*a^3*b^2)*e^3)*x^6 + 1/5*(5*(2* 
B*a^2*b^3 + A*a*b^4)*d^3 + 30*(B*a^3*b^2 + A*a^2*b^3)*d^2*e + 15*(B*a^4*b 
+ 2*A*a^3*b^2)*d*e^2 + (B*a^5 + 5*A*a^4*b)*e^3)*x^5 + 1/4*(A*a^5*e^3 + 10* 
(B*a^3*b^2 + A*a^2*b^3)*d^3 + 15*(B*a^4*b + 2*A*a^3*b^2)*d^2*e + 3*(B*a^5 
+ 5*A*a^4*b)*d*e^2)*x^4 + 1/3*(3*A*a^5*d*e^2 + 5*(B*a^4*b + 2*A*a^3*b^2)*d 
^3 + 3*(B*a^5 + 5*A*a^4*b)*d^2*e)*x^3 + 1/2*(3*A*a^5*d^2*e + (B*a^5 + 5*A* 
a^4*b)*d^3)*x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26879 vs. \(2 (207) = 414\).

Time = 1.76 (sec) , antiderivative size = 26879, normalized size of antiderivative = 103.78 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)**3*(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**3*x**9/10 + x**8*(A 
*b**6*e**3 + 41*B*a*b**5*e**3/10 + 3*B*b**6*d*e**2)/(9*b**2) + x**7*(6*A*a 
*b**5*e**3 + 3*A*b**6*d*e**2 + 141*B*a**2*b**4*e**3/10 + 18*B*a*b**5*d*e** 
2 + 3*B*b**6*d**2*e - 17*a*(A*b**6*e**3 + 41*B*a*b**5*e**3/10 + 3*B*b**6*d 
*e**2)/(9*b))/(8*b**2) + x**6*(15*A*a**2*b**4*e**3 + 18*A*a*b**5*d*e**2 + 
3*A*b**6*d**2*e + 20*B*a**3*b**3*e**3 + 45*B*a**2*b**4*d*e**2 + 18*B*a*b** 
5*d**2*e + B*b**6*d**3 - 8*a**2*(A*b**6*e**3 + 41*B*a*b**5*e**3/10 + 3*B*b 
**6*d*e**2)/(9*b**2) - 15*a*(6*A*a*b**5*e**3 + 3*A*b**6*d*e**2 + 141*B*a** 
2*b**4*e**3/10 + 18*B*a*b**5*d*e**2 + 3*B*b**6*d**2*e - 17*a*(A*b**6*e**3 
+ 41*B*a*b**5*e**3/10 + 3*B*b**6*d*e**2)/(9*b))/(8*b))/(7*b**2) + x**5*(20 
*A*a**3*b**3*e**3 + 45*A*a**2*b**4*d*e**2 + 18*A*a*b**5*d**2*e + A*b**6*d* 
*3 + 15*B*a**4*b**2*e**3 + 60*B*a**3*b**3*d*e**2 + 45*B*a**2*b**4*d**2*e + 
 6*B*a*b**5*d**3 - 7*a**2*(6*A*a*b**5*e**3 + 3*A*b**6*d*e**2 + 141*B*a**2* 
b**4*e**3/10 + 18*B*a*b**5*d*e**2 + 3*B*b**6*d**2*e - 17*a*(A*b**6*e**3 + 
41*B*a*b**5*e**3/10 + 3*B*b**6*d*e**2)/(9*b))/(8*b**2) - 13*a*(15*A*a**2*b 
**4*e**3 + 18*A*a*b**5*d*e**2 + 3*A*b**6*d**2*e + 20*B*a**3*b**3*e**3 + 45 
*B*a**2*b**4*d*e**2 + 18*B*a*b**5*d**2*e + B*b**6*d**3 - 8*a**2*(A*b**6*e* 
*3 + 41*B*a*b**5*e**3/10 + 3*B*b**6*d*e**2)/(9*b**2) - 15*a*(6*A*a*b**5*e* 
*3 + 3*A*b**6*d*e**2 + 141*B*a**2*b**4*e**3/10 + 18*B*a*b**5*d*e**2 + 3*B* 
b**6*d**2*e - 17*a*(A*b**6*e**3 + 41*B*a*b**5*e**3/10 + 3*B*b**6*d*e**2...
 

Maxima [B] (verification not implemented)

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

Time = 0.05 (sec) , antiderivative size = 698, normalized size of antiderivative = 2.69 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxi 
ma")
 

Output:

1/10*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*e^3*x^3/b^2 + 1/6*(b^2*x^2 + 2*a*b* 
x + a^2)^(5/2)*A*d^3*x + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^4*e^3*x/b 
^4 - 13/90*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a*e^3*x^2/b^3 + 1/6*(b^2*x^2 
+ 2*a*b*x + a^2)^(5/2)*A*a*d^3/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a 
^5*e^3/b^5 + 29/180*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^2*e^3*x/b^4 - 209/ 
1260*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^3*e^3/b^5 - 1/6*(3*B*d*e^2 + A*e^ 
3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*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)*a^2*x/b^2 - 1/6*(B*d^3 + 3*A*d^2*e)*(b^2*x^2 
+ 2*a*b*x + a^2)^(5/2)*a*x/b + 1/9*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x 
+ a^2)^(7/2)*x^2/b^2 - 1/6*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^( 
5/2)*a^4/b^4 + 1/2*(B*d^2*e + A*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3 
/b^3 - 1/6*(B*d^3 + 3*A*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^2 - 1 
1/72*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x/b^3 + 3/8*(B* 
d^2*e + A*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x/b^2 + 83/504*(3*B*d*e^2 
 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2/b^4 - 27/56*(B*d^2*e + A*d*e 
^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a/b^3 + 1/7*(B*d^3 + 3*A*d^2*e)*(b^2*x 
^2 + 2*a*b*x + a^2)^(7/2)/b^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1051 vs. \(2 (194) = 388\).

Time = 0.20 (sec) , antiderivative size = 1051, normalized size of antiderivative = 4.06 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac 
")
 

Output:

1/10*B*b^5*e^3*x^10*sgn(b*x + a) + 1/3*B*b^5*d*e^2*x^9*sgn(b*x + a) + 5/9* 
B*a*b^4*e^3*x^9*sgn(b*x + a) + 1/9*A*b^5*e^3*x^9*sgn(b*x + a) + 3/8*B*b^5* 
d^2*e*x^8*sgn(b*x + a) + 15/8*B*a*b^4*d*e^2*x^8*sgn(b*x + a) + 3/8*A*b^5*d 
*e^2*x^8*sgn(b*x + a) + 5/4*B*a^2*b^3*e^3*x^8*sgn(b*x + a) + 5/8*A*a*b^4*e 
^3*x^8*sgn(b*x + a) + 1/7*B*b^5*d^3*x^7*sgn(b*x + a) + 15/7*B*a*b^4*d^2*e* 
x^7*sgn(b*x + a) + 3/7*A*b^5*d^2*e*x^7*sgn(b*x + a) + 30/7*B*a^2*b^3*d*e^2 
*x^7*sgn(b*x + a) + 15/7*A*a*b^4*d*e^2*x^7*sgn(b*x + a) + 10/7*B*a^3*b^2*e 
^3*x^7*sgn(b*x + a) + 10/7*A*a^2*b^3*e^3*x^7*sgn(b*x + a) + 5/6*B*a*b^4*d^ 
3*x^6*sgn(b*x + a) + 1/6*A*b^5*d^3*x^6*sgn(b*x + a) + 5*B*a^2*b^3*d^2*e*x^ 
6*sgn(b*x + a) + 5/2*A*a*b^4*d^2*e*x^6*sgn(b*x + a) + 5*B*a^3*b^2*d*e^2*x^ 
6*sgn(b*x + a) + 5*A*a^2*b^3*d*e^2*x^6*sgn(b*x + a) + 5/6*B*a^4*b*e^3*x^6* 
sgn(b*x + a) + 5/3*A*a^3*b^2*e^3*x^6*sgn(b*x + a) + 2*B*a^2*b^3*d^3*x^5*sg 
n(b*x + a) + A*a*b^4*d^3*x^5*sgn(b*x + a) + 6*B*a^3*b^2*d^2*e*x^5*sgn(b*x 
+ a) + 6*A*a^2*b^3*d^2*e*x^5*sgn(b*x + a) + 3*B*a^4*b*d*e^2*x^5*sgn(b*x + 
a) + 6*A*a^3*b^2*d*e^2*x^5*sgn(b*x + a) + 1/5*B*a^5*e^3*x^5*sgn(b*x + a) + 
 A*a^4*b*e^3*x^5*sgn(b*x + a) + 5/2*B*a^3*b^2*d^3*x^4*sgn(b*x + a) + 5/2*A 
*a^2*b^3*d^3*x^4*sgn(b*x + a) + 15/4*B*a^4*b*d^2*e*x^4*sgn(b*x + a) + 15/2 
*A*a^3*b^2*d^2*e*x^4*sgn(b*x + a) + 3/4*B*a^5*d*e^2*x^4*sgn(b*x + a) + 15/ 
4*A*a^4*b*d*e^2*x^4*sgn(b*x + a) + 1/4*A*a^5*e^3*x^4*sgn(b*x + a) + 5/3*B* 
a^4*b*d^3*x^3*sgn(b*x + a) + 10/3*A*a^3*b^2*d^3*x^3*sgn(b*x + a) + B*a^...
 

Mupad [F(-1)]

Timed out. \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \] Input:

int((A + B*x)*(d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
 

Output:

int((A + B*x)*(d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.40 \[ \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \left (84 b^{6} e^{3} x^{9}+560 a \,b^{5} e^{3} x^{8}+280 b^{6} d \,e^{2} x^{8}+1575 a^{2} b^{4} e^{3} x^{7}+1890 a \,b^{5} d \,e^{2} x^{7}+315 b^{6} d^{2} e \,x^{7}+2400 a^{3} b^{3} e^{3} x^{6}+5400 a^{2} b^{4} d \,e^{2} x^{6}+2160 a \,b^{5} d^{2} e \,x^{6}+120 b^{6} d^{3} x^{6}+2100 a^{4} b^{2} e^{3} x^{5}+8400 a^{3} b^{3} d \,e^{2} x^{5}+6300 a^{2} b^{4} d^{2} e \,x^{5}+840 a \,b^{5} d^{3} x^{5}+1008 a^{5} b \,e^{3} x^{4}+7560 a^{4} b^{2} d \,e^{2} x^{4}+10080 a^{3} b^{3} d^{2} e \,x^{4}+2520 a^{2} b^{4} d^{3} x^{4}+210 a^{6} e^{3} x^{3}+3780 a^{5} b d \,e^{2} x^{3}+9450 a^{4} b^{2} d^{2} e \,x^{3}+4200 a^{3} b^{3} d^{3} x^{3}+840 a^{6} d \,e^{2} x^{2}+5040 a^{5} b \,d^{2} e \,x^{2}+4200 a^{4} b^{2} d^{3} x^{2}+1260 a^{6} d^{2} e x +2520 a^{5} b \,d^{3} x +840 a^{6} d^{3}\right )}{840} \] Input:

int((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
 

Output:

(x*(840*a**6*d**3 + 1260*a**6*d**2*e*x + 840*a**6*d*e**2*x**2 + 210*a**6*e 
**3*x**3 + 2520*a**5*b*d**3*x + 5040*a**5*b*d**2*e*x**2 + 3780*a**5*b*d*e* 
*2*x**3 + 1008*a**5*b*e**3*x**4 + 4200*a**4*b**2*d**3*x**2 + 9450*a**4*b** 
2*d**2*e*x**3 + 7560*a**4*b**2*d*e**2*x**4 + 2100*a**4*b**2*e**3*x**5 + 42 
00*a**3*b**3*d**3*x**3 + 10080*a**3*b**3*d**2*e*x**4 + 8400*a**3*b**3*d*e* 
*2*x**5 + 2400*a**3*b**3*e**3*x**6 + 2520*a**2*b**4*d**3*x**4 + 6300*a**2* 
b**4*d**2*e*x**5 + 5400*a**2*b**4*d*e**2*x**6 + 1575*a**2*b**4*e**3*x**7 + 
 840*a*b**5*d**3*x**5 + 2160*a*b**5*d**2*e*x**6 + 1890*a*b**5*d*e**2*x**7 
+ 560*a*b**5*e**3*x**8 + 120*b**6*d**3*x**6 + 315*b**6*d**2*e*x**7 + 280*b 
**6*d*e**2*x**8 + 84*b**6*e**3*x**9))/840