\(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{12}} \, dx\) [426]

Optimal result

Integrand size = 33, antiderivative size = 342 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\frac {b^4 (A b-a B) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 (b d-a e)^6 (d+e x)^6}+\frac {(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{11 (b d-a e)^2 (d+e x)^{11}}+\frac {(4 b B d-15 A b e+11 a B e) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{110 (b d-a e)^3 (d+e x)^{10}}+\frac {b (12 b B d-155 A b e+143 a B e) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{990 (b d-a e)^4 (d+e x)^9}+\frac {b^2 (6 b B d-325 A b e+319 a B e) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{1980 (b d-a e)^5 (d+e x)^8}+\frac {b^3 (6 b B d-2305 A b e+2299 a B e) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{13860 (b d-a e)^6 (d+e x)^7} \] Output:

1/6*b^4*(A*b-B*a)*(b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(-a*e+b*d)^6/(e*x+d) 
^6+1/11*(-A*e+B*d)*(b^2*x^2+2*a*b*x+a^2)^(7/2)/(-a*e+b*d)^2/(e*x+d)^11+1/1 
10*(-15*A*b*e+11*B*a*e+4*B*b*d)*(b^2*x^2+2*a*b*x+a^2)^(7/2)/(-a*e+b*d)^3/( 
e*x+d)^10+1/990*b*(-155*A*b*e+143*B*a*e+12*B*b*d)*(b^2*x^2+2*a*b*x+a^2)^(7 
/2)/(-a*e+b*d)^4/(e*x+d)^9+1/1980*b^2*(-325*A*b*e+319*B*a*e+6*B*b*d)*(b^2* 
x^2+2*a*b*x+a^2)^(7/2)/(-a*e+b*d)^5/(e*x+d)^8+1/13860*b^3*(-2305*A*b*e+229 
9*B*a*e+6*B*b*d)*(b^2*x^2+2*a*b*x+a^2)^(7/2)/(-a*e+b*d)^6/(e*x+d)^7
 

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.38 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (126 a^5 e^5 (10 A e+B (d+11 e x))+70 a^4 b e^4 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+35 a^3 b^2 e^3 \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )\right )+15 a^2 b^3 e^2 \left (7 A e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )+5 a b^4 e \left (6 A e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 B \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )+b^5 \left (5 A e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+6 B \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )\right )}{13860 e^7 (a+b x) (d+e x)^{11}} \] Input:

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^12,x]
 

Output:

-1/13860*(Sqrt[(a + b*x)^2]*(126*a^5*e^5*(10*A*e + B*(d + 11*e*x)) + 70*a^ 
4*b*e^4*(9*A*e*(d + 11*e*x) + 2*B*(d^2 + 11*d*e*x + 55*e^2*x^2)) + 35*a^3* 
b^2*e^3*(8*A*e*(d^2 + 11*d*e*x + 55*e^2*x^2) + 3*B*(d^3 + 11*d^2*e*x + 55* 
d*e^2*x^2 + 165*e^3*x^3)) + 15*a^2*b^3*e^2*(7*A*e*(d^3 + 11*d^2*e*x + 55*d 
*e^2*x^2 + 165*e^3*x^3) + 4*B*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e 
^3*x^3 + 330*e^4*x^4)) + 5*a*b^4*e*(6*A*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x 
^2 + 165*d*e^3*x^3 + 330*e^4*x^4) + 5*B*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 
 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5)) + b^5*(5*A*e*(d^5 + 11* 
d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5) 
+ 6*B*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x 
^4 + 462*d*e^5*x^5 + 462*e^6*x^6))))/(e^7*(a + b*x)*(d + e*x)^11)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.81, 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.

\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{(d+e x)^{12}} \, dx\)

\(\Big \downarrow \) 1187

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{(d+e x)^{12}}dx}{b^5 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{(d+e x)^{12}}dx}{a+b x}\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B b^5}{e^6 (d+e x)^6}+\frac {(-6 b B d+A b e+5 a B e) b^4}{e^6 (d+e x)^7}-\frac {5 (b d-a e) (-3 b B d+A b e+2 a B e) b^3}{e^6 (d+e x)^8}+\frac {10 (b d-a e)^2 (-2 b B d+A b e+a B e) b^2}{e^6 (d+e x)^9}-\frac {5 (b d-a e)^3 (-3 b B d+2 A b e+a B e) b}{e^6 (d+e x)^{10}}+\frac {(a e-b d)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{11}}+\frac {(a e-b d)^5 (A e-B d)}{e^6 (d+e x)^{12}}\right )dx}{a+b x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {b^4 (-5 a B e-A b e+6 b B d)}{6 e^7 (d+e x)^6}-\frac {5 b^3 (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7 (d+e x)^7}+\frac {5 b^2 (b d-a e)^2 (-a B e-A b e+2 b B d)}{4 e^7 (d+e x)^8}-\frac {5 b (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (d+e x)^9}+\frac {(b d-a e)^4 (-a B e-5 A b e+6 b B d)}{10 e^7 (d+e x)^{10}}-\frac {(b d-a e)^5 (B d-A e)}{11 e^7 (d+e x)^{11}}-\frac {b^5 B}{5 e^7 (d+e x)^5}\right )}{a+b x}\)

Input:

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^12,x]
 

Output:

(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/11*((b*d - a*e)^5*(B*d - A*e))/(e^7*(d 
+ e*x)^11) + ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e))/(10*e^7*(d + e*x) 
^10) - (5*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e))/(9*e^7*(d + e*x)^9) 
 + (5*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e))/(4*e^7*(d + e*x)^8) - ( 
5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e))/(7*e^7*(d + e*x)^7) + (b^4* 
(6*b*B*d - A*b*e - 5*a*B*e))/(6*e^7*(d + e*x)^6) - (b^5*B)/(5*e^7*(d + e*x 
)^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 [A] (verified)

Time = 11.13 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.76

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} B \,x^{6}}{5 e}-\frac {b^{4} \left (5 A b e +25 B a e +6 B b d \right ) x^{5}}{30 e^{2}}-\frac {b^{3} \left (30 A a b \,e^{2}+5 A \,b^{2} d e +60 B \,e^{2} a^{2}+25 B a b d e +6 B \,b^{2} d^{2}\right ) x^{4}}{42 e^{3}}-\frac {b^{2} \left (105 A \,a^{2} b \,e^{3}+30 A a \,b^{2} d \,e^{2}+5 A \,b^{3} d^{2} e +105 B \,e^{3} a^{3}+60 B \,a^{2} b d \,e^{2}+25 B a \,b^{2} d^{2} e +6 B \,b^{3} d^{3}\right ) x^{3}}{84 e^{4}}-\frac {b \left (280 A \,a^{3} b \,e^{4}+105 A \,a^{2} b^{2} d \,e^{3}+30 A a \,b^{3} d^{2} e^{2}+5 A \,b^{4} d^{3} e +140 B \,e^{4} a^{4}+105 B \,a^{3} b d \,e^{3}+60 B \,a^{2} b^{2} d^{2} e^{2}+25 B a \,b^{3} d^{3} e +6 B \,b^{4} d^{4}\right ) x^{2}}{252 e^{5}}-\frac {\left (630 A \,a^{4} b \,e^{5}+280 A \,a^{3} b^{2} d \,e^{4}+105 A \,a^{2} b^{3} d^{2} e^{3}+30 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +126 B \,a^{5} e^{5}+140 B \,a^{4} b d \,e^{4}+105 B \,a^{3} b^{2} d^{2} e^{3}+60 B \,a^{2} b^{3} d^{3} e^{2}+25 B a \,b^{4} d^{4} e +6 B \,b^{5} d^{5}\right ) x}{1260 e^{6}}-\frac {1260 A \,a^{5} e^{6}+630 A \,a^{4} b d \,e^{5}+280 A \,a^{3} b^{2} d^{2} e^{4}+105 A \,a^{2} b^{3} d^{3} e^{3}+30 A a \,b^{4} d^{4} e^{2}+5 A \,b^{5} d^{5} e +126 B \,a^{5} d \,e^{5}+140 B \,a^{4} b \,d^{2} e^{4}+105 B \,a^{3} b^{2} d^{3} e^{3}+60 B \,a^{2} b^{3} d^{4} e^{2}+25 B a \,b^{4} d^{5} e +6 b^{5} B \,d^{6}}{13860 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{11}}\)	\(602\)
gosper	\(-\frac {\left (2772 B \,b^{5} e^{6} x^{6}+2310 A \,b^{5} e^{6} x^{5}+11550 B a \,b^{4} e^{6} x^{5}+2772 B \,b^{5} d \,e^{5} x^{5}+9900 A a \,b^{4} e^{6} x^{4}+1650 A \,b^{5} d \,e^{5} x^{4}+19800 B \,a^{2} b^{3} e^{6} x^{4}+8250 B a \,b^{4} d \,e^{5} x^{4}+1980 B \,b^{5} d^{2} e^{4} x^{4}+17325 A \,a^{2} b^{3} e^{6} x^{3}+4950 A a \,b^{4} d \,e^{5} x^{3}+825 A \,b^{5} d^{2} e^{4} x^{3}+17325 B \,a^{3} b^{2} e^{6} x^{3}+9900 B \,a^{2} b^{3} d \,e^{5} x^{3}+4125 B a \,b^{4} d^{2} e^{4} x^{3}+990 B \,b^{5} d^{3} e^{3} x^{3}+15400 A \,a^{3} b^{2} e^{6} x^{2}+5775 A \,a^{2} b^{3} d \,e^{5} x^{2}+1650 A a \,b^{4} d^{2} e^{4} x^{2}+275 A \,b^{5} d^{3} e^{3} x^{2}+7700 B \,a^{4} b \,e^{6} x^{2}+5775 B \,a^{3} b^{2} d \,e^{5} x^{2}+3300 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1375 B a \,b^{4} d^{3} e^{3} x^{2}+330 B \,b^{5} d^{4} e^{2} x^{2}+6930 A \,a^{4} b \,e^{6} x +3080 A \,a^{3} b^{2} d \,e^{5} x +1155 A \,a^{2} b^{3} d^{2} e^{4} x +330 A a \,b^{4} d^{3} e^{3} x +55 A \,b^{5} d^{4} e^{2} x +1386 B \,a^{5} e^{6} x +1540 B \,a^{4} b d \,e^{5} x +1155 B \,a^{3} b^{2} d^{2} e^{4} x +660 B \,a^{2} b^{3} d^{3} e^{3} x +275 B a \,b^{4} d^{4} e^{2} x +66 B \,b^{5} d^{5} e x +1260 A \,a^{5} e^{6}+630 A \,a^{4} b d \,e^{5}+280 A \,a^{3} b^{2} d^{2} e^{4}+105 A \,a^{2} b^{3} d^{3} e^{3}+30 A a \,b^{4} d^{4} e^{2}+5 A \,b^{5} d^{5} e +126 B \,a^{5} d \,e^{5}+140 B \,a^{4} b \,d^{2} e^{4}+105 B \,a^{3} b^{2} d^{3} e^{3}+60 B \,a^{2} b^{3} d^{4} e^{2}+25 B a \,b^{4} d^{5} e +6 b^{5} B \,d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{13860 e^{7} \left (e x +d \right )^{11} \left (b x +a \right )^{5}}\)	\(689\)
default	\(-\frac {\left (2772 B \,b^{5} e^{6} x^{6}+2310 A \,b^{5} e^{6} x^{5}+11550 B a \,b^{4} e^{6} x^{5}+2772 B \,b^{5} d \,e^{5} x^{5}+9900 A a \,b^{4} e^{6} x^{4}+1650 A \,b^{5} d \,e^{5} x^{4}+19800 B \,a^{2} b^{3} e^{6} x^{4}+8250 B a \,b^{4} d \,e^{5} x^{4}+1980 B \,b^{5} d^{2} e^{4} x^{4}+17325 A \,a^{2} b^{3} e^{6} x^{3}+4950 A a \,b^{4} d \,e^{5} x^{3}+825 A \,b^{5} d^{2} e^{4} x^{3}+17325 B \,a^{3} b^{2} e^{6} x^{3}+9900 B \,a^{2} b^{3} d \,e^{5} x^{3}+4125 B a \,b^{4} d^{2} e^{4} x^{3}+990 B \,b^{5} d^{3} e^{3} x^{3}+15400 A \,a^{3} b^{2} e^{6} x^{2}+5775 A \,a^{2} b^{3} d \,e^{5} x^{2}+1650 A a \,b^{4} d^{2} e^{4} x^{2}+275 A \,b^{5} d^{3} e^{3} x^{2}+7700 B \,a^{4} b \,e^{6} x^{2}+5775 B \,a^{3} b^{2} d \,e^{5} x^{2}+3300 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1375 B a \,b^{4} d^{3} e^{3} x^{2}+330 B \,b^{5} d^{4} e^{2} x^{2}+6930 A \,a^{4} b \,e^{6} x +3080 A \,a^{3} b^{2} d \,e^{5} x +1155 A \,a^{2} b^{3} d^{2} e^{4} x +330 A a \,b^{4} d^{3} e^{3} x +55 A \,b^{5} d^{4} e^{2} x +1386 B \,a^{5} e^{6} x +1540 B \,a^{4} b d \,e^{5} x +1155 B \,a^{3} b^{2} d^{2} e^{4} x +660 B \,a^{2} b^{3} d^{3} e^{3} x +275 B a \,b^{4} d^{4} e^{2} x +66 B \,b^{5} d^{5} e x +1260 A \,a^{5} e^{6}+630 A \,a^{4} b d \,e^{5}+280 A \,a^{3} b^{2} d^{2} e^{4}+105 A \,a^{2} b^{3} d^{3} e^{3}+30 A a \,b^{4} d^{4} e^{2}+5 A \,b^{5} d^{5} e +126 B \,a^{5} d \,e^{5}+140 B \,a^{4} b \,d^{2} e^{4}+105 B \,a^{3} b^{2} d^{3} e^{3}+60 B \,a^{2} b^{3} d^{4} e^{2}+25 B a \,b^{4} d^{5} e +6 b^{5} B \,d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{13860 e^{7} \left (e x +d \right )^{11} \left (b x +a \right )^{5}}\)	\(689\)
orering	\(-\frac {\left (2772 B \,b^{5} e^{6} x^{6}+2310 A \,b^{5} e^{6} x^{5}+11550 B a \,b^{4} e^{6} x^{5}+2772 B \,b^{5} d \,e^{5} x^{5}+9900 A a \,b^{4} e^{6} x^{4}+1650 A \,b^{5} d \,e^{5} x^{4}+19800 B \,a^{2} b^{3} e^{6} x^{4}+8250 B a \,b^{4} d \,e^{5} x^{4}+1980 B \,b^{5} d^{2} e^{4} x^{4}+17325 A \,a^{2} b^{3} e^{6} x^{3}+4950 A a \,b^{4} d \,e^{5} x^{3}+825 A \,b^{5} d^{2} e^{4} x^{3}+17325 B \,a^{3} b^{2} e^{6} x^{3}+9900 B \,a^{2} b^{3} d \,e^{5} x^{3}+4125 B a \,b^{4} d^{2} e^{4} x^{3}+990 B \,b^{5} d^{3} e^{3} x^{3}+15400 A \,a^{3} b^{2} e^{6} x^{2}+5775 A \,a^{2} b^{3} d \,e^{5} x^{2}+1650 A a \,b^{4} d^{2} e^{4} x^{2}+275 A \,b^{5} d^{3} e^{3} x^{2}+7700 B \,a^{4} b \,e^{6} x^{2}+5775 B \,a^{3} b^{2} d \,e^{5} x^{2}+3300 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1375 B a \,b^{4} d^{3} e^{3} x^{2}+330 B \,b^{5} d^{4} e^{2} x^{2}+6930 A \,a^{4} b \,e^{6} x +3080 A \,a^{3} b^{2} d \,e^{5} x +1155 A \,a^{2} b^{3} d^{2} e^{4} x +330 A a \,b^{4} d^{3} e^{3} x +55 A \,b^{5} d^{4} e^{2} x +1386 B \,a^{5} e^{6} x +1540 B \,a^{4} b d \,e^{5} x +1155 B \,a^{3} b^{2} d^{2} e^{4} x +660 B \,a^{2} b^{3} d^{3} e^{3} x +275 B a \,b^{4} d^{4} e^{2} x +66 B \,b^{5} d^{5} e x +1260 A \,a^{5} e^{6}+630 A \,a^{4} b d \,e^{5}+280 A \,a^{3} b^{2} d^{2} e^{4}+105 A \,a^{2} b^{3} d^{3} e^{3}+30 A a \,b^{4} d^{4} e^{2}+5 A \,b^{5} d^{5} e +126 B \,a^{5} d \,e^{5}+140 B \,a^{4} b \,d^{2} e^{4}+105 B \,a^{3} b^{2} d^{3} e^{3}+60 B \,a^{2} b^{3} d^{4} e^{2}+25 B a \,b^{4} d^{5} e +6 b^{5} B \,d^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{13860 e^{7} \left (b x +a \right )^{5} \left (e x +d \right )^{11}}\)	\(698\)

Input:

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x,method=_RETURNVERBOSE 
)
 

Output:

((b*x+a)^2)^(1/2)/(b*x+a)*(-1/5*b^5*B/e*x^6-1/30*b^4/e^2*(5*A*b*e+25*B*a*e 
+6*B*b*d)*x^5-1/42/e^3*b^3*(30*A*a*b*e^2+5*A*b^2*d*e+60*B*a^2*e^2+25*B*a*b 
*d*e+6*B*b^2*d^2)*x^4-1/84*b^2/e^4*(105*A*a^2*b*e^3+30*A*a*b^2*d*e^2+5*A*b 
^3*d^2*e+105*B*a^3*e^3+60*B*a^2*b*d*e^2+25*B*a*b^2*d^2*e+6*B*b^3*d^3)*x^3- 
1/252*b/e^5*(280*A*a^3*b*e^4+105*A*a^2*b^2*d*e^3+30*A*a*b^3*d^2*e^2+5*A*b^ 
4*d^3*e+140*B*a^4*e^4+105*B*a^3*b*d*e^3+60*B*a^2*b^2*d^2*e^2+25*B*a*b^3*d^ 
3*e+6*B*b^4*d^4)*x^2-1/1260/e^6*(630*A*a^4*b*e^5+280*A*a^3*b^2*d*e^4+105*A 
*a^2*b^3*d^2*e^3+30*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+126*B*a^5*e^5+140*B*a^4* 
b*d*e^4+105*B*a^3*b^2*d^2*e^3+60*B*a^2*b^3*d^3*e^2+25*B*a*b^4*d^4*e+6*B*b^ 
5*d^5)*x-1/13860/e^7*(1260*A*a^5*e^6+630*A*a^4*b*d*e^5+280*A*a^3*b^2*d^2*e 
^4+105*A*a^2*b^3*d^3*e^3+30*A*a*b^4*d^4*e^2+5*A*b^5*d^5*e+126*B*a^5*d*e^5+ 
140*B*a^4*b*d^2*e^4+105*B*a^3*b^2*d^3*e^3+60*B*a^2*b^3*d^4*e^2+25*B*a*b^4* 
d^5*e+6*B*b^5*d^6))/(e*x+d)^11
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (318) = 636\).

Time = 0.10 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.97 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=-\frac {2772 \, B b^{5} e^{6} x^{6} + 6 \, B b^{5} d^{6} + 1260 \, A a^{5} e^{6} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 126 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 462 \, {\left (6 \, B b^{5} d e^{5} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 330 \, {\left (6 \, B b^{5} d^{2} e^{4} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 165 \, {\left (6 \, B b^{5} d^{3} e^{3} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 55 \, {\left (6 \, B b^{5} d^{4} e^{2} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 11 \, {\left (6 \, B b^{5} d^{5} e + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 126 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{13860 \, {\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\right )}} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x, algorithm="fri 
cas")
 

Output:

-1/13860*(2772*B*b^5*e^6*x^6 + 6*B*b^5*d^6 + 1260*A*a^5*e^6 + 5*(5*B*a*b^4 
 + A*b^5)*d^5*e + 30*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 105*(B*a^3*b^2 + A* 
a^2*b^3)*d^3*e^3 + 140*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 126*(B*a^5 + 5*A* 
a^4*b)*d*e^5 + 462*(6*B*b^5*d*e^5 + 5*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 330*( 
6*B*b^5*d^2*e^4 + 5*(5*B*a*b^4 + A*b^5)*d*e^5 + 30*(2*B*a^2*b^3 + A*a*b^4) 
*e^6)*x^4 + 165*(6*B*b^5*d^3*e^3 + 5*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 30*(2*B 
*a^2*b^3 + A*a*b^4)*d*e^5 + 105*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 55*(6*B 
*b^5*d^4*e^2 + 5*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 30*(2*B*a^2*b^3 + A*a*b^4)* 
d^2*e^4 + 105*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 140*(B*a^4*b + 2*A*a^3*b^2)* 
e^6)*x^2 + 11*(6*B*b^5*d^5*e + 5*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 30*(2*B*a^2 
*b^3 + A*a*b^4)*d^3*e^3 + 105*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 140*(B*a^4 
*b + 2*A*a^3*b^2)*d*e^5 + 126*(B*a^5 + 5*A*a^4*b)*e^6)*x)/(e^18*x^11 + 11* 
d*e^17*x^10 + 55*d^2*e^16*x^9 + 165*d^3*e^15*x^8 + 330*d^4*e^14*x^7 + 462* 
d^5*e^13*x^6 + 462*d^6*e^12*x^5 + 330*d^7*e^11*x^4 + 165*d^8*e^10*x^3 + 55 
*d^9*e^9*x^2 + 11*d^10*e^8*x + d^11*e^7)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**12,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x, algorithm="max 
ima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1076 vs. \(2 (318) = 636\).

Time = 0.19 (sec) , antiderivative size = 1076, normalized size of antiderivative = 3.15 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x, algorithm="gia 
c")
 

Output:

1/13860*(6*B*b^11*d - 11*B*a*b^10*e + 5*A*b^11*e)*sgn(b*x + a)/(b^6*d^6*e^ 
7 - 6*a*b^5*d^5*e^8 + 15*a^2*b^4*d^4*e^9 - 20*a^3*b^3*d^3*e^10 + 15*a^4*b^ 
2*d^2*e^11 - 6*a^5*b*d*e^12 + a^6*e^13) - 1/13860*(2772*B*b^5*e^6*x^6*sgn( 
b*x + a) + 2772*B*b^5*d*e^5*x^5*sgn(b*x + a) + 11550*B*a*b^4*e^6*x^5*sgn(b 
*x + a) + 2310*A*b^5*e^6*x^5*sgn(b*x + a) + 1980*B*b^5*d^2*e^4*x^4*sgn(b*x 
 + a) + 8250*B*a*b^4*d*e^5*x^4*sgn(b*x + a) + 1650*A*b^5*d*e^5*x^4*sgn(b*x 
 + a) + 19800*B*a^2*b^3*e^6*x^4*sgn(b*x + a) + 9900*A*a*b^4*e^6*x^4*sgn(b* 
x + a) + 990*B*b^5*d^3*e^3*x^3*sgn(b*x + a) + 4125*B*a*b^4*d^2*e^4*x^3*sgn 
(b*x + a) + 825*A*b^5*d^2*e^4*x^3*sgn(b*x + a) + 9900*B*a^2*b^3*d*e^5*x^3* 
sgn(b*x + a) + 4950*A*a*b^4*d*e^5*x^3*sgn(b*x + a) + 17325*B*a^3*b^2*e^6*x 
^3*sgn(b*x + a) + 17325*A*a^2*b^3*e^6*x^3*sgn(b*x + a) + 330*B*b^5*d^4*e^2 
*x^2*sgn(b*x + a) + 1375*B*a*b^4*d^3*e^3*x^2*sgn(b*x + a) + 275*A*b^5*d^3* 
e^3*x^2*sgn(b*x + a) + 3300*B*a^2*b^3*d^2*e^4*x^2*sgn(b*x + a) + 1650*A*a* 
b^4*d^2*e^4*x^2*sgn(b*x + a) + 5775*B*a^3*b^2*d*e^5*x^2*sgn(b*x + a) + 577 
5*A*a^2*b^3*d*e^5*x^2*sgn(b*x + a) + 7700*B*a^4*b*e^6*x^2*sgn(b*x + a) + 1 
5400*A*a^3*b^2*e^6*x^2*sgn(b*x + a) + 66*B*b^5*d^5*e*x*sgn(b*x + a) + 275* 
B*a*b^4*d^4*e^2*x*sgn(b*x + a) + 55*A*b^5*d^4*e^2*x*sgn(b*x + a) + 660*B*a 
^2*b^3*d^3*e^3*x*sgn(b*x + a) + 330*A*a*b^4*d^3*e^3*x*sgn(b*x + a) + 1155* 
B*a^3*b^2*d^2*e^4*x*sgn(b*x + a) + 1155*A*a^2*b^3*d^2*e^4*x*sgn(b*x + a) + 
 1540*B*a^4*b*d*e^5*x*sgn(b*x + a) + 3080*A*a^3*b^2*d*e^5*x*sgn(b*x + a...
 

Mupad [B] (verification not implemented)

Time = 11.66 (sec) , antiderivative size = 1489, normalized size of antiderivative = 4.35 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^12,x)
 

Output:

- (((10*B*b^5*d^2 - 4*A*b^5*d*e + 5*A*a*b^4*e^2 + 10*B*a^2*b^3*e^2 - 20*B* 
a*b^4*d*e)/(7*e^7) - (d*((b^4*(A*b*e + 5*B*a*e - 4*B*b*d))/(7*e^6) - (B*b^ 
5*d)/(7*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) 
 - (((A*b^5*e - 5*B*b^5*d + 5*B*a*b^4*e)/(6*e^7) - (B*b^5*d)/(6*e^7))*(a^2 
 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) - (((A*a^5)/(11*e) - 
(d*((B*a^5 + 5*A*a^4*b)/(11*e) + (d*((d*((d*((d*((A*b^5 + 5*B*a*b^4)/(11*e 
) - (B*b^5*d)/(11*e^2)))/e - (5*a*b^3*(A*b + 2*B*a))/(11*e)))/e + (10*a^2* 
b^2*(A*b + B*a))/(11*e)))/e - (5*a^3*b*(2*A*b + B*a))/(11*e)))/e))/e)*(a^2 
 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^11) - (((6*A*b^5*d^2*e - 
 10*B*b^5*d^3 + 10*A*a^2*b^3*e^3 + 10*B*a^3*b^2*e^3 - 30*B*a^2*b^3*d*e^2 - 
 15*A*a*b^4*d*e^2 + 30*B*a*b^4*d^2*e)/(8*e^7) - (d*((5*A*a*b^4*e^3 - 3*A*b 
^5*d*e^2 + 6*B*b^5*d^2*e + 10*B*a^2*b^3*e^3 - 15*B*a*b^4*d*e^2)/(8*e^7) - 
(d*((b^4*(A*b*e + 5*B*a*e - 3*B*b*d))/(8*e^5) - (B*b^5*d)/(8*e^5)))/e))/e) 
*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) - (((B*a^5*e^5 - 
 B*b^5*d^5 + 5*A*a^4*b*e^5 + A*b^5*d^4*e - 5*A*a*b^4*d^3*e^2 - 10*A*a^3*b^ 
2*d*e^4 + 10*A*a^2*b^3*d^2*e^3 - 10*B*a^2*b^3*d^3*e^2 + 10*B*a^3*b^2*d^2*e 
^3 + 5*B*a*b^4*d^4*e - 5*B*a^4*b*d*e^4)/(10*e^7) - (d*((5*B*a^4*b*e^5 + B* 
b^5*d^4*e + 10*A*a^3*b^2*e^5 - A*b^5*d^3*e^2 + 5*A*a*b^4*d^2*e^3 - 10*A*a^ 
2*b^3*d*e^4 - 5*B*a*b^4*d^3*e^2 - 10*B*a^3*b^2*d*e^4 + 10*B*a^2*b^3*d^2*e^ 
3)/(10*e^7) - (d*((10*A*a^2*b^3*e^5 + 10*B*a^3*b^2*e^5 + A*b^5*d^2*e^3 ...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\frac {-462 b^{6} e^{6} x^{6}-2310 a \,b^{5} e^{6} x^{5}-462 b^{6} d \,e^{5} x^{5}-4950 a^{2} b^{4} e^{6} x^{4}-1650 a \,b^{5} d \,e^{5} x^{4}-330 b^{6} d^{2} e^{4} x^{4}-5775 a^{3} b^{3} e^{6} x^{3}-2475 a^{2} b^{4} d \,e^{5} x^{3}-825 a \,b^{5} d^{2} e^{4} x^{3}-165 b^{6} d^{3} e^{3} x^{3}-3850 a^{4} b^{2} e^{6} x^{2}-1925 a^{3} b^{3} d \,e^{5} x^{2}-825 a^{2} b^{4} d^{2} e^{4} x^{2}-275 a \,b^{5} d^{3} e^{3} x^{2}-55 b^{6} d^{4} e^{2} x^{2}-1386 a^{5} b \,e^{6} x -770 a^{4} b^{2} d \,e^{5} x -385 a^{3} b^{3} d^{2} e^{4} x -165 a^{2} b^{4} d^{3} e^{3} x -55 a \,b^{5} d^{4} e^{2} x -11 b^{6} d^{5} e x -210 a^{6} e^{6}-126 a^{5} b d \,e^{5}-70 a^{4} b^{2} d^{2} e^{4}-35 a^{3} b^{3} d^{3} e^{3}-15 a^{2} b^{4} d^{4} e^{2}-5 a \,b^{5} d^{5} e -b^{6} d^{6}}{2310 e^{7} \left (e^{11} x^{11}+11 d \,e^{10} x^{10}+55 d^{2} e^{9} x^{9}+165 d^{3} e^{8} x^{8}+330 d^{4} e^{7} x^{7}+462 d^{5} e^{6} x^{6}+462 d^{6} e^{5} x^{5}+330 d^{7} e^{4} x^{4}+165 d^{8} e^{3} x^{3}+55 d^{9} e^{2} x^{2}+11 d^{10} e x +d^{11}\right )} \] Input:

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x)
 

Output:

( - 210*a**6*e**6 - 126*a**5*b*d*e**5 - 1386*a**5*b*e**6*x - 70*a**4*b**2* 
d**2*e**4 - 770*a**4*b**2*d*e**5*x - 3850*a**4*b**2*e**6*x**2 - 35*a**3*b* 
*3*d**3*e**3 - 385*a**3*b**3*d**2*e**4*x - 1925*a**3*b**3*d*e**5*x**2 - 57 
75*a**3*b**3*e**6*x**3 - 15*a**2*b**4*d**4*e**2 - 165*a**2*b**4*d**3*e**3* 
x - 825*a**2*b**4*d**2*e**4*x**2 - 2475*a**2*b**4*d*e**5*x**3 - 4950*a**2* 
b**4*e**6*x**4 - 5*a*b**5*d**5*e - 55*a*b**5*d**4*e**2*x - 275*a*b**5*d**3 
*e**3*x**2 - 825*a*b**5*d**2*e**4*x**3 - 1650*a*b**5*d*e**5*x**4 - 2310*a* 
b**5*e**6*x**5 - b**6*d**6 - 11*b**6*d**5*e*x - 55*b**6*d**4*e**2*x**2 - 1 
65*b**6*d**3*e**3*x**3 - 330*b**6*d**2*e**4*x**4 - 462*b**6*d*e**5*x**5 - 
462*b**6*e**6*x**6)/(2310*e**7*(d**11 + 11*d**10*e*x + 55*d**9*e**2*x**2 + 
 165*d**8*e**3*x**3 + 330*d**7*e**4*x**4 + 462*d**6*e**5*x**5 + 462*d**5*e 
**6*x**6 + 330*d**4*e**7*x**7 + 165*d**3*e**8*x**8 + 55*d**2*e**9*x**9 + 1 
1*d*e**10*x**10 + e**11*x**11))