\(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^{14}} \, dx\) [187]

Optimal result

Integrand size = 27, antiderivative size = 143 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{14}} \, dx=-\frac {a^6 A}{13 x^{13}}-\frac {a^5 (6 A b+a B)}{12 x^{12}}-\frac {3 a^4 b (5 A b+2 a B)}{11 x^{11}}-\frac {a^3 b^2 (4 A b+3 a B)}{2 x^{10}}-\frac {5 a^2 b^3 (3 A b+4 a B)}{9 x^9}-\frac {3 a b^4 (2 A b+5 a B)}{8 x^8}-\frac {b^5 (A b+6 a B)}{7 x^7}-\frac {b^6 B}{6 x^6} \] Output:

-1/13*a^6*A/x^13-1/12*a^5*(6*A*b+B*a)/x^12-3/11*a^4*b*(5*A*b+2*B*a)/x^11-1 
/2*a^3*b^2*(4*A*b+3*B*a)/x^10-5/9*a^2*b^3*(3*A*b+4*B*a)/x^9-3/8*a*b^4*(2*A 
*b+5*B*a)/x^8-1/7*b^5*(A*b+6*B*a)/x^7-1/6*b^6*B/x^6
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{14}} \, dx=-\frac {1716 b^6 x^6 (6 A+7 B x)+7722 a b^5 x^5 (7 A+8 B x)+15015 a^2 b^4 x^4 (8 A+9 B x)+16016 a^3 b^3 x^3 (9 A+10 B x)+9828 a^4 b^2 x^2 (10 A+11 B x)+3276 a^5 b x (11 A+12 B x)+462 a^6 (12 A+13 B x)}{72072 x^{13}} \] Input:

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^14,x]
 

Output:

-1/72072*(1716*b^6*x^6*(6*A + 7*B*x) + 7722*a*b^5*x^5*(7*A + 8*B*x) + 1501 
5*a^2*b^4*x^4*(8*A + 9*B*x) + 16016*a^3*b^3*x^3*(9*A + 10*B*x) + 9828*a^4* 
b^2*x^2*(10*A + 11*B*x) + 3276*a^5*b*x*(11*A + 12*B*x) + 462*a^6*(12*A + 1 
3*B*x))/x^13
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1184, 27, 85, 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.

\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x)}{x^{14}} \, dx\)

\(\Big \downarrow \) 1184

\(\displaystyle \frac {\int \frac {b^6 (a+b x)^6 (A+B x)}{x^{14}}dx}{b^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{x^{14}}dx\)

\(\Big \downarrow \) 85

\(\displaystyle \int \left (\frac {a^6 A}{x^{14}}+\frac {a^5 (a B+6 A b)}{x^{13}}+\frac {3 a^4 b (2 a B+5 A b)}{x^{12}}+\frac {5 a^3 b^2 (3 a B+4 A b)}{x^{11}}+\frac {5 a^2 b^3 (4 a B+3 A b)}{x^{10}}+\frac {b^5 (6 a B+A b)}{x^8}+\frac {3 a b^4 (5 a B+2 A b)}{x^9}+\frac {b^6 B}{x^7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^6 A}{13 x^{13}}-\frac {a^5 (a B+6 A b)}{12 x^{12}}-\frac {3 a^4 b (2 a B+5 A b)}{11 x^{11}}-\frac {a^3 b^2 (3 a B+4 A b)}{2 x^{10}}-\frac {5 a^2 b^3 (4 a B+3 A b)}{9 x^9}-\frac {b^5 (6 a B+A b)}{7 x^7}-\frac {3 a b^4 (5 a B+2 A b)}{8 x^8}-\frac {b^6 B}{6 x^6}\)

Input:

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^14,x]
 

Output:

-1/13*(a^6*A)/x^13 - (a^5*(6*A*b + a*B))/(12*x^12) - (3*a^4*b*(5*A*b + 2*a 
*B))/(11*x^11) - (a^3*b^2*(4*A*b + 3*a*B))/(2*x^10) - (5*a^2*b^3*(3*A*b + 
4*a*B))/(9*x^9) - (3*a*b^4*(2*A*b + 5*a*B))/(8*x^8) - (b^5*(A*b + 6*a*B))/ 
(7*x^7) - (b^6*B)/(6*x^6)
 

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 85

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : 
> Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, 
 d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* 
f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n 
 + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 
1])
 

rule 1184

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^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}, x] && E 
qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

rule 2009

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

Maple [A] (warning: unable to verify)

Time = 0.93 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90

method	result	size
default	\(-\frac {a^{6} A}{13 x^{13}}-\frac {a^{5} \left (6 A b +B a \right )}{12 x^{12}}-\frac {3 a^{4} b \left (5 A b +2 B a \right )}{11 x^{11}}-\frac {a^{3} b^{2} \left (4 A b +3 B a \right )}{2 x^{10}}-\frac {5 a^{2} b^{3} \left (3 A b +4 B a \right )}{9 x^{9}}-\frac {3 a \,b^{4} \left (2 A b +5 B a \right )}{8 x^{8}}-\frac {b^{5} \left (A b +6 B a \right )}{7 x^{7}}-\frac {b^{6} B}{6 x^{6}}\)	\(128\)
norman	\(\frac {-\frac {b^{6} B \,x^{7}}{6}+\left (-\frac {1}{7} A \,b^{6}-\frac {6}{7} B a \,b^{5}\right ) x^{6}+\left (-\frac {3}{4} A a \,b^{5}-\frac {15}{8} B \,a^{2} b^{4}\right ) x^{5}+\left (-\frac {5}{3} A \,a^{2} b^{4}-\frac {20}{9} B \,a^{3} b^{3}\right ) x^{4}+\left (-2 A \,a^{3} b^{3}-\frac {3}{2} B \,a^{4} b^{2}\right ) x^{3}+\left (-\frac {15}{11} A \,a^{4} b^{2}-\frac {6}{11} B \,a^{5} b \right ) x^{2}+\left (-\frac {1}{2} A \,a^{5} b -\frac {1}{12} B \,a^{6}\right ) x -\frac {A \,a^{6}}{13}}{x^{13}}\)	\(143\)
risch	\(\frac {-\frac {b^{6} B \,x^{7}}{6}+\left (-\frac {1}{7} A \,b^{6}-\frac {6}{7} B a \,b^{5}\right ) x^{6}+\left (-\frac {3}{4} A a \,b^{5}-\frac {15}{8} B \,a^{2} b^{4}\right ) x^{5}+\left (-\frac {5}{3} A \,a^{2} b^{4}-\frac {20}{9} B \,a^{3} b^{3}\right ) x^{4}+\left (-2 A \,a^{3} b^{3}-\frac {3}{2} B \,a^{4} b^{2}\right ) x^{3}+\left (-\frac {15}{11} A \,a^{4} b^{2}-\frac {6}{11} B \,a^{5} b \right ) x^{2}+\left (-\frac {1}{2} A \,a^{5} b -\frac {1}{12} B \,a^{6}\right ) x -\frac {A \,a^{6}}{13}}{x^{13}}\)	\(143\)
gosper	\(-\frac {12012 b^{6} B \,x^{7}+10296 A \,b^{6} x^{6}+61776 B a \,b^{5} x^{6}+54054 A a \,b^{5} x^{5}+135135 B \,a^{2} b^{4} x^{5}+120120 A \,a^{2} b^{4} x^{4}+160160 B \,a^{3} b^{3} x^{4}+144144 A \,a^{3} b^{3} x^{3}+108108 B \,a^{4} b^{2} x^{3}+98280 A \,a^{4} b^{2} x^{2}+39312 B \,a^{5} b \,x^{2}+36036 A \,a^{5} b x +6006 B \,a^{6} x +5544 A \,a^{6}}{72072 x^{13}}\)	\(148\)
parallelrisch	\(-\frac {12012 b^{6} B \,x^{7}+10296 A \,b^{6} x^{6}+61776 B a \,b^{5} x^{6}+54054 A a \,b^{5} x^{5}+135135 B \,a^{2} b^{4} x^{5}+120120 A \,a^{2} b^{4} x^{4}+160160 B \,a^{3} b^{3} x^{4}+144144 A \,a^{3} b^{3} x^{3}+108108 B \,a^{4} b^{2} x^{3}+98280 A \,a^{4} b^{2} x^{2}+39312 B \,a^{5} b \,x^{2}+36036 A \,a^{5} b x +6006 B \,a^{6} x +5544 A \,a^{6}}{72072 x^{13}}\)	\(148\)
orering	\(-\frac {\left (12012 b^{6} B \,x^{7}+10296 A \,b^{6} x^{6}+61776 B a \,b^{5} x^{6}+54054 A a \,b^{5} x^{5}+135135 B \,a^{2} b^{4} x^{5}+120120 A \,a^{2} b^{4} x^{4}+160160 B \,a^{3} b^{3} x^{4}+144144 A \,a^{3} b^{3} x^{3}+108108 B \,a^{4} b^{2} x^{3}+98280 A \,a^{4} b^{2} x^{2}+39312 B \,a^{5} b \,x^{2}+36036 A \,a^{5} b x +6006 B \,a^{6} x +5544 A \,a^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}{72072 x^{13} \left (b x +a \right )^{6}}\)	\(173\)

Input:

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^14,x,method=_RETURNVERBOSE)
 

Output:

-1/13*a^6*A/x^13-1/12*a^5*(6*A*b+B*a)/x^12-3/11*a^4*b*(5*A*b+2*B*a)/x^11-1 
/2*a^3*b^2*(4*A*b+3*B*a)/x^10-5/9*a^2*b^3*(3*A*b+4*B*a)/x^9-3/8*a*b^4*(2*A 
*b+5*B*a)/x^8-1/7*b^5*(A*b+6*B*a)/x^7-1/6*b^6*B/x^6
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{14}} \, dx=-\frac {12012 \, B b^{6} x^{7} + 5544 \, A a^{6} + 10296 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 27027 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 40040 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 36036 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 19656 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 6006 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{72072 \, x^{13}} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^14,x, algorithm="fricas")
 

Output:

-1/72072*(12012*B*b^6*x^7 + 5544*A*a^6 + 10296*(6*B*a*b^5 + A*b^6)*x^6 + 2 
7027*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 40040*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 
 + 36036*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 19656*(2*B*a^5*b + 5*A*a^4*b^2) 
*x^2 + 6006*(B*a^6 + 6*A*a^5*b)*x)/x^13
 

Sympy [A] (verification not implemented)

Time = 10.15 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{14}} \, dx=\frac {- 5544 A a^{6} - 12012 B b^{6} x^{7} + x^{6} \left (- 10296 A b^{6} - 61776 B a b^{5}\right ) + x^{5} \left (- 54054 A a b^{5} - 135135 B a^{2} b^{4}\right ) + x^{4} \left (- 120120 A a^{2} b^{4} - 160160 B a^{3} b^{3}\right ) + x^{3} \left (- 144144 A a^{3} b^{3} - 108108 B a^{4} b^{2}\right ) + x^{2} \left (- 98280 A a^{4} b^{2} - 39312 B a^{5} b\right ) + x \left (- 36036 A a^{5} b - 6006 B a^{6}\right )}{72072 x^{13}} \] Input:

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x**14,x)
 

Output:

(-5544*A*a**6 - 12012*B*b**6*x**7 + x**6*(-10296*A*b**6 - 61776*B*a*b**5) 
+ x**5*(-54054*A*a*b**5 - 135135*B*a**2*b**4) + x**4*(-120120*A*a**2*b**4 
- 160160*B*a**3*b**3) + x**3*(-144144*A*a**3*b**3 - 108108*B*a**4*b**2) + 
x**2*(-98280*A*a**4*b**2 - 39312*B*a**5*b) + x*(-36036*A*a**5*b - 6006*B*a 
**6))/(72072*x**13)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{14}} \, dx=-\frac {12012 \, B b^{6} x^{7} + 5544 \, A a^{6} + 10296 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 27027 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 40040 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 36036 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 19656 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 6006 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{72072 \, x^{13}} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^14,x, algorithm="maxima")
 

Output:

-1/72072*(12012*B*b^6*x^7 + 5544*A*a^6 + 10296*(6*B*a*b^5 + A*b^6)*x^6 + 2 
7027*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 40040*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 
 + 36036*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 19656*(2*B*a^5*b + 5*A*a^4*b^2) 
*x^2 + 6006*(B*a^6 + 6*A*a^5*b)*x)/x^13
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{14}} \, dx=-\frac {12012 \, B b^{6} x^{7} + 61776 \, B a b^{5} x^{6} + 10296 \, A b^{6} x^{6} + 135135 \, B a^{2} b^{4} x^{5} + 54054 \, A a b^{5} x^{5} + 160160 \, B a^{3} b^{3} x^{4} + 120120 \, A a^{2} b^{4} x^{4} + 108108 \, B a^{4} b^{2} x^{3} + 144144 \, A a^{3} b^{3} x^{3} + 39312 \, B a^{5} b x^{2} + 98280 \, A a^{4} b^{2} x^{2} + 6006 \, B a^{6} x + 36036 \, A a^{5} b x + 5544 \, A a^{6}}{72072 \, x^{13}} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^14,x, algorithm="giac")
 

Output:

-1/72072*(12012*B*b^6*x^7 + 61776*B*a*b^5*x^6 + 10296*A*b^6*x^6 + 135135*B 
*a^2*b^4*x^5 + 54054*A*a*b^5*x^5 + 160160*B*a^3*b^3*x^4 + 120120*A*a^2*b^4 
*x^4 + 108108*B*a^4*b^2*x^3 + 144144*A*a^3*b^3*x^3 + 39312*B*a^5*b*x^2 + 9 
8280*A*a^4*b^2*x^2 + 6006*B*a^6*x + 36036*A*a^5*b*x + 5544*A*a^6)/x^13
 

Mupad [B] (verification not implemented)

Time = 10.84 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{14}} \, dx=-\frac {x\,\left (\frac {B\,a^6}{12}+\frac {A\,b\,a^5}{2}\right )+\frac {A\,a^6}{13}+x^5\,\left (\frac {15\,B\,a^2\,b^4}{8}+\frac {3\,A\,a\,b^5}{4}\right )+x^2\,\left (\frac {6\,B\,a^5\,b}{11}+\frac {15\,A\,a^4\,b^2}{11}\right )+x^6\,\left (\frac {A\,b^6}{7}+\frac {6\,B\,a\,b^5}{7}\right )+x^3\,\left (\frac {3\,B\,a^4\,b^2}{2}+2\,A\,a^3\,b^3\right )+x^4\,\left (\frac {20\,B\,a^3\,b^3}{9}+\frac {5\,A\,a^2\,b^4}{3}\right )+\frac {B\,b^6\,x^7}{6}}{x^{13}} \] Input:

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3)/x^14,x)
 

Output:

-(x*((B*a^6)/12 + (A*a^5*b)/2) + (A*a^6)/13 + x^5*((15*B*a^2*b^4)/8 + (3*A 
*a*b^5)/4) + x^2*((15*A*a^4*b^2)/11 + (6*B*a^5*b)/11) + x^6*((A*b^6)/7 + ( 
6*B*a*b^5)/7) + x^3*(2*A*a^3*b^3 + (3*B*a^4*b^2)/2) + x^4*((5*A*a^2*b^4)/3 
 + (20*B*a^3*b^3)/9) + (B*b^6*x^7)/6)/x^13
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.55 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{14}} \, dx=\frac {-1716 b^{7} x^{7}-10296 a \,b^{6} x^{6}-27027 a^{2} b^{5} x^{5}-40040 a^{3} b^{4} x^{4}-36036 a^{4} b^{3} x^{3}-19656 a^{5} b^{2} x^{2}-6006 a^{6} b x -792 a^{7}}{10296 x^{13}} \] Input:

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^14,x)
 

Output:

( - 792*a**7 - 6006*a**6*b*x - 19656*a**5*b**2*x**2 - 36036*a**4*b**3*x**3 
 - 40040*a**3*b**4*x**4 - 27027*a**2*b**5*x**5 - 10296*a*b**6*x**6 - 1716* 
b**7*x**7)/(10296*x**13)