\(\int \frac {(A+B x) (a+b x+c x^2)^3}{x^{11}} \, dx\) [38]

Optimal result

Integrand size = 21, antiderivative size = 166 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11}} \, dx=-\frac {a^3 A}{10 x^{10}}-\frac {a^2 (3 A b+a B)}{9 x^9}-\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{8 x^8}-\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{7 x^7}-\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{6 x^6}-\frac {3 c \left (b^2 B+A b c+a B c\right )}{5 x^5}-\frac {c^2 (3 b B+A c)}{4 x^4}-\frac {B c^3}{3 x^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.06 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11}} \, dx=-\frac {28 a^3 (9 A+10 B x)+15 a^2 x (9 B x (7 b+8 c x)+7 A (8 b+9 c x))+9 a x^2 \left (8 B x \left (15 b^2+35 b c x+21 c^2 x^2\right )+5 A \left (21 b^2+48 b c x+28 c^2 x^2\right )\right )+6 x^3 \left (7 B x \left (10 b^3+36 b^2 c x+45 b c^2 x^2+20 c^3 x^3\right )+3 A \left (20 b^3+70 b^2 c x+84 b c^2 x^2+35 c^3 x^3\right )\right )}{2520 x^{10}} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^11,x]
 

Output:

-1/2520*(28*a^3*(9*A + 10*B*x) + 15*a^2*x*(9*B*x*(7*b + 8*c*x) + 7*A*(8*b 
+ 9*c*x)) + 9*a*x^2*(8*B*x*(15*b^2 + 35*b*c*x + 21*c^2*x^2) + 5*A*(21*b^2 
+ 48*b*c*x + 28*c^2*x^2)) + 6*x^3*(7*B*x*(10*b^3 + 36*b^2*c*x + 45*b*c^2*x 
^2 + 20*c^3*x^3) + 3*A*(20*b^3 + 70*b^2*c*x + 84*b*c^2*x^2 + 35*c^3*x^3))) 
/x^10
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1195, 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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11}} \, dx\)

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^11,x]
 

Output:

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

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

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

Maple [A] (verified)

Time = 0.96 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.93

method	result	size
default	\(-\frac {3 c \left (A b c +a B c +B \,b^{2}\right )}{5 x^{5}}-\frac {3 A a \,c^{2}+3 A \,b^{2} c +6 B a b c +B \,b^{3}}{6 x^{6}}-\frac {B \,c^{3}}{3 x^{3}}-\frac {3 a \left (A a c +b^{2} A +a b B \right )}{8 x^{8}}-\frac {c^{2} \left (A c +3 B b \right )}{4 x^{4}}-\frac {a^{2} \left (3 A b +B a \right )}{9 x^{9}}-\frac {6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}}{7 x^{7}}-\frac {a^{3} A}{10 x^{10}}\)	\(154\)
norman	\(\frac {-\frac {B \,c^{3} x^{7}}{3}+\left (-\frac {1}{4} A \,c^{3}-\frac {3}{4} B b \,c^{2}\right ) x^{6}+\left (-\frac {3}{5} A b \,c^{2}-\frac {3}{5} B a \,c^{2}-\frac {3}{5} B \,b^{2} c \right ) x^{5}+\left (-\frac {1}{2} A a \,c^{2}-\frac {1}{2} A \,b^{2} c -B a b c -\frac {1}{6} B \,b^{3}\right ) x^{4}+\left (-\frac {6}{7} A a b c -\frac {1}{7} A \,b^{3}-\frac {3}{7} B \,a^{2} c -\frac {3}{7} B a \,b^{2}\right ) x^{3}+\left (-\frac {3}{8} a^{2} A c -\frac {3}{8} A a \,b^{2}-\frac {3}{8} B \,a^{2} b \right ) x^{2}+\left (-\frac {1}{3} A \,a^{2} b -\frac {1}{9} B \,a^{3}\right ) x -\frac {a^{3} A}{10}}{x^{10}}\)	\(169\)
risch	\(\frac {-\frac {B \,c^{3} x^{7}}{3}+\left (-\frac {1}{4} A \,c^{3}-\frac {3}{4} B b \,c^{2}\right ) x^{6}+\left (-\frac {3}{5} A b \,c^{2}-\frac {3}{5} B a \,c^{2}-\frac {3}{5} B \,b^{2} c \right ) x^{5}+\left (-\frac {1}{2} A a \,c^{2}-\frac {1}{2} A \,b^{2} c -B a b c -\frac {1}{6} B \,b^{3}\right ) x^{4}+\left (-\frac {6}{7} A a b c -\frac {1}{7} A \,b^{3}-\frac {3}{7} B \,a^{2} c -\frac {3}{7} B a \,b^{2}\right ) x^{3}+\left (-\frac {3}{8} a^{2} A c -\frac {3}{8} A a \,b^{2}-\frac {3}{8} B \,a^{2} b \right ) x^{2}+\left (-\frac {1}{3} A \,a^{2} b -\frac {1}{9} B \,a^{3}\right ) x -\frac {a^{3} A}{10}}{x^{10}}\)	\(169\)
gosper	\(-\frac {840 B \,c^{3} x^{7}+630 A \,c^{3} x^{6}+1890 B b \,c^{2} x^{6}+1512 A b \,c^{2} x^{5}+1512 B a \,c^{2} x^{5}+1512 B \,b^{2} c \,x^{5}+1260 A a \,c^{2} x^{4}+1260 A \,b^{2} c \,x^{4}+2520 B a b c \,x^{4}+420 B \,b^{3} x^{4}+2160 A a b c \,x^{3}+360 A \,b^{3} x^{3}+1080 B \,a^{2} c \,x^{3}+1080 B a \,b^{2} x^{3}+945 A \,a^{2} c \,x^{2}+945 A a \,b^{2} x^{2}+945 B \,a^{2} b \,x^{2}+840 A \,a^{2} b x +280 B \,a^{3} x +252 a^{3} A}{2520 x^{10}}\)	\(192\)
parallelrisch	\(-\frac {840 B \,c^{3} x^{7}+630 A \,c^{3} x^{6}+1890 B b \,c^{2} x^{6}+1512 A b \,c^{2} x^{5}+1512 B a \,c^{2} x^{5}+1512 B \,b^{2} c \,x^{5}+1260 A a \,c^{2} x^{4}+1260 A \,b^{2} c \,x^{4}+2520 B a b c \,x^{4}+420 B \,b^{3} x^{4}+2160 A a b c \,x^{3}+360 A \,b^{3} x^{3}+1080 B \,a^{2} c \,x^{3}+1080 B a \,b^{2} x^{3}+945 A \,a^{2} c \,x^{2}+945 A a \,b^{2} x^{2}+945 B \,a^{2} b \,x^{2}+840 A \,a^{2} b x +280 B \,a^{3} x +252 a^{3} A}{2520 x^{10}}\)	\(192\)
orering	\(-\frac {840 B \,c^{3} x^{7}+630 A \,c^{3} x^{6}+1890 B b \,c^{2} x^{6}+1512 A b \,c^{2} x^{5}+1512 B a \,c^{2} x^{5}+1512 B \,b^{2} c \,x^{5}+1260 A a \,c^{2} x^{4}+1260 A \,b^{2} c \,x^{4}+2520 B a b c \,x^{4}+420 B \,b^{3} x^{4}+2160 A a b c \,x^{3}+360 A \,b^{3} x^{3}+1080 B \,a^{2} c \,x^{3}+1080 B a \,b^{2} x^{3}+945 A \,a^{2} c \,x^{2}+945 A a \,b^{2} x^{2}+945 B \,a^{2} b \,x^{2}+840 A \,a^{2} b x +280 B \,a^{3} x +252 a^{3} A}{2520 x^{10}}\)	\(192\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^3/x^11,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11}} \, dx=-\frac {840 \, B c^{3} x^{7} + 630 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1512 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 420 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 252 \, A a^{3} + 360 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 945 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 280 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{10}} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^11,x, algorithm="fricas")
 

Output:

-1/2520*(840*B*c^3*x^7 + 630*(3*B*b*c^2 + A*c^3)*x^6 + 1512*(B*b^2*c + (B* 
a + A*b)*c^2)*x^5 + 420*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 
252*A*a^3 + 360*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 945*(B*a 
^2*b + A*a*b^2 + A*a^2*c)*x^2 + 280*(B*a^3 + 3*A*a^2*b)*x)/x^10
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**3/x**11,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11}} \, dx=-\frac {840 \, B c^{3} x^{7} + 630 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1512 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 420 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 252 \, A a^{3} + 360 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 945 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 280 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{10}} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^11,x, algorithm="maxima")
 

Output:

-1/2520*(840*B*c^3*x^7 + 630*(3*B*b*c^2 + A*c^3)*x^6 + 1512*(B*b^2*c + (B* 
a + A*b)*c^2)*x^5 + 420*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 
252*A*a^3 + 360*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 945*(B*a 
^2*b + A*a*b^2 + A*a^2*c)*x^2 + 280*(B*a^3 + 3*A*a^2*b)*x)/x^10
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.15 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11}} \, dx=-\frac {840 \, B c^{3} x^{7} + 1890 \, B b c^{2} x^{6} + 630 \, A c^{3} x^{6} + 1512 \, B b^{2} c x^{5} + 1512 \, B a c^{2} x^{5} + 1512 \, A b c^{2} x^{5} + 420 \, B b^{3} x^{4} + 2520 \, B a b c x^{4} + 1260 \, A b^{2} c x^{4} + 1260 \, A a c^{2} x^{4} + 1080 \, B a b^{2} x^{3} + 360 \, A b^{3} x^{3} + 1080 \, B a^{2} c x^{3} + 2160 \, A a b c x^{3} + 945 \, B a^{2} b x^{2} + 945 \, A a b^{2} x^{2} + 945 \, A a^{2} c x^{2} + 280 \, B a^{3} x + 840 \, A a^{2} b x + 252 \, A a^{3}}{2520 \, x^{10}} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^11,x, algorithm="giac")
 

Output:

-1/2520*(840*B*c^3*x^7 + 1890*B*b*c^2*x^6 + 630*A*c^3*x^6 + 1512*B*b^2*c*x 
^5 + 1512*B*a*c^2*x^5 + 1512*A*b*c^2*x^5 + 420*B*b^3*x^4 + 2520*B*a*b*c*x^ 
4 + 1260*A*b^2*c*x^4 + 1260*A*a*c^2*x^4 + 1080*B*a*b^2*x^3 + 360*A*b^3*x^3 
 + 1080*B*a^2*c*x^3 + 2160*A*a*b*c*x^3 + 945*B*a^2*b*x^2 + 945*A*a*b^2*x^2 
 + 945*A*a^2*c*x^2 + 280*B*a^3*x + 840*A*a^2*b*x + 252*A*a^3)/x^10
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.01 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11}} \, dx=-\frac {x^3\,\left (\frac {3\,B\,c\,a^2}{7}+\frac {3\,B\,a\,b^2}{7}+\frac {6\,A\,c\,a\,b}{7}+\frac {A\,b^3}{7}\right )+x^4\,\left (\frac {B\,b^3}{6}+\frac {A\,b^2\,c}{2}+B\,a\,b\,c+\frac {A\,a\,c^2}{2}\right )+x\,\left (\frac {B\,a^3}{9}+\frac {A\,b\,a^2}{3}\right )+\frac {A\,a^3}{10}+x^6\,\left (\frac {A\,c^3}{4}+\frac {3\,B\,b\,c^2}{4}\right )+x^2\,\left (\frac {3\,B\,a^2\,b}{8}+\frac {3\,A\,c\,a^2}{8}+\frac {3\,A\,a\,b^2}{8}\right )+x^5\,\left (\frac {3\,B\,b^2\,c}{5}+\frac {3\,A\,b\,c^2}{5}+\frac {3\,B\,a\,c^2}{5}\right )+\frac {B\,c^3\,x^7}{3}}{x^{10}} \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^3)/x^11,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.81 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11}} \, dx=\frac {-840 b \,c^{3} x^{7}-630 a \,c^{3} x^{6}-1890 b^{2} c^{2} x^{6}-3024 a b \,c^{2} x^{5}-1512 b^{3} c \,x^{5}-1260 a^{2} c^{2} x^{4}-3780 a \,b^{2} c \,x^{4}-420 b^{4} x^{4}-3240 a^{2} b c \,x^{3}-1440 a \,b^{3} x^{3}-945 a^{3} c \,x^{2}-1890 a^{2} b^{2} x^{2}-1120 a^{3} b x -252 a^{4}}{2520 x^{10}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^3/x^11,x)
 

Output:

( - 252*a**4 - 1120*a**3*b*x - 945*a**3*c*x**2 - 1890*a**2*b**2*x**2 - 324 
0*a**2*b*c*x**3 - 1260*a**2*c**2*x**4 - 1440*a*b**3*x**3 - 3780*a*b**2*c*x 
**4 - 3024*a*b*c**2*x**5 - 630*a*c**3*x**6 - 420*b**4*x**4 - 1512*b**3*c*x 
**5 - 1890*b**2*c**2*x**6 - 840*b*c**3*x**7)/(2520*x**10)