\(\int \frac {(A+B x) (a+b x+c x^2)^3}{x^{10}} \, dx\) [37]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{10}} \, dx=-\frac {35 a^3 (8 A+9 B x)+45 a^2 x (4 B x (6 b+7 c x)+3 A (7 b+8 c x))+18 a x^2 \left (7 B x \left (10 b^2+24 b c x+15 c^2 x^2\right )+4 A \left (15 b^2+35 b c x+21 c^2 x^2\right )\right )+42 x^3 \left (3 B x \left (4 b^3+15 b^2 c x+20 b c^2 x^2+10 c^3 x^3\right )+A \left (10 b^3+36 b^2 c x+45 b c^2 x^2+20 c^3 x^3\right )\right )}{2520 x^9} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^10,x]
 

Output:

-1/2520*(35*a^3*(8*A + 9*B*x) + 45*a^2*x*(4*B*x*(6*b + 7*c*x) + 3*A*(7*b + 
 8*c*x)) + 18*a*x^2*(7*B*x*(10*b^2 + 24*b*c*x + 15*c^2*x^2) + 4*A*(15*b^2 
+ 35*b*c*x + 21*c^2*x^2)) + 42*x^3*(3*B*x*(4*b^3 + 15*b^2*c*x + 20*b*c^2*x 
^2 + 10*c^3*x^3) + A*(10*b^3 + 36*b^2*c*x + 45*b*c^2*x^2 + 20*c^3*x^3)))/x 
^9
 

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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{10}} \, dx\)

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^10,x]
 

Output:

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

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	\(-\frac {3 A a \,c^{2}+3 A \,b^{2} c +6 B a b c +B \,b^{3}}{5 x^{5}}-\frac {6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}}{6 x^{6}}-\frac {c^{2} \left (A c +3 B b \right )}{3 x^{3}}-\frac {B \,c^{3}}{2 x^{2}}-\frac {a^{2} \left (3 A b +B a \right )}{8 x^{8}}-\frac {3 c \left (A b c +a B c +B \,b^{2}\right )}{4 x^{4}}-\frac {a^{3} A}{9 x^{9}}-\frac {3 a \left (A a c +b^{2} A +a b B \right )}{7 x^{7}}\)	\(154\)
norman	\(\frac {-\frac {B \,c^{3} x^{7}}{2}+\left (-\frac {1}{3} A \,c^{3}-B b \,c^{2}\right ) x^{6}+\left (-\frac {3}{4} A b \,c^{2}-\frac {3}{4} B a \,c^{2}-\frac {3}{4} B \,b^{2} c \right ) x^{5}+\left (-\frac {3}{5} A a \,c^{2}-\frac {3}{5} A \,b^{2} c -\frac {6}{5} B a b c -\frac {1}{5} B \,b^{3}\right ) x^{4}+\left (-A a b c -\frac {1}{6} A \,b^{3}-\frac {1}{2} B \,a^{2} c -\frac {1}{2} B a \,b^{2}\right ) x^{3}+\left (-\frac {3}{7} a^{2} A c -\frac {3}{7} A a \,b^{2}-\frac {3}{7} B \,a^{2} b \right ) x^{2}+\left (-\frac {3}{8} A \,a^{2} b -\frac {1}{8} B \,a^{3}\right ) x -\frac {a^{3} A}{9}}{x^{9}}\)	\(169\)
risch	\(\frac {-\frac {B \,c^{3} x^{7}}{2}+\left (-\frac {1}{3} A \,c^{3}-B b \,c^{2}\right ) x^{6}+\left (-\frac {3}{4} A b \,c^{2}-\frac {3}{4} B a \,c^{2}-\frac {3}{4} B \,b^{2} c \right ) x^{5}+\left (-\frac {3}{5} A a \,c^{2}-\frac {3}{5} A \,b^{2} c -\frac {6}{5} B a b c -\frac {1}{5} B \,b^{3}\right ) x^{4}+\left (-A a b c -\frac {1}{6} A \,b^{3}-\frac {1}{2} B \,a^{2} c -\frac {1}{2} B a \,b^{2}\right ) x^{3}+\left (-\frac {3}{7} a^{2} A c -\frac {3}{7} A a \,b^{2}-\frac {3}{7} B \,a^{2} b \right ) x^{2}+\left (-\frac {3}{8} A \,a^{2} b -\frac {1}{8} B \,a^{3}\right ) x -\frac {a^{3} A}{9}}{x^{9}}\)	\(169\)
gosper	\(-\frac {1260 B \,c^{3} x^{7}+840 A \,c^{3} x^{6}+2520 B b \,c^{2} x^{6}+1890 A b \,c^{2} x^{5}+1890 B a \,c^{2} x^{5}+1890 B \,b^{2} c \,x^{5}+1512 A a \,c^{2} x^{4}+1512 A \,b^{2} c \,x^{4}+3024 B a b c \,x^{4}+504 B \,b^{3} x^{4}+2520 A a b c \,x^{3}+420 A \,b^{3} x^{3}+1260 B \,a^{2} c \,x^{3}+1260 B a \,b^{2} x^{3}+1080 A \,a^{2} c \,x^{2}+1080 A a \,b^{2} x^{2}+1080 B \,a^{2} b \,x^{2}+945 A \,a^{2} b x +315 B \,a^{3} x +280 a^{3} A}{2520 x^{9}}\)	\(192\)
parallelrisch	\(-\frac {1260 B \,c^{3} x^{7}+840 A \,c^{3} x^{6}+2520 B b \,c^{2} x^{6}+1890 A b \,c^{2} x^{5}+1890 B a \,c^{2} x^{5}+1890 B \,b^{2} c \,x^{5}+1512 A a \,c^{2} x^{4}+1512 A \,b^{2} c \,x^{4}+3024 B a b c \,x^{4}+504 B \,b^{3} x^{4}+2520 A a b c \,x^{3}+420 A \,b^{3} x^{3}+1260 B \,a^{2} c \,x^{3}+1260 B a \,b^{2} x^{3}+1080 A \,a^{2} c \,x^{2}+1080 A a \,b^{2} x^{2}+1080 B \,a^{2} b \,x^{2}+945 A \,a^{2} b x +315 B \,a^{3} x +280 a^{3} A}{2520 x^{9}}\)	\(192\)
orering	\(-\frac {1260 B \,c^{3} x^{7}+840 A \,c^{3} x^{6}+2520 B b \,c^{2} x^{6}+1890 A b \,c^{2} x^{5}+1890 B a \,c^{2} x^{5}+1890 B \,b^{2} c \,x^{5}+1512 A a \,c^{2} x^{4}+1512 A \,b^{2} c \,x^{4}+3024 B a b c \,x^{4}+504 B \,b^{3} x^{4}+2520 A a b c \,x^{3}+420 A \,b^{3} x^{3}+1260 B \,a^{2} c \,x^{3}+1260 B a \,b^{2} x^{3}+1080 A \,a^{2} c \,x^{2}+1080 A a \,b^{2} x^{2}+1080 B \,a^{2} b \,x^{2}+945 A \,a^{2} b x +315 B \,a^{3} x +280 a^{3} A}{2520 x^{9}}\)	\(192\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^3/x^10,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{10}} \, dx=-\frac {1260 \, B c^{3} x^{7} + 840 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1890 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 504 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 280 \, A a^{3} + 420 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 1080 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 315 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^10,x, algorithm="fricas")
 

Output:

-1/2520*(1260*B*c^3*x^7 + 840*(3*B*b*c^2 + A*c^3)*x^6 + 1890*(B*b^2*c + (B 
*a + A*b)*c^2)*x^5 + 504*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 
 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 1080*(B 
*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 315*(B*a^3 + 3*A*a^2*b)*x)/x^9
 

Sympy [A] (verification not implemented)

Time = 121.61 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.18 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{10}} \, dx=\frac {- 280 A a^{3} - 1260 B c^{3} x^{7} + x^{6} \left (- 840 A c^{3} - 2520 B b c^{2}\right ) + x^{5} \left (- 1890 A b c^{2} - 1890 B a c^{2} - 1890 B b^{2} c\right ) + x^{4} \left (- 1512 A a c^{2} - 1512 A b^{2} c - 3024 B a b c - 504 B b^{3}\right ) + x^{3} \left (- 2520 A a b c - 420 A b^{3} - 1260 B a^{2} c - 1260 B a b^{2}\right ) + x^{2} \left (- 1080 A a^{2} c - 1080 A a b^{2} - 1080 B a^{2} b\right ) + x \left (- 945 A a^{2} b - 315 B a^{3}\right )}{2520 x^{9}} \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**3/x**10,x)
 

Output:

(-280*A*a**3 - 1260*B*c**3*x**7 + x**6*(-840*A*c**3 - 2520*B*b*c**2) + x** 
5*(-1890*A*b*c**2 - 1890*B*a*c**2 - 1890*B*b**2*c) + x**4*(-1512*A*a*c**2 
- 1512*A*b**2*c - 3024*B*a*b*c - 504*B*b**3) + x**3*(-2520*A*a*b*c - 420*A 
*b**3 - 1260*B*a**2*c - 1260*B*a*b**2) + x**2*(-1080*A*a**2*c - 1080*A*a*b 
**2 - 1080*B*a**2*b) + x*(-945*A*a**2*b - 315*B*a**3))/(2520*x**9)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{10}} \, dx=-\frac {1260 \, B c^{3} x^{7} + 840 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1890 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 504 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 280 \, A a^{3} + 420 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 1080 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 315 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^10,x, algorithm="maxima")
 

Output:

-1/2520*(1260*B*c^3*x^7 + 840*(3*B*b*c^2 + A*c^3)*x^6 + 1890*(B*b^2*c + (B 
*a + A*b)*c^2)*x^5 + 504*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 
 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 1080*(B 
*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 315*(B*a^3 + 3*A*a^2*b)*x)/x^9
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.15 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{10}} \, dx=-\frac {1260 \, B c^{3} x^{7} + 2520 \, B b c^{2} x^{6} + 840 \, A c^{3} x^{6} + 1890 \, B b^{2} c x^{5} + 1890 \, B a c^{2} x^{5} + 1890 \, A b c^{2} x^{5} + 504 \, B b^{3} x^{4} + 3024 \, B a b c x^{4} + 1512 \, A b^{2} c x^{4} + 1512 \, A a c^{2} x^{4} + 1260 \, B a b^{2} x^{3} + 420 \, A b^{3} x^{3} + 1260 \, B a^{2} c x^{3} + 2520 \, A a b c x^{3} + 1080 \, B a^{2} b x^{2} + 1080 \, A a b^{2} x^{2} + 1080 \, A a^{2} c x^{2} + 315 \, B a^{3} x + 945 \, A a^{2} b x + 280 \, A a^{3}}{2520 \, x^{9}} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^10,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

-1/2520*(1260*B*c^3*x^7 + 2520*B*b*c^2*x^6 + 840*A*c^3*x^6 + 1890*B*b^2*c* 
x^5 + 1890*B*a*c^2*x^5 + 1890*A*b*c^2*x^5 + 504*B*b^3*x^4 + 3024*B*a*b*c*x 
^4 + 1512*A*b^2*c*x^4 + 1512*A*a*c^2*x^4 + 1260*B*a*b^2*x^3 + 420*A*b^3*x^ 
3 + 1260*B*a^2*c*x^3 + 2520*A*a*b*c*x^3 + 1080*B*a^2*b*x^2 + 1080*A*a*b^2* 
x^2 + 1080*A*a^2*c*x^2 + 315*B*a^3*x + 945*A*a^2*b*x + 280*A*a^3)/x^9
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.01 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{10}} \, dx=-\frac {x^3\,\left (\frac {B\,c\,a^2}{2}+\frac {B\,a\,b^2}{2}+A\,c\,a\,b+\frac {A\,b^3}{6}\right )+x^4\,\left (\frac {B\,b^3}{5}+\frac {3\,A\,b^2\,c}{5}+\frac {6\,B\,a\,b\,c}{5}+\frac {3\,A\,a\,c^2}{5}\right )+x\,\left (\frac {B\,a^3}{8}+\frac {3\,A\,b\,a^2}{8}\right )+\frac {A\,a^3}{9}+x^6\,\left (\frac {A\,c^3}{3}+B\,b\,c^2\right )+x^2\,\left (\frac {3\,B\,a^2\,b}{7}+\frac {3\,A\,c\,a^2}{7}+\frac {3\,A\,a\,b^2}{7}\right )+x^5\,\left (\frac {3\,B\,b^2\,c}{4}+\frac {3\,A\,b\,c^2}{4}+\frac {3\,B\,a\,c^2}{4}\right )+\frac {B\,c^3\,x^7}{2}}{x^9} \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^3)/x^10,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.81 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{10}} \, dx=\frac {-630 b \,c^{3} x^{7}-420 a \,c^{3} x^{6}-1260 b^{2} c^{2} x^{6}-1890 a b \,c^{2} x^{5}-945 b^{3} c \,x^{5}-756 a^{2} c^{2} x^{4}-2268 a \,b^{2} c \,x^{4}-252 b^{4} x^{4}-1890 a^{2} b c \,x^{3}-840 a \,b^{3} x^{3}-540 a^{3} c \,x^{2}-1080 a^{2} b^{2} x^{2}-630 a^{3} b x -140 a^{4}}{1260 x^{9}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^3/x^10,x)
 

Output:

( - 140*a**4 - 630*a**3*b*x - 540*a**3*c*x**2 - 1080*a**2*b**2*x**2 - 1890 
*a**2*b*c*x**3 - 756*a**2*c**2*x**4 - 840*a*b**3*x**3 - 2268*a*b**2*c*x**4 
 - 1890*a*b*c**2*x**5 - 420*a*c**3*x**6 - 252*b**4*x**4 - 945*b**3*c*x**5 
- 1260*b**2*c**2*x**6 - 630*b*c**3*x**7)/(1260*x**9)