\(\int \frac {(A+B x) (a+b x+c x^2)^3}{x^8} \, dx\) [35]

Optimal result

Integrand size = 21, antiderivative size = 160 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=-\frac {a^3 A}{7 x^7}-\frac {a^2 (3 A b+a B)}{6 x^6}-\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{5 x^5}-\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{4 x^4}-\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{3 x^3}-\frac {3 c \left (b^2 B+A b c+a B c\right )}{2 x^2}-\frac {c^2 (3 b B+A c)}{x}+B c^3 \log (x) \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=-\frac {10 a^3 (6 A+7 B x)+21 a^2 x (3 B x (4 b+5 c x)+2 A (5 b+6 c x))+21 a x^2 \left (5 B x \left (3 b^2+8 b c x+6 c^2 x^2\right )+2 A \left (6 b^2+15 b c x+10 c^2 x^2\right )\right )+35 x^3 \left (2 b B x \left (2 b^2+9 b c x+18 c^2 x^2\right )+3 A \left (b^3+4 b^2 c x+6 b c^2 x^2+4 c^3 x^3\right )\right )-420 B c^3 x^7 \log (x)}{420 x^7} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^8,x]
 

Output:

-1/420*(10*a^3*(6*A + 7*B*x) + 21*a^2*x*(3*B*x*(4*b + 5*c*x) + 2*A*(5*b + 
6*c*x)) + 21*a*x^2*(5*B*x*(3*b^2 + 8*b*c*x + 6*c^2*x^2) + 2*A*(6*b^2 + 15* 
b*c*x + 10*c^2*x^2)) + 35*x^3*(2*b*B*x*(2*b^2 + 9*b*c*x + 18*c^2*x^2) + 3* 
A*(b^3 + 4*b^2*c*x + 6*b*c^2*x^2 + 4*c^3*x^3)) - 420*B*c^3*x^7*Log[x])/x^7
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 160, 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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx\)

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^8,x]
 

Output:

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

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 = 0.94 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.95

method	result	size
default	\(-\frac {3 a \left (A a c +b^{2} A +a b B \right )}{5 x^{5}}-\frac {a^{2} \left (3 A b +B a \right )}{6 x^{6}}-\frac {3 A a \,c^{2}+3 A \,b^{2} c +6 B a b c +B \,b^{3}}{3 x^{3}}-\frac {3 c \left (A b c +a B c +B \,b^{2}\right )}{2 x^{2}}-\frac {6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}}{4 x^{4}}+B \,c^{3} \ln \left (x \right )-\frac {a^{3} A}{7 x^{7}}-\frac {c^{2} \left (A c +3 B b \right )}{x}\)	\(152\)
norman	\(\frac {\left (-\frac {1}{2} A \,a^{2} b -\frac {1}{6} B \,a^{3}\right ) x +\left (-\frac {3}{2} A b \,c^{2}-\frac {3}{2} B a \,c^{2}-\frac {3}{2} B \,b^{2} c \right ) x^{5}+\left (-\frac {3}{5} a^{2} A c -\frac {3}{5} A a \,b^{2}-\frac {3}{5} B \,a^{2} b \right ) x^{2}+\left (-A a \,c^{2}-A \,b^{2} c -2 B a b c -\frac {1}{3} B \,b^{3}\right ) x^{4}+\left (-\frac {3}{2} A a b c -\frac {1}{4} A \,b^{3}-\frac {3}{4} B \,a^{2} c -\frac {3}{4} B a \,b^{2}\right ) x^{3}+\left (-A \,c^{3}-3 B b \,c^{2}\right ) x^{6}-\frac {a^{3} A}{7}}{x^{7}}+B \,c^{3} \ln \left (x \right )\)	\(168\)
risch	\(\frac {\left (-\frac {1}{2} A \,a^{2} b -\frac {1}{6} B \,a^{3}\right ) x +\left (-\frac {3}{2} A b \,c^{2}-\frac {3}{2} B a \,c^{2}-\frac {3}{2} B \,b^{2} c \right ) x^{5}+\left (-\frac {3}{5} a^{2} A c -\frac {3}{5} A a \,b^{2}-\frac {3}{5} B \,a^{2} b \right ) x^{2}+\left (-A a \,c^{2}-A \,b^{2} c -2 B a b c -\frac {1}{3} B \,b^{3}\right ) x^{4}+\left (-\frac {3}{2} A a b c -\frac {1}{4} A \,b^{3}-\frac {3}{4} B \,a^{2} c -\frac {3}{4} B a \,b^{2}\right ) x^{3}+\left (-A \,c^{3}-3 B b \,c^{2}\right ) x^{6}-\frac {a^{3} A}{7}}{x^{7}}+B \,c^{3} \ln \left (x \right )\)	\(168\)
parallelrisch	\(-\frac {-420 B \,c^{3} \ln \left (x \right ) x^{7}+420 A \,c^{3} x^{6}+1260 B b \,c^{2} x^{6}+630 A b \,c^{2} x^{5}+630 B a \,c^{2} x^{5}+630 B \,b^{2} c \,x^{5}+420 A a \,c^{2} x^{4}+420 A \,b^{2} c \,x^{4}+840 B a b c \,x^{4}+140 B \,b^{3} x^{4}+630 A a b c \,x^{3}+105 A \,b^{3} x^{3}+315 B \,a^{2} c \,x^{3}+315 B a \,b^{2} x^{3}+252 A \,a^{2} c \,x^{2}+252 A a \,b^{2} x^{2}+252 B \,a^{2} b \,x^{2}+210 A \,a^{2} b x +70 B \,a^{3} x +60 a^{3} A}{420 x^{7}}\)	\(194\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^3/x^8,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=\frac {420 \, B c^{3} x^{7} \log \left (x\right ) - 420 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} - 630 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} - 140 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 60 \, A a^{3} - 105 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 252 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^8,x, algorithm="fricas")
 

Output:

1/420*(420*B*c^3*x^7*log(x) - 420*(3*B*b*c^2 + A*c^3)*x^6 - 630*(B*b^2*c + 
 (B*a + A*b)*c^2)*x^5 - 140*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^ 
4 - 60*A*a^3 - 105*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 - 252*( 
B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 - 70*(B*a^3 + 3*A*a^2*b)*x)/x^7
 

Sympy [A] (verification not implemented)

Time = 26.69 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.21 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=B c^{3} \log {\left (x \right )} + \frac {- 60 A a^{3} + x^{6} \left (- 420 A c^{3} - 1260 B b c^{2}\right ) + x^{5} \left (- 630 A b c^{2} - 630 B a c^{2} - 630 B b^{2} c\right ) + x^{4} \left (- 420 A a c^{2} - 420 A b^{2} c - 840 B a b c - 140 B b^{3}\right ) + x^{3} \left (- 630 A a b c - 105 A b^{3} - 315 B a^{2} c - 315 B a b^{2}\right ) + x^{2} \left (- 252 A a^{2} c - 252 A a b^{2} - 252 B a^{2} b\right ) + x \left (- 210 A a^{2} b - 70 B a^{3}\right )}{420 x^{7}} \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**3/x**8,x)
                                                                                    
                                                                                    
 

Output:

B*c**3*log(x) + (-60*A*a**3 + x**6*(-420*A*c**3 - 1260*B*b*c**2) + x**5*(- 
630*A*b*c**2 - 630*B*a*c**2 - 630*B*b**2*c) + x**4*(-420*A*a*c**2 - 420*A* 
b**2*c - 840*B*a*b*c - 140*B*b**3) + x**3*(-630*A*a*b*c - 105*A*b**3 - 315 
*B*a**2*c - 315*B*a*b**2) + x**2*(-252*A*a**2*c - 252*A*a*b**2 - 252*B*a** 
2*b) + x*(-210*A*a**2*b - 70*B*a**3))/(420*x**7)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=B c^{3} \log \left (x\right ) - \frac {420 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 630 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 140 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 60 \, A a^{3} + 105 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 252 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^8,x, algorithm="maxima")
 

Output:

B*c^3*log(x) - 1/420*(420*(3*B*b*c^2 + A*c^3)*x^6 + 630*(B*b^2*c + (B*a + 
A*b)*c^2)*x^5 + 140*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 60*A 
*a^3 + 105*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 252*(B*a^2*b 
+ A*a*b^2 + A*a^2*c)*x^2 + 70*(B*a^3 + 3*A*a^2*b)*x)/x^7
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=B c^{3} \log \left ({\left | x \right |}\right ) - \frac {420 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 630 \, {\left (B b^{2} c + B a c^{2} + A b c^{2}\right )} x^{5} + 140 \, {\left (B b^{3} + 6 \, B a b c + 3 \, A b^{2} c + 3 \, A a c^{2}\right )} x^{4} + 60 \, A a^{3} + 105 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, B a^{2} c + 6 \, A a b c\right )} x^{3} + 252 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^8,x, algorithm="giac")
 

Output:

B*c^3*log(abs(x)) - 1/420*(420*(3*B*b*c^2 + A*c^3)*x^6 + 630*(B*b^2*c + B* 
a*c^2 + A*b*c^2)*x^5 + 140*(B*b^3 + 6*B*a*b*c + 3*A*b^2*c + 3*A*a*c^2)*x^4 
 + 60*A*a^3 + 105*(3*B*a*b^2 + A*b^3 + 3*B*a^2*c + 6*A*a*b*c)*x^3 + 252*(B 
*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 70*(B*a^3 + 3*A*a^2*b)*x)/x^7
 

Mupad [B] (verification not implemented)

Time = 10.46 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=B\,c^3\,\ln \left (x\right )-\frac {x^3\,\left (\frac {3\,B\,c\,a^2}{4}+\frac {3\,B\,a\,b^2}{4}+\frac {3\,A\,c\,a\,b}{2}+\frac {A\,b^3}{4}\right )+x^4\,\left (\frac {B\,b^3}{3}+A\,b^2\,c+2\,B\,a\,b\,c+A\,a\,c^2\right )+x\,\left (\frac {B\,a^3}{6}+\frac {A\,b\,a^2}{2}\right )+\frac {A\,a^3}{7}+x^6\,\left (A\,c^3+3\,B\,b\,c^2\right )+x^2\,\left (\frac {3\,B\,a^2\,b}{5}+\frac {3\,A\,c\,a^2}{5}+\frac {3\,A\,a\,b^2}{5}\right )+x^5\,\left (\frac {3\,B\,b^2\,c}{2}+\frac {3\,A\,b\,c^2}{2}+\frac {3\,B\,a\,c^2}{2}\right )}{x^7} \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^3)/x^8,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.85 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=\frac {420 \,\mathrm {log}\left (x \right ) b \,c^{3} x^{7}-60 a^{4}-280 a^{3} b x -252 a^{3} c \,x^{2}-504 a^{2} b^{2} x^{2}-945 a^{2} b c \,x^{3}-420 a^{2} c^{2} x^{4}-420 a \,b^{3} x^{3}-1260 a \,b^{2} c \,x^{4}-1260 a b \,c^{2} x^{5}-420 a \,c^{3} x^{6}-140 b^{4} x^{4}-630 b^{3} c \,x^{5}-1260 b^{2} c^{2} x^{6}}{420 x^{7}} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^3/x^8,x)
 

Output:

(420*log(x)*b*c**3*x**7 - 60*a**4 - 280*a**3*b*x - 252*a**3*c*x**2 - 504*a 
**2*b**2*x**2 - 945*a**2*b*c*x**3 - 420*a**2*c**2*x**4 - 420*a*b**3*x**3 - 
 1260*a*b**2*c*x**4 - 1260*a*b*c**2*x**5 - 420*a*c**3*x**6 - 140*b**4*x**4 
 - 630*b**3*c*x**5 - 1260*b**2*c**2*x**6)/(420*x**7)