Integrand size = 21, antiderivative size = 101 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^8} \, dx=-\frac {a^2 A}{7 x^7}-\frac {a (2 A b+a B)}{6 x^6}-\frac {2 a b B+A \left (b^2+2 a c\right )}{5 x^5}-\frac {b^2 B+2 A b c+2 a B c}{4 x^4}-\frac {c (2 b B+A c)}{3 x^3}-\frac {B c^2}{2 x^2} \] Output:
-1/7*a^2*A/x^7-1/6*a*(2*A*b+B*a)/x^6-1/5*(2*a*b*B+A*(2*a*c+b^2))/x^5-1/4*( 2*A*b*c+2*B*a*c+B*b^2)/x^4-1/3*c*(A*c+2*B*b)/x^3-1/2*B*c^2/x^2
Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^8} \, dx=-\frac {10 a^2 (6 A+7 B x)+14 a x (3 B x (4 b+5 c x)+2 A (5 b+6 c x))+7 x^2 \left (5 B x \left (3 b^2+8 b c x+6 c^2 x^2\right )+2 A \left (6 b^2+15 b c x+10 c^2 x^2\right )\right )}{420 x^7} \] Input:
Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/x^8,x]
Output:
-1/420*(10*a^2*(6*A + 7*B*x) + 14*a*x*(3*B*x*(4*b + 5*c*x) + 2*A*(5*b + 6* c*x)) + 7*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)))/x^7
Time = 0.25 (sec) , antiderivative size = 101, 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.
\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^8} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {a^2 A}{x^8}+\frac {A \left (2 a c+b^2\right )+2 a b B}{x^6}+\frac {2 a B c+2 A b c+b^2 B}{x^5}+\frac {a (a B+2 A b)}{x^7}+\frac {c (A c+2 b B)}{x^4}+\frac {B c^2}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^2 A}{7 x^7}-\frac {A \left (2 a c+b^2\right )+2 a b B}{5 x^5}-\frac {2 a B c+2 A b c+b^2 B}{4 x^4}-\frac {a (a B+2 A b)}{6 x^6}-\frac {c (A c+2 b B)}{3 x^3}-\frac {B c^2}{2 x^2}\) |
Input:
Int[((A + B*x)*(a + b*x + c*x^2)^2)/x^8,x]
Output:
-1/7*(a^2*A)/x^7 - (a*(2*A*b + a*B))/(6*x^6) - (2*a*b*B + A*(b^2 + 2*a*c)) /(5*x^5) - (b^2*B + 2*A*b*c + 2*a*B*c)/(4*x^4) - (c*(2*b*B + A*c))/(3*x^3) - (B*c^2)/(2*x^2)
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]
Time = 0.97 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {2 A a c +b^{2} A +2 a b B}{5 x^{5}}-\frac {a \left (2 A b +B a \right )}{6 x^{6}}-\frac {c \left (A c +2 B b \right )}{3 x^{3}}-\frac {B \,c^{2}}{2 x^{2}}-\frac {2 A b c +2 a B c +B \,b^{2}}{4 x^{4}}-\frac {a^{2} A}{7 x^{7}}\) | \(90\) |
norman | \(\frac {-\frac {B \,c^{2} x^{5}}{2}+\left (-\frac {1}{3} A \,c^{2}-\frac {2}{3} B b c \right ) x^{4}+\left (-\frac {1}{2} A b c -\frac {1}{2} a B c -\frac {1}{4} B \,b^{2}\right ) x^{3}+\left (-\frac {2}{5} A a c -\frac {1}{5} b^{2} A -\frac {2}{5} a b B \right ) x^{2}+\left (-\frac {1}{3} a b A -\frac {1}{6} a^{2} B \right ) x -\frac {a^{2} A}{7}}{x^{7}}\) | \(93\) |
risch | \(\frac {-\frac {B \,c^{2} x^{5}}{2}+\left (-\frac {1}{3} A \,c^{2}-\frac {2}{3} B b c \right ) x^{4}+\left (-\frac {1}{2} A b c -\frac {1}{2} a B c -\frac {1}{4} B \,b^{2}\right ) x^{3}+\left (-\frac {2}{5} A a c -\frac {1}{5} b^{2} A -\frac {2}{5} a b B \right ) x^{2}+\left (-\frac {1}{3} a b A -\frac {1}{6} a^{2} B \right ) x -\frac {a^{2} A}{7}}{x^{7}}\) | \(93\) |
gosper | \(-\frac {210 B \,c^{2} x^{5}+140 x^{4} A \,c^{2}+280 x^{4} B b c +210 x^{3} A b c +210 B a c \,x^{3}+105 x^{3} B \,b^{2}+168 A a c \,x^{2}+84 x^{2} b^{2} A +168 B a \,x^{2} b +140 a b A x +70 a^{2} B x +60 a^{2} A}{420 x^{7}}\) | \(102\) |
parallelrisch | \(-\frac {210 B \,c^{2} x^{5}+140 x^{4} A \,c^{2}+280 x^{4} B b c +210 x^{3} A b c +210 B a c \,x^{3}+105 x^{3} B \,b^{2}+168 A a c \,x^{2}+84 x^{2} b^{2} A +168 B a \,x^{2} b +140 a b A x +70 a^{2} B x +60 a^{2} A}{420 x^{7}}\) | \(102\) |
orering | \(-\frac {210 B \,c^{2} x^{5}+140 x^{4} A \,c^{2}+280 x^{4} B b c +210 x^{3} A b c +210 B a c \,x^{3}+105 x^{3} B \,b^{2}+168 A a c \,x^{2}+84 x^{2} b^{2} A +168 B a \,x^{2} b +140 a b A x +70 a^{2} B x +60 a^{2} A}{420 x^{7}}\) | \(102\) |
Input:
int((B*x+A)*(c*x^2+b*x+a)^2/x^8,x,method=_RETURNVERBOSE)
Output:
-1/5*(2*A*a*c+A*b^2+2*B*a*b)/x^5-1/6*a*(2*A*b+B*a)/x^6-1/3*c*(A*c+2*B*b)/x ^3-1/2*B*c^2/x^2-1/4*(2*A*b*c+2*B*a*c+B*b^2)/x^4-1/7*a^2*A/x^7
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^8} \, dx=-\frac {210 \, B c^{2} x^{5} + 140 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 105 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 60 \, A a^{2} + 84 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 70 \, {\left (B a^{2} + 2 \, A a b\right )} x}{420 \, x^{7}} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^2/x^8,x, algorithm="fricas")
Output:
-1/420*(210*B*c^2*x^5 + 140*(2*B*b*c + A*c^2)*x^4 + 105*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 60*A*a^2 + 84*(2*B*a*b + A*b^2 + 2*A*a*c)*x^2 + 70*(B*a^2 + 2*A*a*b)*x)/x^7
Time = 10.85 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.06 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^8} \, dx=\frac {- 60 A a^{2} - 210 B c^{2} x^{5} + x^{4} \left (- 140 A c^{2} - 280 B b c\right ) + x^{3} \left (- 210 A b c - 210 B a c - 105 B b^{2}\right ) + x^{2} \left (- 168 A a c - 84 A b^{2} - 168 B a b\right ) + x \left (- 140 A a b - 70 B a^{2}\right )}{420 x^{7}} \] Input:
integrate((B*x+A)*(c*x**2+b*x+a)**2/x**8,x)
Output:
(-60*A*a**2 - 210*B*c**2*x**5 + x**4*(-140*A*c**2 - 280*B*b*c) + x**3*(-21 0*A*b*c - 210*B*a*c - 105*B*b**2) + x**2*(-168*A*a*c - 84*A*b**2 - 168*B*a *b) + x*(-140*A*a*b - 70*B*a**2))/(420*x**7)
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^8} \, dx=-\frac {210 \, B c^{2} x^{5} + 140 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 105 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 60 \, A a^{2} + 84 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 70 \, {\left (B a^{2} + 2 \, A a b\right )} x}{420 \, x^{7}} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^2/x^8,x, algorithm="maxima")
Output:
-1/420*(210*B*c^2*x^5 + 140*(2*B*b*c + A*c^2)*x^4 + 105*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 60*A*a^2 + 84*(2*B*a*b + A*b^2 + 2*A*a*c)*x^2 + 70*(B*a^2 + 2*A*a*b)*x)/x^7
Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^8} \, dx=-\frac {210 \, B c^{2} x^{5} + 280 \, B b c x^{4} + 140 \, A c^{2} x^{4} + 105 \, B b^{2} x^{3} + 210 \, B a c x^{3} + 210 \, A b c x^{3} + 168 \, B a b x^{2} + 84 \, A b^{2} x^{2} + 168 \, A a c x^{2} + 70 \, B a^{2} x + 140 \, A a b x + 60 \, A a^{2}}{420 \, x^{7}} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^2/x^8,x, algorithm="giac")
Output:
-1/420*(210*B*c^2*x^5 + 280*B*b*c*x^4 + 140*A*c^2*x^4 + 105*B*b^2*x^3 + 21 0*B*a*c*x^3 + 210*A*b*c*x^3 + 168*B*a*b*x^2 + 84*A*b^2*x^2 + 168*A*a*c*x^2 + 70*B*a^2*x + 140*A*a*b*x + 60*A*a^2)/x^7
Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^8} \, dx=-\frac {x^4\,\left (\frac {A\,c^2}{3}+\frac {2\,B\,b\,c}{3}\right )+\frac {A\,a^2}{7}+x^2\,\left (\frac {A\,b^2}{5}+\frac {2\,B\,a\,b}{5}+\frac {2\,A\,a\,c}{5}\right )+x^3\,\left (\frac {B\,b^2}{4}+\frac {A\,c\,b}{2}+\frac {B\,a\,c}{2}\right )+x\,\left (\frac {B\,a^2}{6}+\frac {A\,b\,a}{3}\right )+\frac {B\,c^2\,x^5}{2}}{x^7} \] Input:
int(((A + B*x)*(a + b*x + c*x^2)^2)/x^8,x)
Output:
-(x^4*((A*c^2)/3 + (2*B*b*c)/3) + (A*a^2)/7 + x^2*((A*b^2)/5 + (2*A*a*c)/5 + (2*B*a*b)/5) + x^3*((B*b^2)/4 + (A*b*c)/2 + (B*a*c)/2) + x*((B*a^2)/6 + (A*a*b)/3) + (B*c^2*x^5)/2)/x^7
Time = 0.24 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.78 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^8} \, dx=\frac {-210 b \,c^{2} x^{5}-140 a \,c^{2} x^{4}-280 b^{2} c \,x^{4}-420 a b c \,x^{3}-105 b^{3} x^{3}-168 a^{2} c \,x^{2}-252 a \,b^{2} x^{2}-210 a^{2} b x -60 a^{3}}{420 x^{7}} \] Input:
int((B*x+A)*(c*x^2+b*x+a)^2/x^8,x)
Output:
( - 60*a**3 - 210*a**2*b*x - 168*a**2*c*x**2 - 252*a*b**2*x**2 - 420*a*b*c *x**3 - 140*a*c**2*x**4 - 105*b**3*x**3 - 280*b**2*c*x**4 - 210*b*c**2*x** 5)/(420*x**7)