Integrand size = 27, antiderivative size = 143 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{12}} \, dx=-\frac {a^6 A}{11 x^{11}}-\frac {a^5 (6 A b+a B)}{10 x^{10}}-\frac {a^4 b (5 A b+2 a B)}{3 x^9}-\frac {5 a^3 b^2 (4 A b+3 a B)}{8 x^8}-\frac {5 a^2 b^3 (3 A b+4 a B)}{7 x^7}-\frac {a b^4 (2 A b+5 a B)}{2 x^6}-\frac {b^5 (A b+6 a B)}{5 x^5}-\frac {b^6 B}{4 x^4} \] Output:
-1/11*a^6*A/x^11-1/10*a^5*(6*A*b+B*a)/x^10-1/3*a^4*b*(5*A*b+2*B*a)/x^9-5/8 *a^3*b^2*(4*A*b+3*B*a)/x^8-5/7*a^2*b^3*(3*A*b+4*B*a)/x^7-1/2*a*b^4*(2*A*b+ 5*B*a)/x^6-1/5*b^5*(A*b+6*B*a)/x^5-1/4*b^6*B/x^4
Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{12}} \, dx=-\frac {462 b^6 x^6 (4 A+5 B x)+1848 a b^5 x^5 (5 A+6 B x)+3300 a^2 b^4 x^4 (6 A+7 B x)+3300 a^3 b^3 x^3 (7 A+8 B x)+1925 a^4 b^2 x^2 (8 A+9 B x)+616 a^5 b x (9 A+10 B x)+84 a^6 (10 A+11 B x)}{9240 x^{11}} \] Input:
Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^12,x]
Output:
-1/9240*(462*b^6*x^6*(4*A + 5*B*x) + 1848*a*b^5*x^5*(5*A + 6*B*x) + 3300*a ^2*b^4*x^4*(6*A + 7*B*x) + 3300*a^3*b^3*x^3*(7*A + 8*B*x) + 1925*a^4*b^2*x ^2*(8*A + 9*B*x) + 616*a^5*b*x*(9*A + 10*B*x) + 84*a^6*(10*A + 11*B*x))/x^ 11
Time = 0.49 (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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x)}{x^{12}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int \frac {b^6 (a+b x)^6 (A+B x)}{x^{12}}dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{x^{12}}dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (\frac {a^6 A}{x^{12}}+\frac {a^5 (a B+6 A b)}{x^{11}}+\frac {3 a^4 b (2 a B+5 A b)}{x^{10}}+\frac {5 a^3 b^2 (3 a B+4 A b)}{x^9}+\frac {5 a^2 b^3 (4 a B+3 A b)}{x^8}+\frac {b^5 (6 a B+A b)}{x^6}+\frac {3 a b^4 (5 a B+2 A b)}{x^7}+\frac {b^6 B}{x^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^6 A}{11 x^{11}}-\frac {a^5 (a B+6 A b)}{10 x^{10}}-\frac {a^4 b (2 a B+5 A b)}{3 x^9}-\frac {5 a^3 b^2 (3 a B+4 A b)}{8 x^8}-\frac {5 a^2 b^3 (4 a B+3 A b)}{7 x^7}-\frac {b^5 (6 a B+A b)}{5 x^5}-\frac {a b^4 (5 a B+2 A b)}{2 x^6}-\frac {b^6 B}{4 x^4}\) |
Input:
Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^12,x]
Output:
-1/11*(a^6*A)/x^11 - (a^5*(6*A*b + a*B))/(10*x^10) - (a^4*b*(5*A*b + 2*a*B ))/(3*x^9) - (5*a^3*b^2*(4*A*b + 3*a*B))/(8*x^8) - (5*a^2*b^3*(3*A*b + 4*a *B))/(7*x^7) - (a*b^4*(2*A*b + 5*a*B))/(2*x^6) - (b^5*(A*b + 6*a*B))/(5*x^ 5) - (b^6*B)/(4*x^4)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
Time = 0.93 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {a^{6} A}{11 x^{11}}-\frac {a^{5} \left (6 A b +B a \right )}{10 x^{10}}-\frac {a^{4} b \left (5 A b +2 B a \right )}{3 x^{9}}-\frac {5 a^{3} b^{2} \left (4 A b +3 B a \right )}{8 x^{8}}-\frac {5 a^{2} b^{3} \left (3 A b +4 B a \right )}{7 x^{7}}-\frac {a \,b^{4} \left (2 A b +5 B a \right )}{2 x^{6}}-\frac {b^{5} \left (A b +6 B a \right )}{5 x^{5}}-\frac {b^{6} B}{4 x^{4}}\) | \(128\) |
norman | \(\frac {-\frac {b^{6} B \,x^{7}}{4}+\left (-\frac {1}{5} A \,b^{6}-\frac {6}{5} B a \,b^{5}\right ) x^{6}+\left (-A a \,b^{5}-\frac {5}{2} B \,a^{2} b^{4}\right ) x^{5}+\left (-\frac {15}{7} A \,a^{2} b^{4}-\frac {20}{7} B \,a^{3} b^{3}\right ) x^{4}+\left (-\frac {5}{2} A \,a^{3} b^{3}-\frac {15}{8} B \,a^{4} b^{2}\right ) x^{3}+\left (-\frac {5}{3} A \,a^{4} b^{2}-\frac {2}{3} B \,a^{5} b \right ) x^{2}+\left (-\frac {3}{5} A \,a^{5} b -\frac {1}{10} B \,a^{6}\right ) x -\frac {A \,a^{6}}{11}}{x^{11}}\) | \(143\) |
risch | \(\frac {-\frac {b^{6} B \,x^{7}}{4}+\left (-\frac {1}{5} A \,b^{6}-\frac {6}{5} B a \,b^{5}\right ) x^{6}+\left (-A a \,b^{5}-\frac {5}{2} B \,a^{2} b^{4}\right ) x^{5}+\left (-\frac {15}{7} A \,a^{2} b^{4}-\frac {20}{7} B \,a^{3} b^{3}\right ) x^{4}+\left (-\frac {5}{2} A \,a^{3} b^{3}-\frac {15}{8} B \,a^{4} b^{2}\right ) x^{3}+\left (-\frac {5}{3} A \,a^{4} b^{2}-\frac {2}{3} B \,a^{5} b \right ) x^{2}+\left (-\frac {3}{5} A \,a^{5} b -\frac {1}{10} B \,a^{6}\right ) x -\frac {A \,a^{6}}{11}}{x^{11}}\) | \(143\) |
gosper | \(-\frac {2310 b^{6} B \,x^{7}+1848 A \,b^{6} x^{6}+11088 B a \,b^{5} x^{6}+9240 A a \,b^{5} x^{5}+23100 B \,a^{2} b^{4} x^{5}+19800 A \,a^{2} b^{4} x^{4}+26400 B \,a^{3} b^{3} x^{4}+23100 A \,a^{3} b^{3} x^{3}+17325 B \,a^{4} b^{2} x^{3}+15400 A \,a^{4} b^{2} x^{2}+6160 B \,a^{5} b \,x^{2}+5544 A \,a^{5} b x +924 B \,a^{6} x +840 A \,a^{6}}{9240 x^{11}}\) | \(148\) |
parallelrisch | \(-\frac {2310 b^{6} B \,x^{7}+1848 A \,b^{6} x^{6}+11088 B a \,b^{5} x^{6}+9240 A a \,b^{5} x^{5}+23100 B \,a^{2} b^{4} x^{5}+19800 A \,a^{2} b^{4} x^{4}+26400 B \,a^{3} b^{3} x^{4}+23100 A \,a^{3} b^{3} x^{3}+17325 B \,a^{4} b^{2} x^{3}+15400 A \,a^{4} b^{2} x^{2}+6160 B \,a^{5} b \,x^{2}+5544 A \,a^{5} b x +924 B \,a^{6} x +840 A \,a^{6}}{9240 x^{11}}\) | \(148\) |
orering | \(-\frac {\left (2310 b^{6} B \,x^{7}+1848 A \,b^{6} x^{6}+11088 B a \,b^{5} x^{6}+9240 A a \,b^{5} x^{5}+23100 B \,a^{2} b^{4} x^{5}+19800 A \,a^{2} b^{4} x^{4}+26400 B \,a^{3} b^{3} x^{4}+23100 A \,a^{3} b^{3} x^{3}+17325 B \,a^{4} b^{2} x^{3}+15400 A \,a^{4} b^{2} x^{2}+6160 B \,a^{5} b \,x^{2}+5544 A \,a^{5} b x +924 B \,a^{6} x +840 A \,a^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}{9240 x^{11} \left (b x +a \right )^{6}}\) | \(173\) |
Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^12,x,method=_RETURNVERBOSE)
Output:
-1/11*a^6*A/x^11-1/10*a^5*(6*A*b+B*a)/x^10-1/3*a^4*b*(5*A*b+2*B*a)/x^9-5/8 *a^3*b^2*(4*A*b+3*B*a)/x^8-5/7*a^2*b^3*(3*A*b+4*B*a)/x^7-1/2*a*b^4*(2*A*b+ 5*B*a)/x^6-1/5*b^5*(A*b+6*B*a)/x^5-1/4*b^6*B/x^4
Time = 0.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{12}} \, dx=-\frac {2310 \, B b^{6} x^{7} + 840 \, A a^{6} + 1848 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 4620 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 6600 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 5775 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 3080 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 924 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{9240 \, x^{11}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^12,x, algorithm="fricas")
Output:
-1/9240*(2310*B*b^6*x^7 + 840*A*a^6 + 1848*(6*B*a*b^5 + A*b^6)*x^6 + 4620* (5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 6600*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 577 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 3080*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 9 24*(B*a^6 + 6*A*a^5*b)*x)/x^11
Time = 6.01 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{12}} \, dx=\frac {- 840 A a^{6} - 2310 B b^{6} x^{7} + x^{6} \left (- 1848 A b^{6} - 11088 B a b^{5}\right ) + x^{5} \left (- 9240 A a b^{5} - 23100 B a^{2} b^{4}\right ) + x^{4} \left (- 19800 A a^{2} b^{4} - 26400 B a^{3} b^{3}\right ) + x^{3} \left (- 23100 A a^{3} b^{3} - 17325 B a^{4} b^{2}\right ) + x^{2} \left (- 15400 A a^{4} b^{2} - 6160 B a^{5} b\right ) + x \left (- 5544 A a^{5} b - 924 B a^{6}\right )}{9240 x^{11}} \] Input:
integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x**12,x)
Output:
(-840*A*a**6 - 2310*B*b**6*x**7 + x**6*(-1848*A*b**6 - 11088*B*a*b**5) + x **5*(-9240*A*a*b**5 - 23100*B*a**2*b**4) + x**4*(-19800*A*a**2*b**4 - 2640 0*B*a**3*b**3) + x**3*(-23100*A*a**3*b**3 - 17325*B*a**4*b**2) + x**2*(-15 400*A*a**4*b**2 - 6160*B*a**5*b) + x*(-5544*A*a**5*b - 924*B*a**6))/(9240* x**11)
Time = 0.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{12}} \, dx=-\frac {2310 \, B b^{6} x^{7} + 840 \, A a^{6} + 1848 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 4620 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 6600 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 5775 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 3080 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 924 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{9240 \, x^{11}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^12,x, algorithm="maxima")
Output:
-1/9240*(2310*B*b^6*x^7 + 840*A*a^6 + 1848*(6*B*a*b^5 + A*b^6)*x^6 + 4620* (5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 6600*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 577 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 3080*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 9 24*(B*a^6 + 6*A*a^5*b)*x)/x^11
Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{12}} \, dx=-\frac {2310 \, B b^{6} x^{7} + 11088 \, B a b^{5} x^{6} + 1848 \, A b^{6} x^{6} + 23100 \, B a^{2} b^{4} x^{5} + 9240 \, A a b^{5} x^{5} + 26400 \, B a^{3} b^{3} x^{4} + 19800 \, A a^{2} b^{4} x^{4} + 17325 \, B a^{4} b^{2} x^{3} + 23100 \, A a^{3} b^{3} x^{3} + 6160 \, B a^{5} b x^{2} + 15400 \, A a^{4} b^{2} x^{2} + 924 \, B a^{6} x + 5544 \, A a^{5} b x + 840 \, A a^{6}}{9240 \, x^{11}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^12,x, algorithm="giac")
Output:
-1/9240*(2310*B*b^6*x^7 + 11088*B*a*b^5*x^6 + 1848*A*b^6*x^6 + 23100*B*a^2 *b^4*x^5 + 9240*A*a*b^5*x^5 + 26400*B*a^3*b^3*x^4 + 19800*A*a^2*b^4*x^4 + 17325*B*a^4*b^2*x^3 + 23100*A*a^3*b^3*x^3 + 6160*B*a^5*b*x^2 + 15400*A*a^4 *b^2*x^2 + 924*B*a^6*x + 5544*A*a^5*b*x + 840*A*a^6)/x^11
Time = 10.86 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{12}} \, dx=-\frac {x\,\left (\frac {B\,a^6}{10}+\frac {3\,A\,b\,a^5}{5}\right )+\frac {A\,a^6}{11}+x^5\,\left (\frac {5\,B\,a^2\,b^4}{2}+A\,a\,b^5\right )+x^2\,\left (\frac {2\,B\,a^5\,b}{3}+\frac {5\,A\,a^4\,b^2}{3}\right )+x^6\,\left (\frac {A\,b^6}{5}+\frac {6\,B\,a\,b^5}{5}\right )+x^3\,\left (\frac {15\,B\,a^4\,b^2}{8}+\frac {5\,A\,a^3\,b^3}{2}\right )+x^4\,\left (\frac {20\,B\,a^3\,b^3}{7}+\frac {15\,A\,a^2\,b^4}{7}\right )+\frac {B\,b^6\,x^7}{4}}{x^{11}} \] Input:
int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3)/x^12,x)
Output:
-(x*((B*a^6)/10 + (3*A*a^5*b)/5) + (A*a^6)/11 + x^5*((5*B*a^2*b^4)/2 + A*a *b^5) + x^2*((5*A*a^4*b^2)/3 + (2*B*a^5*b)/3) + x^6*((A*b^6)/5 + (6*B*a*b^ 5)/5) + x^3*((5*A*a^3*b^3)/2 + (15*B*a^4*b^2)/8) + x^4*((15*A*a^2*b^4)/7 + (20*B*a^3*b^3)/7) + (B*b^6*x^7)/4)/x^11
Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.55 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{12}} \, dx=\frac {-330 b^{7} x^{7}-1848 a \,b^{6} x^{6}-4620 a^{2} b^{5} x^{5}-6600 a^{3} b^{4} x^{4}-5775 a^{4} b^{3} x^{3}-3080 a^{5} b^{2} x^{2}-924 a^{6} b x -120 a^{7}}{1320 x^{11}} \] Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^12,x)
Output:
( - 120*a**7 - 924*a**6*b*x - 3080*a**5*b**2*x**2 - 5775*a**4*b**3*x**3 - 6600*a**3*b**4*x**4 - 4620*a**2*b**5*x**5 - 1848*a*b**6*x**6 - 330*b**7*x* *7)/(1320*x**11)