Integrand size = 27, antiderivative size = 143 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{13}} \, dx=-\frac {a^6 A}{12 x^{12}}-\frac {a^5 (6 A b+a B)}{11 x^{11}}-\frac {3 a^4 b (5 A b+2 a B)}{10 x^{10}}-\frac {5 a^3 b^2 (4 A b+3 a B)}{9 x^9}-\frac {5 a^2 b^3 (3 A b+4 a B)}{8 x^8}-\frac {3 a b^4 (2 A b+5 a B)}{7 x^7}-\frac {b^5 (A b+6 a B)}{6 x^6}-\frac {b^6 B}{5 x^5} \]
-1/12*a^6*A/x^12-1/11*a^5*(6*A*b+B*a)/x^11-3/10*a^4*b*(5*A*b+2*B*a)/x^10-5 /9*a^3*b^2*(4*A*b+3*B*a)/x^9-5/8*a^2*b^3*(3*A*b+4*B*a)/x^8-3/7*a*b^4*(2*A* b+5*B*a)/x^7-1/6*b^5*(A*b+6*B*a)/x^6-1/5*b^6*B/x^5
Time = 0.02 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{13}} \, dx=-\frac {924 b^6 x^6 (5 A+6 B x)+3960 a b^5 x^5 (6 A+7 B x)+7425 a^2 b^4 x^4 (7 A+8 B x)+7700 a^3 b^3 x^3 (8 A+9 B x)+4620 a^4 b^2 x^2 (9 A+10 B x)+1512 a^5 b x (10 A+11 B x)+210 a^6 (11 A+12 B x)}{27720 x^{12}} \]
-1/27720*(924*b^6*x^6*(5*A + 6*B*x) + 3960*a*b^5*x^5*(6*A + 7*B*x) + 7425* a^2*b^4*x^4*(7*A + 8*B*x) + 7700*a^3*b^3*x^3*(8*A + 9*B*x) + 4620*a^4*b^2* x^2*(9*A + 10*B*x) + 1512*a^5*b*x*(10*A + 11*B*x) + 210*a^6*(11*A + 12*B*x ))/x^12
Time = 0.30 (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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x)}{x^{13}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int \frac {b^6 (a+b x)^6 (A+B x)}{x^{13}}dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{x^{13}}dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (\frac {a^6 A}{x^{13}}+\frac {a^5 (a B+6 A b)}{x^{12}}+\frac {3 a^4 b (2 a B+5 A b)}{x^{11}}+\frac {5 a^3 b^2 (3 a B+4 A b)}{x^{10}}+\frac {5 a^2 b^3 (4 a B+3 A b)}{x^9}+\frac {b^5 (6 a B+A b)}{x^7}+\frac {3 a b^4 (5 a B+2 A b)}{x^8}+\frac {b^6 B}{x^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^6 A}{12 x^{12}}-\frac {a^5 (a B+6 A b)}{11 x^{11}}-\frac {3 a^4 b (2 a B+5 A b)}{10 x^{10}}-\frac {5 a^3 b^2 (3 a B+4 A b)}{9 x^9}-\frac {5 a^2 b^3 (4 a B+3 A b)}{8 x^8}-\frac {b^5 (6 a B+A b)}{6 x^6}-\frac {3 a b^4 (5 a B+2 A b)}{7 x^7}-\frac {b^6 B}{5 x^5}\) |
-1/12*(a^6*A)/x^12 - (a^5*(6*A*b + a*B))/(11*x^11) - (3*a^4*b*(5*A*b + 2*a *B))/(10*x^10) - (5*a^3*b^2*(4*A*b + 3*a*B))/(9*x^9) - (5*a^2*b^3*(3*A*b + 4*a*B))/(8*x^8) - (3*a*b^4*(2*A*b + 5*a*B))/(7*x^7) - (b^5*(A*b + 6*a*B)) /(6*x^6) - (b^6*B)/(5*x^5)
3.6.57.3.1 Defintions of rubi rules used
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 = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {a^{6} A}{12 x^{12}}-\frac {a^{5} \left (6 A b +B a \right )}{11 x^{11}}-\frac {3 a^{4} b \left (5 A b +2 B a \right )}{10 x^{10}}-\frac {5 a^{3} b^{2} \left (4 A b +3 B a \right )}{9 x^{9}}-\frac {5 a^{2} b^{3} \left (3 A b +4 B a \right )}{8 x^{8}}-\frac {3 a \,b^{4} \left (2 A b +5 B a \right )}{7 x^{7}}-\frac {b^{5} \left (A b +6 B a \right )}{6 x^{6}}-\frac {b^{6} B}{5 x^{5}}\) | \(128\) |
norman | \(\frac {-\frac {b^{6} B \,x^{7}}{5}+\left (-\frac {1}{6} A \,b^{6}-B a \,b^{5}\right ) x^{6}+\left (-\frac {6}{7} A a \,b^{5}-\frac {15}{7} B \,b^{4} a^{2}\right ) x^{5}+\left (-\frac {15}{8} A \,b^{4} a^{2}-\frac {5}{2} B \,a^{3} b^{3}\right ) x^{4}+\left (-\frac {20}{9} A \,a^{3} b^{3}-\frac {5}{3} B \,a^{4} b^{2}\right ) x^{3}+\left (-\frac {3}{2} A \,a^{4} b^{2}-\frac {3}{5} B \,a^{5} b \right ) x^{2}+\left (-\frac {6}{11} A \,a^{5} b -\frac {1}{11} B \,a^{6}\right ) x -\frac {A \,a^{6}}{12}}{x^{12}}\) | \(143\) |
risch | \(\frac {-\frac {b^{6} B \,x^{7}}{5}+\left (-\frac {1}{6} A \,b^{6}-B a \,b^{5}\right ) x^{6}+\left (-\frac {6}{7} A a \,b^{5}-\frac {15}{7} B \,b^{4} a^{2}\right ) x^{5}+\left (-\frac {15}{8} A \,b^{4} a^{2}-\frac {5}{2} B \,a^{3} b^{3}\right ) x^{4}+\left (-\frac {20}{9} A \,a^{3} b^{3}-\frac {5}{3} B \,a^{4} b^{2}\right ) x^{3}+\left (-\frac {3}{2} A \,a^{4} b^{2}-\frac {3}{5} B \,a^{5} b \right ) x^{2}+\left (-\frac {6}{11} A \,a^{5} b -\frac {1}{11} B \,a^{6}\right ) x -\frac {A \,a^{6}}{12}}{x^{12}}\) | \(143\) |
gosper | \(-\frac {5544 b^{6} B \,x^{7}+4620 A \,b^{6} x^{6}+27720 x^{6} B a \,b^{5}+23760 a A \,b^{5} x^{5}+59400 x^{5} B \,b^{4} a^{2}+51975 a^{2} A \,b^{4} x^{4}+69300 x^{4} B \,a^{3} b^{3}+61600 a^{3} A \,b^{3} x^{3}+46200 x^{3} B \,a^{4} b^{2}+41580 a^{4} A \,b^{2} x^{2}+16632 x^{2} B \,a^{5} b +15120 a^{5} A b x +2520 x B \,a^{6}+2310 A \,a^{6}}{27720 x^{12}}\) | \(148\) |
parallelrisch | \(-\frac {5544 b^{6} B \,x^{7}+4620 A \,b^{6} x^{6}+27720 x^{6} B a \,b^{5}+23760 a A \,b^{5} x^{5}+59400 x^{5} B \,b^{4} a^{2}+51975 a^{2} A \,b^{4} x^{4}+69300 x^{4} B \,a^{3} b^{3}+61600 a^{3} A \,b^{3} x^{3}+46200 x^{3} B \,a^{4} b^{2}+41580 a^{4} A \,b^{2} x^{2}+16632 x^{2} B \,a^{5} b +15120 a^{5} A b x +2520 x B \,a^{6}+2310 A \,a^{6}}{27720 x^{12}}\) | \(148\) |
-1/12*a^6*A/x^12-1/11*a^5*(6*A*b+B*a)/x^11-3/10*a^4*b*(5*A*b+2*B*a)/x^10-5 /9*a^3*b^2*(4*A*b+3*B*a)/x^9-5/8*a^2*b^3*(3*A*b+4*B*a)/x^8-3/7*a*b^4*(2*A* b+5*B*a)/x^7-1/6*b^5*(A*b+6*B*a)/x^6-1/5*b^6*B/x^5
Time = 0.31 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{13}} \, dx=-\frac {5544 \, B b^{6} x^{7} + 2310 \, A a^{6} + 4620 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 11880 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 17325 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 15400 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 8316 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 2520 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{27720 \, x^{12}} \]
-1/27720*(5544*B*b^6*x^7 + 2310*A*a^6 + 4620*(6*B*a*b^5 + A*b^6)*x^6 + 118 80*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 17325*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 15400*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 8316*(2*B*a^5*b + 5*A*a^4*b^2)*x^ 2 + 2520*(B*a^6 + 6*A*a^5*b)*x)/x^12
Time = 41.16 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{13}} \, dx=\frac {- 2310 A a^{6} - 5544 B b^{6} x^{7} + x^{6} \left (- 4620 A b^{6} - 27720 B a b^{5}\right ) + x^{5} \left (- 23760 A a b^{5} - 59400 B a^{2} b^{4}\right ) + x^{4} \left (- 51975 A a^{2} b^{4} - 69300 B a^{3} b^{3}\right ) + x^{3} \left (- 61600 A a^{3} b^{3} - 46200 B a^{4} b^{2}\right ) + x^{2} \left (- 41580 A a^{4} b^{2} - 16632 B a^{5} b\right ) + x \left (- 15120 A a^{5} b - 2520 B a^{6}\right )}{27720 x^{12}} \]
(-2310*A*a**6 - 5544*B*b**6*x**7 + x**6*(-4620*A*b**6 - 27720*B*a*b**5) + x**5*(-23760*A*a*b**5 - 59400*B*a**2*b**4) + x**4*(-51975*A*a**2*b**4 - 69 300*B*a**3*b**3) + x**3*(-61600*A*a**3*b**3 - 46200*B*a**4*b**2) + x**2*(- 41580*A*a**4*b**2 - 16632*B*a**5*b) + x*(-15120*A*a**5*b - 2520*B*a**6))/( 27720*x**12)
Time = 0.23 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{13}} \, dx=-\frac {5544 \, B b^{6} x^{7} + 2310 \, A a^{6} + 4620 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 11880 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 17325 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 15400 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 8316 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 2520 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{27720 \, x^{12}} \]
-1/27720*(5544*B*b^6*x^7 + 2310*A*a^6 + 4620*(6*B*a*b^5 + A*b^6)*x^6 + 118 80*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 17325*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 15400*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 8316*(2*B*a^5*b + 5*A*a^4*b^2)*x^ 2 + 2520*(B*a^6 + 6*A*a^5*b)*x)/x^12
Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{13}} \, dx=-\frac {5544 \, B b^{6} x^{7} + 27720 \, B a b^{5} x^{6} + 4620 \, A b^{6} x^{6} + 59400 \, B a^{2} b^{4} x^{5} + 23760 \, A a b^{5} x^{5} + 69300 \, B a^{3} b^{3} x^{4} + 51975 \, A a^{2} b^{4} x^{4} + 46200 \, B a^{4} b^{2} x^{3} + 61600 \, A a^{3} b^{3} x^{3} + 16632 \, B a^{5} b x^{2} + 41580 \, A a^{4} b^{2} x^{2} + 2520 \, B a^{6} x + 15120 \, A a^{5} b x + 2310 \, A a^{6}}{27720 \, x^{12}} \]
-1/27720*(5544*B*b^6*x^7 + 27720*B*a*b^5*x^6 + 4620*A*b^6*x^6 + 59400*B*a^ 2*b^4*x^5 + 23760*A*a*b^5*x^5 + 69300*B*a^3*b^3*x^4 + 51975*A*a^2*b^4*x^4 + 46200*B*a^4*b^2*x^3 + 61600*A*a^3*b^3*x^3 + 16632*B*a^5*b*x^2 + 41580*A* a^4*b^2*x^2 + 2520*B*a^6*x + 15120*A*a^5*b*x + 2310*A*a^6)/x^12
Time = 0.08 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{13}} \, dx=-\frac {x\,\left (\frac {B\,a^6}{11}+\frac {6\,A\,b\,a^5}{11}\right )+\frac {A\,a^6}{12}+x^2\,\left (\frac {3\,B\,a^5\,b}{5}+\frac {3\,A\,a^4\,b^2}{2}\right )+x^5\,\left (\frac {15\,B\,a^2\,b^4}{7}+\frac {6\,A\,a\,b^5}{7}\right )+x^6\,\left (\frac {A\,b^6}{6}+B\,a\,b^5\right )+x^4\,\left (\frac {5\,B\,a^3\,b^3}{2}+\frac {15\,A\,a^2\,b^4}{8}\right )+x^3\,\left (\frac {5\,B\,a^4\,b^2}{3}+\frac {20\,A\,a^3\,b^3}{9}\right )+\frac {B\,b^6\,x^7}{5}}{x^{12}} \]