Integrand size = 20, antiderivative size = 115 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{17}} \, dx=-\frac {a^5 A}{16 x^{16}}-\frac {a^4 (5 A b+a B)}{13 x^{13}}-\frac {a^3 b (2 A b+a B)}{2 x^{10}}-\frac {10 a^2 b^2 (A b+a B)}{7 x^7}-\frac {5 a b^3 (A b+2 a B)}{4 x^4}-\frac {b^4 (A b+5 a B)}{x}+\frac {1}{2} b^5 B x^2 \]
-1/16*a^5*A/x^16-1/13*a^4*(5*A*b+B*a)/x^13-1/2*a^3*b*(2*A*b+B*a)/x^10-10/7 *a^2*b^2*(A*b+B*a)/x^7-5/4*a*b^3*(A*b+2*B*a)/x^4-b^4*(A*b+5*B*a)/x+1/2*b^5 *B*x^2
Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{17}} \, dx=-\frac {-728 b^5 x^{15} \left (-2 A+B x^3\right )+1820 a b^4 x^{12} \left (A+4 B x^3\right )+520 a^2 b^3 x^9 \left (4 A+7 B x^3\right )+208 a^3 b^2 x^6 \left (7 A+10 B x^3\right )+56 a^4 b x^3 \left (10 A+13 B x^3\right )+7 a^5 \left (13 A+16 B x^3\right )}{1456 x^{16}} \]
-1/1456*(-728*b^5*x^15*(-2*A + B*x^3) + 1820*a*b^4*x^12*(A + 4*B*x^3) + 52 0*a^2*b^3*x^9*(4*A + 7*B*x^3) + 208*a^3*b^2*x^6*(7*A + 10*B*x^3) + 56*a^4* b*x^3*(10*A + 13*B*x^3) + 7*a^5*(13*A + 16*B*x^3))/x^16
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {950, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{17}} \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (\frac {a^5 A}{x^{17}}+\frac {a^4 (a B+5 A b)}{x^{14}}+\frac {5 a^3 b (a B+2 A b)}{x^{11}}+\frac {10 a^2 b^2 (a B+A b)}{x^8}+\frac {b^4 (5 a B+A b)}{x^2}+\frac {5 a b^3 (2 a B+A b)}{x^5}+b^5 B x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 A}{16 x^{16}}-\frac {a^4 (a B+5 A b)}{13 x^{13}}-\frac {a^3 b (a B+2 A b)}{2 x^{10}}-\frac {10 a^2 b^2 (a B+A b)}{7 x^7}-\frac {b^4 (5 a B+A b)}{x}-\frac {5 a b^3 (2 a B+A b)}{4 x^4}+\frac {1}{2} b^5 B x^2\) |
-1/16*(a^5*A)/x^16 - (a^4*(5*A*b + a*B))/(13*x^13) - (a^3*b*(2*A*b + a*B)) /(2*x^10) - (10*a^2*b^2*(A*b + a*B))/(7*x^7) - (5*a*b^3*(A*b + 2*a*B))/(4* x^4) - (b^4*(A*b + 5*a*B))/x + (b^5*B*x^2)/2
3.1.49.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^ n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGt Q[p, 0] && IGtQ[q, 0]
Time = 4.34 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {a^{5} A}{16 x^{16}}-\frac {a^{4} \left (5 A b +B a \right )}{13 x^{13}}-\frac {a^{3} b \left (2 A b +B a \right )}{2 x^{10}}-\frac {10 a^{2} b^{2} \left (A b +B a \right )}{7 x^{7}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{4 x^{4}}-\frac {b^{4} \left (A b +5 B a \right )}{x}+\frac {b^{5} B \,x^{2}}{2}\) | \(104\) |
norman | \(\frac {-\frac {a^{5} A}{16}+\left (-\frac {5}{13} a^{4} b A -\frac {1}{13} a^{5} B \right ) x^{3}+\left (-a^{3} b^{2} A -\frac {1}{2} a^{4} b B \right ) x^{6}+\left (-\frac {10}{7} a^{2} b^{3} A -\frac {10}{7} a^{3} b^{2} B \right ) x^{9}+\left (-\frac {5}{4} a \,b^{4} A -\frac {5}{2} a^{2} b^{3} B \right ) x^{12}+\left (-b^{5} A -5 a \,b^{4} B \right ) x^{15}+\frac {b^{5} B \,x^{18}}{2}}{x^{16}}\) | \(122\) |
risch | \(\frac {b^{5} B \,x^{2}}{2}+\frac {-\frac {a^{5} A}{16}+\left (-\frac {5}{13} a^{4} b A -\frac {1}{13} a^{5} B \right ) x^{3}+\left (-a^{3} b^{2} A -\frac {1}{2} a^{4} b B \right ) x^{6}+\left (-\frac {10}{7} a^{2} b^{3} A -\frac {10}{7} a^{3} b^{2} B \right ) x^{9}+\left (-\frac {5}{4} a \,b^{4} A -\frac {5}{2} a^{2} b^{3} B \right ) x^{12}+\left (-b^{5} A -5 a \,b^{4} B \right ) x^{15}}{x^{16}}\) | \(123\) |
gosper | \(-\frac {-728 b^{5} B \,x^{18}+1456 A \,b^{5} x^{15}+7280 B a \,b^{4} x^{15}+1820 a A \,b^{4} x^{12}+3640 B \,a^{2} b^{3} x^{12}+2080 a^{2} A \,b^{3} x^{9}+2080 B \,a^{3} b^{2} x^{9}+1456 a^{3} A \,b^{2} x^{6}+728 B \,a^{4} b \,x^{6}+560 a^{4} A b \,x^{3}+112 B \,a^{5} x^{3}+91 a^{5} A}{1456 x^{16}}\) | \(128\) |
parallelrisch | \(-\frac {-728 b^{5} B \,x^{18}+1456 A \,b^{5} x^{15}+7280 B a \,b^{4} x^{15}+1820 a A \,b^{4} x^{12}+3640 B \,a^{2} b^{3} x^{12}+2080 a^{2} A \,b^{3} x^{9}+2080 B \,a^{3} b^{2} x^{9}+1456 a^{3} A \,b^{2} x^{6}+728 B \,a^{4} b \,x^{6}+560 a^{4} A b \,x^{3}+112 B \,a^{5} x^{3}+91 a^{5} A}{1456 x^{16}}\) | \(128\) |
-1/16*a^5*A/x^16-1/13*a^4*(5*A*b+B*a)/x^13-1/2*a^3*b*(2*A*b+B*a)/x^10-10/7 *a^2*b^2*(A*b+B*a)/x^7-5/4*a*b^3*(A*b+2*B*a)/x^4-b^4*(A*b+5*B*a)/x+1/2*b^5 *B*x^2
Time = 0.24 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{17}} \, dx=\frac {728 \, B b^{5} x^{18} - 1456 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 1820 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 2080 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 728 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 91 \, A a^{5} - 112 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{1456 \, x^{16}} \]
1/1456*(728*B*b^5*x^18 - 1456*(5*B*a*b^4 + A*b^5)*x^15 - 1820*(2*B*a^2*b^3 + A*a*b^4)*x^12 - 2080*(B*a^3*b^2 + A*a^2*b^3)*x^9 - 728*(B*a^4*b + 2*A*a ^3*b^2)*x^6 - 91*A*a^5 - 112*(B*a^5 + 5*A*a^4*b)*x^3)/x^16
Timed out. \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{17}} \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{17}} \, dx=\frac {1}{2} \, B b^{5} x^{2} - \frac {1456 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 1820 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 2080 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 728 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 91 \, A a^{5} + 112 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{1456 \, x^{16}} \]
1/2*B*b^5*x^2 - 1/1456*(1456*(5*B*a*b^4 + A*b^5)*x^15 + 1820*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 2080*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 728*(B*a^4*b + 2*A*a^ 3*b^2)*x^6 + 91*A*a^5 + 112*(B*a^5 + 5*A*a^4*b)*x^3)/x^16
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{17}} \, dx=\frac {1}{2} \, B b^{5} x^{2} - \frac {7280 \, B a b^{4} x^{15} + 1456 \, A b^{5} x^{15} + 3640 \, B a^{2} b^{3} x^{12} + 1820 \, A a b^{4} x^{12} + 2080 \, B a^{3} b^{2} x^{9} + 2080 \, A a^{2} b^{3} x^{9} + 728 \, B a^{4} b x^{6} + 1456 \, A a^{3} b^{2} x^{6} + 112 \, B a^{5} x^{3} + 560 \, A a^{4} b x^{3} + 91 \, A a^{5}}{1456 \, x^{16}} \]
1/2*B*b^5*x^2 - 1/1456*(7280*B*a*b^4*x^15 + 1456*A*b^5*x^15 + 3640*B*a^2*b ^3*x^12 + 1820*A*a*b^4*x^12 + 2080*B*a^3*b^2*x^9 + 2080*A*a^2*b^3*x^9 + 72 8*B*a^4*b*x^6 + 1456*A*a^3*b^2*x^6 + 112*B*a^5*x^3 + 560*A*a^4*b*x^3 + 91* A*a^5)/x^16
Time = 6.59 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{17}} \, dx=\frac {B\,b^5\,x^2}{2}-\frac {\frac {A\,a^5}{16}+x^6\,\left (\frac {B\,a^4\,b}{2}+A\,a^3\,b^2\right )+x^{12}\,\left (\frac {5\,B\,a^2\,b^3}{2}+\frac {5\,A\,a\,b^4}{4}\right )+x^3\,\left (\frac {B\,a^5}{13}+\frac {5\,A\,b\,a^4}{13}\right )+x^{15}\,\left (A\,b^5+5\,B\,a\,b^4\right )+x^9\,\left (\frac {10\,B\,a^3\,b^2}{7}+\frac {10\,A\,a^2\,b^3}{7}\right )}{x^{16}} \]