Integrand size = 20, antiderivative size = 110 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^8} \, dx=-\frac {a^5 A}{7 x^7}-\frac {a^4 (5 A b+a B)}{4 x^4}-\frac {5 a^3 b (2 A b+a B)}{x}+5 a^2 b^2 (A b+a B) x^2+a b^3 (A b+2 a B) x^5+\frac {1}{8} b^4 (A b+5 a B) x^8+\frac {1}{11} b^5 B x^{11} \]
-1/7*a^5*A/x^7-1/4*a^4*(5*A*b+B*a)/x^4-5*a^3*b*(2*A*b+B*a)/x+5*a^2*b^2*(A* b+B*a)*x^2+a*b^3*(A*b+2*B*a)*x^5+1/8*b^4*(A*b+5*B*a)*x^8+1/11*b^5*B*x^11
Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^8} \, dx=-\frac {a^5 A}{7 x^7}-\frac {a^4 (5 A b+a B)}{4 x^4}-\frac {5 a^3 b (2 A b+a B)}{x}+5 a^2 b^2 (A b+a B) x^2+a b^3 (A b+2 a B) x^5+\frac {1}{8} b^4 (A b+5 a B) x^8+\frac {1}{11} b^5 B x^{11} \]
-1/7*(a^5*A)/x^7 - (a^4*(5*A*b + a*B))/(4*x^4) - (5*a^3*b*(2*A*b + a*B))/x + 5*a^2*b^2*(A*b + a*B)*x^2 + a*b^3*(A*b + 2*a*B)*x^5 + (b^4*(A*b + 5*a*B )*x^8)/8 + (b^5*B*x^11)/11
Time = 0.26 (sec) , antiderivative size = 110, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^8} \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (\frac {a^5 A}{x^8}+\frac {a^4 (a B+5 A b)}{x^5}+\frac {5 a^3 b (a B+2 A b)}{x^2}+10 a^2 b^2 x (a B+A b)+b^4 x^7 (5 a B+A b)+5 a b^3 x^4 (2 a B+A b)+b^5 B x^{10}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 A}{7 x^7}-\frac {a^4 (a B+5 A b)}{4 x^4}-\frac {5 a^3 b (a B+2 A b)}{x}+5 a^2 b^2 x^2 (a B+A b)+\frac {1}{8} b^4 x^8 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac {1}{11} b^5 B x^{11}\) |
-1/7*(a^5*A)/x^7 - (a^4*(5*A*b + a*B))/(4*x^4) - (5*a^3*b*(2*A*b + a*B))/x + 5*a^2*b^2*(A*b + a*B)*x^2 + a*b^3*(A*b + 2*a*B)*x^5 + (b^4*(A*b + 5*a*B )*x^8)/8 + (b^5*B*x^11)/11
3.1.40.3.1 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 = 4.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {b^{5} B \,x^{11}}{11}+\frac {A \,b^{5} x^{8}}{8}+\frac {5 B a \,b^{4} x^{8}}{8}+A a \,b^{4} x^{5}+2 B \,a^{2} b^{3} x^{5}+5 A \,a^{2} b^{3} x^{2}+5 B \,a^{3} b^{2} x^{2}-\frac {a^{5} A}{7 x^{7}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{x}-\frac {a^{4} \left (5 A b +B a \right )}{4 x^{4}}\) | \(117\) |
norman | \(\frac {-\frac {a^{5} A}{7}+\left (-\frac {5}{4} a^{4} b A -\frac {1}{4} a^{5} B \right ) x^{3}+\left (-10 a^{3} b^{2} A -5 a^{4} b B \right ) x^{6}+\left (5 a^{2} b^{3} A +5 a^{3} b^{2} B \right ) x^{9}+\left (a \,b^{4} A +2 a^{2} b^{3} B \right ) x^{12}+\left (\frac {1}{8} b^{5} A +\frac {5}{8} a \,b^{4} B \right ) x^{15}+\frac {b^{5} B \,x^{18}}{11}}{x^{7}}\) | \(121\) |
risch | \(\frac {b^{5} B \,x^{11}}{11}+\frac {A \,b^{5} x^{8}}{8}+\frac {5 B a \,b^{4} x^{8}}{8}+A a \,b^{4} x^{5}+2 B \,a^{2} b^{3} x^{5}+5 A \,a^{2} b^{3} x^{2}+5 B \,a^{3} b^{2} x^{2}+\frac {\left (-10 a^{3} b^{2} A -5 a^{4} b B \right ) x^{6}+\left (-\frac {5}{4} a^{4} b A -\frac {1}{4} a^{5} B \right ) x^{3}-\frac {a^{5} A}{7}}{x^{7}}\) | \(125\) |
gosper | \(-\frac {-56 b^{5} B \,x^{18}-77 A \,b^{5} x^{15}-385 B a \,b^{4} x^{15}-616 a A \,b^{4} x^{12}-1232 B \,a^{2} b^{3} x^{12}-3080 a^{2} A \,b^{3} x^{9}-3080 B \,a^{3} b^{2} x^{9}+6160 a^{3} A \,b^{2} x^{6}+3080 B \,a^{4} b \,x^{6}+770 a^{4} A b \,x^{3}+154 B \,a^{5} x^{3}+88 a^{5} A}{616 x^{7}}\) | \(128\) |
parallelrisch | \(\frac {56 b^{5} B \,x^{18}+77 A \,b^{5} x^{15}+385 B a \,b^{4} x^{15}+616 a A \,b^{4} x^{12}+1232 B \,a^{2} b^{3} x^{12}+3080 a^{2} A \,b^{3} x^{9}+3080 B \,a^{3} b^{2} x^{9}-6160 a^{3} A \,b^{2} x^{6}-3080 B \,a^{4} b \,x^{6}-770 a^{4} A b \,x^{3}-154 B \,a^{5} x^{3}-88 a^{5} A}{616 x^{7}}\) | \(128\) |
1/11*b^5*B*x^11+1/8*A*b^5*x^8+5/8*B*a*b^4*x^8+A*a*b^4*x^5+2*B*a^2*b^3*x^5+ 5*A*a^2*b^3*x^2+5*B*a^3*b^2*x^2-1/7*a^5*A/x^7-5*a^3*b*(2*A*b+B*a)/x-1/4*a^ 4*(5*A*b+B*a)/x^4
Time = 0.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^8} \, dx=\frac {56 \, B b^{5} x^{18} + 77 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 616 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 3080 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 3080 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 88 \, A a^{5} - 154 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{7}} \]
1/616*(56*B*b^5*x^18 + 77*(5*B*a*b^4 + A*b^5)*x^15 + 616*(2*B*a^2*b^3 + A* a*b^4)*x^12 + 3080*(B*a^3*b^2 + A*a^2*b^3)*x^9 - 3080*(B*a^4*b + 2*A*a^3*b ^2)*x^6 - 88*A*a^5 - 154*(B*a^5 + 5*A*a^4*b)*x^3)/x^7
Time = 0.59 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^8} \, dx=\frac {B b^{5} x^{11}}{11} + x^{8} \left (\frac {A b^{5}}{8} + \frac {5 B a b^{4}}{8}\right ) + x^{5} \left (A a b^{4} + 2 B a^{2} b^{3}\right ) + x^{2} \cdot \left (5 A a^{2} b^{3} + 5 B a^{3} b^{2}\right ) + \frac {- 4 A a^{5} + x^{6} \left (- 280 A a^{3} b^{2} - 140 B a^{4} b\right ) + x^{3} \left (- 35 A a^{4} b - 7 B a^{5}\right )}{28 x^{7}} \]
B*b**5*x**11/11 + x**8*(A*b**5/8 + 5*B*a*b**4/8) + x**5*(A*a*b**4 + 2*B*a* *2*b**3) + x**2*(5*A*a**2*b**3 + 5*B*a**3*b**2) + (-4*A*a**5 + x**6*(-280* A*a**3*b**2 - 140*B*a**4*b) + x**3*(-35*A*a**4*b - 7*B*a**5))/(28*x**7)
Time = 0.24 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^8} \, dx=\frac {1}{11} \, B b^{5} x^{11} + \frac {1}{8} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} - \frac {140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 4 \, A a^{5} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{28 \, x^{7}} \]
1/11*B*b^5*x^11 + 1/8*(5*B*a*b^4 + A*b^5)*x^8 + (2*B*a^2*b^3 + A*a*b^4)*x^ 5 + 5*(B*a^3*b^2 + A*a^2*b^3)*x^2 - 1/28*(140*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 4*A*a^5 + 7*(B*a^5 + 5*A*a^4*b)*x^3)/x^7
Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^8} \, dx=\frac {1}{11} \, B b^{5} x^{11} + \frac {5}{8} \, B a b^{4} x^{8} + \frac {1}{8} \, A b^{5} x^{8} + 2 \, B a^{2} b^{3} x^{5} + A a b^{4} x^{5} + 5 \, B a^{3} b^{2} x^{2} + 5 \, A a^{2} b^{3} x^{2} - \frac {140 \, B a^{4} b x^{6} + 280 \, A a^{3} b^{2} x^{6} + 7 \, B a^{5} x^{3} + 35 \, A a^{4} b x^{3} + 4 \, A a^{5}}{28 \, x^{7}} \]
1/11*B*b^5*x^11 + 5/8*B*a*b^4*x^8 + 1/8*A*b^5*x^8 + 2*B*a^2*b^3*x^5 + A*a* b^4*x^5 + 5*B*a^3*b^2*x^2 + 5*A*a^2*b^3*x^2 - 1/28*(140*B*a^4*b*x^6 + 280* A*a^3*b^2*x^6 + 7*B*a^5*x^3 + 35*A*a^4*b*x^3 + 4*A*a^5)/x^7
Time = 6.75 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^8} \, dx=x^8\,\left (\frac {A\,b^5}{8}+\frac {5\,B\,a\,b^4}{8}\right )-\frac {\frac {A\,a^5}{7}+x^6\,\left (5\,B\,a^4\,b+10\,A\,a^3\,b^2\right )+x^3\,\left (\frac {B\,a^5}{4}+\frac {5\,A\,b\,a^4}{4}\right )}{x^7}+\frac {B\,b^5\,x^{11}}{11}+5\,a^2\,b^2\,x^2\,\left (A\,b+B\,a\right )+a\,b^3\,x^5\,\left (A\,b+2\,B\,a\right ) \]