Integrand size = 20, antiderivative size = 115 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{11}} \, dx=-\frac {a^5 A}{10 x^{10}}-\frac {a^4 (5 A b+a B)}{7 x^7}-\frac {5 a^3 b (2 A b+a B)}{4 x^4}-\frac {10 a^2 b^2 (A b+a B)}{x}+\frac {5}{2} a b^3 (A b+2 a B) x^2+\frac {1}{5} b^4 (A b+5 a B) x^5+\frac {1}{8} b^5 B x^8 \] Output:
-1/10*a^5*A/x^10-1/7*a^4*(5*A*b+B*a)/x^7-5/4*a^3*b*(2*A*b+B*a)/x^4-10*a^2* b^2*(A*b+B*a)/x+5/2*a*b^3*(A*b+2*B*a)*x^2+1/5*b^4*(A*b+5*B*a)*x^5+1/8*b^5* B*x^8
Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{11}} \, dx=\frac {1400 a^2 b^3 x^9 \left (-2 A+B x^3\right )+140 a b^4 x^{12} \left (5 A+2 B x^3\right )-700 a^3 b^2 x^6 \left (A+4 B x^3\right )+7 b^5 x^{15} \left (8 A+5 B x^3\right )-50 a^4 b x^3 \left (4 A+7 B x^3\right )-4 a^5 \left (7 A+10 B x^3\right )}{280 x^{10}} \] Input:
Integrate[((a + b*x^3)^5*(A + B*x^3))/x^11,x]
Output:
(1400*a^2*b^3*x^9*(-2*A + B*x^3) + 140*a*b^4*x^12*(5*A + 2*B*x^3) - 700*a^ 3*b^2*x^6*(A + 4*B*x^3) + 7*b^5*x^15*(8*A + 5*B*x^3) - 50*a^4*b*x^3*(4*A + 7*B*x^3) - 4*a^5*(7*A + 10*B*x^3))/(280*x^10)
Time = 0.44 (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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{11}} \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (\frac {a^5 A}{x^{11}}+\frac {a^4 (a B+5 A b)}{x^8}+\frac {5 a^3 b (a B+2 A b)}{x^5}+\frac {10 a^2 b^2 (a B+A b)}{x^2}+b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+b^5 B x^7\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 A}{10 x^{10}}-\frac {a^4 (a B+5 A b)}{7 x^7}-\frac {5 a^3 b (a B+2 A b)}{4 x^4}-\frac {10 a^2 b^2 (a B+A b)}{x}+\frac {1}{5} b^4 x^5 (5 a B+A b)+\frac {5}{2} a b^3 x^2 (2 a B+A b)+\frac {1}{8} b^5 B x^8\) |
Input:
Int[((a + b*x^3)^5*(A + B*x^3))/x^11,x]
Output:
-1/10*(a^5*A)/x^10 - (a^4*(5*A*b + a*B))/(7*x^7) - (5*a^3*b*(2*A*b + a*B)) /(4*x^4) - (10*a^2*b^2*(A*b + a*B))/x + (5*a*b^3*(A*b + 2*a*B)*x^2)/2 + (b ^4*(A*b + 5*a*B)*x^5)/5 + (b^5*B*x^8)/8
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 = 0.57 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {b^{5} B \,x^{8}}{8}+\frac {A \,b^{5} x^{5}}{5}+B a \,b^{4} x^{5}+\frac {5 A a \,b^{4} x^{2}}{2}+5 B \,a^{2} b^{3} x^{2}-\frac {a^{4} \left (5 A b +B a \right )}{7 x^{7}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{4 x^{4}}-\frac {a^{5} A}{10 x^{10}}-\frac {10 a^{2} b^{2} \left (A b +B a \right )}{x}\) | \(111\) |
norman | \(\frac {-\frac {a^{5} A}{10}+\left (-\frac {5}{7} a^{4} b A -\frac {1}{7} a^{5} B \right ) x^{3}+\left (-\frac {5}{2} a^{3} b^{2} A -\frac {5}{4} a^{4} b B \right ) x^{6}+\left (-10 a^{2} b^{3} A -10 a^{3} b^{2} B \right ) x^{9}+\left (\frac {5}{2} a \,b^{4} A +5 a^{2} b^{3} B \right ) x^{12}+\left (\frac {1}{5} b^{5} A +a \,b^{4} B \right ) x^{15}+\frac {b^{5} B \,x^{18}}{8}}{x^{10}}\) | \(121\) |
risch | \(\frac {b^{5} B \,x^{8}}{8}+\frac {A \,b^{5} x^{5}}{5}+B a \,b^{4} x^{5}+\frac {5 A a \,b^{4} x^{2}}{2}+5 B \,a^{2} b^{3} x^{2}+\frac {\left (-10 a^{2} b^{3} A -10 a^{3} b^{2} B \right ) x^{9}+\left (-\frac {5}{2} a^{3} b^{2} A -\frac {5}{4} a^{4} b B \right ) x^{6}+\left (-\frac {5}{7} a^{4} b A -\frac {1}{7} a^{5} B \right ) x^{3}-\frac {a^{5} A}{10}}{x^{10}}\) | \(124\) |
gosper | \(-\frac {-35 b^{5} B \,x^{18}-56 A \,b^{5} x^{15}-280 B a \,b^{4} x^{15}-700 a A \,b^{4} x^{12}-1400 B \,a^{2} b^{3} x^{12}+2800 a^{2} A \,b^{3} x^{9}+2800 B \,a^{3} b^{2} x^{9}+700 a^{3} A \,b^{2} x^{6}+350 B \,a^{4} b \,x^{6}+200 a^{4} A b \,x^{3}+40 B \,a^{5} x^{3}+28 a^{5} A}{280 x^{10}}\) | \(128\) |
parallelrisch | \(\frac {35 b^{5} B \,x^{18}+56 A \,b^{5} x^{15}+280 B a \,b^{4} x^{15}+700 a A \,b^{4} x^{12}+1400 B \,a^{2} b^{3} x^{12}-2800 a^{2} A \,b^{3} x^{9}-2800 B \,a^{3} b^{2} x^{9}-700 a^{3} A \,b^{2} x^{6}-350 B \,a^{4} b \,x^{6}-200 a^{4} A b \,x^{3}-40 B \,a^{5} x^{3}-28 a^{5} A}{280 x^{10}}\) | \(128\) |
orering | \(-\frac {-35 b^{5} B \,x^{18}-56 A \,b^{5} x^{15}-280 B a \,b^{4} x^{15}-700 a A \,b^{4} x^{12}-1400 B \,a^{2} b^{3} x^{12}+2800 a^{2} A \,b^{3} x^{9}+2800 B \,a^{3} b^{2} x^{9}+700 a^{3} A \,b^{2} x^{6}+350 B \,a^{4} b \,x^{6}+200 a^{4} A b \,x^{3}+40 B \,a^{5} x^{3}+28 a^{5} A}{280 x^{10}}\) | \(128\) |
Input:
int((b*x^3+a)^5*(B*x^3+A)/x^11,x,method=_RETURNVERBOSE)
Output:
1/8*b^5*B*x^8+1/5*A*b^5*x^5+B*a*b^4*x^5+5/2*A*a*b^4*x^2+5*B*a^2*b^3*x^2-1/ 7*a^4*(5*A*b+B*a)/x^7-5/4*a^3*b*(2*A*b+B*a)/x^4-1/10*a^5*A/x^10-10*a^2*b^2 *(A*b+B*a)/x
Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{11}} \, dx=\frac {35 \, B b^{5} x^{18} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 700 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 2800 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 350 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 28 \, A a^{5} - 40 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{280 \, x^{10}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^11,x, algorithm="fricas")
Output:
1/280*(35*B*b^5*x^18 + 56*(5*B*a*b^4 + A*b^5)*x^15 + 700*(2*B*a^2*b^3 + A* a*b^4)*x^12 - 2800*(B*a^3*b^2 + A*a^2*b^3)*x^9 - 350*(B*a^4*b + 2*A*a^3*b^ 2)*x^6 - 28*A*a^5 - 40*(B*a^5 + 5*A*a^4*b)*x^3)/x^10
Time = 1.82 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{11}} \, dx=\frac {B b^{5} x^{8}}{8} + x^{5} \left (\frac {A b^{5}}{5} + B a b^{4}\right ) + x^{2} \cdot \left (\frac {5 A a b^{4}}{2} + 5 B a^{2} b^{3}\right ) + \frac {- 14 A a^{5} + x^{9} \left (- 1400 A a^{2} b^{3} - 1400 B a^{3} b^{2}\right ) + x^{6} \left (- 350 A a^{3} b^{2} - 175 B a^{4} b\right ) + x^{3} \left (- 100 A a^{4} b - 20 B a^{5}\right )}{140 x^{10}} \] Input:
integrate((b*x**3+a)**5*(B*x**3+A)/x**11,x)
Output:
B*b**5*x**8/8 + x**5*(A*b**5/5 + B*a*b**4) + x**2*(5*A*a*b**4/2 + 5*B*a**2 *b**3) + (-14*A*a**5 + x**9*(-1400*A*a**2*b**3 - 1400*B*a**3*b**2) + x**6* (-350*A*a**3*b**2 - 175*B*a**4*b) + x**3*(-100*A*a**4*b - 20*B*a**5))/(140 *x**10)
Time = 0.03 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{11}} \, dx=\frac {1}{8} \, B b^{5} x^{8} + \frac {1}{5} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac {5}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{2} - \frac {1400 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 175 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 14 \, A a^{5} + 20 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{140 \, x^{10}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^11,x, algorithm="maxima")
Output:
1/8*B*b^5*x^8 + 1/5*(5*B*a*b^4 + A*b^5)*x^5 + 5/2*(2*B*a^2*b^3 + A*a*b^4)* x^2 - 1/140*(1400*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 175*(B*a^4*b + 2*A*a^3*b^2 )*x^6 + 14*A*a^5 + 20*(B*a^5 + 5*A*a^4*b)*x^3)/x^10
Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{11}} \, dx=\frac {1}{8} \, B b^{5} x^{8} + B a b^{4} x^{5} + \frac {1}{5} \, A b^{5} x^{5} + 5 \, B a^{2} b^{3} x^{2} + \frac {5}{2} \, A a b^{4} x^{2} - \frac {1400 \, B a^{3} b^{2} x^{9} + 1400 \, A a^{2} b^{3} x^{9} + 175 \, B a^{4} b x^{6} + 350 \, A a^{3} b^{2} x^{6} + 20 \, B a^{5} x^{3} + 100 \, A a^{4} b x^{3} + 14 \, A a^{5}}{140 \, x^{10}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^11,x, algorithm="giac")
Output:
1/8*B*b^5*x^8 + B*a*b^4*x^5 + 1/5*A*b^5*x^5 + 5*B*a^2*b^3*x^2 + 5/2*A*a*b^ 4*x^2 - 1/140*(1400*B*a^3*b^2*x^9 + 1400*A*a^2*b^3*x^9 + 175*B*a^4*b*x^6 + 350*A*a^3*b^2*x^6 + 20*B*a^5*x^3 + 100*A*a^4*b*x^3 + 14*A*a^5)/x^10
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{11}} \, dx=x^5\,\left (\frac {A\,b^5}{5}+B\,a\,b^4\right )-\frac {\frac {A\,a^5}{10}+x^6\,\left (\frac {5\,B\,a^4\,b}{4}+\frac {5\,A\,a^3\,b^2}{2}\right )+x^3\,\left (\frac {B\,a^5}{7}+\frac {5\,A\,b\,a^4}{7}\right )+x^9\,\left (10\,B\,a^3\,b^2+10\,A\,a^2\,b^3\right )}{x^{10}}+\frac {B\,b^5\,x^8}{8}+\frac {5\,a\,b^3\,x^2\,\left (A\,b+2\,B\,a\right )}{2} \] Input:
int(((A + B*x^3)*(a + b*x^3)^5)/x^11,x)
Output:
x^5*((A*b^5)/5 + B*a*b^4) - ((A*a^5)/10 + x^6*((5*A*a^3*b^2)/2 + (5*B*a^4* b)/4) + x^3*((B*a^5)/7 + (5*A*a^4*b)/7) + x^9*(10*A*a^2*b^3 + 10*B*a^3*b^2 ))/x^10 + (B*b^5*x^8)/8 + (5*a*b^3*x^2*(A*b + 2*B*a))/2
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.61 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{11}} \, dx=\frac {35 b^{6} x^{18}+336 a \,b^{5} x^{15}+2100 a^{2} b^{4} x^{12}-5600 a^{3} b^{3} x^{9}-1050 a^{4} b^{2} x^{6}-240 a^{5} b \,x^{3}-28 a^{6}}{280 x^{10}} \] Input:
int((b*x^3+a)^5*(B*x^3+A)/x^11,x)
Output:
( - 28*a**6 - 240*a**5*b*x**3 - 1050*a**4*b**2*x**6 - 5600*a**3*b**3*x**9 + 2100*a**2*b**4*x**12 + 336*a*b**5*x**15 + 35*b**6*x**18)/(280*x**10)