Integrand size = 20, antiderivative size = 110 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{18}} \, dx=-\frac {a^5 A}{17 x^{17}}-\frac {a^4 (5 A b+a B)}{14 x^{14}}-\frac {5 a^3 b (2 A b+a B)}{11 x^{11}}-\frac {5 a^2 b^2 (A b+a B)}{4 x^8}-\frac {a b^3 (A b+2 a B)}{x^5}-\frac {b^4 (A b+5 a B)}{2 x^2}+b^5 B x \] Output:
-1/17*a^5*A/x^17-1/14*a^4*(5*A*b+B*a)/x^14-5/11*a^3*b*(2*A*b+B*a)/x^11-5/4 *a^2*b^2*(A*b+B*a)/x^8-a*b^3*(A*b+2*B*a)/x^5-1/2*b^4*(A*b+5*B*a)/x^2+b^5*B *x
Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{18}} \, dx=-\frac {a^5 A}{17 x^{17}}-\frac {a^4 (5 A b+a B)}{14 x^{14}}-\frac {5 a^3 b (2 A b+a B)}{11 x^{11}}-\frac {5 a^2 b^2 (A b+a B)}{4 x^8}-\frac {a b^3 (A b+2 a B)}{x^5}-\frac {b^4 (A b+5 a B)}{2 x^2}+b^5 B x \] Input:
Integrate[((a + b*x^3)^5*(A + B*x^3))/x^18,x]
Output:
-1/17*(a^5*A)/x^17 - (a^4*(5*A*b + a*B))/(14*x^14) - (5*a^3*b*(2*A*b + a*B ))/(11*x^11) - (5*a^2*b^2*(A*b + a*B))/(4*x^8) - (a*b^3*(A*b + 2*a*B))/x^5 - (b^4*(A*b + 5*a*B))/(2*x^2) + b^5*B*x
Time = 0.42 (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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{18}} \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (\frac {a^5 A}{x^{18}}+\frac {a^4 (a B+5 A b)}{x^{15}}+\frac {5 a^3 b (a B+2 A b)}{x^{12}}+\frac {10 a^2 b^2 (a B+A b)}{x^9}+\frac {b^4 (5 a B+A b)}{x^3}+\frac {5 a b^3 (2 a B+A b)}{x^6}+b^5 B\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 A}{17 x^{17}}-\frac {a^4 (a B+5 A b)}{14 x^{14}}-\frac {5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac {5 a^2 b^2 (a B+A b)}{4 x^8}-\frac {b^4 (5 a B+A b)}{2 x^2}-\frac {a b^3 (2 a B+A b)}{x^5}+b^5 B x\) |
Input:
Int[((a + b*x^3)^5*(A + B*x^3))/x^18,x]
Output:
-1/17*(a^5*A)/x^17 - (a^4*(5*A*b + a*B))/(14*x^14) - (5*a^3*b*(2*A*b + a*B ))/(11*x^11) - (5*a^2*b^2*(A*b + a*B))/(4*x^8) - (a*b^3*(A*b + 2*a*B))/x^5 - (b^4*(A*b + 5*a*B))/(2*x^2) + b^5*B*x
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 = 0.55 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {a^{5} A}{17 x^{17}}-\frac {a^{4} \left (5 A b +B a \right )}{14 x^{14}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{11 x^{11}}-\frac {5 a^{2} b^{2} \left (A b +B a \right )}{4 x^{8}}-\frac {a \,b^{3} \left (A b +2 B a \right )}{x^{5}}-\frac {b^{4} \left (A b +5 B a \right )}{2 x^{2}}+b^{5} B x\) | \(101\) |
risch | \(b^{5} B x +\frac {-\frac {a^{5} A}{17}+\left (-\frac {5}{14} a^{4} b A -\frac {1}{14} a^{5} B \right ) x^{3}+\left (-\frac {10}{11} a^{3} b^{2} A -\frac {5}{11} a^{4} b B \right ) x^{6}+\left (-\frac {5}{4} a^{2} b^{3} A -\frac {5}{4} a^{3} b^{2} B \right ) x^{9}+\left (-a \,b^{4} A -2 a^{2} b^{3} B \right ) x^{12}+\left (-\frac {1}{2} b^{5} A -\frac {5}{2} a \,b^{4} B \right ) x^{15}}{x^{17}}\) | \(120\) |
norman | \(\frac {-\frac {a^{5} A}{17}+\left (-\frac {5}{14} a^{4} b A -\frac {1}{14} a^{5} B \right ) x^{3}+\left (-\frac {10}{11} a^{3} b^{2} A -\frac {5}{11} a^{4} b B \right ) x^{6}+\left (-\frac {5}{4} a^{2} b^{3} A -\frac {5}{4} a^{3} b^{2} B \right ) x^{9}+\left (-a \,b^{4} A -2 a^{2} b^{3} B \right ) x^{12}+\left (-\frac {1}{2} b^{5} A -\frac {5}{2} a \,b^{4} B \right ) x^{15}+b^{5} B \,x^{18}}{x^{17}}\) | \(121\) |
gosper | \(-\frac {-5236 b^{5} B \,x^{18}+2618 A \,b^{5} x^{15}+13090 B a \,b^{4} x^{15}+5236 a A \,b^{4} x^{12}+10472 B \,a^{2} b^{3} x^{12}+6545 a^{2} A \,b^{3} x^{9}+6545 B \,a^{3} b^{2} x^{9}+4760 a^{3} A \,b^{2} x^{6}+2380 B \,a^{4} b \,x^{6}+1870 a^{4} A b \,x^{3}+374 B \,a^{5} x^{3}+308 a^{5} A}{5236 x^{17}}\) | \(128\) |
parallelrisch | \(-\frac {-5236 b^{5} B \,x^{18}+2618 A \,b^{5} x^{15}+13090 B a \,b^{4} x^{15}+5236 a A \,b^{4} x^{12}+10472 B \,a^{2} b^{3} x^{12}+6545 a^{2} A \,b^{3} x^{9}+6545 B \,a^{3} b^{2} x^{9}+4760 a^{3} A \,b^{2} x^{6}+2380 B \,a^{4} b \,x^{6}+1870 a^{4} A b \,x^{3}+374 B \,a^{5} x^{3}+308 a^{5} A}{5236 x^{17}}\) | \(128\) |
orering | \(-\frac {-5236 b^{5} B \,x^{18}+2618 A \,b^{5} x^{15}+13090 B a \,b^{4} x^{15}+5236 a A \,b^{4} x^{12}+10472 B \,a^{2} b^{3} x^{12}+6545 a^{2} A \,b^{3} x^{9}+6545 B \,a^{3} b^{2} x^{9}+4760 a^{3} A \,b^{2} x^{6}+2380 B \,a^{4} b \,x^{6}+1870 a^{4} A b \,x^{3}+374 B \,a^{5} x^{3}+308 a^{5} A}{5236 x^{17}}\) | \(128\) |
Input:
int((b*x^3+a)^5*(B*x^3+A)/x^18,x,method=_RETURNVERBOSE)
Output:
-1/17*a^5*A/x^17-1/14*a^4*(5*A*b+B*a)/x^14-5/11*a^3*b*(2*A*b+B*a)/x^11-5/4 *a^2*b^2*(A*b+B*a)/x^8-a*b^3*(A*b+2*B*a)/x^5-1/2*b^4*(A*b+5*B*a)/x^2+b^5*B *x
Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{18}} \, dx=\frac {5236 \, B b^{5} x^{18} - 2618 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 5236 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 6545 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 2380 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 308 \, A a^{5} - 374 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{5236 \, x^{17}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^18,x, algorithm="fricas")
Output:
1/5236*(5236*B*b^5*x^18 - 2618*(5*B*a*b^4 + A*b^5)*x^15 - 5236*(2*B*a^2*b^ 3 + A*a*b^4)*x^12 - 6545*(B*a^3*b^2 + A*a^2*b^3)*x^9 - 2380*(B*a^4*b + 2*A *a^3*b^2)*x^6 - 308*A*a^5 - 374*(B*a^5 + 5*A*a^4*b)*x^3)/x^17
Time = 102.68 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{18}} \, dx=B b^{5} x + \frac {- 308 A a^{5} + x^{15} \left (- 2618 A b^{5} - 13090 B a b^{4}\right ) + x^{12} \left (- 5236 A a b^{4} - 10472 B a^{2} b^{3}\right ) + x^{9} \left (- 6545 A a^{2} b^{3} - 6545 B a^{3} b^{2}\right ) + x^{6} \left (- 4760 A a^{3} b^{2} - 2380 B a^{4} b\right ) + x^{3} \left (- 1870 A a^{4} b - 374 B a^{5}\right )}{5236 x^{17}} \] Input:
integrate((b*x**3+a)**5*(B*x**3+A)/x**18,x)
Output:
B*b**5*x + (-308*A*a**5 + x**15*(-2618*A*b**5 - 13090*B*a*b**4) + x**12*(- 5236*A*a*b**4 - 10472*B*a**2*b**3) + x**9*(-6545*A*a**2*b**3 - 6545*B*a**3 *b**2) + x**6*(-4760*A*a**3*b**2 - 2380*B*a**4*b) + x**3*(-1870*A*a**4*b - 374*B*a**5))/(5236*x**17)
Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{18}} \, dx=B b^{5} x - \frac {2618 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 5236 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 6545 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 2380 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 308 \, A a^{5} + 374 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{5236 \, x^{17}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^18,x, algorithm="maxima")
Output:
B*b^5*x - 1/5236*(2618*(5*B*a*b^4 + A*b^5)*x^15 + 5236*(2*B*a^2*b^3 + A*a* b^4)*x^12 + 6545*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 2380*(B*a^4*b + 2*A*a^3*b^2 )*x^6 + 308*A*a^5 + 374*(B*a^5 + 5*A*a^4*b)*x^3)/x^17
Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{18}} \, dx=B b^{5} x - \frac {13090 \, B a b^{4} x^{15} + 2618 \, A b^{5} x^{15} + 10472 \, B a^{2} b^{3} x^{12} + 5236 \, A a b^{4} x^{12} + 6545 \, B a^{3} b^{2} x^{9} + 6545 \, A a^{2} b^{3} x^{9} + 2380 \, B a^{4} b x^{6} + 4760 \, A a^{3} b^{2} x^{6} + 374 \, B a^{5} x^{3} + 1870 \, A a^{4} b x^{3} + 308 \, A a^{5}}{5236 \, x^{17}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^18,x, algorithm="giac")
Output:
B*b^5*x - 1/5236*(13090*B*a*b^4*x^15 + 2618*A*b^5*x^15 + 10472*B*a^2*b^3*x ^12 + 5236*A*a*b^4*x^12 + 6545*B*a^3*b^2*x^9 + 6545*A*a^2*b^3*x^9 + 2380*B *a^4*b*x^6 + 4760*A*a^3*b^2*x^6 + 374*B*a^5*x^3 + 1870*A*a^4*b*x^3 + 308*A *a^5)/x^17
Time = 0.73 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{18}} \, dx=B\,b^5\,x-\frac {\frac {A\,a^5}{17}+x^{12}\,\left (2\,B\,a^2\,b^3+A\,a\,b^4\right )+x^6\,\left (\frac {5\,B\,a^4\,b}{11}+\frac {10\,A\,a^3\,b^2}{11}\right )+x^3\,\left (\frac {B\,a^5}{14}+\frac {5\,A\,b\,a^4}{14}\right )+x^{15}\,\left (\frac {A\,b^5}{2}+\frac {5\,B\,a\,b^4}{2}\right )+x^9\,\left (\frac {5\,B\,a^3\,b^2}{4}+\frac {5\,A\,a^2\,b^3}{4}\right )}{x^{17}} \] Input:
int(((A + B*x^3)*(a + b*x^3)^5)/x^18,x)
Output:
B*b^5*x - ((A*a^5)/17 + x^12*(2*B*a^2*b^3 + A*a*b^4) + x^6*((10*A*a^3*b^2) /11 + (5*B*a^4*b)/11) + x^3*((B*a^5)/14 + (5*A*a^4*b)/14) + x^15*((A*b^5)/ 2 + (5*B*a*b^4)/2) + x^9*((5*A*a^2*b^3)/4 + (5*B*a^3*b^2)/4))/x^17
Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{18}} \, dx=\frac {2618 b^{6} x^{18}-7854 a \,b^{5} x^{15}-7854 a^{2} b^{4} x^{12}-6545 a^{3} b^{3} x^{9}-3570 a^{4} b^{2} x^{6}-1122 a^{5} b \,x^{3}-154 a^{6}}{2618 x^{17}} \] Input:
int((b*x^3+a)^5*(B*x^3+A)/x^18,x)
Output:
( - 154*a**6 - 1122*a**5*b*x**3 - 3570*a**4*b**2*x**6 - 6545*a**3*b**3*x** 9 - 7854*a**2*b**4*x**12 - 7854*a*b**5*x**15 + 2618*b**6*x**18)/(2618*x**1 7)