Integrand size = 20, antiderivative size = 113 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^6} \, dx=-\frac {a^5 A}{5 x^5}-\frac {a^4 (5 A b+a B)}{2 x^2}+5 a^3 b (2 A b+a B) x+\frac {5}{2} a^2 b^2 (A b+a B) x^4+\frac {5}{7} a b^3 (A b+2 a B) x^7+\frac {1}{10} b^4 (A b+5 a B) x^{10}+\frac {1}{13} b^5 B x^{13} \] Output:
-1/5*a^5*A/x^5-1/2*a^4*(5*A*b+B*a)/x^2+5*a^3*b*(2*A*b+B*a)*x+5/2*a^2*b^2*( A*b+B*a)*x^4+5/7*a*b^3*(A*b+2*B*a)*x^7+1/10*b^4*(A*b+5*B*a)*x^10+1/13*b^5* B*x^13
Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^6} \, dx=-\frac {a^5 A}{5 x^5}-\frac {a^4 (5 A b+a B)}{2 x^2}+5 a^3 b (2 A b+a B) x+\frac {5}{2} a^2 b^2 (A b+a B) x^4+\frac {5}{7} a b^3 (A b+2 a B) x^7+\frac {1}{10} b^4 (A b+5 a B) x^{10}+\frac {1}{13} b^5 B x^{13} \] Input:
Integrate[((a + b*x^3)^5*(A + B*x^3))/x^6,x]
Output:
-1/5*(a^5*A)/x^5 - (a^4*(5*A*b + a*B))/(2*x^2) + 5*a^3*b*(2*A*b + a*B)*x + (5*a^2*b^2*(A*b + a*B)*x^4)/2 + (5*a*b^3*(A*b + 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)*x^10)/10 + (b^5*B*x^13)/13
Time = 0.43 (sec) , antiderivative size = 113, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^6} \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (\frac {a^5 A}{x^6}+\frac {a^4 (a B+5 A b)}{x^3}+5 a^3 b (a B+2 A b)+10 a^2 b^2 x^3 (a B+A b)+b^4 x^9 (5 a B+A b)+5 a b^3 x^6 (2 a B+A b)+b^5 B x^{12}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 A}{5 x^5}-\frac {a^4 (a B+5 A b)}{2 x^2}+5 a^3 b x (a B+2 A b)+\frac {5}{2} a^2 b^2 x^4 (a B+A b)+\frac {1}{10} b^4 x^{10} (5 a B+A b)+\frac {5}{7} a b^3 x^7 (2 a B+A b)+\frac {1}{13} b^5 B x^{13}\) |
Input:
Int[((a + b*x^3)^5*(A + B*x^3))/x^6,x]
Output:
-1/5*(a^5*A)/x^5 - (a^4*(5*A*b + a*B))/(2*x^2) + 5*a^3*b*(2*A*b + a*B)*x + (5*a^2*b^2*(A*b + a*B)*x^4)/2 + (5*a*b^3*(A*b + 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)*x^10)/10 + (b^5*B*x^13)/13
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 = 119, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {b^{5} B \,x^{13}}{13}+\frac {A \,b^{5} x^{10}}{10}+\frac {B a \,b^{4} x^{10}}{2}+\frac {5 A a \,b^{4} x^{7}}{7}+\frac {10 B \,a^{2} b^{3} x^{7}}{7}+\frac {5 a^{2} A \,b^{3} x^{4}}{2}+\frac {5 B \,a^{3} b^{2} x^{4}}{2}+10 a^{3} b^{2} A x +5 a^{4} b B x -\frac {a^{5} A}{5 x^{5}}-\frac {a^{4} \left (5 A b +B a \right )}{2 x^{2}}\) | \(119\) |
norman | \(\frac {-\frac {a^{5} A}{5}+\left (-\frac {5}{2} a^{4} b A -\frac {1}{2} a^{5} B \right ) x^{3}+\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{6}+\left (\frac {5}{2} a^{2} b^{3} A +\frac {5}{2} a^{3} b^{2} B \right ) x^{9}+\left (\frac {5}{7} a \,b^{4} A +\frac {10}{7} a^{2} b^{3} B \right ) x^{12}+\left (\frac {1}{10} b^{5} A +\frac {1}{2} a \,b^{4} B \right ) x^{15}+\frac {b^{5} B \,x^{18}}{13}}{x^{5}}\) | \(122\) |
risch | \(\frac {b^{5} B \,x^{13}}{13}+\frac {A \,b^{5} x^{10}}{10}+\frac {B a \,b^{4} x^{10}}{2}+\frac {5 A a \,b^{4} x^{7}}{7}+\frac {10 B \,a^{2} b^{3} x^{7}}{7}+\frac {5 a^{2} A \,b^{3} x^{4}}{2}+\frac {5 B \,a^{3} b^{2} x^{4}}{2}+10 a^{3} b^{2} A x +5 a^{4} b B x +\frac {\left (-\frac {5}{2} a^{4} b A -\frac {1}{2} a^{5} B \right ) x^{3}-\frac {a^{5} A}{5}}{x^{5}}\) | \(123\) |
gosper | \(-\frac {-70 b^{5} B \,x^{18}-91 A \,b^{5} x^{15}-455 B a \,b^{4} x^{15}-650 a A \,b^{4} x^{12}-1300 B \,a^{2} b^{3} x^{12}-2275 a^{2} A \,b^{3} x^{9}-2275 B \,a^{3} b^{2} x^{9}-9100 a^{3} A \,b^{2} x^{6}-4550 B \,a^{4} b \,x^{6}+2275 a^{4} A b \,x^{3}+455 B \,a^{5} x^{3}+182 a^{5} A}{910 x^{5}}\) | \(128\) |
parallelrisch | \(\frac {70 b^{5} B \,x^{18}+91 A \,b^{5} x^{15}+455 B a \,b^{4} x^{15}+650 a A \,b^{4} x^{12}+1300 B \,a^{2} b^{3} x^{12}+2275 a^{2} A \,b^{3} x^{9}+2275 B \,a^{3} b^{2} x^{9}+9100 a^{3} A \,b^{2} x^{6}+4550 B \,a^{4} b \,x^{6}-2275 a^{4} A b \,x^{3}-455 B \,a^{5} x^{3}-182 a^{5} A}{910 x^{5}}\) | \(128\) |
orering | \(-\frac {-70 b^{5} B \,x^{18}-91 A \,b^{5} x^{15}-455 B a \,b^{4} x^{15}-650 a A \,b^{4} x^{12}-1300 B \,a^{2} b^{3} x^{12}-2275 a^{2} A \,b^{3} x^{9}-2275 B \,a^{3} b^{2} x^{9}-9100 a^{3} A \,b^{2} x^{6}-4550 B \,a^{4} b \,x^{6}+2275 a^{4} A b \,x^{3}+455 B \,a^{5} x^{3}+182 a^{5} A}{910 x^{5}}\) | \(128\) |
Input:
int((b*x^3+a)^5*(B*x^3+A)/x^6,x,method=_RETURNVERBOSE)
Output:
1/13*b^5*B*x^13+1/10*A*b^5*x^10+1/2*B*a*b^4*x^10+5/7*A*a*b^4*x^7+10/7*B*a^ 2*b^3*x^7+5/2*a^2*A*b^3*x^4+5/2*B*a^3*b^2*x^4+10*a^3*b^2*A*x+5*a^4*b*B*x-1 /5*a^5*A/x^5-1/2*a^4*(5*A*b+B*a)/x^2
Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^6} \, dx=\frac {70 \, B b^{5} x^{18} + 91 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 650 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 2275 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 4550 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 182 \, A a^{5} - 455 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{910 \, x^{5}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^6,x, algorithm="fricas")
Output:
1/910*(70*B*b^5*x^18 + 91*(5*B*a*b^4 + A*b^5)*x^15 + 650*(2*B*a^2*b^3 + A* a*b^4)*x^12 + 2275*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 4550*(B*a^4*b + 2*A*a^3*b ^2)*x^6 - 182*A*a^5 - 455*(B*a^5 + 5*A*a^4*b)*x^3)/x^5
Time = 0.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^6} \, dx=\frac {B b^{5} x^{13}}{13} + x^{10} \left (\frac {A b^{5}}{10} + \frac {B a b^{4}}{2}\right ) + x^{7} \cdot \left (\frac {5 A a b^{4}}{7} + \frac {10 B a^{2} b^{3}}{7}\right ) + x^{4} \cdot \left (\frac {5 A a^{2} b^{3}}{2} + \frac {5 B a^{3} b^{2}}{2}\right ) + x \left (10 A a^{3} b^{2} + 5 B a^{4} b\right ) + \frac {- 2 A a^{5} + x^{3} \left (- 25 A a^{4} b - 5 B a^{5}\right )}{10 x^{5}} \] Input:
integrate((b*x**3+a)**5*(B*x**3+A)/x**6,x)
Output:
B*b**5*x**13/13 + x**10*(A*b**5/10 + B*a*b**4/2) + x**7*(5*A*a*b**4/7 + 10 *B*a**2*b**3/7) + x**4*(5*A*a**2*b**3/2 + 5*B*a**3*b**2/2) + x*(10*A*a**3* b**2 + 5*B*a**4*b) + (-2*A*a**5 + x**3*(-25*A*a**4*b - 5*B*a**5))/(10*x**5 )
Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^6} \, dx=\frac {1}{13} \, B b^{5} x^{13} + \frac {1}{10} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac {5}{7} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + \frac {5}{2} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{4} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x - \frac {2 \, A a^{5} + 5 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{10 \, x^{5}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^6,x, algorithm="maxima")
Output:
1/13*B*b^5*x^13 + 1/10*(5*B*a*b^4 + A*b^5)*x^10 + 5/7*(2*B*a^2*b^3 + A*a*b ^4)*x^7 + 5/2*(B*a^3*b^2 + A*a^2*b^3)*x^4 + 5*(B*a^4*b + 2*A*a^3*b^2)*x - 1/10*(2*A*a^5 + 5*(B*a^5 + 5*A*a^4*b)*x^3)/x^5
Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^6} \, dx=\frac {1}{13} \, B b^{5} x^{13} + \frac {1}{2} \, B a b^{4} x^{10} + \frac {1}{10} \, A b^{5} x^{10} + \frac {10}{7} \, B a^{2} b^{3} x^{7} + \frac {5}{7} \, A a b^{4} x^{7} + \frac {5}{2} \, B a^{3} b^{2} x^{4} + \frac {5}{2} \, A a^{2} b^{3} x^{4} + 5 \, B a^{4} b x + 10 \, A a^{3} b^{2} x - \frac {5 \, B a^{5} x^{3} + 25 \, A a^{4} b x^{3} + 2 \, A a^{5}}{10 \, x^{5}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^6,x, algorithm="giac")
Output:
1/13*B*b^5*x^13 + 1/2*B*a*b^4*x^10 + 1/10*A*b^5*x^10 + 10/7*B*a^2*b^3*x^7 + 5/7*A*a*b^4*x^7 + 5/2*B*a^3*b^2*x^4 + 5/2*A*a^2*b^3*x^4 + 5*B*a^4*b*x + 10*A*a^3*b^2*x - 1/10*(5*B*a^5*x^3 + 25*A*a^4*b*x^3 + 2*A*a^5)/x^5
Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^6} \, dx=x^{10}\,\left (\frac {A\,b^5}{10}+\frac {B\,a\,b^4}{2}\right )-\frac {\frac {A\,a^5}{5}+x^3\,\left (\frac {B\,a^5}{2}+\frac {5\,A\,b\,a^4}{2}\right )}{x^5}+\frac {B\,b^5\,x^{13}}{13}+\frac {5\,a^2\,b^2\,x^4\,\left (A\,b+B\,a\right )}{2}+5\,a^3\,b\,x\,\left (2\,A\,b+B\,a\right )+\frac {5\,a\,b^3\,x^7\,\left (A\,b+2\,B\,a\right )}{7} \] Input:
int(((A + B*x^3)*(a + b*x^3)^5)/x^6,x)
Output:
x^10*((A*b^5)/10 + (B*a*b^4)/2) - ((A*a^5)/5 + x^3*((B*a^5)/2 + (5*A*a^4*b )/2))/x^5 + (B*b^5*x^13)/13 + (5*a^2*b^2*x^4*(A*b + B*a))/2 + 5*a^3*b*x*(2 *A*b + B*a) + (5*a*b^3*x^7*(A*b + 2*B*a))/7
Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^6} \, dx=\frac {35 b^{6} x^{18}+273 a \,b^{5} x^{15}+975 a^{2} b^{4} x^{12}+2275 a^{3} b^{3} x^{9}+6825 a^{4} b^{2} x^{6}-1365 a^{5} b \,x^{3}-91 a^{6}}{455 x^{5}} \] Input:
int((b*x^3+a)^5*(B*x^3+A)/x^6,x)
Output:
( - 91*a**6 - 1365*a**5*b*x**3 + 6825*a**4*b**2*x**6 + 2275*a**3*b**3*x**9 + 975*a**2*b**4*x**12 + 273*a*b**5*x**15 + 35*b**6*x**18)/(455*x**5)