Integrand size = 20, antiderivative size = 113 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^5} \, dx=-\frac {a^5 A}{4 x^4}-\frac {a^4 (5 A b+a B)}{x}+\frac {5}{2} a^3 b (2 A b+a B) x^2+2 a^2 b^2 (A b+a B) x^5+\frac {5}{8} a b^3 (A b+2 a B) x^8+\frac {1}{11} b^4 (A b+5 a B) x^{11}+\frac {1}{14} b^5 B x^{14} \] Output:
-1/4*a^5*A/x^4-a^4*(5*A*b+B*a)/x+5/2*a^3*b*(2*A*b+B*a)*x^2+2*a^2*b^2*(A*b+ B*a)*x^5+5/8*a*b^3*(A*b+2*B*a)*x^8+1/11*b^4*(A*b+5*B*a)*x^11+1/14*b^5*B*x^ 14
Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^5} \, dx=-\frac {a^5 A}{4 x^4}+\frac {-5 a^4 A b-a^5 B}{x}+\frac {5}{2} a^3 b (2 A b+a B) x^2+2 a^2 b^2 (A b+a B) x^5+\frac {5}{8} a b^3 (A b+2 a B) x^8+\frac {1}{11} b^4 (A b+5 a B) x^{11}+\frac {1}{14} b^5 B x^{14} \] Input:
Integrate[((a + b*x^3)^5*(A + B*x^3))/x^5,x]
Output:
-1/4*(a^5*A)/x^4 + (-5*a^4*A*b - a^5*B)/x + (5*a^3*b*(2*A*b + a*B)*x^2)/2 + 2*a^2*b^2*(A*b + a*B)*x^5 + (5*a*b^3*(A*b + 2*a*B)*x^8)/8 + (b^4*(A*b + 5*a*B)*x^11)/11 + (b^5*B*x^14)/14
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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^5} \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (\frac {a^5 A}{x^5}+\frac {a^4 (a B+5 A b)}{x^2}+5 a^3 b x (a B+2 A b)+10 a^2 b^2 x^4 (a B+A b)+b^4 x^{10} (5 a B+A b)+5 a b^3 x^7 (2 a B+A b)+b^5 B x^{13}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 A}{4 x^4}-\frac {a^4 (a B+5 A b)}{x}+\frac {5}{2} a^3 b x^2 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac {1}{11} b^4 x^{11} (5 a B+A b)+\frac {5}{8} a b^3 x^8 (2 a B+A b)+\frac {1}{14} b^5 B x^{14}\) |
Input:
Int[((a + b*x^3)^5*(A + B*x^3))/x^5,x]
Output:
-1/4*(a^5*A)/x^4 - (a^4*(5*A*b + a*B))/x + (5*a^3*b*(2*A*b + a*B)*x^2)/2 + 2*a^2*b^2*(A*b + a*B)*x^5 + (5*a*b^3*(A*b + 2*a*B)*x^8)/8 + (b^4*(A*b + 5 *a*B)*x^11)/11 + (b^5*B*x^14)/14
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 = 122, normalized size of antiderivative = 1.08
method | result | size |
norman | \(\frac {-\frac {a^{5} A}{4}+\left (-5 a^{4} b A -a^{5} B \right ) x^{3}+\left (5 a^{3} b^{2} A +\frac {5}{2} a^{4} b B \right ) x^{6}+\left (2 a^{2} b^{3} A +2 a^{3} b^{2} B \right ) x^{9}+\left (\frac {5}{8} a \,b^{4} A +\frac {5}{4} a^{2} b^{3} B \right ) x^{12}+\left (\frac {1}{11} b^{5} A +\frac {5}{11} a \,b^{4} B \right ) x^{15}+\frac {b^{5} B \,x^{18}}{14}}{x^{4}}\) | \(122\) |
default | \(\frac {b^{5} B \,x^{14}}{14}+\frac {A \,b^{5} x^{11}}{11}+\frac {5 B a \,b^{4} x^{11}}{11}+\frac {5 a A \,b^{4} x^{8}}{8}+\frac {5 B \,a^{2} b^{3} x^{8}}{4}+2 a^{2} A \,b^{3} x^{5}+2 B \,a^{3} b^{2} x^{5}+5 a^{3} A \,b^{2} x^{2}+\frac {5 B \,a^{4} b \,x^{2}}{2}-\frac {a^{5} A}{4 x^{4}}-\frac {a^{4} \left (5 A b +B a \right )}{x}\) | \(123\) |
risch | \(\frac {b^{5} B \,x^{14}}{14}+\frac {A \,b^{5} x^{11}}{11}+\frac {5 B a \,b^{4} x^{11}}{11}+\frac {5 a A \,b^{4} x^{8}}{8}+\frac {5 B \,a^{2} b^{3} x^{8}}{4}+2 a^{2} A \,b^{3} x^{5}+2 B \,a^{3} b^{2} x^{5}+5 a^{3} A \,b^{2} x^{2}+\frac {5 B \,a^{4} b \,x^{2}}{2}+\frac {\left (-5 a^{4} b A -a^{5} B \right ) x^{3}-\frac {a^{5} A}{4}}{x^{4}}\) | \(127\) |
gosper | \(-\frac {-44 b^{5} B \,x^{18}-56 A \,b^{5} x^{15}-280 B a \,b^{4} x^{15}-385 a A \,b^{4} x^{12}-770 B \,a^{2} b^{3} x^{12}-1232 a^{2} A \,b^{3} x^{9}-1232 B \,a^{3} b^{2} x^{9}-3080 a^{3} A \,b^{2} x^{6}-1540 B \,a^{4} b \,x^{6}+3080 a^{4} A b \,x^{3}+616 B \,a^{5} x^{3}+154 a^{5} A}{616 x^{4}}\) | \(128\) |
parallelrisch | \(\frac {44 b^{5} B \,x^{18}+56 A \,b^{5} x^{15}+280 B a \,b^{4} x^{15}+385 a A \,b^{4} x^{12}+770 B \,a^{2} b^{3} x^{12}+1232 a^{2} A \,b^{3} x^{9}+1232 B \,a^{3} b^{2} x^{9}+3080 a^{3} A \,b^{2} x^{6}+1540 B \,a^{4} b \,x^{6}-3080 a^{4} A b \,x^{3}-616 B \,a^{5} x^{3}-154 a^{5} A}{616 x^{4}}\) | \(128\) |
orering | \(-\frac {-44 b^{5} B \,x^{18}-56 A \,b^{5} x^{15}-280 B a \,b^{4} x^{15}-385 a A \,b^{4} x^{12}-770 B \,a^{2} b^{3} x^{12}-1232 a^{2} A \,b^{3} x^{9}-1232 B \,a^{3} b^{2} x^{9}-3080 a^{3} A \,b^{2} x^{6}-1540 B \,a^{4} b \,x^{6}+3080 a^{4} A b \,x^{3}+616 B \,a^{5} x^{3}+154 a^{5} A}{616 x^{4}}\) | \(128\) |
Input:
int((b*x^3+a)^5*(B*x^3+A)/x^5,x,method=_RETURNVERBOSE)
Output:
1/x^4*(-1/4*a^5*A+(-5*A*a^4*b-B*a^5)*x^3+(5*a^3*b^2*A+5/2*a^4*b*B)*x^6+(2* A*a^2*b^3+2*B*a^3*b^2)*x^9+(5/8*a*b^4*A+5/4*a^2*b^3*B)*x^12+(1/11*b^5*A+5/ 11*a*b^4*B)*x^15+1/14*b^5*B*x^18)
Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^5} \, dx=\frac {44 \, B b^{5} x^{18} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 385 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 1232 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 1540 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 154 \, A a^{5} - 616 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{4}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^5,x, algorithm="fricas")
Output:
1/616*(44*B*b^5*x^18 + 56*(5*B*a*b^4 + A*b^5)*x^15 + 385*(2*B*a^2*b^3 + A* a*b^4)*x^12 + 1232*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 1540*(B*a^4*b + 2*A*a^3*b ^2)*x^6 - 154*A*a^5 - 616*(B*a^5 + 5*A*a^4*b)*x^3)/x^4
Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^5} \, dx=\frac {B b^{5} x^{14}}{14} + x^{11} \left (\frac {A b^{5}}{11} + \frac {5 B a b^{4}}{11}\right ) + x^{8} \cdot \left (\frac {5 A a b^{4}}{8} + \frac {5 B a^{2} b^{3}}{4}\right ) + x^{5} \cdot \left (2 A a^{2} b^{3} + 2 B a^{3} b^{2}\right ) + x^{2} \cdot \left (5 A a^{3} b^{2} + \frac {5 B a^{4} b}{2}\right ) + \frac {- A a^{5} + x^{3} \left (- 20 A a^{4} b - 4 B a^{5}\right )}{4 x^{4}} \] Input:
integrate((b*x**3+a)**5*(B*x**3+A)/x**5,x)
Output:
B*b**5*x**14/14 + x**11*(A*b**5/11 + 5*B*a*b**4/11) + x**8*(5*A*a*b**4/8 + 5*B*a**2*b**3/4) + x**5*(2*A*a**2*b**3 + 2*B*a**3*b**2) + x**2*(5*A*a**3* b**2 + 5*B*a**4*b/2) + (-A*a**5 + x**3*(-20*A*a**4*b - 4*B*a**5))/(4*x**4)
Time = 0.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^5} \, dx=\frac {1}{14} \, B b^{5} x^{14} + \frac {1}{11} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac {5}{8} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} + \frac {5}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - \frac {A a^{5} + 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{4 \, x^{4}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^5,x, algorithm="maxima")
Output:
1/14*B*b^5*x^14 + 1/11*(5*B*a*b^4 + A*b^5)*x^11 + 5/8*(2*B*a^2*b^3 + A*a*b ^4)*x^8 + 2*(B*a^3*b^2 + A*a^2*b^3)*x^5 + 5/2*(B*a^4*b + 2*A*a^3*b^2)*x^2 - 1/4*(A*a^5 + 4*(B*a^5 + 5*A*a^4*b)*x^3)/x^4
Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^5} \, dx=\frac {1}{14} \, B b^{5} x^{14} + \frac {5}{11} \, B a b^{4} x^{11} + \frac {1}{11} \, A b^{5} x^{11} + \frac {5}{4} \, B a^{2} b^{3} x^{8} + \frac {5}{8} \, A a b^{4} x^{8} + 2 \, B a^{3} b^{2} x^{5} + 2 \, A a^{2} b^{3} x^{5} + \frac {5}{2} \, B a^{4} b x^{2} + 5 \, A a^{3} b^{2} x^{2} - \frac {4 \, B a^{5} x^{3} + 20 \, A a^{4} b x^{3} + A a^{5}}{4 \, x^{4}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^5,x, algorithm="giac")
Output:
1/14*B*b^5*x^14 + 5/11*B*a*b^4*x^11 + 1/11*A*b^5*x^11 + 5/4*B*a^2*b^3*x^8 + 5/8*A*a*b^4*x^8 + 2*B*a^3*b^2*x^5 + 2*A*a^2*b^3*x^5 + 5/2*B*a^4*b*x^2 + 5*A*a^3*b^2*x^2 - 1/4*(4*B*a^5*x^3 + 20*A*a^4*b*x^3 + A*a^5)/x^4
Time = 0.05 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^5} \, dx=x^{11}\,\left (\frac {A\,b^5}{11}+\frac {5\,B\,a\,b^4}{11}\right )-\frac {\frac {A\,a^5}{4}+x^3\,\left (B\,a^5+5\,A\,b\,a^4\right )}{x^4}+\frac {B\,b^5\,x^{14}}{14}+2\,a^2\,b^2\,x^5\,\left (A\,b+B\,a\right )+\frac {5\,a^3\,b\,x^2\,\left (2\,A\,b+B\,a\right )}{2}+\frac {5\,a\,b^3\,x^8\,\left (A\,b+2\,B\,a\right )}{8} \] Input:
int(((A + B*x^3)*(a + b*x^3)^5)/x^5,x)
Output:
x^11*((A*b^5)/11 + (5*B*a*b^4)/11) - ((A*a^5)/4 + x^3*(B*a^5 + 5*A*a^4*b)) /x^4 + (B*b^5*x^14)/14 + 2*a^2*b^2*x^5*(A*b + B*a) + (5*a^3*b*x^2*(2*A*b + B*a))/2 + (5*a*b^3*x^8*(A*b + 2*B*a))/8
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^5} \, dx=\frac {44 b^{6} x^{18}+336 a \,b^{5} x^{15}+1155 a^{2} b^{4} x^{12}+2464 a^{3} b^{3} x^{9}+4620 a^{4} b^{2} x^{6}-3696 a^{5} b \,x^{3}-154 a^{6}}{616 x^{4}} \] Input:
int((b*x^3+a)^5*(B*x^3+A)/x^5,x)
Output:
( - 154*a**6 - 3696*a**5*b*x**3 + 4620*a**4*b**2*x**6 + 2464*a**3*b**3*x** 9 + 1155*a**2*b**4*x**12 + 336*a*b**5*x**15 + 44*b**6*x**18)/(616*x**4)