Integrand size = 20, antiderivative size = 109 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{12}} \, dx=-\frac {a^5 A}{11 x^{11}}-\frac {a^4 (5 A b+a B)}{8 x^8}-\frac {a^3 b (2 A b+a B)}{x^5}-\frac {5 a^2 b^2 (A b+a B)}{x^2}+5 a b^3 (A b+2 a B) x+\frac {1}{4} b^4 (A b+5 a B) x^4+\frac {1}{7} b^5 B x^7 \] Output:
-1/11*a^5*A/x^11-1/8*a^4*(5*A*b+B*a)/x^8-a^3*b*(2*A*b+B*a)/x^5-5*a^2*b^2*( A*b+B*a)/x^2+5*a*b^3*(A*b+2*B*a)*x+1/4*b^4*(A*b+5*B*a)*x^4+1/7*b^5*B*x^7
Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{12}} \, dx=-\frac {a^5 A}{11 x^{11}}-\frac {a^4 (5 A b+a B)}{8 x^8}-\frac {a^3 b (2 A b+a B)}{x^5}-\frac {5 a^2 b^2 (A b+a B)}{x^2}+5 a b^3 (A b+2 a B) x+\frac {1}{4} b^4 (A b+5 a B) x^4+\frac {1}{7} b^5 B x^7 \] Input:
Integrate[((a + b*x^3)^5*(A + B*x^3))/x^12,x]
Output:
-1/11*(a^5*A)/x^11 - (a^4*(5*A*b + a*B))/(8*x^8) - (a^3*b*(2*A*b + a*B))/x ^5 - (5*a^2*b^2*(A*b + a*B))/x^2 + 5*a*b^3*(A*b + 2*a*B)*x + (b^4*(A*b + 5 *a*B)*x^4)/4 + (b^5*B*x^7)/7
Time = 0.44 (sec) , antiderivative size = 109, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{12}} \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (\frac {a^5 A}{x^{12}}+\frac {a^4 (a B+5 A b)}{x^9}+\frac {5 a^3 b (a B+2 A b)}{x^6}+\frac {10 a^2 b^2 (a B+A b)}{x^3}+b^4 x^3 (5 a B+A b)+5 a b^3 (2 a B+A b)+b^5 B x^6\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 A}{11 x^{11}}-\frac {a^4 (a B+5 A b)}{8 x^8}-\frac {a^3 b (a B+2 A b)}{x^5}-\frac {5 a^2 b^2 (a B+A b)}{x^2}+\frac {1}{4} b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac {1}{7} b^5 B x^7\) |
Input:
Int[((a + b*x^3)^5*(A + B*x^3))/x^12,x]
Output:
-1/11*(a^5*A)/x^11 - (a^4*(5*A*b + a*B))/(8*x^8) - (a^3*b*(2*A*b + a*B))/x ^5 - (5*a^2*b^2*(A*b + a*B))/x^2 + 5*a*b^3*(A*b + 2*a*B)*x + (b^4*(A*b + 5 *a*B)*x^4)/4 + (b^5*B*x^7)/7
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 = 108, normalized size of antiderivative = 0.99
method | result | size |
default | \(\frac {b^{5} B \,x^{7}}{7}+\frac {A \,b^{5} x^{4}}{4}+\frac {5 B a \,b^{4} x^{4}}{4}+5 a \,b^{4} A x +10 a^{2} b^{3} B x -\frac {a^{3} b \left (2 A b +B a \right )}{x^{5}}-\frac {a^{5} A}{11 x^{11}}-\frac {5 a^{2} b^{2} \left (A b +B a \right )}{x^{2}}-\frac {a^{4} \left (5 A b +B a \right )}{8 x^{8}}\) | \(108\) |
risch | \(\frac {b^{5} B \,x^{7}}{7}+\frac {A \,b^{5} x^{4}}{4}+\frac {5 B a \,b^{4} x^{4}}{4}+5 a \,b^{4} A x +10 a^{2} b^{3} B x +\frac {\left (-5 a^{2} b^{3} A -5 a^{3} b^{2} B \right ) x^{9}+\left (-2 a^{3} b^{2} A -a^{4} b B \right ) x^{6}+\left (-\frac {5}{8} a^{4} b A -\frac {1}{8} a^{5} B \right ) x^{3}-\frac {a^{5} A}{11}}{x^{11}}\) | \(121\) |
norman | \(\frac {-\frac {a^{5} A}{11}+\left (-\frac {5}{8} a^{4} b A -\frac {1}{8} a^{5} B \right ) x^{3}+\left (-2 a^{3} b^{2} A -a^{4} b B \right ) x^{6}+\left (-5 a^{2} b^{3} A -5 a^{3} b^{2} B \right ) x^{9}+\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{12}+\left (\frac {1}{4} b^{5} A +\frac {5}{4} a \,b^{4} B \right ) x^{15}+\frac {b^{5} B \,x^{18}}{7}}{x^{11}}\) | \(122\) |
gosper | \(-\frac {-88 b^{5} B \,x^{18}-154 A \,b^{5} x^{15}-770 B a \,b^{4} x^{15}-3080 a A \,b^{4} x^{12}-6160 B \,a^{2} b^{3} x^{12}+3080 a^{2} A \,b^{3} x^{9}+3080 B \,a^{3} b^{2} x^{9}+1232 a^{3} A \,b^{2} x^{6}+616 B \,a^{4} b \,x^{6}+385 a^{4} A b \,x^{3}+77 B \,a^{5} x^{3}+56 a^{5} A}{616 x^{11}}\) | \(128\) |
parallelrisch | \(\frac {88 b^{5} B \,x^{18}+154 A \,b^{5} x^{15}+770 B a \,b^{4} x^{15}+3080 a A \,b^{4} x^{12}+6160 B \,a^{2} b^{3} x^{12}-3080 a^{2} A \,b^{3} x^{9}-3080 B \,a^{3} b^{2} x^{9}-1232 a^{3} A \,b^{2} x^{6}-616 B \,a^{4} b \,x^{6}-385 a^{4} A b \,x^{3}-77 B \,a^{5} x^{3}-56 a^{5} A}{616 x^{11}}\) | \(128\) |
orering | \(-\frac {-88 b^{5} B \,x^{18}-154 A \,b^{5} x^{15}-770 B a \,b^{4} x^{15}-3080 a A \,b^{4} x^{12}-6160 B \,a^{2} b^{3} x^{12}+3080 a^{2} A \,b^{3} x^{9}+3080 B \,a^{3} b^{2} x^{9}+1232 a^{3} A \,b^{2} x^{6}+616 B \,a^{4} b \,x^{6}+385 a^{4} A b \,x^{3}+77 B \,a^{5} x^{3}+56 a^{5} A}{616 x^{11}}\) | \(128\) |
Input:
int((b*x^3+a)^5*(B*x^3+A)/x^12,x,method=_RETURNVERBOSE)
Output:
1/7*b^5*B*x^7+1/4*A*b^5*x^4+5/4*B*a*b^4*x^4+5*a*b^4*A*x+10*a^2*b^3*B*x-a^3 *b*(2*A*b+B*a)/x^5-1/11*a^5*A/x^11-5*a^2*b^2*(A*b+B*a)/x^2-1/8*a^4*(5*A*b+ B*a)/x^8
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{12}} \, dx=\frac {88 \, B b^{5} x^{18} + 154 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 3080 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 3080 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 616 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 56 \, A a^{5} - 77 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{11}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^12,x, algorithm="fricas")
Output:
1/616*(88*B*b^5*x^18 + 154*(5*B*a*b^4 + A*b^5)*x^15 + 3080*(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 - 616*(B*a^4*b + 2*A*a^3* b^2)*x^6 - 56*A*a^5 - 77*(B*a^5 + 5*A*a^4*b)*x^3)/x^11
Time = 2.15 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{12}} \, dx=\frac {B b^{5} x^{7}}{7} + x^{4} \left (\frac {A b^{5}}{4} + \frac {5 B a b^{4}}{4}\right ) + x \left (5 A a b^{4} + 10 B a^{2} b^{3}\right ) + \frac {- 8 A a^{5} + x^{9} \left (- 440 A a^{2} b^{3} - 440 B a^{3} b^{2}\right ) + x^{6} \left (- 176 A a^{3} b^{2} - 88 B a^{4} b\right ) + x^{3} \left (- 55 A a^{4} b - 11 B a^{5}\right )}{88 x^{11}} \] Input:
integrate((b*x**3+a)**5*(B*x**3+A)/x**12,x)
Output:
B*b**5*x**7/7 + x**4*(A*b**5/4 + 5*B*a*b**4/4) + x*(5*A*a*b**4 + 10*B*a**2 *b**3) + (-8*A*a**5 + x**9*(-440*A*a**2*b**3 - 440*B*a**3*b**2) + x**6*(-1 76*A*a**3*b**2 - 88*B*a**4*b) + x**3*(-55*A*a**4*b - 11*B*a**5))/(88*x**11 )
Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{12}} \, dx=\frac {1}{7} \, B b^{5} x^{7} + \frac {1}{4} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{4} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x - \frac {440 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 88 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 8 \, A a^{5} + 11 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{88 \, x^{11}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^12,x, algorithm="maxima")
Output:
1/7*B*b^5*x^7 + 1/4*(5*B*a*b^4 + A*b^5)*x^4 + 5*(2*B*a^2*b^3 + A*a*b^4)*x - 1/88*(440*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 88*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 8*A*a^5 + 11*(B*a^5 + 5*A*a^4*b)*x^3)/x^11
Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{12}} \, dx=\frac {1}{7} \, B b^{5} x^{7} + \frac {5}{4} \, B a b^{4} x^{4} + \frac {1}{4} \, A b^{5} x^{4} + 10 \, B a^{2} b^{3} x + 5 \, A a b^{4} x - \frac {440 \, B a^{3} b^{2} x^{9} + 440 \, A a^{2} b^{3} x^{9} + 88 \, B a^{4} b x^{6} + 176 \, A a^{3} b^{2} x^{6} + 11 \, B a^{5} x^{3} + 55 \, A a^{4} b x^{3} + 8 \, A a^{5}}{88 \, x^{11}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^12,x, algorithm="giac")
Output:
1/7*B*b^5*x^7 + 5/4*B*a*b^4*x^4 + 1/4*A*b^5*x^4 + 10*B*a^2*b^3*x + 5*A*a*b ^4*x - 1/88*(440*B*a^3*b^2*x^9 + 440*A*a^2*b^3*x^9 + 88*B*a^4*b*x^6 + 176* A*a^3*b^2*x^6 + 11*B*a^5*x^3 + 55*A*a^4*b*x^3 + 8*A*a^5)/x^11
Time = 0.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{12}} \, dx=x^4\,\left (\frac {A\,b^5}{4}+\frac {5\,B\,a\,b^4}{4}\right )-\frac {\frac {A\,a^5}{11}+x^6\,\left (B\,a^4\,b+2\,A\,a^3\,b^2\right )+x^3\,\left (\frac {B\,a^5}{8}+\frac {5\,A\,b\,a^4}{8}\right )+x^9\,\left (5\,B\,a^3\,b^2+5\,A\,a^2\,b^3\right )}{x^{11}}+\frac {B\,b^5\,x^7}{7}+5\,a\,b^3\,x\,\left (A\,b+2\,B\,a\right ) \] Input:
int(((A + B*x^3)*(a + b*x^3)^5)/x^12,x)
Output:
x^4*((A*b^5)/4 + (5*B*a*b^4)/4) - ((A*a^5)/11 + x^6*(2*A*a^3*b^2 + B*a^4*b ) + x^3*((B*a^5)/8 + (5*A*a^4*b)/8) + x^9*(5*A*a^2*b^3 + 5*B*a^3*b^2))/x^1 1 + (B*b^5*x^7)/7 + 5*a*b^3*x*(A*b + 2*B*a)
Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{12}} \, dx=\frac {44 b^{6} x^{18}+462 a \,b^{5} x^{15}+4620 a^{2} b^{4} x^{12}-3080 a^{3} b^{3} x^{9}-924 a^{4} b^{2} x^{6}-231 a^{5} b \,x^{3}-28 a^{6}}{308 x^{11}} \] Input:
int((b*x^3+a)^5*(B*x^3+A)/x^12,x)
Output:
( - 28*a**6 - 231*a**5*b*x**3 - 924*a**4*b**2*x**6 - 3080*a**3*b**3*x**9 + 4620*a**2*b**4*x**12 + 462*a*b**5*x**15 + 44*b**6*x**18)/(308*x**11)