Integrand size = 20, antiderivative size = 110 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{15}} \, dx=-\frac {a^5 A}{14 x^{14}}-\frac {a^4 (5 A b+a B)}{11 x^{11}}-\frac {5 a^3 b (2 A b+a B)}{8 x^8}-\frac {2 a^2 b^2 (A b+a B)}{x^5}-\frac {5 a b^3 (A b+2 a B)}{2 x^2}+b^4 (A b+5 a B) x+\frac {1}{4} b^5 B x^4 \] Output:
-1/14*a^5*A/x^14-1/11*a^4*(5*A*b+B*a)/x^11-5/8*a^3*b*(2*A*b+B*a)/x^8-2*a^2 *b^2*(A*b+B*a)/x^5-5/2*a*b^3*(A*b+2*B*a)/x^2+b^4*(A*b+5*B*a)*x+1/4*b^5*B*x ^4
Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{15}} \, dx=-\frac {a^5 A}{14 x^{14}}-\frac {a^4 (5 A b+a B)}{11 x^{11}}-\frac {5 a^3 b (2 A b+a B)}{8 x^8}-\frac {2 a^2 b^2 (A b+a B)}{x^5}-\frac {5 a b^3 (A b+2 a B)}{2 x^2}+b^4 (A b+5 a B) x+\frac {1}{4} b^5 B x^4 \] Input:
Integrate[((a + b*x^3)^5*(A + B*x^3))/x^15,x]
Output:
-1/14*(a^5*A)/x^14 - (a^4*(5*A*b + a*B))/(11*x^11) - (5*a^3*b*(2*A*b + a*B ))/(8*x^8) - (2*a^2*b^2*(A*b + a*B))/x^5 - (5*a*b^3*(A*b + 2*a*B))/(2*x^2) + b^4*(A*b + 5*a*B)*x + (b^5*B*x^4)/4
Time = 0.45 (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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{15}} \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (\frac {a^5 A}{x^{15}}+\frac {a^4 (a B+5 A b)}{x^{12}}+\frac {5 a^3 b (a B+2 A b)}{x^9}+\frac {10 a^2 b^2 (a B+A b)}{x^6}+b^4 (5 a B+A b)+\frac {5 a b^3 (2 a B+A b)}{x^3}+b^5 B x^3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 A}{14 x^{14}}-\frac {a^4 (a B+5 A b)}{11 x^{11}}-\frac {5 a^3 b (a B+2 A b)}{8 x^8}-\frac {2 a^2 b^2 (a B+A b)}{x^5}+b^4 x (5 a B+A b)-\frac {5 a b^3 (2 a B+A b)}{2 x^2}+\frac {1}{4} b^5 B x^4\) |
Input:
Int[((a + b*x^3)^5*(A + B*x^3))/x^15,x]
Output:
-1/14*(a^5*A)/x^14 - (a^4*(5*A*b + a*B))/(11*x^11) - (5*a^3*b*(2*A*b + a*B ))/(8*x^8) - (2*a^2*b^2*(A*b + a*B))/x^5 - (5*a*b^3*(A*b + 2*a*B))/(2*x^2) + b^4*(A*b + 5*a*B)*x + (b^5*B*x^4)/4
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 = 102, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {b^{5} B \,x^{4}}{4}+b^{5} A x +5 a \,b^{4} B x -\frac {2 a^{2} b^{2} \left (A b +B a \right )}{x^{5}}-\frac {a^{4} \left (5 A b +B a \right )}{11 x^{11}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{2 x^{2}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{8 x^{8}}-\frac {a^{5} A}{14 x^{14}}\) | \(102\) |
risch | \(\frac {b^{5} B \,x^{4}}{4}+b^{5} A x +5 a \,b^{4} B x +\frac {\left (-\frac {5}{2} a \,b^{4} A -5 a^{2} b^{3} B \right ) x^{12}+\left (-2 a^{2} b^{3} A -2 a^{3} b^{2} B \right ) x^{9}+\left (-\frac {5}{4} a^{3} b^{2} A -\frac {5}{8} a^{4} b B \right ) x^{6}+\left (-\frac {5}{11} a^{4} b A -\frac {1}{11} a^{5} B \right ) x^{3}-\frac {a^{5} A}{14}}{x^{14}}\) | \(119\) |
norman | \(\frac {-\frac {a^{5} A}{14}+\left (-\frac {5}{11} a^{4} b A -\frac {1}{11} a^{5} B \right ) x^{3}+\left (-\frac {5}{4} a^{3} b^{2} A -\frac {5}{8} a^{4} b B \right ) x^{6}+\left (-2 a^{2} b^{3} A -2 a^{3} b^{2} B \right ) x^{9}+\left (-\frac {5}{2} a \,b^{4} A -5 a^{2} b^{3} B \right ) x^{12}+\left (b^{5} A +5 a \,b^{4} B \right ) x^{15}+\frac {b^{5} B \,x^{18}}{4}}{x^{14}}\) | \(121\) |
gosper | \(-\frac {-154 b^{5} B \,x^{18}-616 A \,b^{5} x^{15}-3080 B a \,b^{4} x^{15}+1540 a A \,b^{4} x^{12}+3080 B \,a^{2} b^{3} x^{12}+1232 a^{2} A \,b^{3} x^{9}+1232 B \,a^{3} b^{2} x^{9}+770 a^{3} A \,b^{2} x^{6}+385 B \,a^{4} b \,x^{6}+280 a^{4} A b \,x^{3}+56 B \,a^{5} x^{3}+44 a^{5} A}{616 x^{14}}\) | \(128\) |
parallelrisch | \(\frac {154 b^{5} B \,x^{18}+616 A \,b^{5} x^{15}+3080 B a \,b^{4} x^{15}-1540 a A \,b^{4} x^{12}-3080 B \,a^{2} b^{3} x^{12}-1232 a^{2} A \,b^{3} x^{9}-1232 B \,a^{3} b^{2} x^{9}-770 a^{3} A \,b^{2} x^{6}-385 B \,a^{4} b \,x^{6}-280 a^{4} A b \,x^{3}-56 B \,a^{5} x^{3}-44 a^{5} A}{616 x^{14}}\) | \(128\) |
orering | \(-\frac {-154 b^{5} B \,x^{18}-616 A \,b^{5} x^{15}-3080 B a \,b^{4} x^{15}+1540 a A \,b^{4} x^{12}+3080 B \,a^{2} b^{3} x^{12}+1232 a^{2} A \,b^{3} x^{9}+1232 B \,a^{3} b^{2} x^{9}+770 a^{3} A \,b^{2} x^{6}+385 B \,a^{4} b \,x^{6}+280 a^{4} A b \,x^{3}+56 B \,a^{5} x^{3}+44 a^{5} A}{616 x^{14}}\) | \(128\) |
Input:
int((b*x^3+a)^5*(B*x^3+A)/x^15,x,method=_RETURNVERBOSE)
Output:
1/4*b^5*B*x^4+b^5*A*x+5*a*b^4*B*x-2*a^2*b^2*(A*b+B*a)/x^5-1/11*a^4*(5*A*b+ B*a)/x^11-5/2*a*b^3*(A*b+2*B*a)/x^2-5/8*a^3*b*(2*A*b+B*a)/x^8-1/14*a^5*A/x ^14
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{15}} \, dx=\frac {154 \, B b^{5} x^{18} + 616 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 1540 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 1232 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 385 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 44 \, A a^{5} - 56 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{14}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^15,x, algorithm="fricas")
Output:
1/616*(154*B*b^5*x^18 + 616*(5*B*a*b^4 + A*b^5)*x^15 - 1540*(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 - 385*(B*a^4*b + 2*A*a^3 *b^2)*x^6 - 44*A*a^5 - 56*(B*a^5 + 5*A*a^4*b)*x^3)/x^14
Time = 15.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{15}} \, dx=\frac {B b^{5} x^{4}}{4} + x \left (A b^{5} + 5 B a b^{4}\right ) + \frac {- 44 A a^{5} + x^{12} \left (- 1540 A a b^{4} - 3080 B a^{2} b^{3}\right ) + x^{9} \left (- 1232 A a^{2} b^{3} - 1232 B a^{3} b^{2}\right ) + x^{6} \left (- 770 A a^{3} b^{2} - 385 B a^{4} b\right ) + x^{3} \left (- 280 A a^{4} b - 56 B a^{5}\right )}{616 x^{14}} \] Input:
integrate((b*x**3+a)**5*(B*x**3+A)/x**15,x)
Output:
B*b**5*x**4/4 + x*(A*b**5 + 5*B*a*b**4) + (-44*A*a**5 + x**12*(-1540*A*a*b **4 - 3080*B*a**2*b**3) + x**9*(-1232*A*a**2*b**3 - 1232*B*a**3*b**2) + x* *6*(-770*A*a**3*b**2 - 385*B*a**4*b) + x**3*(-280*A*a**4*b - 56*B*a**5))/( 616*x**14)
Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{15}} \, dx=\frac {1}{4} \, B b^{5} x^{4} + {\left (5 \, B a b^{4} + A b^{5}\right )} x - \frac {1540 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 1232 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 385 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 44 \, A a^{5} + 56 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{14}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^15,x, algorithm="maxima")
Output:
1/4*B*b^5*x^4 + (5*B*a*b^4 + A*b^5)*x - 1/616*(1540*(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 + 385*(B*a^4*b + 2*A*a^3*b^2)*x^ 6 + 44*A*a^5 + 56*(B*a^5 + 5*A*a^4*b)*x^3)/x^14
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{15}} \, dx=\frac {1}{4} \, B b^{5} x^{4} + 5 \, B a b^{4} x + A b^{5} x - \frac {3080 \, B a^{2} b^{3} x^{12} + 1540 \, A a b^{4} x^{12} + 1232 \, B a^{3} b^{2} x^{9} + 1232 \, A a^{2} b^{3} x^{9} + 385 \, B a^{4} b x^{6} + 770 \, A a^{3} b^{2} x^{6} + 56 \, B a^{5} x^{3} + 280 \, A a^{4} b x^{3} + 44 \, A a^{5}}{616 \, x^{14}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^15,x, algorithm="giac")
Output:
1/4*B*b^5*x^4 + 5*B*a*b^4*x + A*b^5*x - 1/616*(3080*B*a^2*b^3*x^12 + 1540* A*a*b^4*x^12 + 1232*B*a^3*b^2*x^9 + 1232*A*a^2*b^3*x^9 + 385*B*a^4*b*x^6 + 770*A*a^3*b^2*x^6 + 56*B*a^5*x^3 + 280*A*a^4*b*x^3 + 44*A*a^5)/x^14
Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{15}} \, dx=x\,\left (A\,b^5+5\,B\,a\,b^4\right )-\frac {\frac {A\,a^5}{14}+x^{12}\,\left (5\,B\,a^2\,b^3+\frac {5\,A\,a\,b^4}{2}\right )+x^6\,\left (\frac {5\,B\,a^4\,b}{8}+\frac {5\,A\,a^3\,b^2}{4}\right )+x^3\,\left (\frac {B\,a^5}{11}+\frac {5\,A\,b\,a^4}{11}\right )+x^9\,\left (2\,B\,a^3\,b^2+2\,A\,a^2\,b^3\right )}{x^{14}}+\frac {B\,b^5\,x^4}{4} \] Input:
int(((A + B*x^3)*(a + b*x^3)^5)/x^15,x)
Output:
x*(A*b^5 + 5*B*a*b^4) - ((A*a^5)/14 + x^12*(5*B*a^2*b^3 + (5*A*a*b^4)/2) + x^6*((5*A*a^3*b^2)/4 + (5*B*a^4*b)/8) + x^3*((B*a^5)/11 + (5*A*a^4*b)/11) + x^9*(2*A*a^2*b^3 + 2*B*a^3*b^2))/x^14 + (B*b^5*x^4)/4
Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{15}} \, dx=\frac {154 b^{6} x^{18}+3696 a \,b^{5} x^{15}-4620 a^{2} b^{4} x^{12}-2464 a^{3} b^{3} x^{9}-1155 a^{4} b^{2} x^{6}-336 a^{5} b \,x^{3}-44 a^{6}}{616 x^{14}} \] Input:
int((b*x^3+a)^5*(B*x^3+A)/x^15,x)
Output:
( - 44*a**6 - 336*a**5*b*x**3 - 1155*a**4*b**2*x**6 - 2464*a**3*b**3*x**9 - 4620*a**2*b**4*x**12 + 3696*a*b**5*x**15 + 154*b**6*x**18)/(616*x**14)