Integrand size = 20, antiderivative size = 114 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{10}} \, dx=-\frac {a^5 A}{9 x^9}-\frac {a^4 (5 A b+a B)}{6 x^6}-\frac {5 a^3 b (2 A b+a B)}{3 x^3}+\frac {5}{3} a b^3 (A b+2 a B) x^3+\frac {1}{6} b^4 (A b+5 a B) x^6+\frac {1}{9} b^5 B x^9+10 a^2 b^2 (A b+a B) \log (x) \] Output:
-1/9*a^5*A/x^9-1/6*a^4*(5*A*b+B*a)/x^6-5/3*a^3*b*(2*A*b+B*a)/x^3+5/3*a*b^3 *(A*b+2*B*a)*x^3+1/6*b^4*(A*b+5*B*a)*x^6+1/9*b^5*B*x^9+10*a^2*b^2*(A*b+B*a )*ln(x)
Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{10}} \, dx=\frac {1}{18} \left (-\frac {2 a^5 A}{x^9}-\frac {3 a^4 (5 A b+a B)}{x^6}-\frac {30 a^3 b (2 A b+a B)}{x^3}+30 a b^3 (A b+2 a B) x^3+3 b^4 (A b+5 a B) x^6+2 b^5 B x^9+180 a^2 b^2 (A b+a B) \log (x)\right ) \] Input:
Integrate[((a + b*x^3)^5*(A + B*x^3))/x^10,x]
Output:
((-2*a^5*A)/x^9 - (3*a^4*(5*A*b + a*B))/x^6 - (30*a^3*b*(2*A*b + a*B))/x^3 + 30*a*b^3*(A*b + 2*a*B)*x^3 + 3*b^4*(A*b + 5*a*B)*x^6 + 2*b^5*B*x^9 + 18 0*a^2*b^2*(A*b + a*B)*Log[x])/18
Time = 0.46 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {948, 85, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{10}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^5 \left (B x^3+A\right )}{x^{12}}dx^3\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \frac {1}{3} \int \left (b^5 B x^6+b^4 (A b+5 a B) x^3+5 a b^3 (A b+2 a B)+\frac {10 a^2 b^2 (A b+a B)}{x^3}+\frac {5 a^3 b (2 A b+a B)}{x^6}+\frac {a^4 (5 A b+a B)}{x^9}+\frac {a^5 A}{x^{12}}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {a^5 A}{3 x^9}-\frac {a^4 (a B+5 A b)}{2 x^6}-\frac {5 a^3 b (a B+2 A b)}{x^3}+10 a^2 b^2 \log \left (x^3\right ) (a B+A b)+\frac {1}{2} b^4 x^6 (5 a B+A b)+5 a b^3 x^3 (2 a B+A b)+\frac {1}{3} b^5 B x^9\right )\) |
Input:
Int[((a + b*x^3)^5*(A + B*x^3))/x^10,x]
Output:
(-1/3*(a^5*A)/x^9 - (a^4*(5*A*b + a*B))/(2*x^6) - (5*a^3*b*(2*A*b + a*B))/ x^3 + 5*a*b^3*(A*b + 2*a*B)*x^3 + (b^4*(A*b + 5*a*B)*x^6)/2 + (b^5*B*x^9)/ 3 + 10*a^2*b^2*(A*b + a*B)*Log[x^3])/3
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.56 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {b^{5} B \,x^{9}}{9}+\frac {b^{5} A \,x^{6}}{6}+\frac {5 B a \,b^{4} x^{6}}{6}+\frac {5 A a \,b^{4} x^{3}}{3}+\frac {10 B \,a^{2} b^{3} x^{3}}{3}-\frac {5 a^{3} b \left (2 A b +B a \right )}{3 x^{3}}+10 a^{2} b^{2} \left (A b +B a \right ) \ln \left (x \right )-\frac {a^{4} \left (5 A b +B a \right )}{6 x^{6}}-\frac {a^{5} A}{9 x^{9}}\) | \(111\) |
norman | \(\frac {\left (\frac {1}{6} b^{5} A +\frac {5}{6} a \,b^{4} B \right ) x^{15}+\left (\frac {5}{3} a \,b^{4} A +\frac {10}{3} a^{2} b^{3} B \right ) x^{12}+\left (-\frac {10}{3} a^{3} b^{2} A -\frac {5}{3} a^{4} b B \right ) x^{6}+\left (-\frac {5}{6} a^{4} b A -\frac {1}{6} a^{5} B \right ) x^{3}-\frac {a^{5} A}{9}+\frac {b^{5} B \,x^{18}}{9}}{x^{9}}+\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) \ln \left (x \right )\) | \(122\) |
risch | \(\frac {b^{5} B \,x^{9}}{9}+\frac {b^{5} A \,x^{6}}{6}+\frac {5 B a \,b^{4} x^{6}}{6}+\frac {5 A a \,b^{4} x^{3}}{3}+\frac {10 B \,a^{2} b^{3} x^{3}}{3}+\frac {\left (-\frac {10}{3} a^{3} b^{2} A -\frac {5}{3} a^{4} b B \right ) x^{6}+\left (-\frac {5}{6} a^{4} b A -\frac {1}{6} a^{5} B \right ) x^{3}-\frac {a^{5} A}{9}}{x^{9}}+10 A \ln \left (x \right ) a^{2} b^{3}+10 B \ln \left (x \right ) a^{3} b^{2}\) | \(124\) |
parallelrisch | \(\frac {2 b^{5} B \,x^{18}+3 A \,b^{5} x^{15}+15 B a \,b^{4} x^{15}+30 a A \,b^{4} x^{12}+60 B \,a^{2} b^{3} x^{12}+180 A \ln \left (x \right ) x^{9} a^{2} b^{3}+180 B \ln \left (x \right ) x^{9} a^{3} b^{2}-60 a^{3} A \,b^{2} x^{6}-30 B \,a^{4} b \,x^{6}-15 a^{4} A b \,x^{3}-3 B \,a^{5} x^{3}-2 a^{5} A}{18 x^{9}}\) | \(132\) |
Input:
int((b*x^3+a)^5*(B*x^3+A)/x^10,x,method=_RETURNVERBOSE)
Output:
1/9*b^5*B*x^9+1/6*b^5*A*x^6+5/6*B*a*b^4*x^6+5/3*A*a*b^4*x^3+10/3*B*a^2*b^3 *x^3-5/3*a^3*b*(2*A*b+B*a)/x^3+10*a^2*b^2*(A*b+B*a)*ln(x)-1/6*a^4*(5*A*b+B *a)/x^6-1/9*a^5*A/x^9
Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{10}} \, dx=\frac {2 \, B b^{5} x^{18} + 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 180 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} \log \left (x\right ) - 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 2 \, A a^{5} - 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{18 \, x^{9}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^10,x, algorithm="fricas")
Output:
1/18*(2*B*b^5*x^18 + 3*(5*B*a*b^4 + A*b^5)*x^15 + 30*(2*B*a^2*b^3 + A*a*b^ 4)*x^12 + 180*(B*a^3*b^2 + A*a^2*b^3)*x^9*log(x) - 30*(B*a^4*b + 2*A*a^3*b ^2)*x^6 - 2*A*a^5 - 3*(B*a^5 + 5*A*a^4*b)*x^3)/x^9
Time = 1.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{10}} \, dx=\frac {B b^{5} x^{9}}{9} + 10 a^{2} b^{2} \left (A b + B a\right ) \log {\left (x \right )} + x^{6} \left (\frac {A b^{5}}{6} + \frac {5 B a b^{4}}{6}\right ) + x^{3} \cdot \left (\frac {5 A a b^{4}}{3} + \frac {10 B a^{2} b^{3}}{3}\right ) + \frac {- 2 A a^{5} + x^{6} \left (- 60 A a^{3} b^{2} - 30 B a^{4} b\right ) + x^{3} \left (- 15 A a^{4} b - 3 B a^{5}\right )}{18 x^{9}} \] Input:
integrate((b*x**3+a)**5*(B*x**3+A)/x**10,x)
Output:
B*b**5*x**9/9 + 10*a**2*b**2*(A*b + B*a)*log(x) + x**6*(A*b**5/6 + 5*B*a*b **4/6) + x**3*(5*A*a*b**4/3 + 10*B*a**2*b**3/3) + (-2*A*a**5 + x**6*(-60*A *a**3*b**2 - 30*B*a**4*b) + x**3*(-15*A*a**4*b - 3*B*a**5))/(18*x**9)
Time = 0.03 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{10}} \, dx=\frac {1}{9} \, B b^{5} x^{9} + \frac {1}{6} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} + \frac {5}{3} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{3} + \frac {10}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} \log \left (x^{3}\right ) - \frac {30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 2 \, A a^{5} + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{18 \, x^{9}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^10,x, algorithm="maxima")
Output:
1/9*B*b^5*x^9 + 1/6*(5*B*a*b^4 + A*b^5)*x^6 + 5/3*(2*B*a^2*b^3 + A*a*b^4)* x^3 + 10/3*(B*a^3*b^2 + A*a^2*b^3)*log(x^3) - 1/18*(30*(B*a^4*b + 2*A*a^3* b^2)*x^6 + 2*A*a^5 + 3*(B*a^5 + 5*A*a^4*b)*x^3)/x^9
Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{10}} \, dx=\frac {1}{9} \, B b^{5} x^{9} + \frac {5}{6} \, B a b^{4} x^{6} + \frac {1}{6} \, A b^{5} x^{6} + \frac {10}{3} \, B a^{2} b^{3} x^{3} + \frac {5}{3} \, A a b^{4} x^{3} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} \log \left ({\left | x \right |}\right ) - \frac {110 \, B a^{3} b^{2} x^{9} + 110 \, A a^{2} b^{3} x^{9} + 30 \, B a^{4} b x^{6} + 60 \, A a^{3} b^{2} x^{6} + 3 \, B a^{5} x^{3} + 15 \, A a^{4} b x^{3} + 2 \, A a^{5}}{18 \, x^{9}} \] Input:
integrate((b*x^3+a)^5*(B*x^3+A)/x^10,x, algorithm="giac")
Output:
1/9*B*b^5*x^9 + 5/6*B*a*b^4*x^6 + 1/6*A*b^5*x^6 + 10/3*B*a^2*b^3*x^3 + 5/3 *A*a*b^4*x^3 + 10*(B*a^3*b^2 + A*a^2*b^3)*log(abs(x)) - 1/18*(110*B*a^3*b^ 2*x^9 + 110*A*a^2*b^3*x^9 + 30*B*a^4*b*x^6 + 60*A*a^3*b^2*x^6 + 3*B*a^5*x^ 3 + 15*A*a^4*b*x^3 + 2*A*a^5)/x^9
Time = 0.91 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{10}} \, dx=x^6\,\left (\frac {A\,b^5}{6}+\frac {5\,B\,a\,b^4}{6}\right )-\frac {\frac {A\,a^5}{9}+x^6\,\left (\frac {5\,B\,a^4\,b}{3}+\frac {10\,A\,a^3\,b^2}{3}\right )+x^3\,\left (\frac {B\,a^5}{6}+\frac {5\,A\,b\,a^4}{6}\right )}{x^9}+\ln \left (x\right )\,\left (10\,B\,a^3\,b^2+10\,A\,a^2\,b^3\right )+\frac {B\,b^5\,x^9}{9}+\frac {5\,a\,b^3\,x^3\,\left (A\,b+2\,B\,a\right )}{3} \] Input:
int(((A + B*x^3)*(a + b*x^3)^5)/x^10,x)
Output:
x^6*((A*b^5)/6 + (5*B*a*b^4)/6) - ((A*a^5)/9 + x^6*((10*A*a^3*b^2)/3 + (5* B*a^4*b)/3) + x^3*((B*a^5)/6 + (5*A*a^4*b)/6))/x^9 + log(x)*(10*A*a^2*b^3 + 10*B*a^3*b^2) + (B*b^5*x^9)/9 + (5*a*b^3*x^3*(A*b + 2*B*a))/3
Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{10}} \, dx=\frac {180 \,\mathrm {log}\left (x \right ) a^{3} b^{3} x^{9}-a^{6}-9 a^{5} b \,x^{3}-45 a^{4} b^{2} x^{6}+45 a^{2} b^{4} x^{12}+9 a \,b^{5} x^{15}+b^{6} x^{18}}{9 x^{9}} \] Input:
int((b*x^3+a)^5*(B*x^3+A)/x^10,x)
Output:
(180*log(x)*a**3*b**3*x**9 - a**6 - 9*a**5*b*x**3 - 45*a**4*b**2*x**6 + 45 *a**2*b**4*x**12 + 9*a*b**5*x**15 + b**6*x**18)/(9*x**9)