Integrand size = 20, antiderivative size = 88 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x} \, dx=\frac {5}{3} a^4 A b x^3+\frac {5}{3} a^3 A b^2 x^6+\frac {10}{9} a^2 A b^3 x^9+\frac {5}{12} a A b^4 x^{12}+\frac {1}{15} A b^5 x^{15}+\frac {B \left (a+b x^3\right )^6}{18 b}+a^5 A \log (x) \]
5/3*a^4*A*b*x^3+5/3*a^3*A*b^2*x^6+10/9*a^2*A*b^3*x^9+5/12*a*A*b^4*x^12+1/1 5*A*b^5*x^15+1/18*B*(b*x^3+a)^6/b+a^5*A*ln(x)
Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x} \, dx=\frac {1}{3} a^4 (5 A b+a B) x^3+\frac {5}{6} a^3 b (2 A b+a B) x^6+\frac {10}{9} a^2 b^2 (A b+a B) x^9+\frac {5}{12} a b^3 (A b+2 a B) x^{12}+\frac {1}{15} b^4 (A b+5 a B) x^{15}+\frac {1}{18} b^5 B x^{18}+a^5 A \log (x) \]
(a^4*(5*A*b + a*B)*x^3)/3 + (5*a^3*b*(2*A*b + a*B)*x^6)/6 + (10*a^2*b^2*(A *b + a*B)*x^9)/9 + (5*a*b^3*(A*b + 2*a*B)*x^12)/12 + (b^4*(A*b + 5*a*B)*x^ 15)/15 + (b^5*B*x^18)/18 + a^5*A*Log[x]
Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {948, 90, 49, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^5 \left (B x^3+A\right )}{x^3}dx^3\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{3} \left (A \int \frac {\left (b x^3+a\right )^5}{x^3}dx^3+\frac {B \left (a+b x^3\right )^6}{6 b}\right )\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{3} \left (A \int \left (b^5 x^{12}+5 a b^4 x^9+10 a^2 b^3 x^6+10 a^3 b^2 x^3+5 a^4 b+\frac {a^5}{x^3}\right )dx^3+\frac {B \left (a+b x^3\right )^6}{6 b}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (A \left (a^5 \log \left (x^3\right )+5 a^4 b x^3+5 a^3 b^2 x^6+\frac {10}{3} a^2 b^3 x^9+\frac {5}{4} a b^4 x^{12}+\frac {b^5 x^{15}}{5}\right )+\frac {B \left (a+b x^3\right )^6}{6 b}\right )\) |
((B*(a + b*x^3)^6)/(6*b) + A*(5*a^4*b*x^3 + 5*a^3*b^2*x^6 + (10*a^2*b^3*x^ 9)/3 + (5*a*b^4*x^12)/4 + (b^5*x^15)/5 + a^5*Log[x^3]))/3
3.1.33.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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 = 3.94 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.35
method | result | size |
norman | \(\left (\frac {1}{15} b^{5} A +\frac {1}{3} a \,b^{4} B \right ) x^{15}+\left (\frac {5}{12} a \,b^{4} A +\frac {5}{6} a^{2} b^{3} B \right ) x^{12}+\left (\frac {10}{9} a^{2} b^{3} A +\frac {10}{9} a^{3} b^{2} B \right ) x^{9}+\left (\frac {5}{3} a^{3} b^{2} A +\frac {5}{6} a^{4} b B \right ) x^{6}+\left (\frac {5}{3} a^{4} b A +\frac {1}{3} a^{5} B \right ) x^{3}+\frac {b^{5} B \,x^{18}}{18}+a^{5} A \ln \left (x \right )\) | \(119\) |
default | \(\frac {b^{5} B \,x^{18}}{18}+\frac {A \,b^{5} x^{15}}{15}+\frac {B a \,b^{4} x^{15}}{3}+\frac {5 a A \,b^{4} x^{12}}{12}+\frac {5 B \,a^{2} b^{3} x^{12}}{6}+\frac {10 a^{2} A \,b^{3} x^{9}}{9}+\frac {10 B \,a^{3} b^{2} x^{9}}{9}+\frac {5 a^{3} A \,b^{2} x^{6}}{3}+\frac {5 B \,a^{4} b \,x^{6}}{6}+\frac {5 a^{4} A b \,x^{3}}{3}+\frac {B \,a^{5} x^{3}}{3}+a^{5} A \ln \left (x \right )\) | \(124\) |
risch | \(\frac {b^{5} B \,x^{18}}{18}+\frac {A \,b^{5} x^{15}}{15}+\frac {B a \,b^{4} x^{15}}{3}+\frac {5 a A \,b^{4} x^{12}}{12}+\frac {5 B \,a^{2} b^{3} x^{12}}{6}+\frac {10 a^{2} A \,b^{3} x^{9}}{9}+\frac {10 B \,a^{3} b^{2} x^{9}}{9}+\frac {5 a^{3} A \,b^{2} x^{6}}{3}+\frac {5 B \,a^{4} b \,x^{6}}{6}+\frac {5 a^{4} A b \,x^{3}}{3}+\frac {B \,a^{5} x^{3}}{3}+a^{5} A \ln \left (x \right )\) | \(124\) |
parallelrisch | \(\frac {b^{5} B \,x^{18}}{18}+\frac {A \,b^{5} x^{15}}{15}+\frac {B a \,b^{4} x^{15}}{3}+\frac {5 a A \,b^{4} x^{12}}{12}+\frac {5 B \,a^{2} b^{3} x^{12}}{6}+\frac {10 a^{2} A \,b^{3} x^{9}}{9}+\frac {10 B \,a^{3} b^{2} x^{9}}{9}+\frac {5 a^{3} A \,b^{2} x^{6}}{3}+\frac {5 B \,a^{4} b \,x^{6}}{6}+\frac {5 a^{4} A b \,x^{3}}{3}+\frac {B \,a^{5} x^{3}}{3}+a^{5} A \ln \left (x \right )\) | \(124\) |
(1/15*b^5*A+1/3*a*b^4*B)*x^15+(5/12*a*b^4*A+5/6*a^2*b^3*B)*x^12+(10/9*a^2* b^3*A+10/9*a^3*b^2*B)*x^9+(5/3*a^3*b^2*A+5/6*a^4*b*B)*x^6+(5/3*a^4*b*A+1/3 *a^5*B)*x^3+1/18*b^5*B*x^18+a^5*A*ln(x)
Time = 0.24 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x} \, dx=\frac {1}{18} \, B b^{5} x^{18} + \frac {1}{15} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + \frac {5}{12} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + \frac {10}{9} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac {5}{6} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + A a^{5} \log \left (x\right ) + \frac {1}{3} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \]
1/18*B*b^5*x^18 + 1/15*(5*B*a*b^4 + A*b^5)*x^15 + 5/12*(2*B*a^2*b^3 + A*a* b^4)*x^12 + 10/9*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 5/6*(B*a^4*b + 2*A*a^3*b^2) *x^6 + A*a^5*log(x) + 1/3*(B*a^5 + 5*A*a^4*b)*x^3
Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x} \, dx=A a^{5} \log {\left (x \right )} + \frac {B b^{5} x^{18}}{18} + x^{15} \left (\frac {A b^{5}}{15} + \frac {B a b^{4}}{3}\right ) + x^{12} \cdot \left (\frac {5 A a b^{4}}{12} + \frac {5 B a^{2} b^{3}}{6}\right ) + x^{9} \cdot \left (\frac {10 A a^{2} b^{3}}{9} + \frac {10 B a^{3} b^{2}}{9}\right ) + x^{6} \cdot \left (\frac {5 A a^{3} b^{2}}{3} + \frac {5 B a^{4} b}{6}\right ) + x^{3} \cdot \left (\frac {5 A a^{4} b}{3} + \frac {B a^{5}}{3}\right ) \]
A*a**5*log(x) + B*b**5*x**18/18 + x**15*(A*b**5/15 + B*a*b**4/3) + x**12*( 5*A*a*b**4/12 + 5*B*a**2*b**3/6) + x**9*(10*A*a**2*b**3/9 + 10*B*a**3*b**2 /9) + x**6*(5*A*a**3*b**2/3 + 5*B*a**4*b/6) + x**3*(5*A*a**4*b/3 + B*a**5/ 3)
Time = 0.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x} \, dx=\frac {1}{18} \, B b^{5} x^{18} + \frac {1}{15} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + \frac {5}{12} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + \frac {10}{9} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac {5}{6} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + \frac {1}{3} \, A a^{5} \log \left (x^{3}\right ) + \frac {1}{3} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \]
1/18*B*b^5*x^18 + 1/15*(5*B*a*b^4 + A*b^5)*x^15 + 5/12*(2*B*a^2*b^3 + A*a* b^4)*x^12 + 10/9*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 5/6*(B*a^4*b + 2*A*a^3*b^2) *x^6 + 1/3*A*a^5*log(x^3) + 1/3*(B*a^5 + 5*A*a^4*b)*x^3
Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x} \, dx=\frac {1}{18} \, B b^{5} x^{18} + \frac {1}{3} \, B a b^{4} x^{15} + \frac {1}{15} \, A b^{5} x^{15} + \frac {5}{6} \, B a^{2} b^{3} x^{12} + \frac {5}{12} \, A a b^{4} x^{12} + \frac {10}{9} \, B a^{3} b^{2} x^{9} + \frac {10}{9} \, A a^{2} b^{3} x^{9} + \frac {5}{6} \, B a^{4} b x^{6} + \frac {5}{3} \, A a^{3} b^{2} x^{6} + \frac {1}{3} \, B a^{5} x^{3} + \frac {5}{3} \, A a^{4} b x^{3} + A a^{5} \log \left ({\left | x \right |}\right ) \]
1/18*B*b^5*x^18 + 1/3*B*a*b^4*x^15 + 1/15*A*b^5*x^15 + 5/6*B*a^2*b^3*x^12 + 5/12*A*a*b^4*x^12 + 10/9*B*a^3*b^2*x^9 + 10/9*A*a^2*b^3*x^9 + 5/6*B*a^4* b*x^6 + 5/3*A*a^3*b^2*x^6 + 1/3*B*a^5*x^3 + 5/3*A*a^4*b*x^3 + A*a^5*log(ab s(x))
Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x} \, dx=x^3\,\left (\frac {B\,a^5}{3}+\frac {5\,A\,b\,a^4}{3}\right )+x^{15}\,\left (\frac {A\,b^5}{15}+\frac {B\,a\,b^4}{3}\right )+\frac {B\,b^5\,x^{18}}{18}+A\,a^5\,\ln \left (x\right )+\frac {10\,a^2\,b^2\,x^9\,\left (A\,b+B\,a\right )}{9}+\frac {5\,a^3\,b\,x^6\,\left (2\,A\,b+B\,a\right )}{6}+\frac {5\,a\,b^3\,x^{12}\,\left (A\,b+2\,B\,a\right )}{12} \]