Integrand size = 20, antiderivative size = 91 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{19}} \, dx=-\frac {a^5 B}{15 x^{15}}-\frac {5 a^4 b B}{12 x^{12}}-\frac {10 a^3 b^2 B}{9 x^9}-\frac {5 a^2 b^3 B}{3 x^6}-\frac {5 a b^4 B}{3 x^3}-\frac {A \left (a+b x^3\right )^6}{18 a x^{18}}+b^5 B \log (x) \]
-1/15*a^5*B/x^15-5/12*a^4*b*B/x^12-10/9*a^3*b^2*B/x^9-5/3*a^2*b^3*B/x^6-5/ 3*a*b^4*B/x^3-1/18*A*(b*x^3+a)^6/a/x^18+b^5*B*ln(x)
Time = 0.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{19}} \, dx=-\frac {60 A b^5 x^{15}+150 a b^4 x^{12} \left (A+2 B x^3\right )+100 a^2 b^3 x^9 \left (2 A+3 B x^3\right )+50 a^3 b^2 x^6 \left (3 A+4 B x^3\right )+15 a^4 b x^3 \left (4 A+5 B x^3\right )+2 a^5 \left (5 A+6 B x^3\right )-180 b^5 B x^{18} \log (x)}{180 x^{18}} \]
-1/180*(60*A*b^5*x^15 + 150*a*b^4*x^12*(A + 2*B*x^3) + 100*a^2*b^3*x^9*(2* A + 3*B*x^3) + 50*a^3*b^2*x^6*(3*A + 4*B*x^3) + 15*a^4*b*x^3*(4*A + 5*B*x^ 3) + 2*a^5*(5*A + 6*B*x^3) - 180*b^5*B*x^18*Log[x])/x^18
Time = 0.23 (sec) , antiderivative size = 90, 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, 87, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{19}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^5 \left (B x^3+A\right )}{x^{21}}dx^3\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{3} \left (B \int \frac {\left (b x^3+a\right )^5}{x^{18}}dx^3-\frac {A \left (a+b x^3\right )^6}{6 a x^{18}}\right )\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{3} \left (B \int \left (\frac {a^5}{x^{18}}+\frac {5 b a^4}{x^{15}}+\frac {10 b^2 a^3}{x^{12}}+\frac {10 b^3 a^2}{x^9}+\frac {5 b^4 a}{x^6}+\frac {b^5}{x^3}\right )dx^3-\frac {A \left (a+b x^3\right )^6}{6 a x^{18}}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (B \left (-\frac {a^5}{5 x^{15}}-\frac {5 a^4 b}{4 x^{12}}-\frac {10 a^3 b^2}{3 x^9}-\frac {5 a^2 b^3}{x^6}-\frac {5 a b^4}{x^3}+b^5 \log \left (x^3\right )\right )-\frac {A \left (a+b x^3\right )^6}{6 a x^{18}}\right )\) |
(-1/6*(A*(a + b*x^3)^6)/(a*x^18) + B*(-1/5*a^5/x^15 - (5*a^4*b)/(4*x^12) - (10*a^3*b^2)/(3*x^9) - (5*a^2*b^3)/x^6 - (5*a*b^4)/x^3 + b^5*Log[x^3]))/3
3.1.51.3.1 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*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 4.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.12
method | result | size |
default | \(b^{5} B \ln \left (x \right )-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{6 x^{6}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{12 x^{12}}-\frac {a^{4} \left (5 A b +B a \right )}{15 x^{15}}-\frac {a^{5} A}{18 x^{18}}-\frac {b^{4} \left (A b +5 B a \right )}{3 x^{3}}-\frac {10 a^{2} b^{2} \left (A b +B a \right )}{9 x^{9}}\) | \(102\) |
norman | \(\frac {\left (-\frac {1}{3} b^{5} A -\frac {5}{3} a \,b^{4} B \right ) x^{15}+\left (-\frac {5}{6} a \,b^{4} A -\frac {5}{3} a^{2} b^{3} B \right ) x^{12}+\left (-\frac {10}{9} a^{2} b^{3} A -\frac {10}{9} a^{3} b^{2} B \right ) x^{9}+\left (-\frac {5}{6} a^{3} b^{2} A -\frac {5}{12} a^{4} b B \right ) x^{6}+\left (-\frac {1}{3} a^{4} b A -\frac {1}{15} a^{5} B \right ) x^{3}-\frac {a^{5} A}{18}}{x^{18}}+b^{5} B \ln \left (x \right )\) | \(121\) |
risch | \(\frac {\left (-\frac {1}{3} b^{5} A -\frac {5}{3} a \,b^{4} B \right ) x^{15}+\left (-\frac {5}{6} a \,b^{4} A -\frac {5}{3} a^{2} b^{3} B \right ) x^{12}+\left (-\frac {10}{9} a^{2} b^{3} A -\frac {10}{9} a^{3} b^{2} B \right ) x^{9}+\left (-\frac {5}{6} a^{3} b^{2} A -\frac {5}{12} a^{4} b B \right ) x^{6}+\left (-\frac {1}{3} a^{4} b A -\frac {1}{15} a^{5} B \right ) x^{3}-\frac {a^{5} A}{18}}{x^{18}}+b^{5} B \ln \left (x \right )\) | \(121\) |
parallelrisch | \(-\frac {-180 b^{5} B \ln \left (x \right ) x^{18}+60 A \,b^{5} x^{15}+300 B a \,b^{4} x^{15}+150 a A \,b^{4} x^{12}+300 B \,a^{2} b^{3} x^{12}+200 a^{2} A \,b^{3} x^{9}+200 B \,a^{3} b^{2} x^{9}+150 a^{3} A \,b^{2} x^{6}+75 B \,a^{4} b \,x^{6}+60 a^{4} A b \,x^{3}+12 B \,a^{5} x^{3}+10 a^{5} A}{180 x^{18}}\) | \(130\) |
b^5*B*ln(x)-5/6*a*b^3*(A*b+2*B*a)/x^6-5/12*a^3*b*(2*A*b+B*a)/x^12-1/15*a^4 *(5*A*b+B*a)/x^15-1/18*a^5*A/x^18-1/3*b^4*(A*b+5*B*a)/x^3-10/9*a^2*b^2*(A* b+B*a)/x^9
Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{19}} \, dx=\frac {180 \, B b^{5} x^{18} \log \left (x\right ) - 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 10 \, A a^{5} - 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{180 \, x^{18}} \]
1/180*(180*B*b^5*x^18*log(x) - 60*(5*B*a*b^4 + A*b^5)*x^15 - 150*(2*B*a^2* b^3 + A*a*b^4)*x^12 - 200*(B*a^3*b^2 + A*a^2*b^3)*x^9 - 75*(B*a^4*b + 2*A* a^3*b^2)*x^6 - 10*A*a^5 - 12*(B*a^5 + 5*A*a^4*b)*x^3)/x^18
Timed out. \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{19}} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{19}} \, dx=\frac {1}{3} \, B b^{5} \log \left (x^{3}\right ) - \frac {60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 10 \, A a^{5} + 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{180 \, x^{18}} \]
1/3*B*b^5*log(x^3) - 1/180*(60*(5*B*a*b^4 + A*b^5)*x^15 + 150*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 200*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 75*(B*a^4*b + 2*A*a^3 *b^2)*x^6 + 10*A*a^5 + 12*(B*a^5 + 5*A*a^4*b)*x^3)/x^18
Time = 0.29 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{19}} \, dx=B b^{5} \log \left ({\left | x \right |}\right ) - \frac {147 \, B b^{5} x^{18} + 300 \, B a b^{4} x^{15} + 60 \, A b^{5} x^{15} + 300 \, B a^{2} b^{3} x^{12} + 150 \, A a b^{4} x^{12} + 200 \, B a^{3} b^{2} x^{9} + 200 \, A a^{2} b^{3} x^{9} + 75 \, B a^{4} b x^{6} + 150 \, A a^{3} b^{2} x^{6} + 12 \, B a^{5} x^{3} + 60 \, A a^{4} b x^{3} + 10 \, A a^{5}}{180 \, x^{18}} \]
B*b^5*log(abs(x)) - 1/180*(147*B*b^5*x^18 + 300*B*a*b^4*x^15 + 60*A*b^5*x^ 15 + 300*B*a^2*b^3*x^12 + 150*A*a*b^4*x^12 + 200*B*a^3*b^2*x^9 + 200*A*a^2 *b^3*x^9 + 75*B*a^4*b*x^6 + 150*A*a^3*b^2*x^6 + 12*B*a^5*x^3 + 60*A*a^4*b* x^3 + 10*A*a^5)/x^18
Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{19}} \, dx=B\,b^5\,\ln \left (x\right )-\frac {\frac {A\,a^5}{18}+x^{12}\,\left (\frac {5\,B\,a^2\,b^3}{3}+\frac {5\,A\,a\,b^4}{6}\right )+x^6\,\left (\frac {5\,B\,a^4\,b}{12}+\frac {5\,A\,a^3\,b^2}{6}\right )+x^3\,\left (\frac {B\,a^5}{15}+\frac {A\,b\,a^4}{3}\right )+x^{15}\,\left (\frac {A\,b^5}{3}+\frac {5\,B\,a\,b^4}{3}\right )+x^9\,\left (\frac {10\,B\,a^3\,b^2}{9}+\frac {10\,A\,a^2\,b^3}{9}\right )}{x^{18}} \]