Integrand size = 16, antiderivative size = 85 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=-\frac {a^5 B}{5 x^5}-\frac {5 a^4 b B}{4 x^4}-\frac {10 a^3 b^2 B}{3 x^3}-\frac {5 a^2 b^3 B}{x^2}-\frac {5 a b^4 B}{x}-\frac {A (a+b x)^6}{6 a x^6}+b^5 B \log (x) \]
-1/5*a^5*B/x^5-5/4*a^4*b*B/x^4-10/3*a^3*b^2*B/x^3-5*a^2*b^3*B/x^2-5*a*b^4* B/x-1/6*A*(b*x+a)^6/a/x^6+b^5*B*ln(x)
Time = 0.02 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=-\frac {60 A b^5 x^5+150 a b^4 x^4 (A+2 B x)+100 a^2 b^3 x^3 (2 A+3 B x)+50 a^3 b^2 x^2 (3 A+4 B x)+15 a^4 b x (4 A+5 B x)+2 a^5 (5 A+6 B x)-60 b^5 B x^6 \log (x)}{60 x^6} \]
-1/60*(60*A*b^5*x^5 + 150*a*b^4*x^4*(A + 2*B*x) + 100*a^2*b^3*x^3*(2*A + 3 *B*x) + 50*a^3*b^2*x^2*(3*A + 4*B*x) + 15*a^4*b*x*(4*A + 5*B*x) + 2*a^5*(5 *A + 6*B*x) - 60*b^5*B*x^6*Log[x])/x^6
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {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.
\(\displaystyle \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle B \int \frac {(a+b x)^5}{x^6}dx-\frac {A (a+b x)^6}{6 a x^6}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle B \int \left (\frac {a^5}{x^6}+\frac {5 b a^4}{x^5}+\frac {10 b^2 a^3}{x^4}+\frac {10 b^3 a^2}{x^3}+\frac {5 b^4 a}{x^2}+\frac {b^5}{x}\right )dx-\frac {A (a+b x)^6}{6 a x^6}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle B \left (-\frac {a^5}{5 x^5}-\frac {5 a^4 b}{4 x^4}-\frac {10 a^3 b^2}{3 x^3}-\frac {5 a^2 b^3}{x^2}-\frac {5 a b^4}{x}+b^5 \log (x)\right )-\frac {A (a+b x)^6}{6 a x^6}\) |
-1/6*(A*(a + b*x)^6)/(a*x^6) + B*(-1/5*a^5/x^5 - (5*a^4*b)/(4*x^4) - (10*a ^3*b^2)/(3*x^3) - (5*a^2*b^3)/x^2 - (5*a*b^4)/x + b^5*Log[x])
3.2.31.3.1 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]))))
Time = 0.40 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.20
method | result | size |
default | \(b^{5} B \ln \left (x \right )-\frac {a^{5} A}{6 x^{6}}-\frac {10 a^{2} b^{2} \left (A b +B a \right )}{3 x^{3}}-\frac {b^{4} \left (A b +5 B a \right )}{x}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{2 x^{2}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{4 x^{4}}-\frac {a^{4} \left (5 A b +B a \right )}{5 x^{5}}\) | \(102\) |
norman | \(\frac {\left (-\frac {5}{2} a \,b^{4} A -5 a^{2} b^{3} B \right ) x^{4}+\left (-\frac {10}{3} a^{2} b^{3} A -\frac {10}{3} a^{3} b^{2} B \right ) x^{3}+\left (-\frac {5}{2} a^{3} b^{2} A -\frac {5}{4} a^{4} b B \right ) x^{2}+\left (-a^{4} b A -\frac {1}{5} a^{5} B \right ) x +\left (-b^{5} A -5 a \,b^{4} B \right ) x^{5}-\frac {a^{5} A}{6}}{x^{6}}+b^{5} B \ln \left (x \right )\) | \(119\) |
risch | \(\frac {\left (-\frac {5}{2} a \,b^{4} A -5 a^{2} b^{3} B \right ) x^{4}+\left (-\frac {10}{3} a^{2} b^{3} A -\frac {10}{3} a^{3} b^{2} B \right ) x^{3}+\left (-\frac {5}{2} a^{3} b^{2} A -\frac {5}{4} a^{4} b B \right ) x^{2}+\left (-a^{4} b A -\frac {1}{5} a^{5} B \right ) x +\left (-b^{5} A -5 a \,b^{4} B \right ) x^{5}-\frac {a^{5} A}{6}}{x^{6}}+b^{5} B \ln \left (x \right )\) | \(119\) |
parallelrisch | \(-\frac {-60 b^{5} B \ln \left (x \right ) x^{6}+60 A \,b^{5} x^{5}+300 B a \,b^{4} x^{5}+150 a A \,b^{4} x^{4}+300 B \,a^{2} b^{3} x^{4}+200 a^{2} A \,b^{3} x^{3}+200 B \,a^{3} b^{2} x^{3}+150 a^{3} A \,b^{2} x^{2}+75 B \,a^{4} b \,x^{2}+60 a^{4} A b x +12 a^{5} B x +10 a^{5} A}{60 x^{6}}\) | \(126\) |
b^5*B*ln(x)-1/6*a^5*A/x^6-10/3*a^2*b^2*(A*b+B*a)/x^3-b^4*(A*b+5*B*a)/x-5/2 *a*b^3*(A*b+2*B*a)/x^2-5/4*a^3*b*(2*A*b+B*a)/x^4-1/5*a^4*(5*A*b+B*a)/x^5
Time = 0.22 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=\frac {60 \, B b^{5} x^{6} \log \left (x\right ) - 10 \, A a^{5} - 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} - 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \]
1/60*(60*B*b^5*x^6*log(x) - 10*A*a^5 - 60*(5*B*a*b^4 + A*b^5)*x^5 - 150*(2 *B*a^2*b^3 + A*a*b^4)*x^4 - 200*(B*a^3*b^2 + A*a^2*b^3)*x^3 - 75*(B*a^4*b + 2*A*a^3*b^2)*x^2 - 12*(B*a^5 + 5*A*a^4*b)*x)/x^6
Time = 1.54 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.54 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=B b^{5} \log {\left (x \right )} + \frac {- 10 A a^{5} + x^{5} \left (- 60 A b^{5} - 300 B a b^{4}\right ) + x^{4} \left (- 150 A a b^{4} - 300 B a^{2} b^{3}\right ) + x^{3} \left (- 200 A a^{2} b^{3} - 200 B a^{3} b^{2}\right ) + x^{2} \left (- 150 A a^{3} b^{2} - 75 B a^{4} b\right ) + x \left (- 60 A a^{4} b - 12 B a^{5}\right )}{60 x^{6}} \]
B*b**5*log(x) + (-10*A*a**5 + x**5*(-60*A*b**5 - 300*B*a*b**4) + x**4*(-15 0*A*a*b**4 - 300*B*a**2*b**3) + x**3*(-200*A*a**2*b**3 - 200*B*a**3*b**2) + x**2*(-150*A*a**3*b**2 - 75*B*a**4*b) + x*(-60*A*a**4*b - 12*B*a**5))/(6 0*x**6)
Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=B b^{5} \log \left (x\right ) - \frac {10 \, A a^{5} + 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \]
B*b^5*log(x) - 1/60*(10*A*a^5 + 60*(5*B*a*b^4 + A*b^5)*x^5 + 150*(2*B*a^2* b^3 + A*a*b^4)*x^4 + 200*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 75*(B*a^4*b + 2*A*a ^3*b^2)*x^2 + 12*(B*a^5 + 5*A*a^4*b)*x)/x^6
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=B b^{5} \log \left ({\left | x \right |}\right ) - \frac {10 \, A a^{5} + 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \]
B*b^5*log(abs(x)) - 1/60*(10*A*a^5 + 60*(5*B*a*b^4 + A*b^5)*x^5 + 150*(2*B *a^2*b^3 + A*a*b^4)*x^4 + 200*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 75*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 12*(B*a^5 + 5*A*a^4*b)*x)/x^6
Time = 0.36 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.38 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=B\,b^5\,\ln \left (x\right )-\frac {x\,\left (\frac {B\,a^5}{5}+A\,b\,a^4\right )+\frac {A\,a^5}{6}+x^4\,\left (5\,B\,a^2\,b^3+\frac {5\,A\,a\,b^4}{2}\right )+x^2\,\left (\frac {5\,B\,a^4\,b}{4}+\frac {5\,A\,a^3\,b^2}{2}\right )+x^5\,\left (A\,b^5+5\,B\,a\,b^4\right )+x^3\,\left (\frac {10\,B\,a^3\,b^2}{3}+\frac {10\,A\,a^2\,b^3}{3}\right )}{x^6} \]