Integrand size = 16, antiderivative size = 108 \[ \int \frac {(a+b x)^5 (A+B x)}{x^3} \, dx=-\frac {a^5 A}{2 x^2}-\frac {a^4 (5 A b+a B)}{x}+10 a^2 b^2 (A b+a B) x+\frac {5}{2} a b^3 (A b+2 a B) x^2+\frac {1}{3} b^4 (A b+5 a B) x^3+\frac {1}{4} b^5 B x^4+5 a^3 b (2 A b+a B) \log (x) \] Output:
-1/2*a^5*A/x^2-a^4*(5*A*b+B*a)/x+10*a^2*b^2*(A*b+B*a)*x+5/2*a*b^3*(A*b+2*B *a)*x^2+1/3*b^4*(A*b+5*B*a)*x^3+1/4*b^5*B*x^4+5*a^3*b*(2*A*b+B*a)*ln(x)
Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^5 (A+B x)}{x^3} \, dx=-\frac {5 a^4 A b}{x}+10 a^3 b^2 B x+5 a^2 b^3 x (2 A+B x)-\frac {a^5 (A+2 B x)}{2 x^2}+\frac {5}{6} a b^4 x^2 (3 A+2 B x)+\frac {1}{12} b^5 x^3 (4 A+3 B x)+5 a^3 b (2 A b+a B) \log (x) \] Input:
Integrate[((a + b*x)^5*(A + B*x))/x^3,x]
Output:
(-5*a^4*A*b)/x + 10*a^3*b^2*B*x + 5*a^2*b^3*x*(2*A + B*x) - (a^5*(A + 2*B* x))/(2*x^2) + (5*a*b^4*x^2*(3*A + 2*B*x))/6 + (b^5*x^3*(4*A + 3*B*x))/12 + 5*a^3*b*(2*A*b + a*B)*Log[x]
Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {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.
\(\displaystyle \int \frac {(a+b x)^5 (A+B x)}{x^3} \, dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (\frac {a^5 A}{x^3}+\frac {a^4 (a B+5 A b)}{x^2}+\frac {5 a^3 b (a B+2 A b)}{x}+10 a^2 b^2 (a B+A b)+b^4 x^2 (5 a B+A b)+5 a b^3 x (2 a B+A b)+b^5 B x^3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 A}{2 x^2}-\frac {a^4 (a B+5 A b)}{x}+5 a^3 b \log (x) (a B+2 A b)+10 a^2 b^2 x (a B+A b)+\frac {1}{3} b^4 x^3 (5 a B+A b)+\frac {5}{2} a b^3 x^2 (2 a B+A b)+\frac {1}{4} b^5 B x^4\) |
Input:
Int[((a + b*x)^5*(A + B*x))/x^3,x]
Output:
-1/2*(a^5*A)/x^2 - (a^4*(5*A*b + a*B))/x + 10*a^2*b^2*(A*b + a*B)*x + (5*a *b^3*(A*b + 2*a*B)*x^2)/2 + (b^4*(A*b + 5*a*B)*x^3)/3 + (b^5*B*x^4)/4 + 5* a^3*b*(2*A*b + a*B)*Log[x]
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])
Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {b^{5} B \,x^{4}}{4}+\frac {A \,b^{5} x^{3}}{3}+\frac {5 B a \,b^{4} x^{3}}{3}+\frac {5 A a \,b^{4} x^{2}}{2}+5 B \,a^{2} b^{3} x^{2}+10 A \,a^{2} b^{3} x +10 B \,a^{3} b^{2} x -\frac {a^{5} A}{2 x^{2}}+5 a^{3} b \left (2 A b +B a \right ) \ln \left (x \right )-\frac {a^{4} \left (5 A b +B a \right )}{x}\) | \(113\) |
risch | \(\frac {b^{5} B \,x^{4}}{4}+\frac {A \,b^{5} x^{3}}{3}+\frac {5 B a \,b^{4} x^{3}}{3}+\frac {5 A a \,b^{4} x^{2}}{2}+5 B \,a^{2} b^{3} x^{2}+10 A \,a^{2} b^{3} x +10 B \,a^{3} b^{2} x +\frac {\left (-5 a^{4} b A -a^{5} B \right ) x -\frac {a^{5} A}{2}}{x^{2}}+10 A \ln \left (x \right ) a^{3} b^{2}+5 B \ln \left (x \right ) a^{4} b\) | \(119\) |
norman | \(\frac {\left (\frac {1}{3} b^{5} A +\frac {5}{3} a \,b^{4} B \right ) x^{5}+\left (\frac {5}{2} a \,b^{4} A +5 a^{2} b^{3} B \right ) x^{4}+\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{3}+\left (-5 a^{4} b A -a^{5} B \right ) x -\frac {a^{5} A}{2}+\frac {B \,b^{5} x^{6}}{4}}{x^{2}}+\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) \ln \left (x \right )\) | \(120\) |
parallelrisch | \(\frac {3 B \,b^{5} x^{6}+4 A \,b^{5} x^{5}+20 B a \,b^{4} x^{5}+30 a A \,b^{4} x^{4}+60 B \,a^{2} b^{3} x^{4}+120 A \ln \left (x \right ) x^{2} a^{3} b^{2}+120 a^{2} A \,b^{3} x^{3}+60 B \ln \left (x \right ) x^{2} a^{4} b +120 B \,a^{3} b^{2} x^{3}-60 a^{4} A b x -12 B \,a^{5} x -6 a^{5} A}{12 x^{2}}\) | \(128\) |
Input:
int((b*x+a)^5*(B*x+A)/x^3,x,method=_RETURNVERBOSE)
Output:
1/4*b^5*B*x^4+1/3*A*b^5*x^3+5/3*B*a*b^4*x^3+5/2*A*a*b^4*x^2+5*B*a^2*b^3*x^ 2+10*A*a^2*b^3*x+10*B*a^3*b^2*x-1/2*a^5*A/x^2+5*a^3*b*(2*A*b+B*a)*ln(x)-a^ 4*(5*A*b+B*a)/x
Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^5 (A+B x)}{x^3} \, dx=\frac {3 \, B b^{5} x^{6} - 6 \, A a^{5} + 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 60 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} \log \left (x\right ) - 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{2}} \] Input:
integrate((b*x+a)^5*(B*x+A)/x^3,x, algorithm="fricas")
Output:
1/12*(3*B*b^5*x^6 - 6*A*a^5 + 4*(5*B*a*b^4 + A*b^5)*x^5 + 30*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 120*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 60*(B*a^4*b + 2*A*a^3*b ^2)*x^2*log(x) - 12*(B*a^5 + 5*A*a^4*b)*x)/x^2
Time = 0.22 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^5 (A+B x)}{x^3} \, dx=\frac {B b^{5} x^{4}}{4} + 5 a^{3} b \left (2 A b + B a\right ) \log {\left (x \right )} + x^{3} \left (\frac {A b^{5}}{3} + \frac {5 B a b^{4}}{3}\right ) + x^{2} \cdot \left (\frac {5 A a b^{4}}{2} + 5 B a^{2} b^{3}\right ) + x \left (10 A a^{2} b^{3} + 10 B a^{3} b^{2}\right ) + \frac {- A a^{5} + x \left (- 10 A a^{4} b - 2 B a^{5}\right )}{2 x^{2}} \] Input:
integrate((b*x+a)**5*(B*x+A)/x**3,x)
Output:
B*b**5*x**4/4 + 5*a**3*b*(2*A*b + B*a)*log(x) + x**3*(A*b**5/3 + 5*B*a*b** 4/3) + x**2*(5*A*a*b**4/2 + 5*B*a**2*b**3) + x*(10*A*a**2*b**3 + 10*B*a**3 *b**2) + (-A*a**5 + x*(-10*A*a**4*b - 2*B*a**5))/(2*x**2)
Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x)^5 (A+B x)}{x^3} \, dx=\frac {1}{4} \, B b^{5} x^{4} + \frac {1}{3} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{3} + \frac {5}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{2} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} \log \left (x\right ) - \frac {A a^{5} + 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2 \, x^{2}} \] Input:
integrate((b*x+a)^5*(B*x+A)/x^3,x, algorithm="maxima")
Output:
1/4*B*b^5*x^4 + 1/3*(5*B*a*b^4 + A*b^5)*x^3 + 5/2*(2*B*a^2*b^3 + A*a*b^4)* x^2 + 10*(B*a^3*b^2 + A*a^2*b^3)*x + 5*(B*a^4*b + 2*A*a^3*b^2)*log(x) - 1/ 2*(A*a^5 + 2*(B*a^5 + 5*A*a^4*b)*x)/x^2
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b x)^5 (A+B x)}{x^3} \, dx=\frac {1}{4} \, B b^{5} x^{4} + \frac {5}{3} \, B a b^{4} x^{3} + \frac {1}{3} \, A b^{5} x^{3} + 5 \, B a^{2} b^{3} x^{2} + \frac {5}{2} \, A a b^{4} x^{2} + 10 \, B a^{3} b^{2} x + 10 \, A a^{2} b^{3} x + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac {A a^{5} + 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2 \, x^{2}} \] Input:
integrate((b*x+a)^5*(B*x+A)/x^3,x, algorithm="giac")
Output:
1/4*B*b^5*x^4 + 5/3*B*a*b^4*x^3 + 1/3*A*b^5*x^3 + 5*B*a^2*b^3*x^2 + 5/2*A* a*b^4*x^2 + 10*B*a^3*b^2*x + 10*A*a^2*b^3*x + 5*(B*a^4*b + 2*A*a^3*b^2)*lo g(abs(x)) - 1/2*(A*a^5 + 2*(B*a^5 + 5*A*a^4*b)*x)/x^2
Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5 (A+B x)}{x^3} \, dx=\ln \left (x\right )\,\left (5\,B\,a^4\,b+10\,A\,a^3\,b^2\right )-\frac {x\,\left (B\,a^5+5\,A\,b\,a^4\right )+\frac {A\,a^5}{2}}{x^2}+x^3\,\left (\frac {A\,b^5}{3}+\frac {5\,B\,a\,b^4}{3}\right )+\frac {B\,b^5\,x^4}{4}+10\,a^2\,b^2\,x\,\left (A\,b+B\,a\right )+\frac {5\,a\,b^3\,x^2\,\left (A\,b+2\,B\,a\right )}{2} \] Input:
int(((A + B*x)*(a + b*x)^5)/x^3,x)
Output:
log(x)*(10*A*a^3*b^2 + 5*B*a^4*b) - (x*(B*a^5 + 5*A*a^4*b) + (A*a^5)/2)/x^ 2 + x^3*((A*b^5)/3 + (5*B*a*b^4)/3) + (B*b^5*x^4)/4 + 10*a^2*b^2*x*(A*b + B*a) + (5*a*b^3*x^2*(A*b + 2*B*a))/2
Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.64 \[ \int \frac {(a+b x)^5 (A+B x)}{x^3} \, dx=\frac {60 \,\mathrm {log}\left (x \right ) a^{4} b^{2} x^{2}-2 a^{6}-24 a^{5} b x +80 a^{3} b^{3} x^{3}+30 a^{2} b^{4} x^{4}+8 a \,b^{5} x^{5}+b^{6} x^{6}}{4 x^{2}} \] Input:
int((b*x+a)^5*(B*x+A)/x^3,x)
Output:
(60*log(x)*a**4*b**2*x**2 - 2*a**6 - 24*a**5*b*x + 80*a**3*b**3*x**3 + 30* a**2*b**4*x**4 + 8*a*b**5*x**5 + b**6*x**6)/(4*x**2)