Integrand size = 27, antiderivative size = 134 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^5} \, dx=-\frac {a^6 A}{4 x^4}-\frac {a^5 (6 A b+a B)}{3 x^3}-\frac {3 a^4 b (5 A b+2 a B)}{2 x^2}-\frac {5 a^3 b^2 (4 A b+3 a B)}{x}+3 a b^4 (2 A b+5 a B) x+\frac {1}{2} b^5 (A b+6 a B) x^2+\frac {1}{3} b^6 B x^3+5 a^2 b^3 (3 A b+4 a B) \log (x) \]
-1/4*a^6*A/x^4-1/3*a^5*(6*A*b+B*a)/x^3-3/2*a^4*b*(5*A*b+2*B*a)/x^2-5*a^3*b ^2*(4*A*b+3*B*a)/x+3*a*b^4*(2*A*b+5*B*a)*x+1/2*b^5*(A*b+6*B*a)*x^2+1/3*b^6 *B*x^3+5*a^2*b^3*(3*A*b+4*B*a)*ln(x)
Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^5} \, dx=-\frac {20 a^3 A b^3}{x}+15 a^2 b^4 B x+3 a b^5 x (2 A+B x)-\frac {15 a^4 b^2 (A+2 B x)}{2 x^2}+\frac {1}{6} b^6 x^2 (3 A+2 B x)-\frac {a^5 b (2 A+3 B x)}{x^3}-\frac {a^6 (3 A+4 B x)}{12 x^4}+5 a^2 b^3 (3 A b+4 a B) \log (x) \]
(-20*a^3*A*b^3)/x + 15*a^2*b^4*B*x + 3*a*b^5*x*(2*A + B*x) - (15*a^4*b^2*( A + 2*B*x))/(2*x^2) + (b^6*x^2*(3*A + 2*B*x))/6 - (a^5*b*(2*A + 3*B*x))/x^ 3 - (a^6*(3*A + 4*B*x))/(12*x^4) + 5*a^2*b^3*(3*A*b + 4*a*B)*Log[x]
Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1184, 27, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x)}{x^5} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int \frac {b^6 (a+b x)^6 (A+B x)}{x^5}dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{x^5}dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (\frac {a^6 A}{x^5}+\frac {a^5 (a B+6 A b)}{x^4}+\frac {3 a^4 b (2 a B+5 A b)}{x^3}+\frac {5 a^3 b^2 (3 a B+4 A b)}{x^2}+\frac {5 a^2 b^3 (4 a B+3 A b)}{x}+b^5 x (6 a B+A b)+3 a b^4 (5 a B+2 A b)+b^6 B x^2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^6 A}{4 x^4}-\frac {a^5 (a B+6 A b)}{3 x^3}-\frac {3 a^4 b (2 a B+5 A b)}{2 x^2}-\frac {5 a^3 b^2 (3 a B+4 A b)}{x}+5 a^2 b^3 \log (x) (4 a B+3 A b)+\frac {1}{2} b^5 x^2 (6 a B+A b)+3 a b^4 x (5 a B+2 A b)+\frac {1}{3} b^6 B x^3\) |
-1/4*(a^6*A)/x^4 - (a^5*(6*A*b + a*B))/(3*x^3) - (3*a^4*b*(5*A*b + 2*a*B)) /(2*x^2) - (5*a^3*b^2*(4*A*b + 3*a*B))/x + 3*a*b^4*(2*A*b + 5*a*B)*x + (b^ 5*(A*b + 6*a*B)*x^2)/2 + (b^6*B*x^3)/3 + 5*a^2*b^3*(3*A*b + 4*a*B)*Log[x]
3.6.49.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {b^{6} B \,x^{3}}{3}+\frac {A \,b^{6} x^{2}}{2}+3 B a \,b^{5} x^{2}+6 A a \,b^{5} x +15 B \,a^{2} b^{4} x -\frac {a^{6} A}{4 x^{4}}+5 a^{2} b^{3} \left (3 A b +4 B a \right ) \ln \left (x \right )-\frac {3 a^{4} b \left (5 A b +2 B a \right )}{2 x^{2}}-\frac {5 a^{3} b^{2} \left (4 A b +3 B a \right )}{x}-\frac {a^{5} \left (6 A b +B a \right )}{3 x^{3}}\) | \(130\) |
risch | \(\frac {b^{6} B \,x^{3}}{3}+\frac {A \,b^{6} x^{2}}{2}+3 B a \,b^{5} x^{2}+6 A a \,b^{5} x +15 B \,a^{2} b^{4} x +\frac {\left (-20 A \,a^{3} b^{3}-15 B \,a^{4} b^{2}\right ) x^{3}+\left (-\frac {15}{2} A \,a^{4} b^{2}-3 B \,a^{5} b \right ) x^{2}+\left (-2 A \,a^{5} b -\frac {1}{3} B \,a^{6}\right ) x -\frac {A \,a^{6}}{4}}{x^{4}}+15 A \ln \left (x \right ) a^{2} b^{4}+20 B \ln \left (x \right ) a^{3} b^{3}\) | \(141\) |
norman | \(\frac {\left (\frac {1}{2} A \,b^{6}+3 B a \,b^{5}\right ) x^{6}+\left (-\frac {15}{2} A \,a^{4} b^{2}-3 B \,a^{5} b \right ) x^{2}+\left (-2 A \,a^{5} b -\frac {1}{3} B \,a^{6}\right ) x +\left (6 A a \,b^{5}+15 B \,b^{4} a^{2}\right ) x^{5}+\left (-20 A \,a^{3} b^{3}-15 B \,a^{4} b^{2}\right ) x^{3}-\frac {A \,a^{6}}{4}+\frac {b^{6} B \,x^{7}}{3}}{x^{4}}+\left (15 A \,b^{4} a^{2}+20 B \,a^{3} b^{3}\right ) \ln \left (x \right )\) | \(143\) |
parallelrisch | \(\frac {4 b^{6} B \,x^{7}+6 A \,b^{6} x^{6}+36 x^{6} B a \,b^{5}+180 A \ln \left (x \right ) x^{4} a^{2} b^{4}+72 a A \,b^{5} x^{5}+240 B \ln \left (x \right ) x^{4} a^{3} b^{3}+180 x^{5} B \,b^{4} a^{2}-240 a^{3} A \,b^{3} x^{3}-180 x^{3} B \,a^{4} b^{2}-90 a^{4} A \,b^{2} x^{2}-36 x^{2} B \,a^{5} b -24 a^{5} A b x -4 x B \,a^{6}-3 A \,a^{6}}{12 x^{4}}\) | \(152\) |
1/3*b^6*B*x^3+1/2*A*b^6*x^2+3*B*a*b^5*x^2+6*A*a*b^5*x+15*B*a^2*b^4*x-1/4*a ^6*A/x^4+5*a^2*b^3*(3*A*b+4*B*a)*ln(x)-3/2*a^4*b*(5*A*b+2*B*a)/x^2-5*a^3*b ^2*(4*A*b+3*B*a)/x-1/3*a^5*(6*A*b+B*a)/x^3
Time = 0.37 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.11 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^5} \, dx=\frac {4 \, B b^{6} x^{7} - 3 \, A a^{6} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 36 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 60 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} \log \left (x\right ) - 60 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \]
1/12*(4*B*b^6*x^7 - 3*A*a^6 + 6*(6*B*a*b^5 + A*b^6)*x^6 + 36*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 60*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4*log(x) - 60*(3*B*a^4 *b^2 + 4*A*a^3*b^3)*x^3 - 18*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 4*(B*a^6 + 6* A*a^5*b)*x)/x^4
Time = 0.83 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.12 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^5} \, dx=\frac {B b^{6} x^{3}}{3} + 5 a^{2} b^{3} \cdot \left (3 A b + 4 B a\right ) \log {\left (x \right )} + x^{2} \left (\frac {A b^{6}}{2} + 3 B a b^{5}\right ) + x \left (6 A a b^{5} + 15 B a^{2} b^{4}\right ) + \frac {- 3 A a^{6} + x^{3} \left (- 240 A a^{3} b^{3} - 180 B a^{4} b^{2}\right ) + x^{2} \left (- 90 A a^{4} b^{2} - 36 B a^{5} b\right ) + x \left (- 24 A a^{5} b - 4 B a^{6}\right )}{12 x^{4}} \]
B*b**6*x**3/3 + 5*a**2*b**3*(3*A*b + 4*B*a)*log(x) + x**2*(A*b**6/2 + 3*B* a*b**5) + x*(6*A*a*b**5 + 15*B*a**2*b**4) + (-3*A*a**6 + x**3*(-240*A*a**3 *b**3 - 180*B*a**4*b**2) + x**2*(-90*A*a**4*b**2 - 36*B*a**5*b) + x*(-24*A *a**5*b - 4*B*a**6))/(12*x**4)
Time = 0.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^5} \, dx=\frac {1}{3} \, B b^{6} x^{3} + \frac {1}{2} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{2} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} \log \left (x\right ) - \frac {3 \, A a^{6} + 60 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \]
1/3*B*b^6*x^3 + 1/2*(6*B*a*b^5 + A*b^6)*x^2 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)* x + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*log(x) - 1/12*(3*A*a^6 + 60*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 18*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 4*(B*a^6 + 6*A*a^ 5*b)*x)/x^4
Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^5} \, dx=\frac {1}{3} \, B b^{6} x^{3} + 3 \, B a b^{5} x^{2} + \frac {1}{2} \, A b^{6} x^{2} + 15 \, B a^{2} b^{4} x + 6 \, A a b^{5} x + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} \log \left ({\left | x \right |}\right ) - \frac {3 \, A a^{6} + 60 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \]
1/3*B*b^6*x^3 + 3*B*a*b^5*x^2 + 1/2*A*b^6*x^2 + 15*B*a^2*b^4*x + 6*A*a*b^5 *x + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*log(abs(x)) - 1/12*(3*A*a^6 + 60*(3*B*a ^4*b^2 + 4*A*a^3*b^3)*x^3 + 18*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 4*(B*a^6 + 6*A*a^5*b)*x)/x^4
Time = 0.06 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^5} \, dx=x^2\,\left (\frac {A\,b^6}{2}+3\,B\,a\,b^5\right )-\frac {x\,\left (\frac {B\,a^6}{3}+2\,A\,b\,a^5\right )+\frac {A\,a^6}{4}+x^2\,\left (3\,B\,a^5\,b+\frac {15\,A\,a^4\,b^2}{2}\right )+x^3\,\left (15\,B\,a^4\,b^2+20\,A\,a^3\,b^3\right )}{x^4}+\ln \left (x\right )\,\left (20\,B\,a^3\,b^3+15\,A\,a^2\,b^4\right )+\frac {B\,b^6\,x^3}{3}+3\,a\,b^4\,x\,\left (2\,A\,b+5\,B\,a\right ) \]