Integrand size = 18, antiderivative size = 150 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)} \, dx=-\frac {c^3}{5 a x^5}+\frac {c^2 (b c-3 a d)}{4 a^2 x^4}-\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^3}+\frac {(b c-a d)^3}{2 a^4 x^2}-\frac {b (b c-a d)^3}{a^5 x}-\frac {b^2 (b c-a d)^3 \log (x)}{a^6}+\frac {b^2 (b c-a d)^3 \log (a+b x)}{a^6} \]
-1/5*c^3/a/x^5+1/4*c^2*(-3*a*d+b*c)/a^2/x^4-1/3*c*(3*a^2*d^2-3*a*b*c*d+b^2 *c^2)/a^3/x^3+1/2*(-a*d+b*c)^3/a^4/x^2-b*(-a*d+b*c)^3/a^5/x-b^2*(-a*d+b*c) ^3*ln(x)/a^6+b^2*(-a*d+b*c)^3*ln(b*x+a)/a^6
Time = 0.05 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.25 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)} \, dx=\frac {-60 a b^4 c^3 x^4+30 a^2 b^3 c^2 x^3 (c+6 d x)-10 a^3 b^2 c x^2 \left (2 c^2+9 c d x+18 d^2 x^2\right )+15 a^4 b x \left (c^3+4 c^2 d x+6 c d^2 x^2+4 d^3 x^3\right )-3 a^5 \left (4 c^3+15 c^2 d x+20 c d^2 x^2+10 d^3 x^3\right )-60 b^2 (b c-a d)^3 x^5 \log (x)+60 b^2 (b c-a d)^3 x^5 \log (a+b x)}{60 a^6 x^5} \]
(-60*a*b^4*c^3*x^4 + 30*a^2*b^3*c^2*x^3*(c + 6*d*x) - 10*a^3*b^2*c*x^2*(2* c^2 + 9*c*d*x + 18*d^2*x^2) + 15*a^4*b*x*(c^3 + 4*c^2*d*x + 6*c*d^2*x^2 + 4*d^3*x^3) - 3*a^5*(4*c^3 + 15*c^2*d*x + 20*c*d^2*x^2 + 10*d^3*x^3) - 60*b ^2*(b*c - a*d)^3*x^5*Log[x] + 60*b^2*(b*c - a*d)^3*x^5*Log[a + b*x])/(60*a ^6*x^5)
Time = 0.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {99, 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.
| \(\displaystyle \int \frac {(c+d x)^3}{x^6 (a+b x)} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {b^3 (a d-b c)^3}{a^6 (a+b x)}+\frac {b^2 (a d-b c)^3}{a^6 x}-\frac {b (a d-b c)^3}{a^5 x^2}+\frac {(a d-b c)^3}{a^4 x^3}+\frac {c^2 (3 a d-b c)}{a^2 x^5}+\frac {c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3 x^4}+\frac {c^3}{a x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^2 \log (x) (b c-a d)^3}{a^6}+\frac {b^2 (b c-a d)^3 \log (a+b x)}{a^6}-\frac {b (b c-a d)^3}{a^5 x}+\frac {(b c-a d)^3}{2 a^4 x^2}+\frac {c^2 (b c-3 a d)}{4 a^2 x^4}-\frac {c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 a^3 x^3}-\frac {c^3}{5 a x^5}\) |
-1/5*c^3/(a*x^5) + (c^2*(b*c - 3*a*d))/(4*a^2*x^4) - (c*(b^2*c^2 - 3*a*b*c *d + 3*a^2*d^2))/(3*a^3*x^3) + (b*c - a*d)^3/(2*a^4*x^2) - (b*(b*c - a*d)^ 3)/(a^5*x) - (b^2*(b*c - a*d)^3*Log[x])/a^6 + (b^2*(b*c - a*d)^3*Log[a + b *x])/a^6
3.3.31.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.46 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.65
| method | result | size |
| default | \(-\frac {c^{3}}{5 a \,x^{5}}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{2 a^{4} x^{2}}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{3 a^{3} x^{3}}-\frac {c^{2} \left (3 a d -b c \right )}{4 a^{2} x^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \ln \left (x \right )}{a^{6}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}{a^{5} x}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \ln \left (b x +a \right )}{a^{6}}\) | \(247\) |
| norman | \(\frac {\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b \,x^{4}}{a^{5}}-\frac {c^{3}}{5 a}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{3}}{2 a^{4}}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{2}}{3 a^{3}}-\frac {c^{2} \left (3 a d -b c \right ) x}{4 a^{2}}}{x^{5}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \ln \left (x \right )}{a^{6}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \ln \left (b x +a \right )}{a^{6}}\) | \(247\) |
| risch | \(\frac {\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b \,x^{4}}{a^{5}}-\frac {c^{3}}{5 a}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{3}}{2 a^{4}}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{2}}{3 a^{3}}-\frac {c^{2} \left (3 a d -b c \right ) x}{4 a^{2}}}{x^{5}}-\frac {b^{2} \ln \left (b x +a \right ) d^{3}}{a^{3}}+\frac {3 b^{3} \ln \left (b x +a \right ) c \,d^{2}}{a^{4}}-\frac {3 b^{4} \ln \left (b x +a \right ) c^{2} d}{a^{5}}+\frac {b^{5} \ln \left (b x +a \right ) c^{3}}{a^{6}}+\frac {b^{2} \ln \left (-x \right ) d^{3}}{a^{3}}-\frac {3 b^{3} \ln \left (-x \right ) c \,d^{2}}{a^{4}}+\frac {3 b^{4} \ln \left (-x \right ) c^{2} d}{a^{5}}-\frac {b^{5} \ln \left (-x \right ) c^{3}}{a^{6}}\) | \(282\) |
| parallelrisch | \(\frac {60 \ln \left (x \right ) x^{5} a^{3} b^{2} d^{3}-180 \ln \left (x \right ) x^{5} a^{2} b^{3} c \,d^{2}+180 \ln \left (x \right ) x^{5} a \,b^{4} c^{2} d -60 \ln \left (x \right ) x^{5} b^{5} c^{3}-60 \ln \left (b x +a \right ) x^{5} a^{3} b^{2} d^{3}+180 \ln \left (b x +a \right ) x^{5} a^{2} b^{3} c \,d^{2}-180 \ln \left (b x +a \right ) x^{5} a \,b^{4} c^{2} d +60 \ln \left (b x +a \right ) x^{5} b^{5} c^{3}+60 a^{4} b \,d^{3} x^{4}-180 a^{3} b^{2} c \,d^{2} x^{4}+180 a^{2} b^{3} c^{2} d \,x^{4}-60 a \,b^{4} c^{3} x^{4}-30 a^{5} d^{3} x^{3}+90 a^{4} b c \,d^{2} x^{3}-90 a^{3} b^{2} c^{2} d \,x^{3}+30 a^{2} b^{3} c^{3} x^{3}-60 a^{5} c \,d^{2} x^{2}+60 a^{4} b \,c^{2} d \,x^{2}-20 a^{3} b^{2} c^{3} x^{2}-45 a^{5} c^{2} d x +15 a^{4} b \,c^{3} x -12 c^{3} a^{5}}{60 a^{6} x^{5}}\) | \(322\) |
-1/5*c^3/a/x^5-1/2*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/a^4/x^2-1 /3*c*(3*a^2*d^2-3*a*b*c*d+b^2*c^2)/a^3/x^3-1/4*c^2*(3*a*d-b*c)/a^2/x^4+(a^ 3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/a^6*b^2*ln(x)+(a^3*d^3-3*a^2*b* c*d^2+3*a*b^2*c^2*d-b^3*c^3)/a^5*b/x-(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d- b^3*c^3)/a^6*b^2*ln(b*x+a)
Time = 0.23 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.77 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)} \, dx=-\frac {12 \, a^{5} c^{3} - 60 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} \log \left (b x + a\right ) + 60 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} \log \left (x\right ) + 60 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} - 30 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 20 \, {\left (a^{3} b^{2} c^{3} - 3 \, a^{4} b c^{2} d + 3 \, a^{5} c d^{2}\right )} x^{2} - 15 \, {\left (a^{4} b c^{3} - 3 \, a^{5} c^{2} d\right )} x}{60 \, a^{6} x^{5}} \]
-1/60*(12*a^5*c^3 - 60*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^ 2*d^3)*x^5*log(b*x + a) + 60*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^5*log(x) + 60*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^ 2 - a^4*b*d^3)*x^4 - 30*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a ^5*d^3)*x^3 + 20*(a^3*b^2*c^3 - 3*a^4*b*c^2*d + 3*a^5*c*d^2)*x^2 - 15*(a^4 *b*c^3 - 3*a^5*c^2*d)*x)/(a^6*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (136) = 272\).
Time = 0.75 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.79 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)} \, dx=\frac {- 12 a^{4} c^{3} + x^{4} \cdot \left (60 a^{3} b d^{3} - 180 a^{2} b^{2} c d^{2} + 180 a b^{3} c^{2} d - 60 b^{4} c^{3}\right ) + x^{3} \left (- 30 a^{4} d^{3} + 90 a^{3} b c d^{2} - 90 a^{2} b^{2} c^{2} d + 30 a b^{3} c^{3}\right ) + x^{2} \left (- 60 a^{4} c d^{2} + 60 a^{3} b c^{2} d - 20 a^{2} b^{2} c^{3}\right ) + x \left (- 45 a^{4} c^{2} d + 15 a^{3} b c^{3}\right )}{60 a^{5} x^{5}} + \frac {b^{2} \left (a d - b c\right )^{3} \log {\left (x + \frac {a^{4} b^{2} d^{3} - 3 a^{3} b^{3} c d^{2} + 3 a^{2} b^{4} c^{2} d - a b^{5} c^{3} - a b^{2} \left (a d - b c\right )^{3}}{2 a^{3} b^{3} d^{3} - 6 a^{2} b^{4} c d^{2} + 6 a b^{5} c^{2} d - 2 b^{6} c^{3}} \right )}}{a^{6}} - \frac {b^{2} \left (a d - b c\right )^{3} \log {\left (x + \frac {a^{4} b^{2} d^{3} - 3 a^{3} b^{3} c d^{2} + 3 a^{2} b^{4} c^{2} d - a b^{5} c^{3} + a b^{2} \left (a d - b c\right )^{3}}{2 a^{3} b^{3} d^{3} - 6 a^{2} b^{4} c d^{2} + 6 a b^{5} c^{2} d - 2 b^{6} c^{3}} \right )}}{a^{6}} \]
(-12*a**4*c**3 + x**4*(60*a**3*b*d**3 - 180*a**2*b**2*c*d**2 + 180*a*b**3* c**2*d - 60*b**4*c**3) + x**3*(-30*a**4*d**3 + 90*a**3*b*c*d**2 - 90*a**2* b**2*c**2*d + 30*a*b**3*c**3) + x**2*(-60*a**4*c*d**2 + 60*a**3*b*c**2*d - 20*a**2*b**2*c**3) + x*(-45*a**4*c**2*d + 15*a**3*b*c**3))/(60*a**5*x**5) + b**2*(a*d - b*c)**3*log(x + (a**4*b**2*d**3 - 3*a**3*b**3*c*d**2 + 3*a* *2*b**4*c**2*d - a*b**5*c**3 - a*b**2*(a*d - b*c)**3)/(2*a**3*b**3*d**3 - 6*a**2*b**4*c*d**2 + 6*a*b**5*c**2*d - 2*b**6*c**3))/a**6 - b**2*(a*d - b* c)**3*log(x + (a**4*b**2*d**3 - 3*a**3*b**3*c*d**2 + 3*a**2*b**4*c**2*d - a*b**5*c**3 + a*b**2*(a*d - b*c)**3)/(2*a**3*b**3*d**3 - 6*a**2*b**4*c*d** 2 + 6*a*b**5*c**2*d - 2*b**6*c**3))/a**6
Time = 0.22 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.74 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)} \, dx=\frac {{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (b x + a\right )}{a^{6}} - \frac {{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (x\right )}{a^{6}} - \frac {12 \, a^{4} c^{3} + 60 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} - 30 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3} + 20 \, {\left (a^{2} b^{2} c^{3} - 3 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 15 \, {\left (a^{3} b c^{3} - 3 \, a^{4} c^{2} d\right )} x}{60 \, a^{5} x^{5}} \]
(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(b*x + a)/a^6 - (b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(x)/a^6 - 1/60*(12*a^4*c^3 + 60*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d ^3)*x^4 - 30*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*x^3 + 20*(a^2*b^2*c^3 - 3*a^3*b*c^2*d + 3*a^4*c*d^2)*x^2 - 15*(a^3*b*c^3 - 3*a^ 4*c^2*d)*x)/(a^5*x^5)
Time = 0.27 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.81 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)} \, dx=-\frac {{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac {12 \, a^{5} c^{3} + 60 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} - 30 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 20 \, {\left (a^{3} b^{2} c^{3} - 3 \, a^{4} b c^{2} d + 3 \, a^{5} c d^{2}\right )} x^{2} - 15 \, {\left (a^{4} b c^{3} - 3 \, a^{5} c^{2} d\right )} x}{60 \, a^{6} x^{5}} \]
-(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(abs(x))/a^6 + (b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*log(abs(b*x + a))/(a^6*b) - 1/60*(12*a^5*c^3 + 60*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3* b^2*c*d^2 - a^4*b*d^3)*x^4 - 30*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c *d^2 - a^5*d^3)*x^3 + 20*(a^3*b^2*c^3 - 3*a^4*b*c^2*d + 3*a^5*c*d^2)*x^2 - 15*(a^4*b*c^3 - 3*a^5*c^2*d)*x)/(a^6*x^5)
Time = 0.50 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.59 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)} \, dx=\frac {2\,b^2\,\mathrm {atanh}\left (\frac {b^2\,{\left (a\,d-b\,c\right )}^3\,\left (a+2\,b\,x\right )}{a\,\left (-a^3\,b^2\,d^3+3\,a^2\,b^3\,c\,d^2-3\,a\,b^4\,c^2\,d+b^5\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^6}-\frac {\frac {c^3}{5\,a}+\frac {x^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,a^4}+\frac {c^2\,x\,\left (3\,a\,d-b\,c\right )}{4\,a^2}+\frac {c\,x^2\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{3\,a^3}-\frac {b\,x^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{a^5}}{x^5} \]
(2*b^2*atanh((b^2*(a*d - b*c)^3*(a + 2*b*x))/(a*(b^5*c^3 - a^3*b^2*d^3 + 3 *a^2*b^3*c*d^2 - 3*a*b^4*c^2*d)))*(a*d - b*c)^3)/a^6 - (c^3/(5*a) + (x^3*( a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(2*a^4) + (c^2*x*(3*a* d - b*c))/(4*a^2) + (c*x^2*(3*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/(3*a^3) - (b *x^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/a^5)/x^5