Integrand size = 18, antiderivative size = 147 \[ \int \frac {x^5}{(a+b x) (c+d x)^2} \, dx=\frac {\left (3 b^2 c^2+2 a b c d+a^2 d^2\right ) x}{b^3 d^4}-\frac {(2 b c+a d) x^2}{2 b^2 d^3}+\frac {x^3}{3 b d^2}-\frac {c^5}{d^5 (b c-a d) (c+d x)}-\frac {a^5 \log (a+b x)}{b^4 (b c-a d)^2}-\frac {c^4 (4 b c-5 a d) \log (c+d x)}{d^5 (b c-a d)^2} \]
(a^2*d^2+2*a*b*c*d+3*b^2*c^2)*x/b^3/d^4-1/2*(a*d+2*b*c)*x^2/b^2/d^3+1/3*x^ 3/b/d^2-c^5/d^5/(-a*d+b*c)/(d*x+c)-a^5*ln(b*x+a)/b^4/(-a*d+b*c)^2-c^4*(-5* a*d+4*b*c)*ln(d*x+c)/d^5/(-a*d+b*c)^2
Time = 0.11 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{(a+b x) (c+d x)^2} \, dx=\frac {\left (3 b^2 c^2+2 a b c d+a^2 d^2\right ) x}{b^3 d^4}-\frac {(2 b c+a d) x^2}{2 b^2 d^3}+\frac {x^3}{3 b d^2}+\frac {c^5}{d^5 (-b c+a d) (c+d x)}-\frac {a^5 \log (a+b x)}{b^4 (b c-a d)^2}+\frac {\left (-4 b c^5+5 a c^4 d\right ) \log (c+d x)}{d^5 (b c-a d)^2} \]
((3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*x)/(b^3*d^4) - ((2*b*c + a*d)*x^2)/(2*b ^2*d^3) + x^3/(3*b*d^2) + c^5/(d^5*(-(b*c) + a*d)*(c + d*x)) - (a^5*Log[a + b*x])/(b^4*(b*c - a*d)^2) + ((-4*b*c^5 + 5*a*c^4*d)*Log[c + d*x])/(d^5*( b*c - a*d)^2)
Time = 0.35 (sec) , antiderivative size = 147, 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.
| \(\displaystyle \int \frac {x^5}{(a+b x) (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {a^5}{b^3 (a+b x) (b c-a d)^2}+\frac {a^2 d^2+2 a b c d+3 b^2 c^2}{b^3 d^4}-\frac {x (a d+2 b c)}{b^2 d^3}-\frac {c^5}{d^4 (c+d x)^2 (a d-b c)}-\frac {c^4 (4 b c-5 a d)}{d^4 (c+d x) (a d-b c)^2}+\frac {x^2}{b d^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^5 \log (a+b x)}{b^4 (b c-a d)^2}+\frac {x \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{b^3 d^4}-\frac {x^2 (a d+2 b c)}{2 b^2 d^3}-\frac {c^5}{d^5 (c+d x) (b c-a d)}-\frac {c^4 (4 b c-5 a d) \log (c+d x)}{d^5 (b c-a d)^2}+\frac {x^3}{3 b d^2}\) |
((3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*x)/(b^3*d^4) - ((2*b*c + a*d)*x^2)/(2*b ^2*d^3) + x^3/(3*b*d^2) - c^5/(d^5*(b*c - a*d)*(c + d*x)) - (a^5*Log[a + b *x])/(b^4*(b*c - a*d)^2) - (c^4*(4*b*c - 5*a*d)*Log[c + d*x])/(d^5*(b*c - a*d)^2)
3.3.42.3.1 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 = 1.25 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(\frac {\frac {1}{3} d^{2} x^{3} b^{2}-\frac {1}{2} x^{2} a b \,d^{2}-x^{2} b^{2} c d +a^{2} d^{2} x +2 a b c d x +3 b^{2} c^{2} x}{b^{3} d^{4}}+\frac {c^{4} \left (5 a d -4 b c \right ) \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )^{2}}+\frac {c^{5}}{d^{5} \left (a d -b c \right ) \left (d x +c \right )}-\frac {a^{5} \ln \left (b x +a \right )}{b^{4} \left (a d -b c \right )^{2}}\) | \(145\) |
| risch | \(\frac {x^{3}}{3 b \,d^{2}}-\frac {x^{2} a}{2 b^{2} d^{2}}-\frac {x^{2} c}{b \,d^{3}}+\frac {a^{2} x}{b^{3} d^{2}}+\frac {2 a c x}{b^{2} d^{3}}+\frac {3 c^{2} x}{b \,d^{4}}+\frac {c^{5}}{d^{5} \left (a d -b c \right ) \left (d x +c \right )}-\frac {a^{5} \ln \left (b x +a \right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{4}}+\frac {5 c^{4} \ln \left (-d x -c \right ) a}{d^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {4 c^{5} \ln \left (-d x -c \right ) b}{d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(214\) |
| norman | \(\frac {\frac {\left (a^{3} c^{2} d^{3}+a^{2} b \,c^{3} d^{2}+a \,b^{2} c^{4} d -4 c^{5} b^{3}\right ) x}{c \,b^{3} d^{4} \left (a d -b c \right )}+\frac {x^{4}}{3 b d}-\frac {\left (3 a d +4 b c \right ) x^{3}}{6 b^{2} d^{2}}+\frac {\left (2 a^{2} d^{2}+3 a b c d +4 b^{2} c^{2}\right ) x^{2}}{2 b^{3} d^{3}}}{d x +c}+\frac {c^{4} \left (5 a d -4 b c \right ) \ln \left (d x +c \right )}{d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {a^{5} \ln \left (b x +a \right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{4}}\) | \(218\) |
| parallelrisch | \(-\frac {6 a^{4} b \,c^{2} d^{4}-30 a \,b^{4} c^{5} d -2 x^{4} a^{2} b^{3} d^{6}-2 x^{4} b^{5} c^{2} d^{4}+3 x^{3} a^{3} b^{2} d^{6}+4 x^{4} a \,b^{4} c \,d^{5}-2 x^{3} a^{2} b^{3} c \,d^{5}-5 x^{3} a \,b^{4} c^{2} d^{4}+3 x^{2} a^{3} b^{2} c \,d^{5}+15 x^{2} a \,b^{4} c^{3} d^{3}+24 \ln \left (d x +c \right ) x \,b^{5} c^{5} d -30 \ln \left (d x +c \right ) a \,b^{4} c^{5} d +4 x^{3} b^{5} c^{3} d^{3}-6 x^{2} a^{4} b \,d^{6}-12 x^{2} b^{5} c^{4} d^{2}+6 \ln \left (b x +a \right ) x \,a^{5} d^{6}+6 \ln \left (b x +a \right ) a^{5} c \,d^{5}-30 \ln \left (d x +c \right ) x a \,b^{4} c^{4} d^{2}+24 \ln \left (d x +c \right ) b^{5} c^{6}+24 b^{5} c^{6}}{6 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (d x +c \right ) b^{4} d^{5}}\) | \(320\) |
1/b^3/d^4*(1/3*d^2*x^3*b^2-1/2*x^2*a*b*d^2-x^2*b^2*c*d+a^2*d^2*x+2*a*b*c*d *x+3*b^2*c^2*x)+1/d^5*c^4*(5*a*d-4*b*c)/(a*d-b*c)^2*ln(d*x+c)+1/d^5*c^5/(a *d-b*c)/(d*x+c)-1/b^4*a^5/(a*d-b*c)^2*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (143) = 286\).
Time = 0.24 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.29 \[ \int \frac {x^5}{(a+b x) (c+d x)^2} \, dx=-\frac {6 \, b^{5} c^{6} - 6 \, a b^{4} c^{5} d - 2 \, {\left (b^{5} c^{2} d^{4} - 2 \, a b^{4} c d^{5} + a^{2} b^{3} d^{6}\right )} x^{4} + {\left (4 \, b^{5} c^{3} d^{3} - 5 \, a b^{4} c^{2} d^{4} - 2 \, a^{2} b^{3} c d^{5} + 3 \, a^{3} b^{2} d^{6}\right )} x^{3} - 3 \, {\left (4 \, b^{5} c^{4} d^{2} - 5 \, a b^{4} c^{3} d^{3} - a^{3} b^{2} c d^{5} + 2 \, a^{4} b d^{6}\right )} x^{2} - 6 \, {\left (3 \, b^{5} c^{5} d - 4 \, a b^{4} c^{4} d^{2} + a^{4} b c d^{5}\right )} x + 6 \, {\left (a^{5} d^{6} x + a^{5} c d^{5}\right )} \log \left (b x + a\right ) + 6 \, {\left (4 \, b^{5} c^{6} - 5 \, a b^{4} c^{5} d + {\left (4 \, b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2}\right )} x\right )} \log \left (d x + c\right )}{6 \, {\left (b^{6} c^{3} d^{5} - 2 \, a b^{5} c^{2} d^{6} + a^{2} b^{4} c d^{7} + {\left (b^{6} c^{2} d^{6} - 2 \, a b^{5} c d^{7} + a^{2} b^{4} d^{8}\right )} x\right )}} \]
-1/6*(6*b^5*c^6 - 6*a*b^4*c^5*d - 2*(b^5*c^2*d^4 - 2*a*b^4*c*d^5 + a^2*b^3 *d^6)*x^4 + (4*b^5*c^3*d^3 - 5*a*b^4*c^2*d^4 - 2*a^2*b^3*c*d^5 + 3*a^3*b^2 *d^6)*x^3 - 3*(4*b^5*c^4*d^2 - 5*a*b^4*c^3*d^3 - a^3*b^2*c*d^5 + 2*a^4*b*d ^6)*x^2 - 6*(3*b^5*c^5*d - 4*a*b^4*c^4*d^2 + a^4*b*c*d^5)*x + 6*(a^5*d^6*x + a^5*c*d^5)*log(b*x + a) + 6*(4*b^5*c^6 - 5*a*b^4*c^5*d + (4*b^5*c^5*d - 5*a*b^4*c^4*d^2)*x)*log(d*x + c))/(b^6*c^3*d^5 - 2*a*b^5*c^2*d^6 + a^2*b^ 4*c*d^7 + (b^6*c^2*d^6 - 2*a*b^5*c*d^7 + a^2*b^4*d^8)*x)
Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (133) = 266\).
Time = 12.49 (sec) , antiderivative size = 464, normalized size of antiderivative = 3.16 \[ \int \frac {x^5}{(a+b x) (c+d x)^2} \, dx=- \frac {a^{5} \log {\left (x + \frac {\frac {a^{8} d^{7}}{b \left (a d - b c\right )^{2}} - \frac {3 a^{7} c d^{6}}{\left (a d - b c\right )^{2}} + \frac {3 a^{6} b c^{2} d^{5}}{\left (a d - b c\right )^{2}} - \frac {a^{5} b^{2} c^{3} d^{4}}{\left (a d - b c\right )^{2}} + a^{5} c d^{4} + 5 a^{2} b^{3} c^{4} d - 4 a b^{4} c^{5}}{a^{5} d^{5} + 5 a b^{4} c^{4} d - 4 b^{5} c^{5}} \right )}}{b^{4} \left (a d - b c\right )^{2}} + \frac {c^{5}}{a c d^{6} - b c^{2} d^{5} + x \left (a d^{7} - b c d^{6}\right )} + \frac {c^{4} \cdot \left (5 a d - 4 b c\right ) \log {\left (x + \frac {a^{5} c d^{4} - \frac {a^{3} b^{3} c^{4} d^{2} \cdot \left (5 a d - 4 b c\right )}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{4} c^{5} d \left (5 a d - 4 b c\right )}{\left (a d - b c\right )^{2}} + 5 a^{2} b^{3} c^{4} d - \frac {3 a b^{5} c^{6} \cdot \left (5 a d - 4 b c\right )}{\left (a d - b c\right )^{2}} - 4 a b^{4} c^{5} + \frac {b^{6} c^{7} \cdot \left (5 a d - 4 b c\right )}{d \left (a d - b c\right )^{2}}}{a^{5} d^{5} + 5 a b^{4} c^{4} d - 4 b^{5} c^{5}} \right )}}{d^{5} \left (a d - b c\right )^{2}} + x^{2} \left (- \frac {a}{2 b^{2} d^{2}} - \frac {c}{b d^{3}}\right ) + x \left (\frac {a^{2}}{b^{3} d^{2}} + \frac {2 a c}{b^{2} d^{3}} + \frac {3 c^{2}}{b d^{4}}\right ) + \frac {x^{3}}{3 b d^{2}} \]
-a**5*log(x + (a**8*d**7/(b*(a*d - b*c)**2) - 3*a**7*c*d**6/(a*d - b*c)**2 + 3*a**6*b*c**2*d**5/(a*d - b*c)**2 - a**5*b**2*c**3*d**4/(a*d - b*c)**2 + a**5*c*d**4 + 5*a**2*b**3*c**4*d - 4*a*b**4*c**5)/(a**5*d**5 + 5*a*b**4* c**4*d - 4*b**5*c**5))/(b**4*(a*d - b*c)**2) + c**5/(a*c*d**6 - b*c**2*d** 5 + x*(a*d**7 - b*c*d**6)) + c**4*(5*a*d - 4*b*c)*log(x + (a**5*c*d**4 - a **3*b**3*c**4*d**2*(5*a*d - 4*b*c)/(a*d - b*c)**2 + 3*a**2*b**4*c**5*d*(5* a*d - 4*b*c)/(a*d - b*c)**2 + 5*a**2*b**3*c**4*d - 3*a*b**5*c**6*(5*a*d - 4*b*c)/(a*d - b*c)**2 - 4*a*b**4*c**5 + b**6*c**7*(5*a*d - 4*b*c)/(d*(a*d - b*c)**2))/(a**5*d**5 + 5*a*b**4*c**4*d - 4*b**5*c**5))/(d**5*(a*d - b*c) **2) + x**2*(-a/(2*b**2*d**2) - c/(b*d**3)) + x*(a**2/(b**3*d**2) + 2*a*c/ (b**2*d**3) + 3*c**2/(b*d**4)) + x**3/(3*b*d**2)
Time = 0.19 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.31 \[ \int \frac {x^5}{(a+b x) (c+d x)^2} \, dx=-\frac {a^{5} \log \left (b x + a\right )}{b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}} - \frac {c^{5}}{b c^{2} d^{5} - a c d^{6} + {\left (b c d^{6} - a d^{7}\right )} x} - \frac {{\left (4 \, b c^{5} - 5 \, a c^{4} d\right )} \log \left (d x + c\right )}{b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}} + \frac {2 \, b^{2} d^{2} x^{3} - 3 \, {\left (2 \, b^{2} c d + a b d^{2}\right )} x^{2} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x}{6 \, b^{3} d^{4}} \]
-a^5*log(b*x + a)/(b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2) - c^5/(b*c^2*d^5 - a*c*d^6 + (b*c*d^6 - a*d^7)*x) - (4*b*c^5 - 5*a*c^4*d)*log(d*x + c)/(b^2* c^2*d^5 - 2*a*b*c*d^6 + a^2*d^7) + 1/6*(2*b^2*d^2*x^3 - 3*(2*b^2*c*d + a*b *d^2)*x^2 + 6*(3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*x)/(b^3*d^4)
Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.67 \[ \int \frac {x^5}{(a+b x) (c+d x)^2} \, dx=-\frac {c^{5} d^{4}}{{\left (b c d^{9} - a d^{10}\right )} {\left (d x + c\right )}} - \frac {a^{5} d \log \left ({\left | b - \frac {b c}{d x + c} + \frac {a d}{d x + c} \right |}\right )}{b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}} + \frac {{\left (2 \, b^{3} - \frac {3 \, {\left (4 \, b^{3} c d + a b^{2} d^{2}\right )}}{{\left (d x + c\right )} d} + \frac {6 \, {\left (6 \, b^{3} c^{2} d^{2} + 3 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )}}{{\left (d x + c\right )}^{2} d^{2}}\right )} {\left (d x + c\right )}^{3}}{6 \, b^{4} d^{5}} + \frac {{\left (4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{b^{4} d^{5}} \]
-c^5*d^4/((b*c*d^9 - a*d^10)*(d*x + c)) - a^5*d*log(abs(b - b*c/(d*x + c) + a*d/(d*x + c)))/(b^6*c^2*d - 2*a*b^5*c*d^2 + a^2*b^4*d^3) + 1/6*(2*b^3 - 3*(4*b^3*c*d + a*b^2*d^2)/((d*x + c)*d) + 6*(6*b^3*c^2*d^2 + 3*a*b^2*c*d^ 3 + a^2*b*d^4)/((d*x + c)^2*d^2))*(d*x + c)^3/(b^4*d^5) + (4*b^3*c^3 + 3*a *b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*log(abs(d*x + c)/((d*x + c)^2*abs(d) ))/(b^4*d^5)
Time = 0.59 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.39 \[ \int \frac {x^5}{(a+b x) (c+d x)^2} \, dx=\frac {x^3}{3\,b\,d^2}-\frac {a^5\,\ln \left (a+b\,x\right )}{a^2\,b^4\,d^2-2\,a\,b^5\,c\,d+b^6\,c^2}-\frac {\ln \left (c+d\,x\right )\,\left (4\,b\,c^5-5\,a\,c^4\,d\right )}{a^2\,d^7-2\,a\,b\,c\,d^6+b^2\,c^2\,d^5}-x\,\left (\frac {b\,c^2+2\,a\,d\,c}{b^2\,d^4}-\frac {{\left (a\,d^2+2\,b\,c\,d\right )}^2}{b^3\,d^6}\right )-\frac {x^2\,\left (a\,d^2+2\,b\,c\,d\right )}{2\,b^2\,d^4}+\frac {b^3\,c^5}{d\,\left (x\,b^3\,d^5+c\,b^3\,d^4\right )\,\left (a\,d-b\,c\right )} \]
x^3/(3*b*d^2) - (a^5*log(a + b*x))/(b^6*c^2 + a^2*b^4*d^2 - 2*a*b^5*c*d) - (log(c + d*x)*(4*b*c^5 - 5*a*c^4*d))/(a^2*d^7 + b^2*c^2*d^5 - 2*a*b*c*d^6 ) - x*((b*c^2 + 2*a*c*d)/(b^2*d^4) - (a*d^2 + 2*b*c*d)^2/(b^3*d^6)) - (x^2 *(a*d^2 + 2*b*c*d))/(2*b^2*d^4) + (b^3*c^5)/(d*(b^3*c*d^4 + b^3*d^5*x)*(a* d - b*c))