Integrand size = 18, antiderivative size = 112 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^2} \, dx=\frac {a^3}{b^2 (b c-a d)^2 (a+b x)}+\frac {c^3}{d^2 (b c-a d)^2 (c+d x)}+\frac {a^2 (3 b c-a d) \log (a+b x)}{b^2 (b c-a d)^3}+\frac {c^2 (b c-3 a d) \log (c+d x)}{d^2 (b c-a d)^3} \]
a^3/b^2/(-a*d+b*c)^2/(b*x+a)+c^3/d^2/(-a*d+b*c)^2/(d*x+c)+a^2*(-a*d+3*b*c) *ln(b*x+a)/b^2/(-a*d+b*c)^3+c^2*(-3*a*d+b*c)*ln(d*x+c)/d^2/(-a*d+b*c)^3
Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.94 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^2} \, dx=\frac {\frac {a^3}{b^2 (a+b x)}+\frac {c^3}{d^2 (c+d x)}}{(b c-a d)^2}+\frac {a^2 (3 b c-a d) \log (a+b x)}{b^2 (b c-a d)^3}+\frac {c^2 (-b c+3 a d) \log (c+d x)}{d^2 (-b c+a d)^3} \]
(a^3/(b^2*(a + b*x)) + c^3/(d^2*(c + d*x)))/(b*c - a*d)^2 + (a^2*(3*b*c - a*d)*Log[a + b*x])/(b^2*(b*c - a*d)^3) + (c^2*(-(b*c) + 3*a*d)*Log[c + d*x ])/(d^2*(-(b*c) + a*d)^3)
Time = 0.29 (sec) , antiderivative size = 112, 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.
| \(\displaystyle \int \frac {x^3}{(a+b x)^2 (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {a^3}{b (a+b x)^2 (b c-a d)^2}-\frac {a^2 (a d-3 b c)}{b (a+b x) (b c-a d)^3}-\frac {c^3}{d (c+d x)^2 (a d-b c)^2}-\frac {c^2 (b c-3 a d)}{d (c+d x) (a d-b c)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3}{b^2 (a+b x) (b c-a d)^2}+\frac {a^2 (3 b c-a d) \log (a+b x)}{b^2 (b c-a d)^3}+\frac {c^3}{d^2 (c+d x) (b c-a d)^2}+\frac {c^2 (b c-3 a d) \log (c+d x)}{d^2 (b c-a d)^3}\) |
a^3/(b^2*(b*c - a*d)^2*(a + b*x)) + c^3/(d^2*(b*c - a*d)^2*(c + d*x)) + (a ^2*(3*b*c - a*d)*Log[a + b*x])/(b^2*(b*c - a*d)^3) + (c^2*(b*c - 3*a*d)*Lo g[c + d*x])/(d^2*(b*c - a*d)^3)
3.3.83.3.1 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 = 0.50 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.01
| method | result | size |
| default | \(\frac {c^{2} \left (3 a d -b c \right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} d^{2}}+\frac {c^{3}}{d^{2} \left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {a^{2} \left (a d -3 b c \right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} b^{2}}+\frac {a^{3}}{b^{2} \left (a d -b c \right )^{2} \left (b x +a \right )}\) | \(113\) |
| norman | \(\frac {\frac {\left (a^{3} d^{3}+b^{3} c^{3}\right ) x}{d^{2} b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) a c}{d^{2} b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {a^{2} \left (a d -3 b c \right ) \ln \left (b x +a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2}}+\frac {c^{2} \left (3 a d -b c \right ) \ln \left (d x +c \right )}{d^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(230\) |
| risch | \(\frac {\frac {\left (a^{3} d^{3}+b^{3} c^{3}\right ) x}{d^{2} b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) a c}{d^{2} b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {3 c^{2} \ln \left (d x +c \right ) a}{d \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {c^{3} \ln \left (d x +c \right ) b}{d^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {a^{3} \ln \left (-b x -a \right ) d}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2}}-\frac {3 a^{2} \ln \left (-b x -a \right ) c}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}\) | \(328\) |
| parallelrisch | \(\frac {a^{4} c \,d^{3}-a \,b^{3} c^{4}-a^{3} b \,c^{2} d^{2}+a^{2} b^{2} c^{3} d +\ln \left (b x +a \right ) x \,a^{4} d^{4}-\ln \left (d x +c \right ) x \,b^{4} c^{4}+\ln \left (b x +a \right ) a^{4} c \,d^{3}-\ln \left (d x +c \right ) a \,b^{3} c^{4}-a^{3} b c \,d^{3} x +a \,b^{3} c^{3} d x +\ln \left (b x +a \right ) x^{2} a^{3} b \,d^{4}-\ln \left (d x +c \right ) x^{2} b^{4} c^{3} d -3 \ln \left (b x +a \right ) a^{3} b \,c^{2} d^{2}+3 \ln \left (d x +c \right ) a^{2} b^{2} c^{3} d -b^{4} c^{4} x +a^{4} d^{4} x -3 \ln \left (b x +a \right ) x^{2} a^{2} b^{2} c \,d^{3}+3 \ln \left (d x +c \right ) x^{2} a \,b^{3} c^{2} d^{2}-2 \ln \left (b x +a \right ) x \,a^{3} b c \,d^{3}-3 \ln \left (b x +a \right ) x \,a^{2} b^{2} c^{2} d^{2}+3 \ln \left (d x +c \right ) x \,a^{2} b^{2} c^{2} d^{2}+2 \ln \left (d x +c \right ) x a \,b^{3} c^{3} d}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right ) \left (b x +a \right ) b^{2} d^{2}}\) | \(386\) |
c^2*(3*a*d-b*c)/(a*d-b*c)^3/d^2*ln(d*x+c)+c^3/d^2/(a*d-b*c)^2/(d*x+c)+a^2* (a*d-3*b*c)/(a*d-b*c)^3/b^2*ln(b*x+a)+a^3/b^2/(a*d-b*c)^2/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (112) = 224\).
Time = 0.24 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.64 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^2} \, dx=\frac {a b^{3} c^{4} - a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (b^{4} c^{4} - a b^{3} c^{3} d + a^{3} b c d^{3} - a^{4} d^{4}\right )} x + {\left (3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} + {\left (3 \, a^{2} b^{2} c^{2} d^{2} + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \log \left (b x + a\right ) + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2}\right )} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{5} c^{4} d^{2} - 3 \, a^{2} b^{4} c^{3} d^{3} + 3 \, a^{3} b^{3} c^{2} d^{4} - a^{4} b^{2} c d^{5} + {\left (b^{6} c^{3} d^{3} - 3 \, a b^{5} c^{2} d^{4} + 3 \, a^{2} b^{4} c d^{5} - a^{3} b^{3} d^{6}\right )} x^{2} + {\left (b^{6} c^{4} d^{2} - 2 \, a b^{5} c^{3} d^{3} + 2 \, a^{3} b^{3} c d^{5} - a^{4} b^{2} d^{6}\right )} x} \]
(a*b^3*c^4 - a^2*b^2*c^3*d + a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^4 - a*b^3* c^3*d + a^3*b*c*d^3 - a^4*d^4)*x + (3*a^3*b*c^2*d^2 - a^4*c*d^3 + (3*a^2*b ^2*c*d^3 - a^3*b*d^4)*x^2 + (3*a^2*b^2*c^2*d^2 + 2*a^3*b*c*d^3 - a^4*d^4)* x)*log(b*x + a) + (a*b^3*c^4 - 3*a^2*b^2*c^3*d + (b^4*c^3*d - 3*a*b^3*c^2* d^2)*x^2 + (b^4*c^4 - 2*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2)*x)*log(d*x + c))/ (a*b^5*c^4*d^2 - 3*a^2*b^4*c^3*d^3 + 3*a^3*b^3*c^2*d^4 - a^4*b^2*c*d^5 + ( b^6*c^3*d^3 - 3*a*b^5*c^2*d^4 + 3*a^2*b^4*c*d^5 - a^3*b^3*d^6)*x^2 + (b^6* c^4*d^2 - 2*a*b^5*c^3*d^3 + 2*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x)
Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (100) = 200\).
Time = 27.89 (sec) , antiderivative size = 627, normalized size of antiderivative = 5.60 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^2} \, dx=\frac {a^{2} \left (a d - 3 b c\right ) \log {\left (x + \frac {\frac {a^{6} d^{5} \left (a d - 3 b c\right )}{b \left (a d - b c\right )^{3}} - \frac {4 a^{5} c d^{4} \left (a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{4} b c^{2} d^{3} \left (a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b^{2} c^{3} d^{2} \left (a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + a^{3} c d^{2} + \frac {a^{2} b^{3} c^{4} d \left (a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - 6 a^{2} b c^{2} d + a b^{2} c^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}} \right )}}{b^{2} \left (a d - b c\right )^{3}} + \frac {c^{2} \cdot \left (3 a d - b c\right ) \log {\left (x + \frac {\frac {a^{4} b c^{2} d^{3} \cdot \left (3 a d - b c\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b^{2} c^{3} d^{2} \cdot \left (3 a d - b c\right )}{\left (a d - b c\right )^{3}} + a^{3} c d^{2} + \frac {6 a^{2} b^{3} c^{4} d \left (3 a d - b c\right )}{\left (a d - b c\right )^{3}} - 6 a^{2} b c^{2} d - \frac {4 a b^{4} c^{5} \cdot \left (3 a d - b c\right )}{\left (a d - b c\right )^{3}} + a b^{2} c^{3} + \frac {b^{5} c^{6} \cdot \left (3 a d - b c\right )}{d \left (a d - b c\right )^{3}}}{a^{3} d^{3} - 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}} \right )}}{d^{2} \left (a d - b c\right )^{3}} + \frac {a^{3} c d^{2} + a b^{2} c^{3} + x \left (a^{3} d^{3} + b^{3} c^{3}\right )}{a^{3} b^{2} c d^{4} - 2 a^{2} b^{3} c^{2} d^{3} + a b^{4} c^{3} d^{2} + x^{2} \left (a^{2} b^{3} d^{5} - 2 a b^{4} c d^{4} + b^{5} c^{2} d^{3}\right ) + x \left (a^{3} b^{2} d^{5} - a^{2} b^{3} c d^{4} - a b^{4} c^{2} d^{3} + b^{5} c^{3} d^{2}\right )} \]
a**2*(a*d - 3*b*c)*log(x + (a**6*d**5*(a*d - 3*b*c)/(b*(a*d - b*c)**3) - 4 *a**5*c*d**4*(a*d - 3*b*c)/(a*d - b*c)**3 + 6*a**4*b*c**2*d**3*(a*d - 3*b* c)/(a*d - b*c)**3 - 4*a**3*b**2*c**3*d**2*(a*d - 3*b*c)/(a*d - b*c)**3 + a **3*c*d**2 + a**2*b**3*c**4*d*(a*d - 3*b*c)/(a*d - b*c)**3 - 6*a**2*b*c**2 *d + a*b**2*c**3)/(a**3*d**3 - 3*a**2*b*c*d**2 - 3*a*b**2*c**2*d + b**3*c* *3))/(b**2*(a*d - b*c)**3) + c**2*(3*a*d - b*c)*log(x + (a**4*b*c**2*d**3* (3*a*d - b*c)/(a*d - b*c)**3 - 4*a**3*b**2*c**3*d**2*(3*a*d - b*c)/(a*d - b*c)**3 + a**3*c*d**2 + 6*a**2*b**3*c**4*d*(3*a*d - b*c)/(a*d - b*c)**3 - 6*a**2*b*c**2*d - 4*a*b**4*c**5*(3*a*d - b*c)/(a*d - b*c)**3 + a*b**2*c**3 + b**5*c**6*(3*a*d - b*c)/(d*(a*d - b*c)**3))/(a**3*d**3 - 3*a**2*b*c*d** 2 - 3*a*b**2*c**2*d + b**3*c**3))/(d**2*(a*d - b*c)**3) + (a**3*c*d**2 + a *b**2*c**3 + x*(a**3*d**3 + b**3*c**3))/(a**3*b**2*c*d**4 - 2*a**2*b**3*c* *2*d**3 + a*b**4*c**3*d**2 + x**2*(a**2*b**3*d**5 - 2*a*b**4*c*d**4 + b**5 *c**2*d**3) + x*(a**3*b**2*d**5 - a**2*b**3*c*d**4 - a*b**4*c**2*d**3 + b* *5*c**3*d**2))
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (112) = 224\).
Time = 0.20 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.54 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^2} \, dx=\frac {{\left (3 \, a^{2} b c - a^{3} d\right )} \log \left (b x + a\right )}{b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}} + \frac {{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}} + \frac {a b^{2} c^{3} + a^{3} c d^{2} + {\left (b^{3} c^{3} + a^{3} d^{3}\right )} x}{a b^{4} c^{3} d^{2} - 2 \, a^{2} b^{3} c^{2} d^{3} + a^{3} b^{2} c d^{4} + {\left (b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} + {\left (b^{5} c^{3} d^{2} - a b^{4} c^{2} d^{3} - a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x} \]
(3*a^2*b*c - a^3*d)*log(b*x + a)/(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) + (b*c^3 - 3*a*c^2*d)*log(d*x + c)/(b^3*c^3*d^2 - 3*a*b^2 *c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5) + (a*b^2*c^3 + a^3*c*d^2 + (b^3*c^3 + a^3*d^3)*x)/(a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 + a^3*b^2*c*d^4 + (b^5*c^2* d^3 - 2*a*b^4*c*d^4 + a^2*b^3*d^5)*x^2 + (b^5*c^3*d^2 - a*b^4*c^2*d^3 - a^ 2*b^3*c*d^4 + a^3*b^2*d^5)*x)
Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.80 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^2} \, dx=\frac {a^{3} b^{2}}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} {\left (b x + a\right )}} + \frac {{\left (b^{2} c^{3} - 3 \, a b c^{2} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}} - \frac {b c^{3}}{{\left (b c - a d\right )}^{3} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d} - \frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{2} d^{2}} \]
a^3*b^2/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*(b*x + a)) + (b^2*c^3 - 3*a *b*c^2*d)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^4*c^3*d^2 - 3*a*b ^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5) - b*c^3/((b*c - a*d)^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d) - log(abs(b*x + a)/((b*x + a)^2*abs(b)))/(b^ 2*d^2)
Time = 0.47 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.30 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^2} \, dx=\frac {\frac {a\,c\,\left (a^2\,d^2+b^2\,c^2\right )}{b^2\,d^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (a\,d+b\,c\right )\,\left (a^2\,d^2-a\,b\,c\,d+b^2\,c^2\right )}{b^2\,d^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c}-\frac {\ln \left (a+b\,x\right )\,\left (a^3\,d-3\,a^2\,b\,c\right )}{-a^3\,b^2\,d^3+3\,a^2\,b^3\,c\,d^2-3\,a\,b^4\,c^2\,d+b^5\,c^3}-\frac {\ln \left (c+d\,x\right )\,\left (b\,c^3-3\,a\,c^2\,d\right )}{a^3\,d^5-3\,a^2\,b\,c\,d^4+3\,a\,b^2\,c^2\,d^3-b^3\,c^3\,d^2} \]
((a*c*(a^2*d^2 + b^2*c^2))/(b^2*d^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x* (a*d + b*c)*(a^2*d^2 + b^2*c^2 - a*b*c*d))/(b^2*d^2*(a^2*d^2 + b^2*c^2 - 2 *a*b*c*d)))/(a*c + x*(a*d + b*c) + b*d*x^2) - (log(a + b*x)*(a^3*d - 3*a^2 *b*c))/(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) - (log(c + d*x)*(b*c^3 - 3*a*c^2*d))/(a^3*d^5 - b^3*c^3*d^2 + 3*a*b^2*c^2*d^3 - 3*a ^2*b*c*d^4)