Integrand size = 18, antiderivative size = 128 \[ \int \frac {x^3}{(a+b x) (c+d x)^3} \, dx=-\frac {c^3}{2 d^3 (b c-a d) (c+d x)^2}+\frac {c^2 (2 b c-3 a d)}{d^3 (b c-a d)^2 (c+d x)}-\frac {a^3 \log (a+b x)}{b (b c-a d)^3}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) \log (c+d x)}{d^3 (b c-a d)^3} \]
-1/2*c^3/d^3/(-a*d+b*c)/(d*x+c)^2+c^2*(-3*a*d+2*b*c)/d^3/(-a*d+b*c)^2/(d*x +c)-a^3*ln(b*x+a)/b/(-a*d+b*c)^3+c*(3*a^2*d^2-3*a*b*c*d+b^2*c^2)*ln(d*x+c) /d^3/(-a*d+b*c)^3
Time = 0.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.05 \[ \int \frac {x^3}{(a+b x) (c+d x)^3} \, dx=\frac {c^3}{2 d^3 (-b c+a d) (c+d x)^2}+\frac {2 b c^3-3 a c^2 d}{d^3 (-b c+a d)^2 (c+d x)}-\frac {a^3 \log (a+b x)}{b (b c-a d)^3}-\frac {\left (-b^2 c^3+3 a b c^2 d-3 a^2 c d^2\right ) \log (c+d x)}{d^3 (b c-a d)^3} \]
c^3/(2*d^3*(-(b*c) + a*d)*(c + d*x)^2) + (2*b*c^3 - 3*a*c^2*d)/(d^3*(-(b*c ) + a*d)^2*(c + d*x)) - (a^3*Log[a + b*x])/(b*(b*c - a*d)^3) - ((-(b^2*c^3 ) + 3*a*b*c^2*d - 3*a^2*c*d^2)*Log[c + d*x])/(d^3*(b*c - a*d)^3)
Time = 0.30 (sec) , antiderivative size = 128, 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.
| \(\displaystyle \int \frac {x^3}{(a+b x) (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {a^3}{(a+b x) (b c-a d)^3}-\frac {c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{d^2 (c+d x) (a d-b c)^3}-\frac {c^3}{d^2 (c+d x)^3 (a d-b c)}-\frac {c^2 (2 b c-3 a d)}{d^2 (c+d x)^2 (a d-b c)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \log (a+b x)}{b (b c-a d)^3}+\frac {c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^3}-\frac {c^3}{2 d^3 (c+d x)^2 (b c-a d)}+\frac {c^2 (2 b c-3 a d)}{d^3 (c+d x) (b c-a d)^2}\) |
-1/2*c^3/(d^3*(b*c - a*d)*(c + d*x)^2) + (c^2*(2*b*c - 3*a*d))/(d^3*(b*c - a*d)^2*(c + d*x)) - (a^3*Log[a + b*x])/(b*(b*c - a*d)^3) + (c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*Log[c + d*x])/(d^3*(b*c - a*d)^3)
3.3.53.3.1 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 = 128, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(-\frac {c^{2} \left (3 a d -2 b c \right )}{d^{3} \left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {c^{3}}{2 d^{3} \left (a d -b c \right ) \left (d x +c \right )^{2}}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} d^{3}}+\frac {a^{3} \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} b}\) | \(128\) |
| norman | \(\frac {\frac {\left (-3 a d c +2 b \,c^{2}\right ) c x}{d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-5 a d c +3 b \,c^{2}\right ) c^{2}}{2 d^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d x +c \right )^{2}}+\frac {a^{3} \ln \left (b x +a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(218\) |
| risch | \(\frac {-\frac {c^{2} \left (3 a d -2 b c \right ) x}{d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {c^{3} \left (5 a d -3 b c \right )}{2 d^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d x +c \right )^{2}}-\frac {3 c \ln \left (d x +c \right ) a^{2}}{d \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 c^{2} \ln \left (d x +c \right ) a b}{d^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {c^{3} \ln \left (d x +c \right ) b^{2}}{d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {a^{3} \ln \left (-b x -a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}\) | \(308\) |
| parallelrisch | \(\frac {-6 x \,a^{2} b \,c^{2} d^{3}+10 x a \,b^{2} c^{3} d^{2}-2 \ln \left (d x +c \right ) x^{2} b^{3} c^{3} d^{2}+4 \ln \left (b x +a \right ) x \,a^{3} c \,d^{4}-4 \ln \left (d x +c \right ) x \,b^{3} c^{4} d -6 \ln \left (d x +c \right ) a^{2} b \,c^{3} d^{2}+6 \ln \left (d x +c \right ) a \,b^{2} c^{4} d -5 a^{2} b \,c^{3} d^{2}+8 a \,b^{2} c^{4} d -3 c^{5} b^{3}-6 \ln \left (d x +c \right ) x^{2} a^{2} b c \,d^{4}-12 \ln \left (d x +c \right ) x \,a^{2} b \,c^{2} d^{3}-4 x \,b^{3} c^{4} d +2 \ln \left (b x +a \right ) x^{2} a^{3} d^{5}+2 \ln \left (b x +a \right ) a^{3} c^{2} d^{3}-2 \ln \left (d x +c \right ) b^{3} c^{5}+6 \ln \left (d x +c \right ) x^{2} a \,b^{2} c^{2} d^{3}+12 \ln \left (d x +c \right ) x a \,b^{2} c^{3} d^{2}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b \left (d x +c \right )^{2} d^{3}}\) | \(333\) |
-c^2*(3*a*d-2*b*c)/d^3/(a*d-b*c)^2/(d*x+c)+1/2/d^3*c^3/(a*d-b*c)/(d*x+c)^2 -c*(3*a^2*d^2-3*a*b*c*d+b^2*c^2)/(a*d-b*c)^3/d^3*ln(d*x+c)+a^3/(a*d-b*c)^3 /b*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (126) = 252\).
Time = 0.24 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.88 \[ \int \frac {x^3}{(a+b x) (c+d x)^3} \, dx=\frac {3 \, b^{3} c^{5} - 8 \, a b^{2} c^{4} d + 5 \, a^{2} b c^{3} d^{2} + 2 \, {\left (2 \, b^{3} c^{4} d - 5 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3}\right )} x - 2 \, {\left (a^{3} d^{5} x^{2} + 2 \, a^{3} c d^{4} x + a^{3} c^{2} d^{3}\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} + {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (b^{4} c^{5} d^{3} - 3 \, a b^{3} c^{4} d^{4} + 3 \, a^{2} b^{2} c^{3} d^{5} - a^{3} b c^{2} d^{6} + {\left (b^{4} c^{3} d^{5} - 3 \, a b^{3} c^{2} d^{6} + 3 \, a^{2} b^{2} c d^{7} - a^{3} b d^{8}\right )} x^{2} + 2 \, {\left (b^{4} c^{4} d^{4} - 3 \, a b^{3} c^{3} d^{5} + 3 \, a^{2} b^{2} c^{2} d^{6} - a^{3} b c d^{7}\right )} x\right )}} \]
1/2*(3*b^3*c^5 - 8*a*b^2*c^4*d + 5*a^2*b*c^3*d^2 + 2*(2*b^3*c^4*d - 5*a*b^ 2*c^3*d^2 + 3*a^2*b*c^2*d^3)*x - 2*(a^3*d^5*x^2 + 2*a^3*c*d^4*x + a^3*c^2* d^3)*log(b*x + a) + 2*(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 + (b^3*c^ 3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4)*x^2 + 2*(b^3*c^4*d - 3*a*b^2*c^3* d^2 + 3*a^2*b*c^2*d^3)*x)*log(d*x + c))/(b^4*c^5*d^3 - 3*a*b^3*c^4*d^4 + 3 *a^2*b^2*c^3*d^5 - a^3*b*c^2*d^6 + (b^4*c^3*d^5 - 3*a*b^3*c^2*d^6 + 3*a^2* b^2*c*d^7 - a^3*b*d^8)*x^2 + 2*(b^4*c^4*d^4 - 3*a*b^3*c^3*d^5 + 3*a^2*b^2* c^2*d^6 - a^3*b*c*d^7)*x)
Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (116) = 232\).
Time = 40.05 (sec) , antiderivative size = 653, normalized size of antiderivative = 5.10 \[ \int \frac {x^3}{(a+b x) (c+d x)^3} \, dx=\frac {a^{3} \log {\left (x + \frac {\frac {a^{7} d^{6}}{b \left (a d - b c\right )^{3}} - \frac {4 a^{6} c d^{5}}{\left (a d - b c\right )^{3}} + \frac {6 a^{5} b c^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac {4 a^{4} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{3}} + \frac {a^{3} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + 4 a^{3} c d^{2} - 3 a^{2} b c^{2} d + a b^{2} c^{3}}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}} \right )}}{b \left (a d - b c\right )^{3}} - \frac {c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {- \frac {a^{4} c d^{3} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c^{2} d^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 4 a^{3} c d^{2} - \frac {6 a^{2} b^{2} c^{3} d \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} - 3 a^{2} b c^{2} d + \frac {4 a b^{3} c^{4} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + a b^{2} c^{3} - \frac {b^{4} c^{5} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{d \left (a d - b c\right )^{3}}}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}} \right )}}{d^{3} \left (a d - b c\right )^{3}} + \frac {- 5 a c^{3} d + 3 b c^{4} + x \left (- 6 a c^{2} d^{2} + 4 b c^{3} d\right )}{2 a^{2} c^{2} d^{5} - 4 a b c^{3} d^{4} + 2 b^{2} c^{4} d^{3} + x^{2} \cdot \left (2 a^{2} d^{7} - 4 a b c d^{6} + 2 b^{2} c^{2} d^{5}\right ) + x \left (4 a^{2} c d^{6} - 8 a b c^{2} d^{5} + 4 b^{2} c^{3} d^{4}\right )} \]
a**3*log(x + (a**7*d**6/(b*(a*d - b*c)**3) - 4*a**6*c*d**5/(a*d - b*c)**3 + 6*a**5*b*c**2*d**4/(a*d - b*c)**3 - 4*a**4*b**2*c**3*d**3/(a*d - b*c)**3 + a**3*b**3*c**4*d**2/(a*d - b*c)**3 + 4*a**3*c*d**2 - 3*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*(a*d - b*c)**3) - c*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)*log(x + (-a** 4*c*d**3*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)/(a*d - b*c)**3 + 4*a**3*b*c **2*d**2*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)/(a*d - b*c)**3 + 4*a**3*c*d **2 - 6*a**2*b**2*c**3*d*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)/(a*d - b*c) **3 - 3*a**2*b*c**2*d + 4*a*b**3*c**4*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2 )/(a*d - b*c)**3 + a*b**2*c**3 - b**4*c**5*(3*a**2*d**2 - 3*a*b*c*d + b**2 *c**2)/(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**3*(a*d - b*c)**3) + (-5*a*c**3*d + 3*b*c**4 + x*(-6*a*c* *2*d**2 + 4*b*c**3*d))/(2*a**2*c**2*d**5 - 4*a*b*c**3*d**4 + 2*b**2*c**4*d **3 + x**2*(2*a**2*d**7 - 4*a*b*c*d**6 + 2*b**2*c**2*d**5) + x*(4*a**2*c*d **6 - 8*a*b*c**2*d**5 + 4*b**2*c**3*d**4))
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (126) = 252\).
Time = 0.22 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.02 \[ \int \frac {x^3}{(a+b x) (c+d x)^3} \, dx=-\frac {a^{3} \log \left (b x + a\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac {{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}} + \frac {3 \, b c^{4} - 5 \, a c^{3} d + 2 \, {\left (2 \, b c^{3} d - 3 \, a c^{2} d^{2}\right )} x}{2 \, {\left (b^{2} c^{4} d^{3} - 2 \, a b c^{3} d^{4} + a^{2} c^{2} d^{5} + {\left (b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d^{4} - 2 \, a b c^{2} d^{5} + a^{2} c d^{6}\right )} x\right )}} \]
-a^3*log(b*x + a)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) + (b^2*c^3 - 3*a*b*c^2*d + 3*a^2*c*d^2)*log(d*x + c)/(b^3*c^3*d^3 - 3*a*b^ 2*c^2*d^4 + 3*a^2*b*c*d^5 - a^3*d^6) + 1/2*(3*b*c^4 - 5*a*c^3*d + 2*(2*b*c ^3*d - 3*a*c^2*d^2)*x)/(b^2*c^4*d^3 - 2*a*b*c^3*d^4 + a^2*c^2*d^5 + (b^2*c ^2*d^5 - 2*a*b*c*d^6 + a^2*d^7)*x^2 + 2*(b^2*c^3*d^4 - 2*a*b*c^2*d^5 + a^2 *c*d^6)*x)
Time = 0.30 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.69 \[ \int \frac {x^3}{(a+b x) (c+d x)^3} \, dx=-\frac {a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac {{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}} + \frac {2 \, {\left (2 \, b^{2} c^{4} - 5 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2}\right )} x + \frac {3 \, b^{2} c^{5} - 8 \, a b c^{4} d + 5 \, a^{2} c^{3} d^{2}}{d}}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2} d^{2}} \]
-a^3*log(abs(b*x + a))/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b* d^3) + (b^2*c^3 - 3*a*b*c^2*d + 3*a^2*c*d^2)*log(abs(d*x + c))/(b^3*c^3*d^ 3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5 - a^3*d^6) + 1/2*(2*(2*b^2*c^4 - 5*a*b *c^3*d + 3*a^2*c^2*d^2)*x + (3*b^2*c^5 - 8*a*b*c^4*d + 5*a^2*c^3*d^2)/d)/( (b*c - a*d)^3*(d*x + c)^2*d^2)
Time = 0.75 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.81 \[ \int \frac {x^3}{(a+b x) (c+d x)^3} \, dx=\frac {\frac {3\,b\,c^4-5\,a\,c^3\,d}{2\,d^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {c^2\,x\,\left (3\,a\,d-2\,b\,c\right )}{d^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{c^2+2\,c\,d\,x+d^2\,x^2}-\frac {a^3\,\ln \left (a+b\,x\right )}{-a^3\,b\,d^3+3\,a^2\,b^2\,c\,d^2-3\,a\,b^3\,c^2\,d+b^4\,c^3}-\frac {\ln \left (c+d\,x\right )\,\left (3\,a^2\,c\,d^2-3\,a\,b\,c^2\,d+b^2\,c^3\right )}{a^3\,d^6-3\,a^2\,b\,c\,d^5+3\,a\,b^2\,c^2\,d^4-b^3\,c^3\,d^3} \]
((3*b*c^4 - 5*a*c^3*d)/(2*d^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (c^2*x*(3 *a*d - 2*b*c))/(d^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(c^2 + d^2*x^2 + 2*c *d*x) - (a^3*log(a + b*x))/(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^ 3*c^2*d) - (log(c + d*x)*(b^2*c^3 + 3*a^2*c*d^2 - 3*a*b*c^2*d))/(a^3*d^6 - b^3*c^3*d^3 + 3*a*b^2*c^2*d^4 - 3*a^2*b*c*d^5)