Integrand size = 15, antiderivative size = 109 \[ \int \frac {1}{(a+b x)^3 (c+d x)^2} \, dx=-\frac {b}{2 (b c-a d)^2 (a+b x)^2}+\frac {2 b d}{(b c-a d)^3 (a+b x)}+\frac {d^2}{(b c-a d)^3 (c+d x)}+\frac {3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac {3 b d^2 \log (c+d x)}{(b c-a d)^4} \]
-1/2*b/(-a*d+b*c)^2/(b*x+a)^2+2*b*d/(-a*d+b*c)^3/(b*x+a)+d^2/(-a*d+b*c)^3/ (d*x+c)+3*b*d^2*ln(b*x+a)/(-a*d+b*c)^4-3*b*d^2*ln(d*x+c)/(-a*d+b*c)^4
Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+b x)^3 (c+d x)^2} \, dx=\frac {-\frac {b (b c-a d)^2}{(a+b x)^2}+\frac {4 b d (b c-a d)}{a+b x}+\frac {2 d^2 (b c-a d)}{c+d x}+6 b d^2 \log (a+b x)-6 b d^2 \log (c+d x)}{2 (b c-a d)^4} \]
(-((b*(b*c - a*d)^2)/(a + b*x)^2) + (4*b*d*(b*c - a*d))/(a + b*x) + (2*d^2 *(b*c - a*d))/(c + d*x) + 6*b*d^2*Log[a + b*x] - 6*b*d^2*Log[c + d*x])/(2* (b*c - a*d)^4)
Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {54, 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.
| \(\displaystyle \int \frac {1}{(a+b x)^3 (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \int \left (\frac {3 b^2 d^2}{(a+b x) (b c-a d)^4}-\frac {2 b^2 d}{(a+b x)^2 (b c-a d)^3}+\frac {b^2}{(a+b x)^3 (b c-a d)^2}-\frac {3 b d^3}{(c+d x) (b c-a d)^4}-\frac {d^3}{(c+d x)^2 (b c-a d)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2}{(c+d x) (b c-a d)^3}+\frac {3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac {3 b d^2 \log (c+d x)}{(b c-a d)^4}+\frac {2 b d}{(a+b x) (b c-a d)^3}-\frac {b}{2 (a+b x)^2 (b c-a d)^2}\) |
-1/2*b/((b*c - a*d)^2*(a + b*x)^2) + (2*b*d)/((b*c - a*d)^3*(a + b*x)) + d ^2/((b*c - a*d)^3*(c + d*x)) + (3*b*d^2*Log[a + b*x])/(b*c - a*d)^4 - (3*b *d^2*Log[c + d*x])/(b*c - a*d)^4
3.14.51.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(-\frac {d^{2}}{\left (a d -b c \right )^{3} \left (d x +c \right )}-\frac {3 d^{2} b \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}-\frac {b}{2 \left (a d -b c \right )^{2} \left (b x +a \right )^{2}}+\frac {3 d^{2} b \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}-\frac {2 b d}{\left (a d -b c \right )^{3} \left (b x +a \right )}\) | \(109\) |
| risch | \(\frac {-\frac {3 b^{2} d^{2} x^{2}}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {3 \left (3 a d +b c \right ) b d x}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {2 a^{2} d^{2}+5 a b c d -b^{2} c^{2}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )}+\frac {3 b \,d^{2} \ln \left (-b x -a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}-\frac {3 b \,d^{2} \ln \left (d x +c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}\) | \(310\) |
| norman | \(\frac {-\frac {3 b^{2} d^{2} x^{2}}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {-2 a^{2} b^{2} d^{3}-5 a \,b^{3} c \,d^{2}+b^{4} c^{2} d}{2 d \,b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (-9 a \,b^{3} d^{3}-3 b^{4} c \,d^{2}\right ) x}{2 d \,b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )}+\frac {3 b \,d^{2} \ln \left (b x +a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}-\frac {3 b \,d^{2} \ln \left (d x +c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}\) | \(335\) |
| parallelrisch | \(\frac {12 \ln \left (b x +a \right ) x a \,b^{4} c \,d^{3}-12 \ln \left (d x +c \right ) x a \,b^{4} c \,d^{3}-3 a^{2} c \,b^{3} d^{3}-2 a^{3} b^{2} d^{4}-b^{5} c^{3} d -9 x \,a^{2} b^{3} d^{4}+3 x \,b^{5} c^{2} d^{2}+6 \ln \left (b x +a \right ) x^{3} b^{5} d^{4}-6 \ln \left (d x +c \right ) x^{3} b^{5} d^{4}-6 x^{2} a \,b^{4} d^{4}+6 x^{2} b^{5} c \,d^{3}+6 a \,b^{4} c^{2} d^{2}+6 \ln \left (b x +a \right ) a^{2} b^{3} c \,d^{3}-6 \ln \left (d x +c \right ) a^{2} b^{3} c \,d^{3}+6 x a \,b^{4} c \,d^{3}+12 \ln \left (b x +a \right ) x^{2} a \,b^{4} d^{4}+6 \ln \left (b x +a \right ) x^{2} b^{5} c \,d^{3}-12 \ln \left (d x +c \right ) x^{2} a \,b^{4} d^{4}-6 \ln \left (d x +c \right ) x^{2} b^{5} c \,d^{3}+6 \ln \left (b x +a \right ) x \,a^{2} b^{3} d^{4}-6 \ln \left (d x +c \right ) x \,a^{2} b^{3} d^{4}}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (d x +c \right ) \left (b x +a \right )^{2} b^{2} d}\) | \(390\) |
-d^2/(a*d-b*c)^3/(d*x+c)-3*d^2/(a*d-b*c)^4*b*ln(d*x+c)-1/2*b/(a*d-b*c)^2/( b*x+a)^2+3*d^2/(a*d-b*c)^4*b*ln(b*x+a)-2*b/(a*d-b*c)^3*d/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (107) = 214\).
Time = 0.23 (sec) , antiderivative size = 494, normalized size of antiderivative = 4.53 \[ \int \frac {1}{(a+b x)^3 (c+d x)^2} \, dx=-\frac {b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x - 6 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{4} c^{5} - 4 \, a^{3} b^{3} c^{4} d + 6 \, a^{4} b^{2} c^{3} d^{2} - 4 \, a^{5} b c^{2} d^{3} + a^{6} c d^{4} + {\left (b^{6} c^{4} d - 4 \, a b^{5} c^{3} d^{2} + 6 \, a^{2} b^{4} c^{2} d^{3} - 4 \, a^{3} b^{3} c d^{4} + a^{4} b^{2} d^{5}\right )} x^{3} + {\left (b^{6} c^{5} - 2 \, a b^{5} c^{4} d - 2 \, a^{2} b^{4} c^{3} d^{2} + 8 \, a^{3} b^{3} c^{2} d^{3} - 7 \, a^{4} b^{2} c d^{4} + 2 \, a^{5} b d^{5}\right )} x^{2} + {\left (2 \, a b^{5} c^{5} - 7 \, a^{2} b^{4} c^{4} d + 8 \, a^{3} b^{3} c^{3} d^{2} - 2 \, a^{4} b^{2} c^{2} d^{3} - 2 \, a^{5} b c d^{4} + a^{6} d^{5}\right )} x\right )}} \]
-1/2*(b^3*c^3 - 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 2*a^3*d^3 - 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 3*(b^3*c^2*d + 2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x - 6*(b^3*d ^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^ 2*b*d^3)*x)*log(b*x + a) + 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a *b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(d*x + c))/(a^2*b^4*c^5 - 4*a^3*b^3*c^4*d + 6*a^4*b^2*c^3*d^2 - 4*a^5*b*c^2*d^3 + a^6*c*d^4 + (b^6 *c^4*d - 4*a*b^5*c^3*d^2 + 6*a^2*b^4*c^2*d^3 - 4*a^3*b^3*c*d^4 + a^4*b^2*d ^5)*x^3 + (b^6*c^5 - 2*a*b^5*c^4*d - 2*a^2*b^4*c^3*d^2 + 8*a^3*b^3*c^2*d^3 - 7*a^4*b^2*c*d^4 + 2*a^5*b*d^5)*x^2 + (2*a*b^5*c^5 - 7*a^2*b^4*c^4*d + 8 *a^3*b^3*c^3*d^2 - 2*a^4*b^2*c^2*d^3 - 2*a^5*b*c*d^4 + a^6*d^5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 634 vs. \(2 (97) = 194\).
Time = 0.99 (sec) , antiderivative size = 634, normalized size of antiderivative = 5.82 \[ \int \frac {1}{(a+b x)^3 (c+d x)^2} \, dx=- \frac {3 b d^{2} \log {\left (x + \frac {- \frac {3 a^{5} b d^{7}}{\left (a d - b c\right )^{4}} + \frac {15 a^{4} b^{2} c d^{6}}{\left (a d - b c\right )^{4}} - \frac {30 a^{3} b^{3} c^{2} d^{5}}{\left (a d - b c\right )^{4}} + \frac {30 a^{2} b^{4} c^{3} d^{4}}{\left (a d - b c\right )^{4}} - \frac {15 a b^{5} c^{4} d^{3}}{\left (a d - b c\right )^{4}} + 3 a b d^{3} + \frac {3 b^{6} c^{5} d^{2}}{\left (a d - b c\right )^{4}} + 3 b^{2} c d^{2}}{6 b^{2} d^{3}} \right )}}{\left (a d - b c\right )^{4}} + \frac {3 b d^{2} \log {\left (x + \frac {\frac {3 a^{5} b d^{7}}{\left (a d - b c\right )^{4}} - \frac {15 a^{4} b^{2} c d^{6}}{\left (a d - b c\right )^{4}} + \frac {30 a^{3} b^{3} c^{2} d^{5}}{\left (a d - b c\right )^{4}} - \frac {30 a^{2} b^{4} c^{3} d^{4}}{\left (a d - b c\right )^{4}} + \frac {15 a b^{5} c^{4} d^{3}}{\left (a d - b c\right )^{4}} + 3 a b d^{3} - \frac {3 b^{6} c^{5} d^{2}}{\left (a d - b c\right )^{4}} + 3 b^{2} c d^{2}}{6 b^{2} d^{3}} \right )}}{\left (a d - b c\right )^{4}} + \frac {- 2 a^{2} d^{2} - 5 a b c d + b^{2} c^{2} - 6 b^{2} d^{2} x^{2} + x \left (- 9 a b d^{2} - 3 b^{2} c d\right )}{2 a^{5} c d^{3} - 6 a^{4} b c^{2} d^{2} + 6 a^{3} b^{2} c^{3} d - 2 a^{2} b^{3} c^{4} + x^{3} \cdot \left (2 a^{3} b^{2} d^{4} - 6 a^{2} b^{3} c d^{3} + 6 a b^{4} c^{2} d^{2} - 2 b^{5} c^{3} d\right ) + x^{2} \cdot \left (4 a^{4} b d^{4} - 10 a^{3} b^{2} c d^{3} + 6 a^{2} b^{3} c^{2} d^{2} + 2 a b^{4} c^{3} d - 2 b^{5} c^{4}\right ) + x \left (2 a^{5} d^{4} - 2 a^{4} b c d^{3} - 6 a^{3} b^{2} c^{2} d^{2} + 10 a^{2} b^{3} c^{3} d - 4 a b^{4} c^{4}\right )} \]
-3*b*d**2*log(x + (-3*a**5*b*d**7/(a*d - b*c)**4 + 15*a**4*b**2*c*d**6/(a* d - b*c)**4 - 30*a**3*b**3*c**2*d**5/(a*d - b*c)**4 + 30*a**2*b**4*c**3*d* *4/(a*d - b*c)**4 - 15*a*b**5*c**4*d**3/(a*d - b*c)**4 + 3*a*b*d**3 + 3*b* *6*c**5*d**2/(a*d - b*c)**4 + 3*b**2*c*d**2)/(6*b**2*d**3))/(a*d - b*c)**4 + 3*b*d**2*log(x + (3*a**5*b*d**7/(a*d - b*c)**4 - 15*a**4*b**2*c*d**6/(a *d - b*c)**4 + 30*a**3*b**3*c**2*d**5/(a*d - b*c)**4 - 30*a**2*b**4*c**3*d **4/(a*d - b*c)**4 + 15*a*b**5*c**4*d**3/(a*d - b*c)**4 + 3*a*b*d**3 - 3*b **6*c**5*d**2/(a*d - b*c)**4 + 3*b**2*c*d**2)/(6*b**2*d**3))/(a*d - b*c)** 4 + (-2*a**2*d**2 - 5*a*b*c*d + b**2*c**2 - 6*b**2*d**2*x**2 + x*(-9*a*b*d **2 - 3*b**2*c*d))/(2*a**5*c*d**3 - 6*a**4*b*c**2*d**2 + 6*a**3*b**2*c**3* d - 2*a**2*b**3*c**4 + x**3*(2*a**3*b**2*d**4 - 6*a**2*b**3*c*d**3 + 6*a*b **4*c**2*d**2 - 2*b**5*c**3*d) + x**2*(4*a**4*b*d**4 - 10*a**3*b**2*c*d**3 + 6*a**2*b**3*c**2*d**2 + 2*a*b**4*c**3*d - 2*b**5*c**4) + x*(2*a**5*d**4 - 2*a**4*b*c*d**3 - 6*a**3*b**2*c**2*d**2 + 10*a**2*b**3*c**3*d - 4*a*b** 4*c**4))
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (107) = 214\).
Time = 0.23 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.54 \[ \int \frac {1}{(a+b x)^3 (c+d x)^2} \, dx=\frac {3 \, b d^{2} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {3 \, b d^{2} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {6 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 5 \, a b c d + 2 \, a^{2} d^{2} + 3 \, {\left (b^{2} c d + 3 \, a b d^{2}\right )} x}{2 \, {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} + {\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{3} + {\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{2} + {\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} x\right )}} \]
3*b*d^2*log(b*x + a)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3* b*c*d^3 + a^4*d^4) - 3*b*d^2*log(d*x + c)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2 *b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) + 1/2*(6*b^2*d^2*x^2 - b^2*c^2 + 5 *a*b*c*d + 2*a^2*d^2 + 3*(b^2*c*d + 3*a*b*d^2)*x)/(a^2*b^3*c^4 - 3*a^3*b^2 *c^3*d + 3*a^4*b*c^2*d^2 - a^5*c*d^3 + (b^5*c^3*d - 3*a*b^4*c^2*d^2 + 3*a^ 2*b^3*c*d^3 - a^3*b^2*d^4)*x^3 + (b^5*c^4 - a*b^4*c^3*d - 3*a^2*b^3*c^2*d^ 2 + 5*a^3*b^2*c*d^3 - 2*a^4*b*d^4)*x^2 + (2*a*b^4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3*b^2*c^2*d^2 + a^4*b*c*d^3 - a^5*d^4)*x)
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (107) = 214\).
Time = 0.31 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.98 \[ \int \frac {1}{(a+b x)^3 (c+d x)^2} \, dx=\frac {3 \, b d^{3} \log \left ({\left | b - \frac {b c}{d x + c} + \frac {a d}{d x + c} \right |}\right )}{b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}} + \frac {d^{5}}{{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} {\left (d x + c\right )}} + \frac {5 \, b^{3} d^{2} - \frac {6 \, {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )}}{{\left (d x + c\right )} d}}{2 \, {\left (b c - a d\right )}^{4} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}^{2}} \]
3*b*d^3*log(abs(b - b*c/(d*x + c) + a*d/(d*x + c)))/(b^4*c^4*d - 4*a*b^3*c ^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4 + a^4*d^5) + d^5/((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)*(d*x + c)) + 1/2*(5*b^3*d^2 - 6*(b^3*c*d^3 - a*b^2*d^4)/((d*x + c)*d))/((b*c - a*d)^4*(b - b*c/(d*x + c ) + a*d/(d*x + c))^2)
Time = 0.45 (sec) , antiderivative size = 330, normalized size of antiderivative = 3.03 \[ \int \frac {1}{(a+b x)^3 (c+d x)^2} \, dx=\frac {6\,b\,d^2\,\mathrm {atanh}\left (\frac {a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4}{{\left (a\,d-b\,c\right )}^4}+\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{{\left (a\,d-b\,c\right )}^4}-\frac {\frac {2\,a^2\,d^2+5\,a\,b\,c\,d-b^2\,c^2}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {3\,d\,x\,\left (c\,b^2+3\,a\,d\,b\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {3\,b^2\,d^2\,x^2}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{x\,\left (d\,a^2+2\,b\,c\,a\right )+a^2\,c+x^2\,\left (c\,b^2+2\,a\,d\,b\right )+b^2\,d\,x^3} \]
(6*b*d^2*atanh((a^4*d^4 - b^4*c^4 + 2*a*b^3*c^3*d - 2*a^3*b*c*d^3)/(a*d - b*c)^4 + (2*b*d*x*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a* d - b*c)^4))/(a*d - b*c)^4 - ((2*a^2*d^2 - b^2*c^2 + 5*a*b*c*d)/(2*(a^3*d^ 3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (3*d*x*(b^2*c + 3*a*b*d))/ (2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (3*b^2*d^2*x^2)/ (a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(x*(a^2*d + 2*a*b*c) + a^2*c + x^2*(b^2*c + 2*a*b*d) + b^2*d*x^3)
Time = 0.00 (sec) , antiderivative size = 821, normalized size of antiderivative = 7.53 \[ \int \frac {1}{(a+b x)^3 (c+d x)^2} \, dx =\text {Too large to display} \]
int(1/(a**3*c**2 + 2*a**3*c*d*x + a**3*d**2*x**2 + 3*a**2*b*c**2*x + 6*a** 2*b*c*d*x**2 + 3*a**2*b*d**2*x**3 + 3*a*b**2*c**2*x**2 + 6*a*b**2*c*d*x**3 + 3*a*b**2*d**2*x**4 + b**3*c**2*x**3 + 2*b**3*c*d*x**4 + b**3*d**2*x**5) ,x)
(12*log(a + b*x)*a**3*b*c*d**3 + 12*log(a + b*x)*a**3*b*d**4*x + 6*log(a + b*x)*a**2*b**2*c**2*d**2 + 30*log(a + b*x)*a**2*b**2*c*d**3*x + 24*log(a + b*x)*a**2*b**2*d**4*x**2 + 12*log(a + b*x)*a*b**3*c**2*d**2*x + 24*log(a + b*x)*a*b**3*c*d**3*x**2 + 12*log(a + b*x)*a*b**3*d**4*x**3 + 6*log(a + b*x)*b**4*c**2*d**2*x**2 + 6*log(a + b*x)*b**4*c*d**3*x**3 - 12*log(c + d* x)*a**3*b*c*d**3 - 12*log(c + d*x)*a**3*b*d**4*x - 6*log(c + d*x)*a**2*b** 2*c**2*d**2 - 30*log(c + d*x)*a**2*b**2*c*d**3*x - 24*log(c + d*x)*a**2*b* *2*d**4*x**2 - 12*log(c + d*x)*a*b**3*c**2*d**2*x - 24*log(c + d*x)*a*b**3 *c*d**3*x**2 - 12*log(c + d*x)*a*b**3*d**4*x**3 - 6*log(c + d*x)*b**4*c**2 *d**2*x**2 - 6*log(c + d*x)*b**4*c*d**3*x**3 - 4*a**4*d**4 - 2*a**3*b*c*d* *3 - 12*a**3*b*d**4*x + 3*a**2*b**2*c**2*d**2 + 9*a**2*b**2*c*d**3*x + 4*a *b**3*c**3*d + 6*a*b**3*d**4*x**3 - b**4*c**4 + 3*b**4*c**3*d*x - 6*b**4*c *d**3*x**3)/(2*(2*a**7*c*d**5 + 2*a**7*d**6*x - 7*a**6*b*c**2*d**4 - 3*a** 6*b*c*d**5*x + 4*a**6*b*d**6*x**2 + 8*a**5*b**2*c**3*d**3 - 6*a**5*b**2*c* *2*d**4*x - 12*a**5*b**2*c*d**5*x**2 + 2*a**5*b**2*d**6*x**3 - 2*a**4*b**3 *c**4*d**2 + 14*a**4*b**3*c**3*d**3*x + 9*a**4*b**3*c**2*d**4*x**2 - 7*a** 4*b**3*c*d**5*x**3 - 2*a**3*b**4*c**5*d - 6*a**3*b**4*c**4*d**2*x + 4*a**3 *b**4*c**3*d**3*x**2 + 8*a**3*b**4*c**2*d**4*x**3 + a**2*b**5*c**6 - 3*a** 2*b**5*c**5*d*x - 6*a**2*b**5*c**4*d**2*x**2 - 2*a**2*b**5*c**3*d**3*x**3 + 2*a*b**6*c**6*x - 2*a*b**6*c**4*d**2*x**3 + b**7*c**6*x**2 + b**7*c**...