Integrand size = 29, antiderivative size = 107 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {1}{3 (b c-a d) (a+b x)^3}+\frac {d}{2 (b c-a d)^2 (a+b x)^2}-\frac {d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3 \log (a+b x)}{(b c-a d)^4}+\frac {d^3 \log (c+d x)}{(b c-a d)^4} \] Output:
-1/3/(-a*d+b*c)/(b*x+a)^3+1/2*d/(-a*d+b*c)^2/(b*x+a)^2-d^2/(-a*d+b*c)^3/(b *x+a)-d^3*ln(b*x+a)/(-a*d+b*c)^4+d^3*ln(d*x+c)/(-a*d+b*c)^4
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {1}{3 (-b c+a d) (a+b x)^3}+\frac {d}{2 (b c-a d)^2 (a+b x)^2}-\frac {d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3 \log (a+b x)}{(b c-a d)^4}+\frac {d^3 \log (c+d x)}{(b c-a d)^4} \] Input:
Integrate[1/((a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)),x]
Output:
1/(3*(-(b*c) + a*d)*(a + b*x)^3) + d/(2*(b*c - a*d)^2*(a + b*x)^2) - d^2/( (b*c - a*d)^3*(a + b*x)) - (d^3*Log[a + b*x])/(b*c - a*d)^4 + (d^3*Log[c + d*x])/(b*c - a*d)^4
Time = 0.46 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1121, 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.
| \(\displaystyle \int \frac {1}{(a+b x)^3 \left (x (a d+b c)+a c+b d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {d^4}{(c+d x) (b c-a d)^4}-\frac {b d^3}{(a+b x) (b c-a d)^4}+\frac {b d^2}{(a+b x)^2 (b c-a d)^3}-\frac {b d}{(a+b x)^3 (b c-a d)^2}+\frac {b}{(a+b x)^4 (b c-a d)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 \log (a+b x)}{(b c-a d)^4}+\frac {d^3 \log (c+d x)}{(b c-a d)^4}-\frac {d^2}{(a+b x) (b c-a d)^3}+\frac {d}{2 (a+b x)^2 (b c-a d)^2}-\frac {1}{3 (a+b x)^3 (b c-a d)}\) |
Input:
Int[1/((a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)),x]
Output:
-1/3*1/((b*c - a*d)*(a + b*x)^3) + d/(2*(b*c - a*d)^2*(a + b*x)^2) - d^2/( (b*c - a*d)^3*(a + b*x)) - (d^3*Log[a + b*x])/(b*c - a*d)^4 + (d^3*Log[c + d*x])/(b*c - a*d)^4
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 1.72 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(\frac {1}{3 \left (a d -b c \right ) \left (b x +a \right )^{3}}+\frac {d}{2 \left (a d -b c \right )^{2} \left (b x +a \right )^{2}}+\frac {d^{2}}{\left (a d -b c \right )^{3} \left (b x +a \right )}-\frac {d^{3} \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}+\frac {d^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}\) | \(103\) |
| risch | \(\frac {\frac {b^{2} d^{2} x^{2}}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {\left (5 a d -b c \right ) b d x}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}+\frac {11 a^{2} d^{2}-7 a b c d +2 b^{2} c^{2}}{6 a^{3} d^{3}-18 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -6 b^{3} c^{3}}}{\left (b x +a \right )^{3}}+\frac {d^{3} \ln \left (-d x -c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}-\frac {d^{3} \ln \left (b x +a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}\) | \(300\) |
| parallelrisch | \(-\frac {18 a^{2} b^{4} c \,d^{2}+2 b^{6} c^{3}+18 \ln \left (b x +a \right ) x \,a^{2} b^{4} d^{3}-18 \ln \left (d x +c \right ) x \,a^{2} b^{4} d^{3}+18 x a \,b^{5} c \,d^{2}+18 \ln \left (b x +a \right ) x^{2} a \,b^{5} d^{3}-18 \ln \left (d x +c \right ) x^{2} a \,b^{5} d^{3}-9 a \,b^{5} c^{2} d +6 x^{2} b^{6} c \,d^{2}+6 \ln \left (b x +a \right ) x^{3} b^{6} d^{3}-6 \ln \left (d x +c \right ) x^{3} b^{6} d^{3}+6 \ln \left (b x +a \right ) a^{3} b^{3} d^{3}-6 \ln \left (d x +c \right ) a^{3} b^{3} d^{3}-6 a \,b^{5} d^{3} x^{2}-15 a^{2} b^{4} d^{3} x -3 b^{6} c^{2} d x -11 d^{3} a^{3} b^{3}}{6 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \left (b x +a \right )^{3} b^{3}}\) | \(303\) |
| norman | \(\frac {\frac {b^{2} d^{2} x^{2}}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {11 d^{2} a^{2} b^{3}-7 c d a \,b^{4}+2 b^{5} c^{2}}{6 b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (5 a \,b^{3} d^{2}-b^{4} c d \right ) x}{2 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right )^{3}}+\frac {d^{3} \ln \left (d x +c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}-\frac {d^{3} \ln \left (b x +a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}\) | \(314\) |
Input:
int(1/(b*x+a)^3/(a*c+(a*d+b*c)*x+b*d*x^2),x,method=_RETURNVERBOSE)
Output:
1/3/(a*d-b*c)/(b*x+a)^3+1/2*d/(a*d-b*c)^2/(b*x+a)^2+d^2/(a*d-b*c)^3/(b*x+a )-d^3/(a*d-b*c)^4*ln(b*x+a)+d^3/(a*d-b*c)^4*ln(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (103) = 206\).
Time = 0.08 (sec) , antiderivative size = 425, normalized size of antiderivative = 3.97 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {2 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x + 6 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (d x + c\right )}{6 \, {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4} + {\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} x^{3} + 3 \, {\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} x\right )}} \] Input:
integrate(1/(b*x+a)^3/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="fricas")
Output:
-1/6*(2*b^3*c^3 - 9*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 11*a^3*d^3 + 6*(b^3*c*d ^2 - a*b^2*d^3)*x^2 - 3*(b^3*c^2*d - 6*a*b^2*c*d^2 + 5*a^2*b*d^3)*x + 6*(b ^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6*( b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c))/(a^ 3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4 + (b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3 *d^4)*x^3 + 3*(a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a^3*b^4*c^2*d^2 - 4*a^4*b^3 *c*d^3 + a^5*b^2*d^4)*x^2 + 3*(a^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 6*a^4*b^3*c ^2*d^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*x)
Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (88) = 176\).
Time = 0.77 (sec) , antiderivative size = 570, normalized size of antiderivative = 5.33 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {d^{3} \log {\left (x + \frac {- \frac {a^{5} d^{8}}{\left (a d - b c\right )^{4}} + \frac {5 a^{4} b c d^{7}}{\left (a d - b c\right )^{4}} - \frac {10 a^{3} b^{2} c^{2} d^{6}}{\left (a d - b c\right )^{4}} + \frac {10 a^{2} b^{3} c^{3} d^{5}}{\left (a d - b c\right )^{4}} - \frac {5 a b^{4} c^{4} d^{4}}{\left (a d - b c\right )^{4}} + a d^{4} + \frac {b^{5} c^{5} d^{3}}{\left (a d - b c\right )^{4}} + b c d^{3}}{2 b d^{4}} \right )}}{\left (a d - b c\right )^{4}} - \frac {d^{3} \log {\left (x + \frac {\frac {a^{5} d^{8}}{\left (a d - b c\right )^{4}} - \frac {5 a^{4} b c d^{7}}{\left (a d - b c\right )^{4}} + \frac {10 a^{3} b^{2} c^{2} d^{6}}{\left (a d - b c\right )^{4}} - \frac {10 a^{2} b^{3} c^{3} d^{5}}{\left (a d - b c\right )^{4}} + \frac {5 a b^{4} c^{4} d^{4}}{\left (a d - b c\right )^{4}} + a d^{4} - \frac {b^{5} c^{5} d^{3}}{\left (a d - b c\right )^{4}} + b c d^{3}}{2 b d^{4}} \right )}}{\left (a d - b c\right )^{4}} + \frac {11 a^{2} d^{2} - 7 a b c d + 2 b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (15 a b d^{2} - 3 b^{2} c d\right )}{6 a^{6} d^{3} - 18 a^{5} b c d^{2} + 18 a^{4} b^{2} c^{2} d - 6 a^{3} b^{3} c^{3} + x^{3} \cdot \left (6 a^{3} b^{3} d^{3} - 18 a^{2} b^{4} c d^{2} + 18 a b^{5} c^{2} d - 6 b^{6} c^{3}\right ) + x^{2} \cdot \left (18 a^{4} b^{2} d^{3} - 54 a^{3} b^{3} c d^{2} + 54 a^{2} b^{4} c^{2} d - 18 a b^{5} c^{3}\right ) + x \left (18 a^{5} b d^{3} - 54 a^{4} b^{2} c d^{2} + 54 a^{3} b^{3} c^{2} d - 18 a^{2} b^{4} c^{3}\right )} \] Input:
integrate(1/(b*x+a)**3/(a*c+(a*d+b*c)*x+b*d*x**2),x)
Output:
d**3*log(x + (-a**5*d**8/(a*d - b*c)**4 + 5*a**4*b*c*d**7/(a*d - b*c)**4 - 10*a**3*b**2*c**2*d**6/(a*d - b*c)**4 + 10*a**2*b**3*c**3*d**5/(a*d - b*c )**4 - 5*a*b**4*c**4*d**4/(a*d - b*c)**4 + a*d**4 + b**5*c**5*d**3/(a*d - b*c)**4 + b*c*d**3)/(2*b*d**4))/(a*d - b*c)**4 - d**3*log(x + (a**5*d**8/( a*d - b*c)**4 - 5*a**4*b*c*d**7/(a*d - b*c)**4 + 10*a**3*b**2*c**2*d**6/(a *d - b*c)**4 - 10*a**2*b**3*c**3*d**5/(a*d - b*c)**4 + 5*a*b**4*c**4*d**4/ (a*d - b*c)**4 + a*d**4 - b**5*c**5*d**3/(a*d - b*c)**4 + b*c*d**3)/(2*b*d **4))/(a*d - b*c)**4 + (11*a**2*d**2 - 7*a*b*c*d + 2*b**2*c**2 + 6*b**2*d* *2*x**2 + x*(15*a*b*d**2 - 3*b**2*c*d))/(6*a**6*d**3 - 18*a**5*b*c*d**2 + 18*a**4*b**2*c**2*d - 6*a**3*b**3*c**3 + x**3*(6*a**3*b**3*d**3 - 18*a**2* b**4*c*d**2 + 18*a*b**5*c**2*d - 6*b**6*c**3) + x**2*(18*a**4*b**2*d**3 - 54*a**3*b**3*c*d**2 + 54*a**2*b**4*c**2*d - 18*a*b**5*c**3) + x*(18*a**5*b *d**3 - 54*a**4*b**2*c*d**2 + 54*a**3*b**3*c**2*d - 18*a**2*b**4*c**3))
Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (103) = 206\).
Time = 0.05 (sec) , antiderivative size = 361, normalized size of antiderivative = 3.37 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {d^{3} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {d^{3} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{6 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x\right )}} \] Input:
integrate(1/(b*x+a)^3/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="maxima")
Output:
-d^3*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) + d^3*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) - 1/6*(6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b* c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/(a^3*b^3*c^3 - 3*a^4*b^2*c^2 *d + 3*a^5*b*c*d^2 - a^6*d^3 + (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)*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^ 4*b^2*d^3)*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5* b*d^3)*x)
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (103) = 206\).
Time = 0.16 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.27 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {b d^{3} \log \left ({\left | b x + a \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac {d^{4} \log \left ({\left | d x + c \right |}\right )}{b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}} - \frac {2 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x}{6 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{3}} \] Input:
integrate(1/(b*x+a)^3/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="giac")
Output:
-b*d^3*log(abs(b*x + a))/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4* a^3*b^2*c*d^3 + a^4*b*d^4) + d^4*log(abs(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) - 1/6*(2*b^3*c^3 - 9* a*b^2*c^2*d + 18*a^2*b*c*d^2 - 11*a^3*d^3 + 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 3*(b^3*c^2*d - 6*a*b^2*c*d^2 + 5*a^2*b*d^3)*x)/((b*c - a*d)^4*(b*x + a)^ 3)
Time = 5.67 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.92 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {\frac {11\,a^2\,d^2-7\,a\,b\,c\,d+2\,b^2\,c^2}{6\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {d\,x\,\left (b^2\,c-5\,a\,b\,d\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b^2\,d^2\,x^2}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3}-\frac {2\,d^3\,\mathrm {atanh}\left (\frac {a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4}{{\left (a\,d-b\,c\right )}^4}+\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{{\left (a\,d-b\,c\right )}^4} \] Input:
int(1/((a + b*x)^3*(a*c + x*(a*d + b*c) + b*d*x^2)),x)
Output:
((11*a^2*d^2 + 2*b^2*c^2 - 7*a*b*c*d)/(6*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2* d - 3*a^2*b*c*d^2)) - (d*x*(b^2*c - 5*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)) + (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))/(a^3 + b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x) - (2*d^3* 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
Time = 0.26 (sec) , antiderivative size = 499, normalized size of antiderivative = 4.66 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {-6 \,\mathrm {log}\left (b x +a \right ) a^{4} d^{3}-18 \,\mathrm {log}\left (b x +a \right ) a^{3} b \,d^{3} x -18 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} d^{3} x^{2}-6 \,\mathrm {log}\left (b x +a \right ) a \,b^{3} d^{3} x^{3}+6 \,\mathrm {log}\left (d x +c \right ) a^{4} d^{3}+18 \,\mathrm {log}\left (d x +c \right ) a^{3} b \,d^{3} x +18 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} d^{3} x^{2}+6 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} d^{3} x^{3}+9 a^{4} d^{3}-16 a^{3} b c \,d^{2}+9 a^{3} b \,d^{3} x +9 a^{2} b^{2} c^{2} d -12 a^{2} b^{2} c \,d^{2} x -2 a \,b^{3} c^{3}+3 a \,b^{3} c^{2} d x -2 a \,b^{3} d^{3} x^{3}+2 b^{4} c \,d^{2} x^{3}}{6 a \left (a^{4} b^{3} d^{4} x^{3}-4 a^{3} b^{4} c \,d^{3} x^{3}+6 a^{2} b^{5} c^{2} d^{2} x^{3}-4 a \,b^{6} c^{3} d \,x^{3}+b^{7} c^{4} x^{3}+3 a^{5} b^{2} d^{4} x^{2}-12 a^{4} b^{3} c \,d^{3} x^{2}+18 a^{3} b^{4} c^{2} d^{2} x^{2}-12 a^{2} b^{5} c^{3} d \,x^{2}+3 a \,b^{6} c^{4} x^{2}+3 a^{6} b \,d^{4} x -12 a^{5} b^{2} c \,d^{3} x +18 a^{4} b^{3} c^{2} d^{2} x -12 a^{3} b^{4} c^{3} d x +3 a^{2} b^{5} c^{4} x +a^{7} d^{4}-4 a^{6} b c \,d^{3}+6 a^{5} b^{2} c^{2} d^{2}-4 a^{4} b^{3} c^{3} d +a^{3} b^{4} c^{4}\right )} \] Input:
int(1/(b*x+a)^3/(a*c+(a*d+b*c)*x+b*d*x^2),x)
Output:
( - 6*log(a + b*x)*a**4*d**3 - 18*log(a + b*x)*a**3*b*d**3*x - 18*log(a + b*x)*a**2*b**2*d**3*x**2 - 6*log(a + b*x)*a*b**3*d**3*x**3 + 6*log(c + d*x )*a**4*d**3 + 18*log(c + d*x)*a**3*b*d**3*x + 18*log(c + d*x)*a**2*b**2*d* *3*x**2 + 6*log(c + d*x)*a*b**3*d**3*x**3 + 9*a**4*d**3 - 16*a**3*b*c*d**2 + 9*a**3*b*d**3*x + 9*a**2*b**2*c**2*d - 12*a**2*b**2*c*d**2*x - 2*a*b**3 *c**3 + 3*a*b**3*c**2*d*x - 2*a*b**3*d**3*x**3 + 2*b**4*c*d**2*x**3)/(6*a* (a**7*d**4 - 4*a**6*b*c*d**3 + 3*a**6*b*d**4*x + 6*a**5*b**2*c**2*d**2 - 1 2*a**5*b**2*c*d**3*x + 3*a**5*b**2*d**4*x**2 - 4*a**4*b**3*c**3*d + 18*a** 4*b**3*c**2*d**2*x - 12*a**4*b**3*c*d**3*x**2 + a**4*b**3*d**4*x**3 + a**3 *b**4*c**4 - 12*a**3*b**4*c**3*d*x + 18*a**3*b**4*c**2*d**2*x**2 - 4*a**3* b**4*c*d**3*x**3 + 3*a**2*b**5*c**4*x - 12*a**2*b**5*c**3*d*x**2 + 6*a**2* b**5*c**2*d**2*x**3 + 3*a*b**6*c**4*x**2 - 4*a*b**6*c**3*d*x**3 + b**7*c** 4*x**3))