Integrand size = 21, antiderivative size = 143 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {b^2}{2 (b c-a d)^3 (a+b x)^2}+\frac {3 b^2 d}{(b c-a d)^4 (a+b x)}+\frac {d^2}{2 (b c-a d)^3 (c+d x)^2}+\frac {3 b d^2}{(b c-a d)^4 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 b^2 d^2 \log (c+d x)}{(b c-a d)^5} \] Output:
-1/2*b^2/(-a*d+b*c)^3/(b*x+a)^2+3*b^2*d/(-a*d+b*c)^4/(b*x+a)+1/2*d^2/(-a*d +b*c)^3/(d*x+c)^2+3*b*d^2/(-a*d+b*c)^4/(d*x+c)+6*b^2*d^2*ln(b*x+a)/(-a*d+b *c)^5-6*b^2*d^2*ln(d*x+c)/(-a*d+b*c)^5
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {-\frac {b^2 (b c-a d)^2}{(a+b x)^2}+\frac {6 b^2 d (b c-a d)}{a+b x}+\frac {d^2 (b c-a d)^2}{(c+d x)^2}+\frac {6 b d^2 (b c-a d)}{c+d x}+12 b^2 d^2 \log (a+b x)-12 b^2 d^2 \log (c+d x)}{2 (b c-a d)^5} \] Input:
Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^(-3),x]
Output:
(-((b^2*(b*c - a*d)^2)/(a + b*x)^2) + (6*b^2*d*(b*c - a*d))/(a + b*x) + (d ^2*(b*c - a*d)^2)/(c + d*x)^2 + (6*b*d^2*(b*c - a*d))/(c + d*x) + 12*b^2*d ^2*Log[a + b*x] - 12*b^2*d^2*Log[c + d*x])/(2*(b*c - a*d)^5)
Time = 0.62 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1084, 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.
| \(\displaystyle \int \frac {1}{\left (x (a d+b c)+a c+b d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1084 |
| \(\displaystyle b^3 d^3 \int \left (-\frac {6}{b (b c-a d)^5 (c+d x)}-\frac {3}{b^2 (b c-a d)^4 (c+d x)^2}-\frac {1}{b^3 (b c-a d)^3 (c+d x)^3}+\frac {6}{d (b c-a d)^5 (a+b x)}-\frac {3}{d^2 (b c-a d)^4 (a+b x)^2}+\frac {1}{d^3 (b c-a d)^3 (a+b x)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b^3 d^3 \left (\frac {1}{2 b^3 d (c+d x)^2 (b c-a d)^3}+\frac {3}{b^2 d (c+d x) (b c-a d)^4}-\frac {1}{2 b d^3 (a+b x)^2 (b c-a d)^3}+\frac {3}{b d^2 (a+b x) (b c-a d)^4}+\frac {6 \log (a+b x)}{b d (b c-a d)^5}-\frac {6 \log (c+d x)}{b d (b c-a d)^5}\right )\) |
Input:
Int[(a*c + (b*c + a*d)*x + b*d*x^2)^(-3),x]
Output:
b^3*d^3*(-1/2*1/(b*d^3*(b*c - a*d)^3*(a + b*x)^2) + 3/(b*d^2*(b*c - a*d)^4 *(a + b*x)) + 1/(2*b^3*d*(b*c - a*d)^3*(c + d*x)^2) + 3/(b^2*d*(b*c - a*d) ^4*(c + d*x)) + (6*Log[a + b*x])/(b*d*(b*c - a*d)^5) - (6*Log[c + d*x])/(b *d*(b*c - a*d)^5))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[(b/2 - q/2 + c*x)^p*(b/2 + q /2 + c*x)^p, x], x], x] /; !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c}, x] && IntegerQ[p] && NiceSqrtQ[b^2 - 4*a*c]
Time = 1.24 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(\frac {b^{2}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {6 b^{2} d^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}+\frac {3 b^{2} d}{\left (a d -b c \right )^{4} \left (b x +a \right )}-\frac {d^{2}}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}+\frac {6 b^{2} d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}+\frac {3 d^{2} b}{\left (a d -b c \right )^{4} \left (d x +c \right )}\) | \(140\) |
| risch | \(\frac {\frac {6 b^{3} d^{3} x^{3}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}+\frac {9 b^{2} d^{2} \left (a d +b c \right ) x^{2}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}+\frac {2 \left (a^{2} d^{2}+7 a b c d +b^{2} c^{2}\right ) b d x}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}-\frac {a^{3} d^{3}-7 a^{2} b c \,d^{2}-7 a \,b^{2} c^{2} d +b^{3} c^{3}}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}}{\left (b d \,x^{2}+a d x +c b x +a c \right )^{2}}+\frac {6 b^{2} d^{2} \ln \left (-d x -c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {6 b^{2} d^{2} \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(483\) |
| norman | \(\frac {\frac {\left (9 a \,b^{4} d^{5}+9 b^{5} c \,d^{4}\right ) x^{2}}{d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {-a^{3} d^{5} b^{2}+7 a^{2} b^{3} c \,d^{4}+7 d^{3} a \,c^{2} b^{4}-c^{3} d^{2} b^{5}}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {6 b^{3} d^{3} x^{3}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}+\frac {2 \left (a^{2} b^{3} d^{5}+7 a \,b^{4} c \,d^{4}+b^{5} c^{2} d^{3}\right ) x}{d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}}{\left (d x +c \right )^{2} \left (b x +a \right )^{2}}-\frac {6 b^{2} d^{2} \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}+\frac {6 b^{2} d^{2} \ln \left (d x +c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(517\) |
| parallelrisch | \(-\frac {-24 x \,a^{2} b^{4} c \,d^{5}+24 x a \,b^{5} c^{2} d^{4}+24 \ln \left (b x +a \right ) x^{3} a \,b^{5} d^{6}+24 \ln \left (b x +a \right ) x^{3} b^{6} c \,d^{5}-24 \ln \left (d x +c \right ) x^{3} a \,b^{5} d^{6}-24 \ln \left (d x +c \right ) x^{3} b^{6} c \,d^{5}+12 \ln \left (b x +a \right ) x^{2} a^{2} b^{4} d^{6}+12 \ln \left (b x +a \right ) x^{2} b^{6} c^{2} d^{4}-12 \ln \left (d x +c \right ) x^{2} a^{2} b^{4} d^{6}-12 \ln \left (d x +c \right ) x^{2} b^{6} c^{2} d^{4}+12 \ln \left (b x +a \right ) a^{2} b^{4} c^{2} d^{4}-12 \ln \left (d x +c \right ) a^{2} b^{4} c^{2} d^{4}-18 x^{2} a^{2} b^{4} d^{6}+18 x^{2} b^{6} c^{2} d^{4}-4 x \,a^{3} b^{3} d^{6}+4 x \,b^{6} c^{3} d^{3}-12 x^{3} a \,b^{5} d^{6}+12 x^{3} b^{6} c \,d^{5}+12 \ln \left (b x +a \right ) x^{4} b^{6} d^{6}-12 \ln \left (d x +c \right ) x^{4} b^{6} d^{6}-8 a^{3} b^{3} c \,d^{5}+8 a \,b^{5} c^{3} d^{3}+48 \ln \left (b x +a \right ) x^{2} a \,b^{5} c \,d^{5}-48 \ln \left (d x +c \right ) x^{2} a \,b^{5} c \,d^{5}+24 \ln \left (b x +a \right ) x \,a^{2} b^{4} c \,d^{5}+24 \ln \left (b x +a \right ) x a \,b^{5} c^{2} d^{4}-24 \ln \left (d x +c \right ) x \,a^{2} b^{4} c \,d^{5}-24 \ln \left (d x +c \right ) x a \,b^{5} c^{2} d^{4}-b^{6} c^{4} d^{2}+a^{4} b^{2} d^{6}}{2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \left (b d \,x^{2}+a d x +c b x +a c \right )^{2} b^{2} d^{2}}\) | \(583\) |
Input:
int(1/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x,method=_RETURNVERBOSE)
Output:
1/2*b^2/(a*d-b*c)^3/(b*x+a)^2-6*b^2/(a*d-b*c)^5*d^2*ln(b*x+a)+3*b^2/(a*d-b *c)^4*d/(b*x+a)-1/2*d^2/(a*d-b*c)^3/(d*x+c)^2+6*b^2/(a*d-b*c)^5*d^2*ln(d*x +c)+3*d^2/(a*d-b*c)^4*b/(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (139) = 278\).
Time = 0.10 (sec) , antiderivative size = 760, normalized size of antiderivative = 5.31 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {b^{4} c^{4} - 8 \, a b^{3} c^{3} d + 8 \, a^{3} b c d^{3} - a^{4} d^{4} - 12 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} - 18 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (b^{4} c^{3} d + 6 \, a b^{3} c^{2} d^{2} - 6 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x - 12 \, {\left (b^{4} d^{4} x^{4} + a^{2} b^{2} c^{2} d^{2} + 2 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} d^{4} x^{4} + a^{2} b^{2} c^{2} d^{2} + 2 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5} + {\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{4} + 2 \, {\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x^{3} + {\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - a^{7} d^{7}\right )} x^{2} + 2 \, {\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} x\right )}} \] Input:
integrate(1/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="fricas")
Output:
-1/2*(b^4*c^4 - 8*a*b^3*c^3*d + 8*a^3*b*c*d^3 - a^4*d^4 - 12*(b^4*c*d^3 - a*b^3*d^4)*x^3 - 18*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^2 - 4*(b^4*c^3*d + 6*a*b ^3*c^2*d^2 - 6*a^2*b^2*c*d^3 - a^3*b*d^4)*x - 12*(b^4*d^4*x^4 + a^2*b^2*c^ 2*d^2 + 2*(b^4*c*d^3 + a*b^3*d^4)*x^3 + (b^4*c^2*d^2 + 4*a*b^3*c*d^3 + a^2 *b^2*d^4)*x^2 + 2*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*x)*log(b*x + a) + 12*(b^ 4*d^4*x^4 + a^2*b^2*c^2*d^2 + 2*(b^4*c*d^3 + a*b^3*d^4)*x^3 + (b^4*c^2*d^2 + 4*a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + 2*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*x) *log(d*x + c))/(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^ 5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5 + (b^7*c^5*d^2 - 5*a*b^6*c^4 *d^3 + 10*a^2*b^5*c^3*d^4 - 10*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2 *d^7)*x^4 + 2*(b^7*c^6*d - 4*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 - 5*a^4*b^3 *c^2*d^5 + 4*a^5*b^2*c*d^6 - a^6*b*d^7)*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a ^2*b^5*c^5*d^2 + 25*a^3*b^4*c^4*d^3 - 25*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d ^5 + a^6*b*c*d^6 - a^7*d^7)*x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a^3*b ^4*c^5*d^2 - 5*a^5*b^2*c^3*d^4 + 4*a^6*b*c^2*d^5 - a^7*c*d^6)*x)
Leaf count of result is larger than twice the leaf count of optimal. 881 vs. \(2 (128) = 256\).
Time = 1.17 (sec) , antiderivative size = 881, normalized size of antiderivative = 6.16 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(1/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
Output:
6*b**2*d**2*log(x + (-6*a**6*b**2*d**8/(a*d - b*c)**5 + 36*a**5*b**3*c*d** 7/(a*d - b*c)**5 - 90*a**4*b**4*c**2*d**6/(a*d - b*c)**5 + 120*a**3*b**5*c **3*d**5/(a*d - b*c)**5 - 90*a**2*b**6*c**4*d**4/(a*d - b*c)**5 + 36*a*b** 7*c**5*d**3/(a*d - b*c)**5 + 6*a*b**2*d**3 - 6*b**8*c**6*d**2/(a*d - b*c)* *5 + 6*b**3*c*d**2)/(12*b**3*d**3))/(a*d - b*c)**5 - 6*b**2*d**2*log(x + ( 6*a**6*b**2*d**8/(a*d - b*c)**5 - 36*a**5*b**3*c*d**7/(a*d - b*c)**5 + 90* a**4*b**4*c**2*d**6/(a*d - b*c)**5 - 120*a**3*b**5*c**3*d**5/(a*d - b*c)** 5 + 90*a**2*b**6*c**4*d**4/(a*d - b*c)**5 - 36*a*b**7*c**5*d**3/(a*d - b*c )**5 + 6*a*b**2*d**3 + 6*b**8*c**6*d**2/(a*d - b*c)**5 + 6*b**3*c*d**2)/(1 2*b**3*d**3))/(a*d - b*c)**5 + (-a**3*d**3 + 7*a**2*b*c*d**2 + 7*a*b**2*c* *2*d - b**3*c**3 + 12*b**3*d**3*x**3 + x**2*(18*a*b**2*d**3 + 18*b**3*c*d* *2) + x*(4*a**2*b*d**3 + 28*a*b**2*c*d**2 + 4*b**3*c**2*d))/(2*a**6*c**2*d **4 - 8*a**5*b*c**3*d**3 + 12*a**4*b**2*c**4*d**2 - 8*a**3*b**3*c**5*d + 2 *a**2*b**4*c**6 + x**4*(2*a**4*b**2*d**6 - 8*a**3*b**3*c*d**5 + 12*a**2*b* *4*c**2*d**4 - 8*a*b**5*c**3*d**3 + 2*b**6*c**4*d**2) + x**3*(4*a**5*b*d** 6 - 12*a**4*b**2*c*d**5 + 8*a**3*b**3*c**2*d**4 + 8*a**2*b**4*c**3*d**3 - 12*a*b**5*c**4*d**2 + 4*b**6*c**5*d) + x**2*(2*a**6*d**6 - 18*a**4*b**2*c* *2*d**4 + 32*a**3*b**3*c**3*d**3 - 18*a**2*b**4*c**4*d**2 + 2*b**6*c**6) + x*(4*a**6*c*d**5 - 12*a**5*b*c**2*d**4 + 8*a**4*b**2*c**3*d**3 + 8*a**3*b **3*c**4*d**2 - 12*a**2*b**4*c**5*d + 4*a*b**5*c**6))
Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (139) = 278\).
Time = 0.05 (sec) , antiderivative size = 594, normalized size of antiderivative = 4.15 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {6 \, b^{2} d^{2} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {6 \, b^{2} d^{2} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac {12 \, b^{3} d^{3} x^{3} - b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} - a^{3} d^{3} + 18 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d + 7 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \] Input:
integrate(1/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="maxima")
Output:
6*b^2*d^2*log(b*x + a)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10* a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) - 6*b^2*d^2*log(d*x + c)/(b^5*c ^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d ^4 - a^5*d^5) + 1/2*(12*b^3*d^3*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c* d^2 - a^3*d^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c^2*d + 7*a*b^2*c* d^2 + a^2*b*d^3)*x)/(a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4 *a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4* c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4* d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6) *x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*d ^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2 *a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (139) = 278\).
Time = 0.11 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.41 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {6 \, b^{3} d^{2} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac {6 \, b^{2} d^{3} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} + \frac {12 \, b^{3} d^{3} x^{3} + 18 \, b^{3} c d^{2} x^{2} + 18 \, a b^{2} d^{3} x^{2} + 4 \, b^{3} c^{2} d x + 28 \, a b^{2} c d^{2} x + 4 \, a^{2} b d^{3} x - b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} - a^{3} d^{3}}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{2}} \] Input:
integrate(1/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="giac")
Output:
6*b^3*d^2*log(abs(b*x + a))/(b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5) - 6*b^2*d^3*log(abs(d* x + c))/(b^5*c^5*d - 5*a*b^4*c^4*d^2 + 10*a^2*b^3*c^3*d^3 - 10*a^3*b^2*c^2 *d^4 + 5*a^4*b*c*d^5 - a^5*d^6) + 1/2*(12*b^3*d^3*x^3 + 18*b^3*c*d^2*x^2 + 18*a*b^2*d^3*x^2 + 4*b^3*c^2*d*x + 28*a*b^2*c*d^2*x + 4*a^2*b*d^3*x - b^3 *c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c*d^2 - a^3*d^3)/((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)*(b*d*x^2 + b*c*x + a*d*x + a*c)^2)
Time = 6.45 (sec) , antiderivative size = 542, normalized size of antiderivative = 3.79 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {\frac {6\,b^3\,d^3\,x^3}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}-\frac {a^3\,d^3-7\,a^2\,b\,c\,d^2-7\,a\,b^2\,c^2\,d+b^3\,c^3}{2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {9\,b\,d\,x^2\,\left (c\,b^2\,d+a\,b\,d^2\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+\frac {2\,b\,d\,x\,\left (a^2\,d^2+7\,a\,b\,c\,d+b^2\,c^2\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4}-\frac {12\,b^2\,d^2\,\mathrm {atanh}\left (\frac {a^5\,d^5-3\,a^4\,b\,c\,d^4+2\,a^3\,b^2\,c^2\,d^3+2\,a^2\,b^3\,c^3\,d^2-3\,a\,b^4\,c^4\,d+b^5\,c^5}{{\left (a\,d-b\,c\right )}^5}+\frac {2\,b\,d\,x\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^5}\right )}{{\left (a\,d-b\,c\right )}^5} \] Input:
int(1/(a*c + x*(a*d + b*c) + b*d*x^2)^3,x)
Output:
((6*b^3*d^3*x^3)/(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3) - (a^3*d^3 + b^3*c^3 - 7*a*b^2*c^2*d - 7*a^2*b*c*d^2)/(2*(a ^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + ( 9*b*d*x^2*(a*b*d^2 + b^2*c*d))/(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4* a*b^3*c^3*d - 4*a^3*b*c*d^3) + (2*b*d*x*(a^2*d^2 + b^2*c^2 + 7*a*b*c*d))/( a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/(x *(2*a*b*c^2 + 2*a^2*c*d) + x^2*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d) + x^3*(2*a* b*d^2 + 2*b^2*c*d) + a^2*c^2 + b^2*d^2*x^4) - (12*b^2*d^2*atanh((a^5*d^5 + b^5*c^5 + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - 3*a*b^4*c^4*d - 3*a^4*b *c*d^4)/(a*d - b*c)^5 + (2*b*d*x*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/(a*d - b*c)^5))/(a*d - b*c)^5
Time = 0.22 (sec) , antiderivative size = 1202, normalized size of antiderivative = 8.41 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x)
Output:
( - 12*log(a + b*x)*a**3*b**2*c**2*d**3 - 24*log(a + b*x)*a**3*b**2*c*d**4 *x - 12*log(a + b*x)*a**3*b**2*d**5*x**2 - 12*log(a + b*x)*a**2*b**3*c**3* d**2 - 48*log(a + b*x)*a**2*b**3*c**2*d**3*x - 60*log(a + b*x)*a**2*b**3*c *d**4*x**2 - 24*log(a + b*x)*a**2*b**3*d**5*x**3 - 24*log(a + b*x)*a*b**4* c**3*d**2*x - 60*log(a + b*x)*a*b**4*c**2*d**3*x**2 - 48*log(a + b*x)*a*b* *4*c*d**4*x**3 - 12*log(a + b*x)*a*b**4*d**5*x**4 - 12*log(a + b*x)*b**5*c **3*d**2*x**2 - 24*log(a + b*x)*b**5*c**2*d**3*x**3 - 12*log(a + b*x)*b**5 *c*d**4*x**4 + 12*log(c + d*x)*a**3*b**2*c**2*d**3 + 24*log(c + d*x)*a**3* b**2*c*d**4*x + 12*log(c + d*x)*a**3*b**2*d**5*x**2 + 12*log(c + d*x)*a**2 *b**3*c**3*d**2 + 48*log(c + d*x)*a**2*b**3*c**2*d**3*x + 60*log(c + d*x)* a**2*b**3*c*d**4*x**2 + 24*log(c + d*x)*a**2*b**3*d**5*x**3 + 24*log(c + d *x)*a*b**4*c**3*d**2*x + 60*log(c + d*x)*a*b**4*c**2*d**3*x**2 + 48*log(c + d*x)*a*b**4*c*d**4*x**3 + 12*log(c + d*x)*a*b**4*d**5*x**4 + 12*log(c + d*x)*b**5*c**3*d**2*x**2 + 24*log(c + d*x)*b**5*c**2*d**3*x**3 + 12*log(c + d*x)*b**5*c*d**4*x**4 - a**5*d**5 + 7*a**4*b*c*d**4 + 4*a**4*b*d**5*x + 2*a**3*b**2*c**2*d**3 + 16*a**3*b**2*c*d**4*x + 12*a**3*b**2*d**5*x**2 - 2 *a**2*b**3*c**3*d**2 - 7*a*b**4*c**4*d - 16*a*b**4*c**3*d**2*x - 6*a*b**4* d**5*x**4 + b**5*c**5 - 4*b**5*c**4*d*x - 12*b**5*c**3*d**2*x**2 + 6*b**5* c*d**4*x**4)/(2*(a**8*c**2*d**6 + 2*a**8*c*d**7*x + a**8*d**8*x**2 - 4*a** 7*b*c**3*d**5 - 6*a**7*b*c**2*d**6*x + 2*a**7*b*d**8*x**3 + 5*a**6*b**2...