Integrand size = 27, antiderivative size = 203 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {b (b e-a f)}{2 (b c-a d)^3 (a+b x)^2}+\frac {b (3 b d e-b c f-2 a d f)}{(b c-a d)^4 (a+b x)}+\frac {d (d e-c f)}{2 (b c-a d)^3 (c+d x)^2}+\frac {d (3 b d e-2 b c f-a d f)}{(b c-a d)^4 (c+d x)}+\frac {3 b d (2 b d e-b c f-a d f) \log (a+b x)}{(b c-a d)^5}-\frac {3 b d (2 b d e-b c f-a d f) \log (c+d x)}{(b c-a d)^5} \] Output:
-1/2*b*(-a*f+b*e)/(-a*d+b*c)^3/(b*x+a)^2+b*(-2*a*d*f-b*c*f+3*b*d*e)/(-a*d+ b*c)^4/(b*x+a)+1/2*d*(-c*f+d*e)/(-a*d+b*c)^3/(d*x+c)^2+d*(-a*d*f-2*b*c*f+3 *b*d*e)/(-a*d+b*c)^4/(d*x+c)+3*b*d*(-a*d*f-b*c*f+2*b*d*e)*ln(b*x+a)/(-a*d+ b*c)^5-3*b*d*(-a*d*f-b*c*f+2*b*d*e)*ln(d*x+c)/(-a*d+b*c)^5
Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.91 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {-\frac {b (b c-a d)^2 (b e-a f)}{(a+b x)^2}-\frac {2 b (b c-a d) (-3 b d e+b c f+2 a d f)}{a+b x}+\frac {d (b c-a d)^2 (d e-c f)}{(c+d x)^2}+\frac {2 d (b c-a d) (3 b d e-2 b c f-a d f)}{c+d x}-6 b d (-2 b d e+b c f+a d f) \log (a+b x)+6 b d (-2 b d e+b c f+a d f) \log (c+d x)}{2 (b c-a d)^5} \] Input:
Integrate[(e + f*x)/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
Output:
(-((b*(b*c - a*d)^2*(b*e - a*f))/(a + b*x)^2) - (2*b*(b*c - a*d)*(-3*b*d*e + b*c*f + 2*a*d*f))/(a + b*x) + (d*(b*c - a*d)^2*(d*e - c*f))/(c + d*x)^2 + (2*d*(b*c - a*d)*(3*b*d*e - 2*b*c*f - a*d*f))/(c + d*x) - 6*b*d*(-2*b*d *e + b*c*f + a*d*f)*Log[a + b*x] + 6*b*d*(-2*b*d*e + b*c*f + a*d*f)*Log[c + d*x])/(2*(b*c - a*d)^5)
Time = 0.51 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.17, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1141, 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.
| \(\displaystyle \int \frac {e+f x}{\left (x (a d+b c)+a c+b d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1141 |
| \(\displaystyle b^3 d^3 \int \left (\frac {b e-a f}{b d^3 (b c-a d)^3 (a+b x)^3}+\frac {3 (2 b d e-b c f-a d f)}{b d^2 (b c-a d)^5 (a+b x)}-\frac {3 (2 b d e-b c f-a d f)}{b^2 d (b c-a d)^5 (c+d x)}-\frac {3 b d e-b c f-2 a d f}{b d^3 (b c-a d)^4 (a+b x)^2}-\frac {3 b d e-2 b c f-a d f}{b^3 d (b c-a d)^4 (c+d x)^2}-\frac {d e-c f}{b^3 d (b c-a d)^3 (c+d x)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b^3 d^3 \left (\frac {-a d f-2 b c f+3 b d e}{b^3 d^2 (c+d x) (b c-a d)^4}+\frac {d e-c f}{2 b^3 d^2 (c+d x)^2 (b c-a d)^3}-\frac {b e-a f}{2 b^2 d^3 (a+b x)^2 (b c-a d)^3}+\frac {-2 a d f-b c f+3 b d e}{b^2 d^3 (a+b x) (b c-a d)^4}+\frac {3 \log (a+b x) (-a d f-b c f+2 b d e)}{b^2 d^2 (b c-a d)^5}-\frac {3 \log (c+d x) (-a d f-b c f+2 b d e)}{b^2 d^2 (b c-a d)^5}\right )\) |
Input:
Int[(e + f*x)/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
Output:
b^3*d^3*(-1/2*(b*e - a*f)/(b^2*d^3*(b*c - a*d)^3*(a + b*x)^2) + (3*b*d*e - b*c*f - 2*a*d*f)/(b^2*d^3*(b*c - a*d)^4*(a + b*x)) + (d*e - c*f)/(2*b^3*d ^2*(b*c - a*d)^3*(c + d*x)^2) + (3*b*d*e - 2*b*c*f - a*d*f)/(b^3*d^2*(b*c - a*d)^4*(c + d*x)) + (3*(2*b*d*e - b*c*f - a*d*f)*Log[a + b*x])/(b^2*d^2* (b*c - a*d)^5) - (3*(2*b*d*e - b*c*f - a*d*f)*Log[c + d*x])/(b^2*d^2*(b*c - a*d)^5))
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[ (d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 1] || !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
Time = 1.06 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {b \left (2 a d f +b c f -3 b d e \right )}{\left (a d -b c \right )^{4} \left (b x +a \right )}-\frac {\left (a f -b e \right ) b}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}+\frac {3 b d \left (a d f +b c f -2 b d e \right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}-\frac {d \left (a d f +2 b c f -3 b d e \right )}{\left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {\left (c f -d e \right ) d}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}-\frac {3 b d \left (a d f +b c f -2 b d e \right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}\) | \(196\) |
| norman | \(\frac {\frac {\left (-3 a \,b^{3} d^{4} f -3 b^{4} c \,d^{3} f +6 b^{4} d^{4} e \right ) x^{3}}{d b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {\left (-a^{3} b^{2} d^{5} f -8 a^{2} b^{3} c \,d^{4} f +2 a^{2} b^{3} d^{5} e -8 a \,b^{4} c^{2} d^{3} f +14 a \,b^{4} c \,d^{4} e -b^{5} c^{3} d^{2} f +2 b^{5} c^{2} d^{3} e \right ) x}{d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {-a^{3} b^{2} c \,d^{4} f -a^{3} b^{2} d^{5} e -10 a^{2} b^{3} c^{2} d^{3} f +7 a^{2} b^{3} c \,d^{4} e -a \,b^{4} c^{3} d^{2} f +7 a \,b^{4} c^{2} d^{3} e -b^{5} c^{3} d^{2} e}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {\left (-9 a^{2} b^{3} d^{5} f -18 a \,b^{4} c \,d^{4} f +18 a \,b^{4} d^{5} e -9 b^{5} c^{2} d^{3} f +18 b^{5} c \,d^{4} e \right ) x^{2}}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}}{\left (d x +c \right )^{2} \left (b x +a \right )^{2}}+\frac {3 b d \left (a d f +b c f -2 b d e \right ) \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {3 b d \left (a d f +b c f -2 b d e \right ) \ln \left (d x +c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(703\) |
| risch | \(\frac {-\frac {3 b^{2} d^{2} \left (a d f +b c f -2 b d e \right ) x^{3}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}-\frac {9 b d \left (a d +b c \right ) \left (a d f +b c f -2 b d e \right ) x^{2}}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}-\frac {\left (a^{3} d^{3} f +8 a^{2} b c \,d^{2} f -2 a^{2} b \,d^{3} e +8 a \,b^{2} c^{2} d f -14 a \,b^{2} c \,d^{2} e +b^{3} c^{3} f -2 b^{3} c^{2} d e \right ) x}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}-\frac {a^{3} c \,d^{2} f +d^{3} e \,a^{3}+10 a^{2} b \,c^{2} d f -7 a^{2} b c \,d^{2} e +a \,b^{2} c^{3} f -7 a \,b^{2} c^{2} d e +e \,b^{3} c^{3}}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}}{\left (b d \,x^{2}+a d x +c b x +a c \right )^{2}}-\frac {3 b \,d^{2} \ln \left (d x +c \right ) a f}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {3 b^{2} d \ln \left (d x +c \right ) c f}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}+\frac {6 b^{2} d^{2} \ln \left (d x +c \right ) e}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}+\frac {3 b \,d^{2} \ln \left (-b x -a \right ) a f}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}+\frac {3 b^{2} d \ln \left (-b x -a \right ) c f}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {6 b^{2} d^{2} \ln \left (-b x -a \right ) e}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(915\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1394\) |
Input:
int((f*x+e)/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x,method=_RETURNVERBOSE)
Output:
-b*(2*a*d*f+b*c*f-3*b*d*e)/(a*d-b*c)^4/(b*x+a)-1/2*(a*f-b*e)*b/(a*d-b*c)^3 /(b*x+a)^2+3*b*d*(a*d*f+b*c*f-2*b*d*e)/(a*d-b*c)^5*ln(b*x+a)-d*(a*d*f+2*b* c*f-3*b*d*e)/(a*d-b*c)^4/(d*x+c)+1/2*(c*f-d*e)*d/(a*d-b*c)^3/(d*x+c)^2-3*b *d*(a*d*f+b*c*f-2*b*d*e)/(a*d-b*c)^5*ln(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 1285 vs. \(2 (199) = 398\).
Time = 0.13 (sec) , antiderivative size = 1285, normalized size of antiderivative = 6.33 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="fricas")
Output:
1/2*(6*(2*(b^4*c*d^3 - a*b^3*d^4)*e - (b^4*c^2*d^2 - a^2*b^2*d^4)*f)*x^3 + 9*(2*(b^4*c^2*d^2 - a^2*b^2*d^4)*e - (b^4*c^3*d + a*b^3*c^2*d^2 - a^2*b^2 *c*d^3 - a^3*b*d^4)*f)*x^2 - (b^4*c^4 - 8*a*b^3*c^3*d + 8*a^3*b*c*d^3 - a^ 4*d^4)*e - (a*b^3*c^4 + 9*a^2*b^2*c^3*d - 9*a^3*b*c^2*d^2 - a^4*c*d^3)*f + 2*(2*(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)*e - (b^4 *c^4 + 7*a*b^3*c^3*d - 7*a^3*b*c*d^3 - a^4*d^4)*f)*x + 6*(2*a^2*b^2*c^2*d^ 2*e + (2*b^4*d^4*e - (b^4*c*d^3 + a*b^3*d^4)*f)*x^4 + 2*(2*(b^4*c*d^3 + a* b^3*d^4)*e - (b^4*c^2*d^2 + 2*a*b^3*c*d^3 + a^2*b^2*d^4)*f)*x^3 + (2*(b^4* c^2*d^2 + 4*a*b^3*c*d^3 + a^2*b^2*d^4)*e - (b^4*c^3*d + 5*a*b^3*c^2*d^2 + 5*a^2*b^2*c*d^3 + a^3*b*d^4)*f)*x^2 - (a^2*b^2*c^3*d + a^3*b*c^2*d^2)*f + 2*(2*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*e - (a*b^3*c^3*d + 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*f)*x)*log(b*x + a) - 6*(2*a^2*b^2*c^2*d^2*e + (2*b^4*d^4*e - (b^4*c*d^3 + a*b^3*d^4)*f)*x^4 + 2*(2*(b^4*c*d^3 + a*b^3*d^4)*e - (b^4*c ^2*d^2 + 2*a*b^3*c*d^3 + a^2*b^2*d^4)*f)*x^3 + (2*(b^4*c^2*d^2 + 4*a*b^3*c *d^3 + a^2*b^2*d^4)*e - (b^4*c^3*d + 5*a*b^3*c^2*d^2 + 5*a^2*b^2*c*d^3 + a ^3*b*d^4)*f)*x^2 - (a^2*b^2*c^3*d + a^3*b*c^2*d^2)*f + 2*(2*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*e - (a*b^3*c^3*d + 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*f)*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...
Leaf count of result is larger than twice the leaf count of optimal. 1431 vs. \(2 (192) = 384\).
Time = 2.70 (sec) , antiderivative size = 1431, normalized size of antiderivative = 7.05 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
Output:
-3*b*d*(a*d*f + b*c*f - 2*b*d*e)*log(x + (-3*a**6*b*d**7*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**5 + 18*a**5*b**2*c*d**6*(a*d*f + b*c*f - 2*b*d*e)/(a *d - b*c)**5 - 45*a**4*b**3*c**2*d**5*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c )**5 + 60*a**3*b**4*c**3*d**4*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**5 - 4 5*a**2*b**5*c**4*d**3*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**5 + 3*a**2*b* d**3*f + 18*a*b**6*c**5*d**2*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**5 + 6* a*b**2*c*d**2*f - 6*a*b**2*d**3*e - 3*b**7*c**6*d*(a*d*f + b*c*f - 2*b*d*e )/(a*d - b*c)**5 + 3*b**3*c**2*d*f - 6*b**3*c*d**2*e)/(6*a*b**2*d**3*f + 6 *b**3*c*d**2*f - 12*b**3*d**3*e))/(a*d - b*c)**5 + 3*b*d*(a*d*f + b*c*f - 2*b*d*e)*log(x + (3*a**6*b*d**7*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**5 - 18*a**5*b**2*c*d**6*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**5 + 45*a**4*b* *3*c**2*d**5*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**5 - 60*a**3*b**4*c**3* d**4*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**5 + 45*a**2*b**5*c**4*d**3*(a* d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**5 + 3*a**2*b*d**3*f - 18*a*b**6*c**5*d **2*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**5 + 6*a*b**2*c*d**2*f - 6*a*b** 2*d**3*e + 3*b**7*c**6*d*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**5 + 3*b**3 *c**2*d*f - 6*b**3*c*d**2*e)/(6*a*b**2*d**3*f + 6*b**3*c*d**2*f - 12*b**3* d**3*e))/(a*d - b*c)**5 + (-a**3*c*d**2*f - a**3*d**3*e - 10*a**2*b*c**2*d *f + 7*a**2*b*c*d**2*e - a*b**2*c**3*f + 7*a*b**2*c**2*d*e - b**3*c**3*e + x**3*(-6*a*b**2*d**3*f - 6*b**3*c*d**2*f + 12*b**3*d**3*e) + x**2*(-9*...
Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (199) = 398\).
Time = 0.06 (sec) , antiderivative size = 766, normalized size of antiderivative = 3.77 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {3 \, {\left (2 \, b^{2} d^{2} e - {\left (b^{2} c d + a b d^{2}\right )} f\right )} \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 {3 \, {\left (2 \, b^{2} d^{2} e - {\left (b^{2} c d + a b d^{2}\right )} f\right )} \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 {6 \, {\left (2 \, b^{3} d^{3} e - {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f\right )} x^{3} + 9 \, {\left (2 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e - {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f\right )} x^{2} - {\left (b^{3} c^{3} - 7 \, a b^{2} c^{2} d - 7 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} e - {\left (a b^{2} c^{3} + 10 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} f + 2 \, {\left (2 \, {\left (b^{3} c^{2} d + 7 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e - {\left (b^{3} c^{3} + 8 \, a b^{2} c^{2} d + 8 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} f\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((f*x+e)/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="maxima")
Output:
3*(2*b^2*d^2*e - (b^2*c*d + a*b*d^2)*f)*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) - 3*(2*b^2*d^2*e - (b^2*c*d + a*b*d^2)*f)*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*(6*(2*b^3*d^3*e - (b^3*c*d^2 + a*b^2*d^3)*f)*x^3 + 9*(2*(b^3*c*d^2 + a*b^2*d^3)*e - (b^3*c^2*d + 2*a*b^2*c*d^2 + a^2*b*d^3)*f)*x^2 - (b^3*c^3 - 7*a*b^2*c^2*d - 7*a^2*b*c*d^2 + a^3*d^3)*e - (a*b^2*c^3 + 10*a^2*b*c^2*d + a^3*c*d^2)*f + 2*(2*(b^3*c^2*d + 7*a*b^2*c*d^2 + a^2*b*d^3)*e - (b^3*c^ 3 + 8*a*b^2*c^2*d + 8*a^2*b*c*d^2 + a^3*d^3)*f)*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. 540 vs. \(2 (199) = 398\).
Time = 0.36 (sec) , antiderivative size = 540, normalized size of antiderivative = 2.66 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {3 \, {\left (2 \, b^{3} d^{2} e - b^{3} c d f - a b^{2} d^{2} f\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac {3 \, {\left (2 \, b^{2} d^{3} e - b^{2} c d^{2} f - a b d^{3} f\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} + \frac {12 \, b^{3} d^{3} e x^{3} - 6 \, b^{3} c d^{2} f x^{3} - 6 \, a b^{2} d^{3} f x^{3} + 18 \, b^{3} c d^{2} e x^{2} + 18 \, a b^{2} d^{3} e x^{2} - 9 \, b^{3} c^{2} d f x^{2} - 18 \, a b^{2} c d^{2} f x^{2} - 9 \, a^{2} b d^{3} f x^{2} + 4 \, b^{3} c^{2} d e x + 28 \, a b^{2} c d^{2} e x + 4 \, a^{2} b d^{3} e x - 2 \, b^{3} c^{3} f x - 16 \, a b^{2} c^{2} d f x - 16 \, a^{2} b c d^{2} f x - 2 \, a^{3} d^{3} f x - b^{3} c^{3} e + 7 \, a b^{2} c^{2} d e + 7 \, a^{2} b c d^{2} e - a^{3} d^{3} e - a b^{2} c^{3} f - 10 \, a^{2} b c^{2} d f - a^{3} c d^{2} f}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{2}} \] Input:
integrate((f*x+e)/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="giac")
Output:
3*(2*b^3*d^2*e - b^3*c*d*f - a*b^2*d^2*f)*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) - 3*(2*b^2*d^3*e - b^2*c*d^2*f - a*b*d^3*f)*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*e*x^3 - 6*b^3*c*d^2*f*x^3 - 6*a*b^ 2*d^3*f*x^3 + 18*b^3*c*d^2*e*x^2 + 18*a*b^2*d^3*e*x^2 - 9*b^3*c^2*d*f*x^2 - 18*a*b^2*c*d^2*f*x^2 - 9*a^2*b*d^3*f*x^2 + 4*b^3*c^2*d*e*x + 28*a*b^2*c* d^2*e*x + 4*a^2*b*d^3*e*x - 2*b^3*c^3*f*x - 16*a*b^2*c^2*d*f*x - 16*a^2*b* c*d^2*f*x - 2*a^3*d^3*f*x - b^3*c^3*e + 7*a*b^2*c^2*d*e + 7*a^2*b*c*d^2*e - a^3*d^3*e - a*b^2*c^3*f - 10*a^2*b*c^2*d*f - a^3*c*d^2*f)/((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 = 5.99 (sec) , antiderivative size = 733, normalized size of antiderivative = 3.61 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {\left (3\,b\,d^2\,\left (a\,f-2\,b\,e\right )+3\,b^2\,c\,d\,f\right )\,\left (a^5\,d^5-3\,a^4\,b\,c\,d^4+2\,a^3\,b^2\,c^2\,d^3+2\,a^2\,b^3\,c^3\,d^2-3\,a\,b^4\,c^4\,d+b^5\,c^5\right )}{{\left (a\,d-b\,c\right )}^5\,\left (-6\,e\,b^2\,d^2+3\,c\,f\,b^2\,d+3\,a\,f\,b\,d^2\right )}+\frac {2\,b\,d\,x\,\left (3\,b\,d^2\,\left (a\,f-2\,b\,e\right )+3\,b^2\,c\,d\,f\right )\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^5\,\left (-6\,e\,b^2\,d^2+3\,c\,f\,b^2\,d+3\,a\,f\,b\,d^2\right )}\right )\,\left (3\,b\,d^2\,\left (a\,f-2\,b\,e\right )+3\,b^2\,c\,d\,f\right )}{{\left (a\,d-b\,c\right )}^5}-\frac {\frac {f\,a^3\,c\,d^2+e\,a^3\,d^3+10\,f\,a^2\,b\,c^2\,d-7\,e\,a^2\,b\,c\,d^2+f\,a\,b^2\,c^3-7\,e\,a\,b^2\,c^2\,d+e\,b^3\,c^3}{2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {x\,\left (a^2\,d^2+7\,a\,b\,c\,d+b^2\,c^2\right )\,\left (a\,d\,f+b\,c\,f-2\,b\,d\,e\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+\frac {9\,x^2\,\left (c\,b^2\,d+a\,b\,d^2\right )\,\left (a\,d\,f+b\,c\,f-2\,b\,d\,e\right )}{2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {3\,b^2\,d^2\,x^3\,\left (a\,d\,f+b\,c\,f-2\,b\,d\,e\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4} \] Input:
int((e + f*x)/(a*c + x*(a*d + b*c) + b*d*x^2)^3,x)
Output:
(2*atanh(((3*b*d^2*(a*f - 2*b*e) + 3*b^2*c*d*f)*(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*(3*a*b*d^2*f - 6*b^2*d^2*e + 3*b^2*c*d*f)) + (2*b*d*x*(3*b*d^2*(a* f - 2*b*e) + 3*b^2*c*d*f)*(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*(3*a*b*d^2*f - 6*b^2*d^2*e + 3*b^2 *c*d*f)))*(3*b*d^2*(a*f - 2*b*e) + 3*b^2*c*d*f))/(a*d - b*c)^5 - ((a^3*d^3 *e + b^3*c^3*e + a*b^2*c^3*f + a^3*c*d^2*f - 7*a*b^2*c^2*d*e - 7*a^2*b*c*d ^2*e + 10*a^2*b*c^2*d*f)/(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)) + (x*(a^2*d^2 + b^2*c^2 + 7*a*b*c*d)*(a*d*f + b *c*f - 2*b*d*e))/(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*x^2*(a*b*d^2 + b^2*c*d)*(a*d*f + b*c*f - 2*b*d*e))/(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)) + (3*b^2*d^2*x^3*(a*d*f + b*c*f - 2*b*d*e))/(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)
Time = 0.21 (sec) , antiderivative size = 2239, normalized size of antiderivative = 11.03 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((f*x+e)/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x)
Output:
(6*log(a + b*x)*a**4*b*c**2*d**3*f + 12*log(a + b*x)*a**4*b*c*d**4*f*x + 6 *log(a + b*x)*a**4*b*d**5*f*x**2 + 12*log(a + b*x)*a**3*b**2*c**3*d**2*f - 12*log(a + b*x)*a**3*b**2*c**2*d**3*e + 36*log(a + b*x)*a**3*b**2*c**2*d* *3*f*x - 24*log(a + b*x)*a**3*b**2*c*d**4*e*x + 36*log(a + b*x)*a**3*b**2* c*d**4*f*x**2 - 12*log(a + b*x)*a**3*b**2*d**5*e*x**2 + 12*log(a + b*x)*a* *3*b**2*d**5*f*x**3 + 6*log(a + b*x)*a**2*b**3*c**4*d*f - 12*log(a + b*x)* a**2*b**3*c**3*d**2*e + 36*log(a + b*x)*a**2*b**3*c**3*d**2*f*x - 48*log(a + b*x)*a**2*b**3*c**2*d**3*e*x + 60*log(a + b*x)*a**2*b**3*c**2*d**3*f*x* *2 - 60*log(a + b*x)*a**2*b**3*c*d**4*e*x**2 + 36*log(a + b*x)*a**2*b**3*c *d**4*f*x**3 - 24*log(a + b*x)*a**2*b**3*d**5*e*x**3 + 6*log(a + b*x)*a**2 *b**3*d**5*f*x**4 + 12*log(a + b*x)*a*b**4*c**4*d*f*x - 24*log(a + b*x)*a* b**4*c**3*d**2*e*x + 36*log(a + b*x)*a*b**4*c**3*d**2*f*x**2 - 60*log(a + b*x)*a*b**4*c**2*d**3*e*x**2 + 36*log(a + b*x)*a*b**4*c**2*d**3*f*x**3 - 4 8*log(a + b*x)*a*b**4*c*d**4*e*x**3 + 12*log(a + b*x)*a*b**4*c*d**4*f*x**4 - 12*log(a + b*x)*a*b**4*d**5*e*x**4 + 6*log(a + b*x)*b**5*c**4*d*f*x**2 - 12*log(a + b*x)*b**5*c**3*d**2*e*x**2 + 12*log(a + b*x)*b**5*c**3*d**2*f *x**3 - 24*log(a + b*x)*b**5*c**2*d**3*e*x**3 + 6*log(a + b*x)*b**5*c**2*d **3*f*x**4 - 12*log(a + b*x)*b**5*c*d**4*e*x**4 - 6*log(c + d*x)*a**4*b*c* *2*d**3*f - 12*log(c + d*x)*a**4*b*c*d**4*f*x - 6*log(c + d*x)*a**4*b*d**5 *f*x**2 - 12*log(c + d*x)*a**3*b**2*c**3*d**2*f + 12*log(c + d*x)*a**3*...