Integrand size = 27, antiderivative size = 122 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {b e-a f}{(b c-a d)^2 (a+b x)}-\frac {d e-c f}{(b c-a d)^2 (c+d x)}-\frac {(2 b d e-b c f-a d f) \log (a+b x)}{(b c-a d)^3}+\frac {(2 b d e-b c f-a d f) \log (c+d x)}{(b c-a d)^3} \] Output:
-(-a*f+b*e)/(-a*d+b*c)^2/(b*x+a)-(-c*f+d*e)/(-a*d+b*c)^2/(d*x+c)-(-a*d*f-b *c*f+2*b*d*e)*ln(b*x+a)/(-a*d+b*c)^3+(-a*d*f-b*c*f+2*b*d*e)*ln(d*x+c)/(-a* d+b*c)^3
Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.87 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {\frac {(b c-a d) (-b e+a f)}{a+b x}+\frac {(b c-a d) (-d e+c f)}{c+d x}-(2 b d e-b c f-a d f) \log (a+b x)-(-2 b d e+b c f+a d f) \log (c+d x)}{(b c-a d)^3} \] Input:
Integrate[(e + f*x)/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
Output:
(((b*c - a*d)*(-(b*e) + a*f))/(a + b*x) + ((b*c - a*d)*(-(d*e) + c*f))/(c + d*x) - (2*b*d*e - b*c*f - a*d*f)*Log[a + b*x] - (-2*b*d*e + b*c*f + a*d* f)*Log[c + d*x])/(b*c - a*d)^3
Time = 0.36 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.25, 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.
| \(\displaystyle \int \frac {e+f x}{\left (x (a d+b c)+a c+b d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1141 |
| \(\displaystyle b^2 d^2 \int \left (\frac {b e-a f}{b d^2 (b c-a d)^2 (a+b x)^2}-\frac {2 b d e-b c f-a d f}{b d^2 (b c-a d)^3 (a+b x)}+\frac {2 b d e-b c f-a d f}{b^2 d (b c-a d)^3 (c+d x)}+\frac {d e-c f}{b^2 d (b c-a d)^2 (c+d x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b^2 d^2 \left (-\frac {b e-a f}{b^2 d^2 (a+b x) (b c-a d)^2}-\frac {d e-c f}{b^2 d^2 (c+d x) (b c-a d)^2}-\frac {\log (a+b x) (-a d f-b c f+2 b d e)}{b^2 d^2 (b c-a d)^3}+\frac {\log (c+d x) (-a d f-b c f+2 b d e)}{b^2 d^2 (b c-a d)^3}\right )\) |
Input:
Int[(e + f*x)/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
Output:
b^2*d^2*(-((b*e - a*f)/(b^2*d^2*(b*c - a*d)^2*(a + b*x))) - (d*e - c*f)/(b ^2*d^2*(b*c - a*d)^2*(c + d*x)) - ((2*b*d*e - b*c*f - a*d*f)*Log[a + b*x]) /(b^2*d^2*(b*c - a*d)^3) + ((2*b*d*e - b*c*f - a*d*f)*Log[c + d*x])/(b^2*d ^2*(b*c - a*d)^3))
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 = 0.97 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(-\frac {\left (a d f +b c f -2 b d e \right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{3}}+\frac {a f -b e}{\left (a d -b c \right )^{2} \left (b x +a \right )}+\frac {\left (a d f +b c f -2 b d e \right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{3}}+\frac {c f -d e}{\left (a d -b c \right )^{2} \left (d x +c \right )}\) | \(117\) |
| norman | \(\frac {\frac {2 a b c d f -a b \,d^{2} e -b^{2} c d e}{d b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (a b \,d^{2} f +b^{2} c d f -2 d^{2} e \,b^{2}\right ) x}{d b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {\left (a d f +b c f -2 b d e \right ) \ln \left (d x +c \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {\left (a d f +b c f -2 b d e \right ) \ln \left (b x +a \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\) | \(246\) |
| risch | \(\frac {\frac {\left (a d f +b c f -2 b d e \right ) x}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {2 a c f -a d e -b c e}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}}{b d \,x^{2}+a d x +c b x +a c}+\frac {\ln \left (-d x -c \right ) a d f}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {\ln \left (-d x -c \right ) b c f}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {2 \ln \left (-d x -c \right ) b d e}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {\ln \left (b x +a \right ) a d f}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {\ln \left (b x +a \right ) b c f}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {2 \ln \left (b x +a \right ) b d e}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\) | \(404\) |
| parallelrisch | \(-\frac {-2 a^{2} b c \,d^{2} f +2 \ln \left (b x +a \right ) x a \,b^{2} c \,d^{2} f -2 \ln \left (d x +c \right ) x a \,b^{2} c \,d^{2} f +2 a \,b^{2} c^{2} d f +a^{2} b \,d^{3} e -b^{3} c^{2} d e -x \,a^{2} b \,d^{3} f +2 x a \,b^{2} d^{3} e -2 \ln \left (b x +a \right ) x^{2} b^{3} d^{3} e +2 \ln \left (d x +c \right ) x^{2} b^{3} d^{3} e +x \,b^{3} c^{2} d f -2 x \,b^{3} c \,d^{2} e -2 \ln \left (b x +a \right ) x \,b^{3} c \,d^{2} e -\ln \left (d x +c \right ) x \,a^{2} b \,d^{3} f +2 \ln \left (d x +c \right ) x a \,b^{2} d^{3} e -\ln \left (d x +c \right ) x \,b^{3} c^{2} d f -\ln \left (d x +c \right ) a \,b^{2} c^{2} d f +2 \ln \left (d x +c \right ) a \,b^{2} c \,d^{2} e +\ln \left (b x +a \right ) x^{2} a \,b^{2} d^{3} f +\ln \left (b x +a \right ) x^{2} b^{3} c \,d^{2} f -\ln \left (d x +c \right ) x^{2} a \,b^{2} d^{3} f -\ln \left (d x +c \right ) x^{2} b^{3} c \,d^{2} f +\ln \left (b x +a \right ) x \,a^{2} b \,d^{3} f -2 \ln \left (b x +a \right ) x a \,b^{2} d^{3} e +\ln \left (b x +a \right ) x \,b^{3} c^{2} d f +\ln \left (b x +a \right ) a \,b^{2} c^{2} d f -2 \ln \left (b x +a \right ) a \,b^{2} c \,d^{2} e -\ln \left (d x +c \right ) a^{2} b c \,d^{2} f +2 \ln \left (d x +c \right ) x \,b^{3} c \,d^{2} e +\ln \left (b x +a \right ) a^{2} b c \,d^{2} f}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b d \,x^{2}+a d x +c b x +a c \right ) b d}\) | \(532\) |
Input:
int((f*x+e)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x,method=_RETURNVERBOSE)
Output:
-(a*d*f+b*c*f-2*b*d*e)/(a*d-b*c)^3*ln(b*x+a)+(a*f-b*e)/(a*d-b*c)^2/(b*x+a) +(a*d*f+b*c*f-2*b*d*e)/(a*d-b*c)^3*ln(d*x+c)+(c*f-d*e)/(a*d-b*c)^2/(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (122) = 244\).
Time = 0.08 (sec) , antiderivative size = 414, normalized size of antiderivative = 3.39 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {{\left (b^{2} c^{2} - a^{2} d^{2}\right )} e - 2 \, {\left (a b c^{2} - a^{2} c d\right )} f + {\left (2 \, {\left (b^{2} c d - a b d^{2}\right )} e - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} f\right )} x + {\left (2 \, a b c d e + {\left (2 \, b^{2} d^{2} e - {\left (b^{2} c d + a b d^{2}\right )} f\right )} x^{2} - {\left (a b c^{2} + a^{2} c d\right )} f + {\left (2 \, {\left (b^{2} c d + a b d^{2}\right )} e - {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} f\right )} x\right )} \log \left (b x + a\right ) - {\left (2 \, a b c d e + {\left (2 \, b^{2} d^{2} e - {\left (b^{2} c d + a b d^{2}\right )} f\right )} x^{2} - {\left (a b c^{2} + a^{2} c d\right )} f + {\left (2 \, {\left (b^{2} c d + a b d^{2}\right )} e - {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} f\right )} x\right )} \log \left (d x + c\right )}{a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x} \] Input:
integrate((f*x+e)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="fricas")
Output:
-((b^2*c^2 - a^2*d^2)*e - 2*(a*b*c^2 - a^2*c*d)*f + (2*(b^2*c*d - a*b*d^2) *e - (b^2*c^2 - a^2*d^2)*f)*x + (2*a*b*c*d*e + (2*b^2*d^2*e - (b^2*c*d + a *b*d^2)*f)*x^2 - (a*b*c^2 + a^2*c*d)*f + (2*(b^2*c*d + a*b*d^2)*e - (b^2*c ^2 + 2*a*b*c*d + a^2*d^2)*f)*x)*log(b*x + a) - (2*a*b*c*d*e + (2*b^2*d^2*e - (b^2*c*d + a*b*d^2)*f)*x^2 - (a*b*c^2 + a^2*c*d)*f + (2*(b^2*c*d + a*b* d^2)*e - (b^2*c^2 + 2*a*b*c*d + a^2*d^2)*f)*x)*log(d*x + c))/(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2*d ^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^2 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b *c*d^3 - a^4*d^4)*x)
Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (104) = 208\).
Time = 1.26 (sec) , antiderivative size = 706, normalized size of antiderivative = 5.79 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {2 a c f - a d e - b c e + x \left (a d f + b c f - 2 b d e\right )}{a^{3} c d^{2} - 2 a^{2} b c^{2} d + a b^{2} c^{3} + x^{2} \left (a^{2} b d^{3} - 2 a b^{2} c d^{2} + b^{3} c^{2} d\right ) + x \left (a^{3} d^{3} - a^{2} b c d^{2} - a b^{2} c^{2} d + b^{3} c^{3}\right )} + \frac {\left (a d f + b c f - 2 b d e\right ) \log {\left (x + \frac {- \frac {a^{4} d^{4} \left (a d f + b c f - 2 b d e\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c d^{3} \left (a d f + b c f - 2 b d e\right )}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{2} c^{2} d^{2} \left (a d f + b c f - 2 b d e\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} f + \frac {4 a b^{3} c^{3} d \left (a d f + b c f - 2 b d e\right )}{\left (a d - b c\right )^{3}} + 2 a b c d f - 2 a b d^{2} e - \frac {b^{4} c^{4} \left (a d f + b c f - 2 b d e\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2} f - 2 b^{2} c d e}{2 a b d^{2} f + 2 b^{2} c d f - 4 b^{2} d^{2} e} \right )}}{\left (a d - b c\right )^{3}} - \frac {\left (a d f + b c f - 2 b d e\right ) \log {\left (x + \frac {\frac {a^{4} d^{4} \left (a d f + b c f - 2 b d e\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b c d^{3} \left (a d f + b c f - 2 b d e\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{2} c^{2} d^{2} \left (a d f + b c f - 2 b d e\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} f - \frac {4 a b^{3} c^{3} d \left (a d f + b c f - 2 b d e\right )}{\left (a d - b c\right )^{3}} + 2 a b c d f - 2 a b d^{2} e + \frac {b^{4} c^{4} \left (a d f + b c f - 2 b d e\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2} f - 2 b^{2} c d e}{2 a b d^{2} f + 2 b^{2} c d f - 4 b^{2} d^{2} e} \right )}}{\left (a d - b c\right )^{3}} \] Input:
integrate((f*x+e)/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
Output:
(2*a*c*f - a*d*e - b*c*e + x*(a*d*f + b*c*f - 2*b*d*e))/(a**3*c*d**2 - 2*a **2*b*c**2*d + a*b**2*c**3 + x**2*(a**2*b*d**3 - 2*a*b**2*c*d**2 + b**3*c* *2*d) + x*(a**3*d**3 - a**2*b*c*d**2 - a*b**2*c**2*d + b**3*c**3)) + (a*d* f + b*c*f - 2*b*d*e)*log(x + (-a**4*d**4*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**3 + 4*a**3*b*c*d**3*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**3 - 6*a** 2*b**2*c**2*d**2*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**3 + a**2*d**2*f + 4*a*b**3*c**3*d*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**3 + 2*a*b*c*d*f - 2 *a*b*d**2*e - b**4*c**4*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**3 + b**2*c* *2*f - 2*b**2*c*d*e)/(2*a*b*d**2*f + 2*b**2*c*d*f - 4*b**2*d**2*e))/(a*d - b*c)**3 - (a*d*f + b*c*f - 2*b*d*e)*log(x + (a**4*d**4*(a*d*f + b*c*f - 2 *b*d*e)/(a*d - b*c)**3 - 4*a**3*b*c*d**3*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**3 + 6*a**2*b**2*c**2*d**2*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**3 + a**2*d**2*f - 4*a*b**3*c**3*d*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b*c)**3 + 2*a*b*c*d*f - 2*a*b*d**2*e + b**4*c**4*(a*d*f + b*c*f - 2*b*d*e)/(a*d - b* c)**3 + b**2*c**2*f - 2*b**2*c*d*e)/(2*a*b*d**2*f + 2*b**2*c*d*f - 4*b**2* d**2*e))/(a*d - b*c)**3
Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (122) = 244\).
Time = 0.05 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.11 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {{\left (2 \, b d e - {\left (b c + a d\right )} f\right )} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {{\left (2 \, b d e - {\left (b c + a d\right )} f\right )} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {2 \, a c f - {\left (b c + a d\right )} e - {\left (2 \, b d e - {\left (b c + a d\right )} f\right )} x}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \] Input:
integrate((f*x+e)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="maxima")
Output:
-(2*b*d*e - (b*c + a*d)*f)*log(b*x + a)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b *c*d^2 - a^3*d^3) + (2*b*d*e - (b*c + a*d)*f)*log(d*x + c)/(b^3*c^3 - 3*a* b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + (2*a*c*f - (b*c + a*d)*e - (2*b*d*e - (b*c + a*d)*f)*x)/(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a ^3*d^3)*x)
Time = 0.33 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.80 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {{\left (2 \, b^{2} d e - b^{2} c f - a b d f\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac {{\left (2 \, b d^{2} e - b c d f - a d^{2} f\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}} - \frac {2 \, b d e x - b c f x - a d f x + b c e + a d e - 2 \, a c f}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}} \] Input:
integrate((f*x+e)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="giac")
Output:
-(2*b^2*d*e - b^2*c*f - a*b*d*f)*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) + (2*b*d^2*e - b*c*d*f - a*d^2*f)*log(abs (d*x + c))/(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4) - (2*b* d*e*x - b*c*f*x - a*d*f*x + b*c*e + a*d*e - 2*a*c*f)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*d*x^2 + b*c*x + a*d*x + a*c))
Time = 5.55 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.12 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {\frac {a\,d\,e-2\,a\,c\,f+b\,c\,e}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}-\frac {x\,\left (a\,d\,f+b\,c\,f-2\,b\,d\,e\right )}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c}-\frac {2\,\mathrm {atanh}\left (\frac {\left (\frac {a^3\,d^3-a^2\,b\,c\,d^2-a\,b^2\,c^2\,d+b^3\,c^3}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}+2\,b\,d\,x\right )\,\left (d\,\left (a\,f-2\,b\,e\right )+b\,c\,f\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3\,\left (a\,d\,f+b\,c\,f-2\,b\,d\,e\right )}\right )\,\left (d\,\left (a\,f-2\,b\,e\right )+b\,c\,f\right )}{{\left (a\,d-b\,c\right )}^3} \] Input:
int((e + f*x)/(a*c + x*(a*d + b*c) + b*d*x^2)^2,x)
Output:
- ((a*d*e - 2*a*c*f + b*c*e)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d) - (x*(a*d*f + b*c*f - 2*b*d*e))/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(a*c + x*(a*d + b*c) + b*d*x^2) - (2*atanh((((a^3*d^3 + b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2)/(a^ 2*d^2 + b^2*c^2 - 2*a*b*c*d) + 2*b*d*x)*(d*(a*f - 2*b*e) + b*c*f)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/((a*d - b*c)^3*(a*d*f + b*c*f - 2*b*d*e)))*(d*(a* f - 2*b*e) + b*c*f))/(a*d - b*c)^3
Time = 0.23 (sec) , antiderivative size = 867, normalized size of antiderivative = 7.11 \[ \int \frac {e+f x}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((f*x+e)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x)
Output:
( - log(a + b*x)*a**3*c*d**2*f - log(a + b*x)*a**3*d**3*f*x - 2*log(a + b* x)*a**2*b*c**2*d*f + 2*log(a + b*x)*a**2*b*c*d**2*e - 3*log(a + b*x)*a**2* b*c*d**2*f*x + 2*log(a + b*x)*a**2*b*d**3*e*x - log(a + b*x)*a**2*b*d**3*f *x**2 - log(a + b*x)*a*b**2*c**3*f + 2*log(a + b*x)*a*b**2*c**2*d*e - 3*lo g(a + b*x)*a*b**2*c**2*d*f*x + 4*log(a + b*x)*a*b**2*c*d**2*e*x - 2*log(a + b*x)*a*b**2*c*d**2*f*x**2 + 2*log(a + b*x)*a*b**2*d**3*e*x**2 - log(a + b*x)*b**3*c**3*f*x + 2*log(a + b*x)*b**3*c**2*d*e*x - log(a + b*x)*b**3*c* *2*d*f*x**2 + 2*log(a + b*x)*b**3*c*d**2*e*x**2 + log(c + d*x)*a**3*c*d**2 *f + log(c + d*x)*a**3*d**3*f*x + 2*log(c + d*x)*a**2*b*c**2*d*f - 2*log(c + d*x)*a**2*b*c*d**2*e + 3*log(c + d*x)*a**2*b*c*d**2*f*x - 2*log(c + d*x )*a**2*b*d**3*e*x + log(c + d*x)*a**2*b*d**3*f*x**2 + log(c + d*x)*a*b**2* c**3*f - 2*log(c + d*x)*a*b**2*c**2*d*e + 3*log(c + d*x)*a*b**2*c**2*d*f*x - 4*log(c + d*x)*a*b**2*c*d**2*e*x + 2*log(c + d*x)*a*b**2*c*d**2*f*x**2 - 2*log(c + d*x)*a*b**2*d**3*e*x**2 + log(c + d*x)*b**3*c**3*f*x - 2*log(c + d*x)*b**3*c**2*d*e*x + log(c + d*x)*b**3*c**2*d*f*x**2 - 2*log(c + d*x) *b**3*c*d**2*e*x**2 + a**3*c*d**2*f - a**3*d**3*e + a**2*b*c*d**2*e - a**2 *b*d**3*f*x**2 - a*b**2*c**3*f - a*b**2*c**2*d*e + 2*a*b**2*d**3*e*x**2 + b**3*c**3*e + b**3*c**2*d*f*x**2 - 2*b**3*c*d**2*e*x**2)/(a**5*c*d**4 + a* *5*d**5*x - 2*a**4*b*c**2*d**3 - a**4*b*c*d**4*x + a**4*b*d**5*x**2 - 2*a* *3*b**2*c**2*d**3*x - 2*a**3*b**2*c*d**4*x**2 + 2*a**2*b**3*c**4*d + 2*...