Integrand size = 33, antiderivative size = 75 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac {e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac {e \log (d+e x)}{\left (c d^2-a e^2\right )^2} \] Output:
-1/(-a*e^2+c*d^2)/(c*d*x+a*e)-e*ln(c*d*x+a*e)/(-a*e^2+c*d^2)^2+e*ln(e*x+d) /(-a*e^2+c*d^2)^2
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {1}{\left (-c d^2+a e^2\right ) (a e+c d x)}-\frac {e \log (a e+c d x)}{\left (-c d^2+a e^2\right )^2}+\frac {e \log (d+e x)}{\left (-c d^2+a e^2\right )^2} \] Input:
Integrate[(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
1/((-(c*d^2) + a*e^2)*(a*e + c*d*x)) - (e*Log[a*e + c*d*x])/(-(c*d^2) + a* e^2)^2 + (e*Log[d + e*x])/(-(c*d^2) + a*e^2)^2
Time = 0.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, 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.
\(\displaystyle \int \frac {d+e x}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (\frac {e^2}{(d+e x) \left (c d^2-a e^2\right )^2}-\frac {c d e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac {c d}{\left (c d^2-a e^2\right ) (a e+c d x)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac {e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac {e \log (d+e x)}{\left (c d^2-a e^2\right )^2}\) |
Input:
Int[(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
-(1/((c*d^2 - a*e^2)*(a*e + c*d*x))) - (e*Log[a*e + c*d*x])/(c*d^2 - a*e^2 )^2 + (e*Log[d + e*x])/(c*d^2 - a*e^2)^2
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 = 1.60 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {1}{\left (a \,e^{2}-c \,d^{2}\right ) \left (c d x +a e \right )}-\frac {e \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {e \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2}}\) | \(75\) |
risch | \(\frac {1}{\left (a \,e^{2}-c \,d^{2}\right ) \left (c d x +a e \right )}-\frac {e \ln \left (c d x +a e \right )}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}+\frac {e \ln \left (-e x -d \right )}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}\) | \(104\) |
parallelrisch | \(\frac {\ln \left (e x +d \right ) x \,c^{2} d^{2} e -\ln \left (c d x +a e \right ) x \,c^{2} d^{2} e +\ln \left (e x +d \right ) a c d \,e^{2}-\ln \left (c d x +a e \right ) a c d \,e^{2}+a d \,e^{2} c -c^{2} d^{3}}{\left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (c d x +a e \right ) c d}\) | \(125\) |
norman | \(\frac {\frac {d}{a \,e^{2}-c \,d^{2}}+\frac {e x}{a \,e^{2}-c \,d^{2}}}{\left (e x +d \right ) \left (c d x +a e \right )}+\frac {e \ln \left (e x +d \right )}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}-\frac {e \ln \left (c d x +a e \right )}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}\) | \(128\) |
Input:
int((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2,x,method=_RETURNVERBOSE)
Output:
1/(a*e^2-c*d^2)/(c*d*x+a*e)-e/(a*e^2-c*d^2)^2*ln(c*d*x+a*e)+e/(a*e^2-c*d^2 )^2*ln(e*x+d)
Time = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.55 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c d^{2} - a e^{2} + {\left (c d e x + a e^{2}\right )} \log \left (c d x + a e\right ) - {\left (c d e x + a e^{2}\right )} \log \left (e x + d\right )}{a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} + {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x} \] Input:
integrate((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fricas ")
Output:
-(c*d^2 - a*e^2 + (c*d*e*x + a*e^2)*log(c*d*x + a*e) - (c*d*e*x + a*e^2)*l og(e*x + d))/(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5 + (c^3*d^5 - 2*a*c^2 *d^3*e^2 + a^2*c*d*e^4)*x)
Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (63) = 126\).
Time = 0.39 (sec) , antiderivative size = 287, normalized size of antiderivative = 3.83 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {e \log {\left (x + \frac {- \frac {a^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac {3 a^{2} c d^{2} e^{5}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac {3 a c^{2} d^{4} e^{3}}{\left (a e^{2} - c d^{2}\right )^{2}} + a e^{3} + \frac {c^{3} d^{6} e}{\left (a e^{2} - c d^{2}\right )^{2}} + c d^{2} e}{2 c d e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac {e \log {\left (x + \frac {\frac {a^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac {3 a^{2} c d^{2} e^{5}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac {3 a c^{2} d^{4} e^{3}}{\left (a e^{2} - c d^{2}\right )^{2}} + a e^{3} - \frac {c^{3} d^{6} e}{\left (a e^{2} - c d^{2}\right )^{2}} + c d^{2} e}{2 c d e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac {1}{a^{2} e^{3} - a c d^{2} e + x \left (a c d e^{2} - c^{2} d^{3}\right )} \] Input:
integrate((e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
Output:
e*log(x + (-a**3*e**7/(a*e**2 - c*d**2)**2 + 3*a**2*c*d**2*e**5/(a*e**2 - c*d**2)**2 - 3*a*c**2*d**4*e**3/(a*e**2 - c*d**2)**2 + a*e**3 + c**3*d**6* e/(a*e**2 - c*d**2)**2 + c*d**2*e)/(2*c*d*e**2))/(a*e**2 - c*d**2)**2 - e* log(x + (a**3*e**7/(a*e**2 - c*d**2)**2 - 3*a**2*c*d**2*e**5/(a*e**2 - c*d **2)**2 + 3*a*c**2*d**4*e**3/(a*e**2 - c*d**2)**2 + a*e**3 - c**3*d**6*e/( a*e**2 - c*d**2)**2 + c*d**2*e)/(2*c*d*e**2))/(a*e**2 - c*d**2)**2 + 1/(a* *2*e**3 - a*c*d**2*e + x*(a*c*d*e**2 - c**2*d**3))
Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.51 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {e \log \left (c d x + a e\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {e \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac {1}{a c d^{2} e - a^{2} e^{3} + {\left (c^{2} d^{3} - a c d e^{2}\right )} x} \] Input:
integrate((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxima ")
Output:
-e*log(c*d*x + a*e)/(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4) + e*log(e*x + d)/( c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4) - 1/(a*c*d^2*e - a^2*e^3 + (c^2*d^3 - a *c*d*e^2)*x)
Time = 0.16 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.49 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c d e \log \left ({\left | c d x + a e \right |}\right )}{c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}} + \frac {e^{2} \log \left ({\left | e x + d \right |}\right )}{c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}} - \frac {1}{{\left (c d^{2} - a e^{2}\right )} {\left (c d x + a e\right )}} \] Input:
integrate((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")
Output:
-c*d*e*log(abs(c*d*x + a*e))/(c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4) + e ^2*log(abs(e*x + d))/(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5) - 1/((c*d^2 - a *e^2)*(c*d*x + a*e))
Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.28 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {1}{\left (a\,e+c\,d\,x\right )\,\left (a\,e^2-c\,d^2\right )}-\frac {2\,e\,\mathrm {atanh}\left (\frac {a^2\,e^4-c^2\,d^4}{{\left (a\,e^2-c\,d^2\right )}^2}+\frac {2\,c\,d\,e\,x}{a\,e^2-c\,d^2}\right )}{{\left (a\,e^2-c\,d^2\right )}^2} \] Input:
int((d + e*x)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)
Output:
1/((a*e + c*d*x)*(a*e^2 - c*d^2)) - (2*e*atanh((a^2*e^4 - c^2*d^4)/(a*e^2 - c*d^2)^2 + (2*c*d*e*x)/(a*e^2 - c*d^2)))/(a*e^2 - c*d^2)^2
Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.99 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {-\mathrm {log}\left (c d x +a e \right ) a^{2} e^{3}-\mathrm {log}\left (c d x +a e \right ) a c d \,e^{2} x +\mathrm {log}\left (e x +d \right ) a^{2} e^{3}+\mathrm {log}\left (e x +d \right ) a c d \,e^{2} x -a c d \,e^{2} x +c^{2} d^{3} x}{a e \left (a^{2} c d \,e^{4} x -2 a \,c^{2} d^{3} e^{2} x +c^{3} d^{5} x +a^{3} e^{5}-2 a^{2} c \,d^{2} e^{3}+a \,c^{2} d^{4} e \right )} \] Input:
int((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
Output:
( - log(a*e + c*d*x)*a**2*e**3 - log(a*e + c*d*x)*a*c*d*e**2*x + log(d + e *x)*a**2*e**3 + log(d + e*x)*a*c*d*e**2*x - a*c*d*e**2*x + c**2*d**3*x)/(a *e*(a**3*e**5 - 2*a**2*c*d**2*e**3 + a**2*c*d*e**4*x + a*c**2*d**4*e - 2*a *c**2*d**3*e**2*x + c**3*d**5*x))