\(\int \frac {1}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [1882]

Optimal result

Integrand size = 27, antiderivative size = 114 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c d^2+a e^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac {2 c d e \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}+\frac {2 c d e \log (d+e x)}{\left (c d^2-a e^2\right )^3} \]

[Out]

(-2*c*d*e*x-a*e^2-c*d^2)/(-a*e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)-2*c*d*e*ln(c*d*x+a*e)/(-a*e^2+c*d^
2)^3+2*c*d*e*ln(e*x+d)/(-a*e^2+c*d^2)^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {628, 630, 31} \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {a e^2+c d^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}-\frac {2 c d e \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}+\frac {2 c d e \log (d+e x)}{\left (c d^2-a e^2\right )^3} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-2),x]

[Out]

-((c*d^2 + a*e^2 + 2*c*d*e*x)/((c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - (2*c*d*e*Log[a*e
+ c*d*x])/(c*d^2 - a*e^2)^3 + (2*c*d*e*Log[d + e*x])/(c*d^2 - a*e^2)^3

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 630

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = -\frac {c d^2+a e^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac {(2 c d e) \int \frac {1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{\left (c d^2-a e^2\right )^2} \\ & = -\frac {c d^2+a e^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}+\frac {\left (2 c^2 d^2 e^2\right ) \int \frac {1}{c d^2+c d e x} \, dx}{\left (c d^2-a e^2\right )^3}-\frac {\left (2 c^2 d^2 e^2\right ) \int \frac {1}{a e^2+c d e x} \, dx}{\left (c d^2-a e^2\right )^3} \\ & = -\frac {c d^2+a e^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac {2 c d e \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}+\frac {2 c d e \log (d+e x)}{\left (c d^2-a e^2\right )^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\frac {\left (c d^2-a e^2\right ) \left (a e^2+c d (d+2 e x)\right )}{(a e+c d x) (d+e x)}+2 c d e \log (a e+c d x)-2 c d e \log (d+e x)}{\left (-c d^2+a e^2\right )^3} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-2),x]

[Out]

(((c*d^2 - a*e^2)*(a*e^2 + c*d*(d + 2*e*x)))/((a*e + c*d*x)*(d + e*x)) + 2*c*d*e*Log[a*e + c*d*x] - 2*c*d*e*Lo
g[d + e*x])/(-(c*d^2) + a*e^2)^3

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {c d}{\left (e^{2} a -c \,d^{2}\right )^{2} \left (c d x +a e \right )}+\frac {2 c d e \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{3}}-\frac {e}{\left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {2 c d e \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{3}}\)	\(107\)
norman	\(\frac {\frac {-a c \,e^{2}-c^{2} d^{2}}{c \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}-\frac {2 c d e x}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}}{\left (c d x +a e \right ) \left (e x +d \right )}-\frac {2 c d e \ln \left (e x +d \right )}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}+\frac {2 c d e \ln \left (c d x +a e \right )}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}\)	\(210\)
risch	\(\frac {-\frac {2 c d e x}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}-\frac {e^{2} a +c \,d^{2}}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}}{c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}-\frac {2 c d e \ln \left (e x +d \right )}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}+\frac {2 c d e \ln \left (-c d x -a e \right )}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}\)	\(214\)
parallelrisch	\(-\frac {2 \ln \left (e x +d \right ) x^{2} c^{3} d^{3} e^{3}-2 \ln \left (c d x +a e \right ) x^{2} c^{3} d^{3} e^{3}+2 \ln \left (e x +d \right ) x a \,c^{2} d^{2} e^{4}+2 \ln \left (e x +d \right ) x \,c^{3} d^{4} e^{2}-2 \ln \left (c d x +a e \right ) x a \,c^{2} d^{2} e^{4}-2 \ln \left (c d x +a e \right ) x \,c^{3} d^{4} e^{2}+2 \ln \left (e x +d \right ) a \,c^{2} d^{3} e^{3}-2 \ln \left (c d x +a e \right ) a \,c^{2} d^{3} e^{3}+2 x a \,c^{2} d^{2} e^{4}-2 x \,c^{3} d^{4} e^{2}+d \,e^{5} a^{2} c -c^{3} d^{5} e}{\left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right ) c d e}\)	\(286\)

[In]

int(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

-c*d/(a*e^2-c*d^2)^2/(c*d*x+a*e)+2*c*d/(a*e^2-c*d^2)^3*e*ln(c*d*x+a*e)-e/(a*e^2-c*d^2)^2/(e*x+d)-2*c*d/(a*e^2-
c*d^2)^3*e*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (114) = 228\).

Time = 0.30 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.43 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c^{2} d^{4} - a^{2} e^{4} + 2 \, {\left (c^{2} d^{3} e - a c d e^{3}\right )} x + 2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (c d x + a e\right ) - 2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{a c^{3} d^{7} e - 3 \, a^{2} c^{2} d^{5} e^{3} + 3 \, a^{3} c d^{3} e^{5} - a^{4} d e^{7} + {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x^{2} + {\left (c^{4} d^{8} - 2 \, a c^{3} d^{6} e^{2} + 2 \, a^{3} c d^{2} e^{6} - a^{4} e^{8}\right )} x} \]

[In]

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fricas")

[Out]

-(c^2*d^4 - a^2*e^4 + 2*(c^2*d^3*e - a*c*d*e^3)*x + 2*(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)
*x)*log(c*d*x + a*e) - 2*(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)*log(e*x + d))/(a*c^3*d^7*
e - 3*a^2*c^2*d^5*e^3 + 3*a^3*c*d^3*e^5 - a^4*d*e^7 + (c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c
*d*e^7)*x^2 + (c^4*d^8 - 2*a*c^3*d^6*e^2 + 2*a^3*c*d^2*e^6 - a^4*e^8)*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (110) = 220\).

Time = 0.66 (sec) , antiderivative size = 486, normalized size of antiderivative = 4.26 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=- \frac {2 c d e \log {\left (x + \frac {- \frac {2 a^{4} c d e^{9}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {8 a^{3} c^{2} d^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {12 a^{2} c^{3} d^{5} e^{5}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {8 a c^{4} d^{7} e^{3}}{\left (a e^{2} - c d^{2}\right )^{3}} + 2 a c d e^{3} - \frac {2 c^{5} d^{9} e}{\left (a e^{2} - c d^{2}\right )^{3}} + 2 c^{2} d^{3} e}{4 c^{2} d^{2} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {2 c d e \log {\left (x + \frac {\frac {2 a^{4} c d e^{9}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {8 a^{3} c^{2} d^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {12 a^{2} c^{3} d^{5} e^{5}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {8 a c^{4} d^{7} e^{3}}{\left (a e^{2} - c d^{2}\right )^{3}} + 2 a c d e^{3} + \frac {2 c^{5} d^{9} e}{\left (a e^{2} - c d^{2}\right )^{3}} + 2 c^{2} d^{3} e}{4 c^{2} d^{2} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {- a e^{2} - c d^{2} - 2 c d e x}{a^{3} d e^{5} - 2 a^{2} c d^{3} e^{3} + a c^{2} d^{5} e + x^{2} \left (a^{2} c d e^{5} - 2 a c^{2} d^{3} e^{3} + c^{3} d^{5} e\right ) + x \left (a^{3} e^{6} - a^{2} c d^{2} e^{4} - a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )} \]

[In]

integrate(1/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

-2*c*d*e*log(x + (-2*a**4*c*d*e**9/(a*e**2 - c*d**2)**3 + 8*a**3*c**2*d**3*e**7/(a*e**2 - c*d**2)**3 - 12*a**2
*c**3*d**5*e**5/(a*e**2 - c*d**2)**3 + 8*a*c**4*d**7*e**3/(a*e**2 - c*d**2)**3 + 2*a*c*d*e**3 - 2*c**5*d**9*e/
(a*e**2 - c*d**2)**3 + 2*c**2*d**3*e)/(4*c**2*d**2*e**2))/(a*e**2 - c*d**2)**3 + 2*c*d*e*log(x + (2*a**4*c*d*e
**9/(a*e**2 - c*d**2)**3 - 8*a**3*c**2*d**3*e**7/(a*e**2 - c*d**2)**3 + 12*a**2*c**3*d**5*e**5/(a*e**2 - c*d**
2)**3 - 8*a*c**4*d**7*e**3/(a*e**2 - c*d**2)**3 + 2*a*c*d*e**3 + 2*c**5*d**9*e/(a*e**2 - c*d**2)**3 + 2*c**2*d
**3*e)/(4*c**2*d**2*e**2))/(a*e**2 - c*d**2)**3 + (-a*e**2 - c*d**2 - 2*c*d*e*x)/(a**3*d*e**5 - 2*a**2*c*d**3*
e**3 + a*c**2*d**5*e + x**2*(a**2*c*d*e**5 - 2*a*c**2*d**3*e**3 + c**3*d**5*e) + x*(a**3*e**6 - a**2*c*d**2*e*
*4 - a*c**2*d**4*e**2 + c**3*d**6))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (114) = 228\).

Time = 0.21 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.07 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {2 \, c d e \log \left (c d x + a e\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} + \frac {2 \, c d e \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} - \frac {2 \, c d e x + c d^{2} + a e^{2}}{a c^{2} d^{5} e - 2 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5} + {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x^{2} + {\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} x} \]

[In]

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxima")

[Out]

-2*c*d*e*log(c*d*x + a*e)/(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6) + 2*c*d*e*log(e*x + d)/(c^3*
d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6) - (2*c*d*e*x + c*d^2 + a*e^2)/(a*c^2*d^5*e - 2*a^2*c*d^3*e^
3 + a^3*d*e^5 + (c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x^2 + (c^3*d^6 - a*c^2*d^4*e^2 - a^2*c*d^2*e^4 + a
^3*e^6)*x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.71 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {2 \, c^{2} d^{2} e \log \left ({\left | c d x + a e \right |}\right )}{c^{4} d^{7} - 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} - a^{3} c d e^{6}} + \frac {2 \, c d e^{2} \log \left ({\left | e x + d \right |}\right )}{c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}} - \frac {2 \, c d e x + c d^{2} + a e^{2}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c d e x^{2} + c d^{2} x + a e^{2} x + a d e\right )}} \]

[In]

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")

[Out]

-2*c^2*d^2*e*log(abs(c*d*x + a*e))/(c^4*d^7 - 3*a*c^3*d^5*e^2 + 3*a^2*c^2*d^3*e^4 - a^3*c*d*e^6) + 2*c*d*e^2*l
og(abs(e*x + d))/(c^3*d^6*e - 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - a^3*e^7) - (2*c*d*e*x + c*d^2 + a*e^2)/((c^2
*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))

Mupad [B] (verification not implemented)

Time = 9.91 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.96 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {4\,c\,d\,e\,\mathrm {atanh}\left (\frac {a^3\,e^6-a^2\,c\,d^2\,e^4-a\,c^2\,d^4\,e^2+c^3\,d^6}{{\left (a\,e^2-c\,d^2\right )}^3}+\frac {2\,c\,d\,e\,x\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{{\left (a\,e^2-c\,d^2\right )}^3}\right )}{{\left (a\,e^2-c\,d^2\right )}^3}-\frac {\frac {c\,d^2+a\,e^2}{a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4}+\frac {2\,c\,d\,e\,x}{a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4}}{c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e} \]

[In]

int(1/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)

[Out]

(4*c*d*e*atanh((a^3*e^6 + c^3*d^6 - a*c^2*d^4*e^2 - a^2*c*d^2*e^4)/(a*e^2 - c*d^2)^3 + (2*c*d*e*x*(a^2*e^4 + c
^2*d^4 - 2*a*c*d^2*e^2))/(a*e^2 - c*d^2)^3))/(a*e^2 - c*d^2)^3 - ((a*e^2 + c*d^2)/(a^2*e^4 + c^2*d^4 - 2*a*c*d
^2*e^2) + (2*c*d*e*x)/(a^2*e^4 + c^2*d^4 - 2*a*c*d^2*e^2))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)