\(\int \frac {1}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx\) [1870]

Optimal result

Integrand size = 27, antiderivative size = 47 \[ \int \frac {1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log (a e+c d x)}{c d^2-a e^2}-\frac {\log (d+e x)}{c d^2-a e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {630, 31} \[ \int \frac {1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log (a e+c d x)}{c d^2-a e^2}-\frac {\log (d+e x)}{c d^2-a e^2} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-1),x]

[Out]

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

Rule 31

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

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 e) \int \frac {1}{c d^2+c d e x} \, dx}{c d^2-a e^2}+\frac {(c d e) \int \frac {1}{a e^2+c d e x} \, dx}{c d^2-a e^2} \\ & = \frac {\log (a e+c d x)}{c d^2-a e^2}-\frac {\log (d+e x)}{c d^2-a e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70 \[ \int \frac {1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log (a e+c d x)-\log (d+e x)}{c d^2-a e^2} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-1),x]

[Out]

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

Maple [A] (verified)

Time = 2.51 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.72

method	result	size
parallelrisch	\(\frac {\ln \left (e x +d \right )-\ln \left (c d x +a e \right )}{e^{2} a -c \,d^{2}}\)	\(34\)
default	\(-\frac {\ln \left (c d x +a e \right )}{e^{2} a -c \,d^{2}}+\frac {\ln \left (e x +d \right )}{e^{2} a -c \,d^{2}}\)	\(48\)
norman	\(-\frac {\ln \left (c d x +a e \right )}{e^{2} a -c \,d^{2}}+\frac {\ln \left (e x +d \right )}{e^{2} a -c \,d^{2}}\)	\(48\)
risch	\(\frac {\ln \left (-e x -d \right )}{e^{2} a -c \,d^{2}}-\frac {\ln \left (c d x +a e \right )}{e^{2} a -c \,d^{2}}\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70 \[ \int \frac {1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log \left (c d x + a e\right ) - \log \left (e x + d\right )}{c d^{2} - a e^{2}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (36) = 72\).

Time = 0.17 (sec) , antiderivative size = 172, normalized size of antiderivative = 3.66 \[ \int \frac {1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log {\left (x + \frac {- \frac {a^{2} e^{4}}{a e^{2} - c d^{2}} + \frac {2 a c d^{2} e^{2}}{a e^{2} - c d^{2}} + a e^{2} - \frac {c^{2} d^{4}}{a e^{2} - c d^{2}} + c d^{2}}{2 c d e} \right )}}{a e^{2} - c d^{2}} - \frac {\log {\left (x + \frac {\frac {a^{2} e^{4}}{a e^{2} - c d^{2}} - \frac {2 a c d^{2} e^{2}}{a e^{2} - c d^{2}} + a e^{2} + \frac {c^{2} d^{4}}{a e^{2} - c d^{2}} + c d^{2}}{2 c d e} \right )}}{a e^{2} - c d^{2}} \]

[In]

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

[Out]

log(x + (-a**2*e**4/(a*e**2 - c*d**2) + 2*a*c*d**2*e**2/(a*e**2 - c*d**2) + a*e**2 - c**2*d**4/(a*e**2 - c*d**
2) + c*d**2)/(2*c*d*e))/(a*e**2 - c*d**2) - log(x + (a**2*e**4/(a*e**2 - c*d**2) - 2*a*c*d**2*e**2/(a*e**2 - c
*d**2) + a*e**2 + c**2*d**4/(a*e**2 - c*d**2) + c*d**2)/(2*c*d*e))/(a*e**2 - c*d**2)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log \left (c d x + a e\right )}{c d^{2} - a e^{2}} - \frac {\log \left (e x + d\right )}{c d^{2} - a e^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.21 \[ \int \frac {1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {c d \log \left ({\left | c d x + a e \right |}\right )}{c^{2} d^{3} - a c d e^{2}} - \frac {e \log \left ({\left | e x + d \right |}\right )}{c d^{2} e - a e^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.09 \[ \int \frac {1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\mathrm {atan}\left (\frac {2{}\mathrm {i}\,c\,d^2+2{}\mathrm {i}\,c\,e\,x\,d}{a\,e^2-c\,d^2}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,e^2-c\,d^2} \]

[In]

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

[Out]

(atan((c*d^2*2i + c*d*e*x*2i)/(a*e^2 - c*d^2) + 1i)*2i)/(a*e^2 - c*d^2)