### 3.1870 $$\int \frac{1}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx$$

Optimal. Leaf size=47 $\frac{\log (a e+c d x)}{c d^2-a e^2}-\frac{\log (d+e x)}{c d^2-a e^2}$

[Out]

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

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Rubi [A]  time = 0.0154168, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.074, Rules used = {616, 31} $\frac{\log (a e+c d x)}{c d^2-a e^2}-\frac{\log (d+e x)}{c d^2-a e^2}$

Antiderivative was successfully veriﬁed.

[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 616

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]

Rule 31

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

Rubi steps

\begin{align*} \int \frac{1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=-\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]  time = 0.0137618, size = 33, normalized size = 0.7 $\frac{\log (a e+c d x)-\log (d+e x)}{c d^2-a e^2}$

Antiderivative was successfully veriﬁed.

[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)

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Maple [A]  time = 0.045, size = 48, normalized size = 1. \begin{align*}{\frac{\ln \left ( ex+d \right ) }{a{e}^{2}-c{d}^{2}}}-{\frac{\ln \left ( cdx+ae \right ) }{a{e}^{2}-c{d}^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06451, size = 63, normalized size = 1.34 \begin{align*} \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.54686, size = 69, normalized size = 1.47 \begin{align*} \frac{\log \left (c d x + a e\right ) - \log \left (e x + d\right )}{c d^{2} - a e^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 0.361001, size = 172, normalized size = 3.66 \begin{align*} \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.28223, size = 101, normalized size = 2.15 \begin{align*} \frac{2 \, \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan((2*c*d*x*e + c*d^2 + a*e^2)/sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4))/sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 -
a^2*e^4)