3.1859 \(\int \frac{1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

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 = 0.0454907, 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 \[ \frac{\log (a e+c d x)}{c d^2-a e^2}-\frac{\log (d+e x)}{c d^2-a e^2} \]

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)

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Rubi in Sympy [A]  time = 6.43775, size = 42, normalized size = 0.89 \[ - \frac{2 \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{a e^{2} - c d^{2}} \right )}}{a e^{2} - c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

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

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Mathematica [A]  time = 0.026676, 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 = 0.009, size = 48, normalized size = 1. \[{\frac{\ln \left ( ex+d \right ) }{a{e}^{2}-c{d}^{2}}}-{\frac{\ln \left ( cdx+ae \right ) }{a{e}^{2}-c{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*e*d+(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 = 0.72516, size = 63, normalized size = 1.34 \[ \frac{\log \left (c d x + a e\right )}{c d^{2} - a e^{2}} - \frac{\log \left (e x + d\right )}{c d^{2} - a e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),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 = 0.212506, size = 45, normalized size = 0.96 \[ \frac{\log \left (c d x + a e\right ) - \log \left (e x + d\right )}{c d^{2} - a e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),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 [A]  time = 1.06362, size = 172, normalized size = 3.66 \[ \frac{\log{\left (x + \frac{- \frac{a^{2} e^{4}}{a e^{2} - c d^{2}} + \frac{2 a c d^{2} e^{2}}{a e^{2} - c d^{2}} + a e^{2} - \frac{c^{2} d^{4}}{a e^{2} - c d^{2}} + c d^{2}}{2 c d e} \right )}}{a e^{2} - c d^{2}} - \frac{\log{\left (x + \frac{\frac{a^{2} e^{4}}{a e^{2} - c d^{2}} - \frac{2 a c d^{2} e^{2}}{a e^{2} - c d^{2}} + a e^{2} + \frac{c^{2} d^{4}}{a e^{2} - c d^{2}} + c d^{2}}{2 c d e} \right )}}{a e^{2} - c d^{2}} \]

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/XCAS [A]  time = 0.212871, size = 101, normalized size = 2.15 \[ \frac{2 \, \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),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))/s
qrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)