3.1858 \(\int \frac{d+e x}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac{\log (a e+c d x)}{c d} \]

[Out]

Log[a*e + c*d*x]/(c*d)

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Rubi [A]  time = 0.0302387, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{\log (a e+c d x)}{c d} \]

Antiderivative was successfully verified.

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

[Out]

Log[a*e + c*d*x]/(c*d)

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Rubi in Sympy [A]  time = 8.60638, size = 12, normalized size = 0.75 \[ \frac{\log{\left (a e + c d x \right )}}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a*e + c*d*x)/(c*d)

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Mathematica [A]  time = 0.00255346, size = 16, normalized size = 1. \[ \frac{\log (a e+c d x)}{c d} \]

Antiderivative was successfully verified.

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

[Out]

Log[a*e + c*d*x]/(c*d)

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Maple [A]  time = 0.002, size = 17, normalized size = 1.1 \[{\frac{\ln \left ( cdx+ae \right ) }{cd}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

ln(c*d*x+a*e)/c/d

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Maxima [A]  time = 0.717634, size = 22, normalized size = 1.38 \[ \frac{\log \left (c d x + a e\right )}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(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)

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Fricas [A]  time = 0.213763, size = 22, normalized size = 1.38 \[ \frac{\log \left (c d x + a e\right )}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(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)/(c*d)

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Sympy [A]  time = 0.146978, size = 12, normalized size = 0.75 \[ \frac{\log{\left (a e + c d x \right )}}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a*e + c*d*x)/(c*d)

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GIAC/XCAS [A]  time = 0.219863, size = 170, normalized size = 10.62 \[ \frac{{\left (c d^{2} - a e^{2}\right )} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c d} + \frac{{\rm ln}\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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