3.1529 \(\int \frac{d+e x}{9+12 x+4 x^2} \, dx\)

Optimal. Leaf size=30 \[ \frac{1}{4} e \log (2 x+3)-\frac{2 d-3 e}{4 (2 x+3)} \]

[Out]

-(2*d - 3*e)/(4*(3 + 2*x)) + (e*Log[3 + 2*x])/4

_______________________________________________________________________________________

Rubi [A]  time = 0.0458462, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{4} e \log (2 x+3)-\frac{2 d-3 e}{4 (2 x+3)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(9 + 12*x + 4*x^2),x]

[Out]

-(2*d - 3*e)/(4*(3 + 2*x)) + (e*Log[3 + 2*x])/4

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.35401, size = 22, normalized size = 0.73 \[ \frac{e \log{\left (2 x + 3 \right )}}{4} - \frac{\frac{d}{2} - \frac{3 e}{4}}{2 x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(4*x**2+12*x+9),x)

[Out]

e*log(2*x + 3)/4 - (d/2 - 3*e/4)/(2*x + 3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.012, size = 30, normalized size = 1. \[ \frac{3 e-2 d}{4 (2 x+3)}+\frac{1}{4} e \log (2 x+3) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(9 + 12*x + 4*x^2),x]

[Out]

(-2*d + 3*e)/(4*(3 + 2*x)) + (e*Log[3 + 2*x])/4

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 31, normalized size = 1. \[{\frac{e\ln \left ( 2\,x+3 \right ) }{4}}-{\frac{d}{4\,x+6}}+{\frac{3\,e}{8\,x+12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(4*x^2+12*x+9),x)

[Out]

1/4*e*ln(2*x+3)-1/2/(2*x+3)*d+3/4*e/(2*x+3)

_______________________________________________________________________________________

Maxima [A]  time = 0.682783, size = 35, normalized size = 1.17 \[ \frac{1}{4} \, e \log \left (2 \, x + 3\right ) - \frac{2 \, d - 3 \, e}{4 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9),x, algorithm="maxima")

[Out]

1/4*e*log(2*x + 3) - 1/4*(2*d - 3*e)/(2*x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.19939, size = 42, normalized size = 1.4 \[ \frac{{\left (2 \, e x + 3 \, e\right )} \log \left (2 \, x + 3\right ) - 2 \, d + 3 \, e}{4 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9),x, algorithm="fricas")

[Out]

1/4*((2*e*x + 3*e)*log(2*x + 3) - 2*d + 3*e)/(2*x + 3)

_______________________________________________________________________________________

Sympy [A]  time = 1.18805, size = 20, normalized size = 0.67 \[ \frac{e \log{\left (2 x + 3 \right )}}{4} - \frac{2 d - 3 e}{8 x + 12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(4*x**2+12*x+9),x)

[Out]

e*log(2*x + 3)/4 - (2*d - 3*e)/(8*x + 12)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211599, size = 39, normalized size = 1.3 \[ \frac{1}{4} \, e{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{2 \, d - 3 \, e}{4 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9),x, algorithm="giac")

[Out]

1/4*e*ln(abs(2*x + 3)) - 1/4*(2*d - 3*e)/(2*x + 3)