3.1606 \(\int \frac{d+e x}{\sqrt{9+12 x+4 x^2}} \, dx\)

Optimal. Leaf size=56 \[ \frac{(2 x+3) (2 d-3 e) \log (2 x+3)}{4 \sqrt{4 x^2+12 x+9}}+\frac{1}{4} e \sqrt{4 x^2+12 x+9} \]

[Out]

(e*Sqrt[9 + 12*x + 4*x^2])/4 + ((2*d - 3*e)*(3 + 2*x)*Log[3 + 2*x])/(4*Sqrt[9 +
12*x + 4*x^2])

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Rubi [A]  time = 0.0510952, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{(2 x+3) (2 d-3 e) \log (2 x+3)}{4 \sqrt{4 x^2+12 x+9}}+\frac{1}{4} e \sqrt{4 x^2+12 x+9} \]

Antiderivative was successfully verified.

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

[Out]

(e*Sqrt[9 + 12*x + 4*x^2])/4 + ((2*d - 3*e)*(3 + 2*x)*Log[3 + 2*x])/(4*Sqrt[9 +
12*x + 4*x^2])

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Rubi in Sympy [A]  time = 5.90873, size = 49, normalized size = 0.88 \[ \frac{e \sqrt{4 x^{2} + 12 x + 9}}{4} + \frac{\left (\frac{d}{4} - \frac{3 e}{8}\right ) \left (4 x + 6\right ) \log{\left (2 x + 3 \right )}}{\sqrt{4 x^{2} + 12 x + 9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(4*x**2+12*x+9)**(1/2),x)

[Out]

e*sqrt(4*x**2 + 12*x + 9)/4 + (d/4 - 3*e/8)*(4*x + 6)*log(2*x + 3)/sqrt(4*x**2 +
12*x + 9)

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Mathematica [A]  time = 0.026932, size = 42, normalized size = 0.75 \[ \frac{(2 x+3) ((2 d-3 e) \log (2 x+3)+e (2 x+3))}{4 \sqrt{(2 x+3)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 40, normalized size = 0.7 \[{\frac{ \left ( 2\,x+3 \right ) \left ( 2\,\ln \left ( 2\,x+3 \right ) d-3\,e\ln \left ( 2\,x+3 \right ) +2\,ex \right ) }{4}{\frac{1}{\sqrt{ \left ( 2\,x+3 \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(4*x^2+12*x+9)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.827972, size = 41, normalized size = 0.73 \[ \frac{1}{2} \, d \log \left (x + \frac{3}{2}\right ) - \frac{3}{4} \, e \log \left (x + \frac{3}{2}\right ) + \frac{1}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} e \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d*log(x + 3/2) - 3/4*e*log(x + 3/2) + 1/4*sqrt(4*x^2 + 12*x + 9)*e

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Fricas [A]  time = 0.204523, size = 27, normalized size = 0.48 \[ \frac{1}{2} \, e x + \frac{1}{4} \,{\left (2 \, d - 3 \, e\right )} \log \left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x}{\sqrt{\left (2 x + 3\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(4*x**2+12*x+9)**(1/2),x)

[Out]

Integral((d + e*x)/sqrt((2*x + 3)**2), x)

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GIAC/XCAS [A]  time = 0.218399, size = 62, normalized size = 1.11 \[ -\frac{1}{4} \,{\left (2 \, d - 3 \, e\right )}{\rm ln}\left ({\left | -2 \, x + \sqrt{4 \, x^{2} + 12 \, x + 9} - 3 \right |}\right ) + \frac{1}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} e \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*d - 3*e)*ln(abs(-2*x + sqrt(4*x^2 + 12*x + 9) - 3)) + 1/4*sqrt(4*x^2 + 1
2*x + 9)*e