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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]