Optimal. Leaf size=53 \[ \frac{\log \left (x+\sqrt{2} \sqrt{3 x+4}+3\right )}{\sqrt{2}}-\frac{\log \left (x-\sqrt{2} \sqrt{3 x+4}+3\right )}{\sqrt{2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0912105, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\log \left (x+\sqrt{2} \sqrt{3 x+4}+3\right )}{\sqrt{2}}-\frac{\log \left (x-\sqrt{2} \sqrt{3 x+4}+3\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(1 - 3*x)/(Sqrt[4 + 3*x]*(1 + x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 18.3944, size = 56, normalized size = 1.06 \[ - \frac{\sqrt{2} \log{\left (3 x - 3 \sqrt{2} \sqrt{3 x + 4} + 9 \right )}}{2} + \frac{\sqrt{2} \log{\left (3 x + 3 \sqrt{2} \sqrt{3 x + 4} + 9 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-3*x)/(x**2+1)/(4+3*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0579998, size = 59, normalized size = 1.11 \[ -\frac{(3+i) \tan ^{-1}\left (\frac{\sqrt{3 x+4}}{\sqrt{-4-3 i}}\right )}{\sqrt{-4-3 i}}-\frac{(3-i) \tan ^{-1}\left (\frac{\sqrt{3 x+4}}{\sqrt{-4+3 i}}\right )}{\sqrt{-4+3 i}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 3*x)/(Sqrt[4 + 3*x]*(1 + x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.023, size = 48, normalized size = 0.9 \[{\frac{\sqrt{2}}{2}\ln \left ( 3\,x+9+3\,\sqrt{2}\sqrt{3\,x+4} \right ) }-{\frac{\sqrt{2}}{2}\ln \left ( 3\,x+9-3\,\sqrt{2}\sqrt{3\,x+4} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-3*x)/(x^2+1)/(3*x+4)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{3 \, x - 1}{{\left (x^{2} + 1\right )} \sqrt{3 \, x + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x - 1)/((x^2 + 1)*sqrt(3*x + 4)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.273106, size = 50, normalized size = 0.94 \[ \frac{1}{2} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{3 \, x + 4}{\left (x + 3\right )} + x^{2} + 12 \, x + 17}{x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x - 1)/((x^2 + 1)*sqrt(3*x + 4)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{3 x}{x^{2} \sqrt{3 x + 4} + \sqrt{3 x + 4}}\, dx - \int \left (- \frac{1}{x^{2} \sqrt{3 x + 4} + \sqrt{3 x + 4}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-3*x)/(x**2+1)/(4+3*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{3 \, x - 1}{{\left (x^{2} + 1\right )} \sqrt{3 \, x + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x - 1)/((x^2 + 1)*sqrt(3*x + 4)),x, algorithm="giac")
[Out]