3.1469 \(\int \frac{1-3 x}{\sqrt{4+3 x} \left (1+x^2\right )} \, dx\)

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]

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

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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]

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

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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]

-sqrt(2)*log(3*x - 3*sqrt(2)*sqrt(3*x + 4) + 9)/2 + sqrt(2)*log(3*x + 3*sqrt(2)*
sqrt(3*x + 4) + 9)/2

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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]

((-3 - I)*ArcTan[Sqrt[4 + 3*x]/Sqrt[-4 - 3*I]])/Sqrt[-4 - 3*I] - ((3 - I)*ArcTan
[Sqrt[4 + 3*x]/Sqrt[-4 + 3*I]])/Sqrt[-4 + 3*I]

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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]

1/2*2^(1/2)*ln(3*x+9+3*2^(1/2)*(3*x+4)^(1/2))-1/2*2^(1/2)*ln(3*x+9-3*2^(1/2)*(3*
x+4)^(1/2))

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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]

-integrate((3*x - 1)/((x^2 + 1)*sqrt(3*x + 4)), x)

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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]

1/2*sqrt(2)*log((2*sqrt(2)*sqrt(3*x + 4)*(x + 3) + x^2 + 12*x + 17)/(x^2 + 1))

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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]

-Integral(3*x/(x**2*sqrt(3*x + 4) + sqrt(3*x + 4)), x) - Integral(-1/(x**2*sqrt(
3*x + 4) + sqrt(3*x + 4)), x)

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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]

integrate(-(3*x - 1)/((x^2 + 1)*sqrt(3*x + 4)), x)