3.28 \(\int \frac{2 a^2+b^2 x^2}{a^3-b^3 x^3} \, dx\)

Optimal. Leaf size=39 \[ \frac{2 \tan ^{-1}\left (\frac{a+2 b x}{\sqrt{3} a}\right )}{\sqrt{3} b}-\frac{\log (a-b x)}{b} \]

[Out]

(2*ArcTan[(a + 2*b*x)/(Sqrt[3]*a)])/(Sqrt[3]*b) - Log[a - b*x]/b

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Rubi [A]  time = 0.0788608, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 \tan ^{-1}\left (\frac{a+2 b x}{\sqrt{3} a}\right )}{\sqrt{3} b}-\frac{\log (a-b x)}{b} \]

Antiderivative was successfully verified.

[In]  Int[(2*a^2 + b^2*x^2)/(a^3 - b^3*x^3),x]

[Out]

(2*ArcTan[(a + 2*b*x)/(Sqrt[3]*a)])/(Sqrt[3]*b) - Log[a - b*x]/b

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Rubi in Sympy [A]  time = 16.7815, size = 36, normalized size = 0.92 \[ - \frac{\log{\left (a - b x \right )}}{b} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{a}{3} + \frac{2 b x}{3}\right )}{a} \right )}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a**2)/(-b**3*x**3+a**3),x)

[Out]

-log(a - b*x)/b + 2*sqrt(3)*atan(sqrt(3)*(a/3 + 2*b*x/3)/a)/(3*b)

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Mathematica [A]  time = 0.0348964, size = 71, normalized size = 1.82 \[ \frac{-\log \left (a^3-b^3 x^3\right )+\log \left (a^2+a b x+b^2 x^2\right )-2 \log (a-b x)+2 \sqrt{3} \tan ^{-1}\left (\frac{a+2 b x}{\sqrt{3} a}\right )}{3 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(2*a^2 + b^2*x^2)/(a^3 - b^3*x^3),x]

[Out]

(2*Sqrt[3]*ArcTan[(a + 2*b*x)/(Sqrt[3]*a)] - 2*Log[a - b*x] + Log[a^2 + a*b*x +
b^2*x^2] - Log[a^3 - b^3*x^3])/(3*b)

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Maple [A]  time = 0.013, size = 45, normalized size = 1.2 \[ -{\frac{\ln \left ( bx-a \right ) }{b}}+{\frac{2\,\sqrt{3}}{3\,b}\arctan \left ({\frac{ \left ( 2\,{b}^{2}x+ab \right ) \sqrt{3}}{3\,ab}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a^2)/(-b^3*x^3+a^3),x)

[Out]

-1/b*ln(b*x-a)+2/3*3^(1/2)/b*arctan(1/3*(2*b^2*x+a*b)*3^(1/2)/a/b)

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Maxima [A]  time = 1.51534, size = 59, normalized size = 1.51 \[ \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, b^{2} x + a b\right )}}{3 \, a b}\right )}{3 \, b} - \frac{\log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b^2*x^2 + 2*a^2)/(b^3*x^3 - a^3),x, algorithm="maxima")

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*b^2*x + a*b)/(a*b))/b - log(b*x - a)/b

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Fricas [A]  time = 0.224573, size = 51, normalized size = 1.31 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (b x - a\right ) - 2 \, \arctan \left (\frac{\sqrt{3}{\left (2 \, b x + a\right )}}{3 \, a}\right )\right )}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b^2*x^2 + 2*a^2)/(b^3*x^3 - a^3),x, algorithm="fricas")

[Out]

-1/3*sqrt(3)*(sqrt(3)*log(b*x - a) - 2*arctan(1/3*sqrt(3)*(2*b*x + a)/a))/b

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Sympy [A]  time = 0.786517, size = 60, normalized size = 1.54 \[ - \frac{\frac{\sqrt{3} i \log{\left (x + \frac{a - \sqrt{3} i a}{2 b} \right )}}{3} - \frac{\sqrt{3} i \log{\left (x + \frac{a + \sqrt{3} i a}{2 b} \right )}}{3} + \log{\left (- \frac{a}{b} + x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a**2)/(-b**3*x**3+a**3),x)

[Out]

-(sqrt(3)*I*log(x + (a - sqrt(3)*I*a)/(2*b))/3 - sqrt(3)*I*log(x + (a + sqrt(3)*
I*a)/(2*b))/3 + log(-a/b + x))/b

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GIAC/XCAS [A]  time = 0.213391, size = 51, normalized size = 1.31 \[ \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, b x + a\right )}}{3 \, a}\right )}{3 \, b} - \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b^2*x^2 + 2*a^2)/(b^3*x^3 - a^3),x, algorithm="giac")

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*b*x + a)/a)/b - ln(abs(b*x - a))/b