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

Optimal. Leaf size=37 \[ \frac{\log (a+b x)}{b}-\frac{2 \tan ^{-1}\left (\frac{a-2 b x}{\sqrt{3} a}\right )}{\sqrt{3} b} \]

[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 = 0.100479, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{\log (a+b x)}{b}-\frac{2 \tan ^{-1}\left (\frac{a-2 b x}{\sqrt{3} a}\right )}{\sqrt{3} b} \]

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 = 15.7949, size = 36, normalized size = 0.97 \[ \frac{\log{\left (a + b x \right )}}{b} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{a}{3} - \frac{2 b x}{3}\right )}{a} \right )}}{3 b} \]

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 = 0.0356842, size = 72, normalized size = 1.95 \[ \frac{\log \left (a^3+b^3 x^3\right )-\log \left (a^2-a b x+b^2 x^2\right )+2 \log (a+b x)+2 \sqrt{3} \tan ^{-1}\left (\frac{2 b x-a}{\sqrt{3} a}\right )}{3 b} \]

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 = 0.011, size = 43, 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]

ln(b*x+a)/b+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 = 1.52875, size = 57, normalized size = 1.54 \[ \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, b^{2} x - a b\right )}}{3 \, a b}\right )}{3 \, b} + \frac{\log \left (b x + a\right )}{b} \]

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 = 0.219281, size = 51, normalized size = 1.38 \[ \frac{\sqrt{3}{\left (\sqrt{3} \log \left (b x + a\right ) + 2 \, \arctan \left (\frac{\sqrt{3}{\left (2 \, b x - a\right )}}{3 \, a}\right )\right )}}{3 \, b} \]

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 = 0.819181, size = 60, normalized size = 1.62 \[ \frac{- \frac{\sqrt{3} i \log{\left (x + \frac{- a - \sqrt{3} i a}{2 b} \right )}}{3} + \frac{\sqrt{3} i \log{\left (x + \frac{- a + \sqrt{3} i a}{2 b} \right )}}{3} + \log{\left (\frac{a}{b} + x \right )}}{b} \]

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 = 0.211481, size = 50, normalized size = 1.35 \[ \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, b x - a\right )}}{3 \, a}\right )}{3 \, b} + \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{b} \]

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