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 = 0.0576785, 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, Rules used = {1868, 31, 617, 204} \[ \frac{\log (a+b x)}{b}-\frac{2 \tan ^{-1}\left (\frac{a-2 b x}{\sqrt{3} a}\right )}{\sqrt{3} b} \]

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

Rule 1868

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, With[{q = Rt[a/b, 3]}, Dist[C/b, Int[1/(q + x), x], x] + Dist[(B + C*q)/b, Int[1/(q^2 - q*x + x^2), x],
x]] /; EqQ[A - Rt[a/b, 3]*B - 2*Rt[a/b, 3]^2*C, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{2 a^2+b^2 x^2}{a^3+b^3 x^3} \, dx &=\frac{a \int \frac{1}{\frac{a^2}{b^2}-\frac{a x}{b}+x^2} \, dx}{b^2}+\frac{\int \frac{1}{\frac{a}{b}+x} \, dx}{b}\\ &=\frac{\log (a+b x)}{b}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 b x}{a}\right )}{b}\\ &=-\frac{2 \tan ^{-1}\left (\frac{a-2 b x}{\sqrt{3} a}\right )}{\sqrt{3} b}+\frac{\log (a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0202841, size = 72, normalized size = 1.95 \[ \frac{-\log \left (a^2-a b x+b^2 x^2\right )+\log \left (a^3+b^3 x^3\right )+2 \log (a+b x)+2 \sqrt{3} \tan ^{-1}\left (\frac{2 b x-a}{\sqrt{3} a}\right )}{3 b} \]

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 = 0.048, size = 43, normalized size = 1.2 \begin{align*}{\frac{2\,\sqrt{3}}{3\,b}\arctan \left ({\frac{ \left ( 2\,{b}^{2}x-ab \right ) \sqrt{3}}{3\,ab}} \right ) }+{\frac{\ln \left ( bx+a \right ) }{b}} \end{align*}

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]

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

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Maxima [A]  time = 1.46757, size = 57, normalized size = 1.54 \begin{align*} \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} \end{align*}

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 = 0.878056, size = 95, normalized size = 2.57 \begin{align*} \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, b x - a\right )}}{3 \, a}\right ) + 3 \, \log \left (b x + a\right )}{3 \, b} \end{align*}

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

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Sympy [C]  time = 0.513175, size = 60, normalized size = 1.62 \begin{align*} \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} \end{align*}

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 [A]  time = 1.08222, size = 50, normalized size = 1.35 \begin{align*} \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, b x - a\right )}}{3 \, a}\right )}{3 \, b} + \frac{\log \left ({\left | b x + a \right |}\right )}{b} \end{align*}

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 + log(abs(b*x + a))/b