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 = 0.0421376, 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, Rules used = {1868, 31, 617, 204} \[ \frac{2 \tan ^{-1}\left (\frac{a+2 b x}{\sqrt{3} a}\right )}{\sqrt{3} b}-\frac{\log (a-b x)}{b} \]

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

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 = 0.0178963, size = 71, normalized size = 1.82 \[ \frac{\log \left (a^2+a b x+b^2 x^2\right )-\log \left (a^3-b^3 x^3\right )-2 \log (a-b x)+2 \sqrt{3} \tan ^{-1}\left (\frac{a+2 b x}{\sqrt{3} a}\right )}{3 b} \]

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 = 0.007, size = 45, 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)-1/b*ln(b*x-a)

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Maxima [A]  time = 1.42968, size = 59, normalized size = 1.51 \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 = 0.998721, size = 95, normalized size = 2.44 \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 = 0.433734, size = 60, normalized size = 1.54 \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 = 1.07399, size = 51, normalized size = 1.31 \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