3.427 \(\int \frac{1}{x (-a^3-b^3 x)^{2/3}} \, dx\)

Optimal. Leaf size=76 \[ \frac{3 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{2 a^2}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt{3} a}\right )}{a^2}-\frac{\log (x)}{2 a^2} \]

[Out]

-((Sqrt[3]*ArcTan[(a - 2*(-a^3 - b^3*x)^(1/3))/(Sqrt[3]*a)])/a^2) - Log[x]/(2*a^2) + (3*Log[a + (-a^3 - b^3*x)
^(1/3)])/(2*a^2)

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Rubi [A]  time = 0.0233589, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {58, 617, 204, 31} \[ \frac{3 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{2 a^2}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt{3} a}\right )}{a^2}-\frac{\log (x)}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(-a^3 - b^3*x)^(2/3)),x]

[Out]

-((Sqrt[3]*ArcTan[(a - 2*(-a^3 - b^3*x)^(1/3))/(Sqrt[3]*a)])/a^2) - Log[x]/(2*a^2) + (3*Log[a + (-a^3 - b^3*x)
^(1/3)])/(2*a^2)

Rule 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, -Sim
p[Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d
*x)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x
] && NegQ[(b*c - a*d)/b]

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])

Rule 31

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

Rubi steps

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

Mathematica [A]  time = 0.122299, size = 112, normalized size = 1.47 \[ \frac{\log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{a^2}-\frac{\log \left (-a \sqrt [3]{-a^3-b^3 x}+\left (-a^3-b^3 x\right )^{2/3}+a^2\right )}{2 a^2}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{-a^3-b^3 x}-a}{\sqrt{3} a}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-a^3 - b^3*x)^(2/3)),x]

[Out]

(Sqrt[3]*ArcTan[(-a + 2*(-a^3 - b^3*x)^(1/3))/(Sqrt[3]*a)])/a^2 + Log[a + (-a^3 - b^3*x)^(1/3)]/a^2 - Log[a^2
- a*(-a^3 - b^3*x)^(1/3) + (-a^3 - b^3*x)^(2/3)]/(2*a^2)

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Maple [A]  time = 0.003, size = 100, normalized size = 1.3 \begin{align*}{\frac{1}{{a}^{2}}\ln \left ( a+\sqrt [3]{-{b}^{3}x-{a}^{3}} \right ) }-{\frac{1}{2\,{a}^{2}}\ln \left ( \left ( -{b}^{3}x-{a}^{3} \right ) ^{{\frac{2}{3}}}-a\sqrt [3]{-{b}^{3}x-{a}^{3}}+{a}^{2} \right ) }+{\frac{\sqrt{3}}{{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3\,a} \left ( 2\,\sqrt [3]{-{b}^{3}x-{a}^{3}}-a \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-b^3*x-a^3)^(2/3),x)

[Out]

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

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Maxima [A]  time = 1.51693, size = 131, normalized size = 1.72 \begin{align*} \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \,{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac{\log \left (a^{2} -{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x - a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a^{2}} + \frac{\log \left (a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-b^3*x-a^3)^(2/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.56733, size = 235, normalized size = 3.09 \begin{align*} \frac{2 \, \sqrt{3} \arctan \left (-\frac{\sqrt{3} a - 2 \, \sqrt{3}{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}}{3 \, a}\right ) - \log \left (a^{2} -{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x - a^{3}\right )}^{\frac{2}{3}}\right ) + 2 \, \log \left (a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}{2 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-b^3*x-a^3)^(2/3),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 2.32263, size = 133, normalized size = 1.75 \begin{align*} \frac{e^{- \frac{2 i \pi }{3}} \log{\left (1 - \frac{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}}{a} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{2} \Gamma \left (\frac{4}{3}\right )} - \frac{e^{- \frac{i \pi }{3}} \log{\left (1 - \frac{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x} e^{\frac{2 i \pi }{3}}}{a} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{2} \Gamma \left (\frac{4}{3}\right )} + \frac{\log{\left (1 - \frac{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x} e^{\frac{4 i \pi }{3}}}{a} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{2} \Gamma \left (\frac{4}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-b**3*x-a**3)**(2/3),x)

[Out]

exp(-2*I*pi/3)*log(1 - b*(a**3/b**3 + x)**(1/3)/a)*gamma(1/3)/(3*a**2*gamma(4/3)) - exp(-I*pi/3)*log(1 - b*(a*
*3/b**3 + x)**(1/3)*exp_polar(2*I*pi/3)/a)*gamma(1/3)/(3*a**2*gamma(4/3)) + log(1 - b*(a**3/b**3 + x)**(1/3)*e
xp_polar(4*I*pi/3)/a)*gamma(1/3)/(3*a**2*gamma(4/3))

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Giac [A]  time = 1.27258, size = 132, normalized size = 1.74 \begin{align*} \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \,{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac{\log \left (a^{2} -{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x - a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a^{2}} + \frac{\log \left ({\left | a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} \right |}\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-b^3*x-a^3)^(2/3),x, algorithm="giac")

[Out]

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