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

Optimal. Leaf size=72 \[ \frac{3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a^2}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt{3} a}\right )}{a^2}-\frac{\log (x)}{2 a^2} \]

[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 = 0.0238681, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {57, 617, 204, 31} \[ \frac{3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a^2}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt{3} a}\right )}{a^2}-\frac{\log (x)}{2 a^2} \]

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 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[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]
&& PosQ[(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 = 0.101855, size = 95, normalized size = 1.32 \[ -\frac{-2 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )+\log \left (a \sqrt [3]{a^3+b^3 x}+\left (a^3+b^3 x\right )^{2/3}+a^2\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt{3} a}\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*Sqrt[3]*ArcTan[(a + 2*(a^3 + b^3*x)^(1/3))/(Sqrt[3]*a)] - 2*Log[a - (a^3 + b^3*x)^(1/3)] + 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 = 0.007, size = 88, normalized size = 1.2 \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 ( a+2\,\sqrt [3]{{b}^{3}x+{a}^{3}} \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]

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

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Maxima [A]  time = 1.51738, size = 117, normalized size = 1.62 \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 = 1.63793, size = 231, normalized size = 3.21 \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 = 2.50924, size = 134, normalized size = 1.86 \begin{align*} \frac{\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{2 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{e^{\frac{2 i \pi }{3}} \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]

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

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Giac [A]  time = 1.23923, size = 119, normalized size = 1.65 \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