3.423 \(\int \frac{1}{x \sqrt [3]{-a^3-b^3 x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0283408, 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 = {56, 617, 204, 31} \[ -\frac{3 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{2 a}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt{3} a}\right )}{a}+\frac{\log (x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(-a^3 - b^3*x)^(1/3)),x]

[Out]

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

Rule 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(
1/3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && Ne
gQ[(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 \sqrt [3]{-a^3-b^3 x}} \, dx &=\frac{\log (x)}{2 a}+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{a^2-a x+x^2} \, dx,x,\sqrt [3]{-a^3-b^3 x}\right )-\frac{3 \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,\sqrt [3]{-a^3-b^3 x}\right )}{2 a}\\ &=\frac{\log (x)}{2 a}-\frac{3 \log \left (a+\sqrt [3]{-a^3-b^3 x}\right )}{2 a}+\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}\\ &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{-a^3-b^3 x}}{a}}{\sqrt{3}}\right )}{a}+\frac{\log (x)}{2 a}-\frac{3 \log \left (a+\sqrt [3]{-a^3-b^3 x}\right )}{2 a}\\ \end{align*}

Mathematica [C]  time = 0.0531762, size = 41, normalized size = 0.54 \[ \frac{3 \left (-a^3-b^3 x\right )^{2/3} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{x b^3}{a^3}+1\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-a^3 - b^3*x)^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, 1 + (b^3*x)/a^3])/(2*a^3)

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Maple [A]  time = 0.004, size = 101, normalized size = 1.3 \begin{align*} -{\frac{1}{a}\ln \left ( a+\sqrt [3]{-{b}^{3}x-{a}^{3}} \right ) }+{\frac{1}{2\,a}\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}\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)^(1/3),x)

[Out]

-ln(a+(-b^3*x-a^3)^(1/3))/a+1/2/a*ln((-b^3*x-a^3)^(2/3)-a*(-b^3*x-a^3)^(1/3)+a^2)+1/a*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 = 1.54618, 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} + \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} - \frac{\log \left (a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.65885, size = 232, normalized size = 3.05 \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [C]  time = 2.37006, size = 139, normalized size = 1.83 \begin{align*} \frac{\log{\left (- \frac{a e^{\frac{2 i \pi }{3}}}{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a \Gamma \left (\frac{2}{3}\right )} - \frac{e^{\frac{i \pi }{3}} \log{\left (- \frac{a e^{\frac{4 i \pi }{3}}}{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a \Gamma \left (\frac{2}{3}\right )} + \frac{e^{\frac{2 i \pi }{3}} \log{\left (- \frac{a e^{2 i \pi }}{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a \Gamma \left (\frac{2}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26894, size = 134, normalized size = 1.76 \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} + \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} - \frac{\log \left ({\left | a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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