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

Optimal. Leaf size=82 \[ -\frac{3 \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{2 \sqrt [3]{a}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b x-a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}+\frac{\log (x)}{2 \sqrt [3]{a}} \]

[Out]

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

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Rubi [A]  time = 0.0325087, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {56, 617, 204, 31} \[ -\frac{3 \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{2 \sqrt [3]{a}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b x-a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}+\frac{\log (x)}{2 \sqrt [3]{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(-a + b*x)^(1/3)),x]

[Out]

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

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

Mathematica [C]  time = 0.0175193, size = 35, normalized size = 0.43 \[ \frac{3 (b x-a)^{2/3} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};1-\frac{b x}{a}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-a + b*x)^(1/3)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 83, normalized size = 1. \begin{align*} -{\ln \left ( \sqrt [3]{a}+\sqrt [3]{bx-a} \right ){\frac{1}{\sqrt [3]{a}}}}+{\frac{1}{2}\ln \left ( \left ( bx-a \right ) ^{{\frac{2}{3}}}-\sqrt [3]{a}\sqrt [3]{bx-a}+{a}^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{a}}}}+{\sqrt{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx-a}}{\sqrt [3]{a}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72384, size = 747, normalized size = 9.11 \begin{align*} \left [\frac{\sqrt{3} a \sqrt{\frac{\left (-a\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b x + \sqrt{3}{\left (2 \,{\left (b x - a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{2}{3}} +{\left (b x - a\right )}^{\frac{1}{3}} a + \left (-a\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a\right )^{\frac{1}{3}}}{a}} - 3 \,{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} - 3 \, a}{x}\right ) + \left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x - a\right )}^{\frac{2}{3}} +{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x - a\right )}^{\frac{1}{3}} - \left (-a\right )^{\frac{1}{3}}\right )}{2 \, a}, \frac{2 \, \sqrt{3} a \sqrt{-\frac{\left (-a\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (b x - a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a\right )^{\frac{1}{3}}}{a}}\right ) + \left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x - a\right )}^{\frac{2}{3}} +{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x - a\right )}^{\frac{1}{3}} - \left (-a\right )^{\frac{1}{3}}\right )}{2 \, a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(3)*a*sqrt((-a)^(1/3)/a)*log((2*b*x + sqrt(3)*(2*(b*x - a)^(2/3)*(-a)^(2/3) + (b*x - a)^(1/3)*a + (-
a)^(1/3)*a)*sqrt((-a)^(1/3)/a) - 3*(b*x - a)^(1/3)*(-a)^(2/3) - 3*a)/x) + (-a)^(2/3)*log((b*x - a)^(2/3) + (b*
x - a)^(1/3)*(-a)^(1/3) + (-a)^(2/3)) - 2*(-a)^(2/3)*log((b*x - a)^(1/3) - (-a)^(1/3)))/a, 1/2*(2*sqrt(3)*a*sq
rt(-(-a)^(1/3)/a)*arctan(1/3*sqrt(3)*(2*(b*x - a)^(1/3) + (-a)^(1/3))*sqrt(-(-a)^(1/3)/a)) + (-a)^(2/3)*log((b
*x - a)^(2/3) + (b*x - a)^(1/3)*(-a)^(1/3) + (-a)^(2/3)) - 2*(-a)^(2/3)*log((b*x - a)^(1/3) - (-a)^(1/3)))/a]

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Sympy [C]  time = 2.74842, size = 160, normalized size = 1.95 \begin{align*} - \frac{2 e^{- \frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{- \frac{a}{b} + x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{2}{3}\right )}{3 \sqrt [3]{a} \Gamma \left (\frac{5}{3}\right )} - \frac{2 \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{- \frac{a}{b} + x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{2}{3}\right )}{3 \sqrt [3]{a} \Gamma \left (\frac{5}{3}\right )} - \frac{2 e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{- \frac{a}{b} + x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{2}{3}\right )}{3 \sqrt [3]{a} \Gamma \left (\frac{5}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*exp(-2*I*pi/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(2/3)/(3*a**(1/3)*gamma(
5/3)) - 2*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(2/3)/(3*a**(1/3)*gamma(5/3)) - 2*
exp(2*I*pi/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(2/3)/(3*a**(1/3)*gamma(5/
3))

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Giac [A]  time = 1.77774, size = 151, normalized size = 1.84 \begin{align*} -\frac{\sqrt{3} \left (-a\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x - a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right )}}{3 \, \left (-a\right )^{\frac{1}{3}}}\right )}{a} + \frac{\left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x - a\right )}^{\frac{2}{3}} +{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right )}{2 \, a} - \frac{\left (-a\right )^{\frac{2}{3}} \log \left ({\left |{\left (b x - a\right )}^{\frac{1}{3}} - \left (-a\right )^{\frac{1}{3}} \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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