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

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

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Rubi [A]  time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 31

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

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 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 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]

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*}

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Mathematica [C]  time = 0.01, size = 35, normalized size = 0.43 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.05, size = 113, normalized size = 1.38 \begin {gather*} \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b x-a}+(b x-a)^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{\sqrt [3]{a}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b x-a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.94, size = 285, normalized size = 3.48 \begin {gather*} \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 {gather*}

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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giac [A]  time = 2.51, size = 112, normalized size = 1.37 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.01, size = 83, normalized size = 1.01 \begin {gather*} \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x -a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{a^{\frac {1}{3}}}-\frac {\ln \left (a^{\frac {1}{3}}+\left (b x -a \right )^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}+\frac {\ln \left (a^{\frac {2}{3}}-\left (b x -a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+\left (b x -a \right )^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}} \end {gather*}

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 [A]  time = 3.01, size = 86, normalized size = 1.05 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x - a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} + \frac {\log \left ({\left (b x - a\right )}^{\frac {2}{3}} - {\left (b x - a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{2 \, a^{\frac {1}{3}}} - \frac {\log \left ({\left (b x - a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.09, size = 117, normalized size = 1.43 \begin {gather*} \frac {\ln \left (9\,{\left (b\,x-a\right )}^{1/3}-9\,{\left (-a\right )}^{1/3}\right )}{{\left (-a\right )}^{1/3}}+\frac {\ln \left (9\,{\left (b\,x-a\right )}^{1/3}-\frac {9\,{\left (-a\right )}^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,{\left (-a\right )}^{1/3}}-\frac {\ln \left (9\,{\left (b\,x-a\right )}^{1/3}-\frac {9\,{\left (-a\right )}^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,{\left (-a\right )}^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b*x - a)^(1/3)),x)

[Out]

log(9*(b*x - a)^(1/3) - 9*(-a)^(1/3))/(-a)^(1/3) + (log(9*(b*x - a)^(1/3) - (9*(-a)^(1/3)*(3^(1/2)*1i - 1)^2)/
4)*(3^(1/2)*1i - 1))/(2*(-a)^(1/3)) - (log(9*(b*x - a)^(1/3) - (9*(-a)^(1/3)*(3^(1/2)*1i + 1)^2)/4)*(3^(1/2)*1
i + 1))/(2*(-a)^(1/3))

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sympy [C]  time = 1.88, size = 160, normalized size = 1.95 \begin {gather*} - \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 {gather*}

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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