3.5.25 \(\int \frac {1}{x (a^3-b^3 x)^{2/3}} \, dx\) [425]

Optimal. Leaf size=74 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {a+2 \sqrt [3]{a^3-b^3 x}}{\sqrt {3} a}\right )}{a^2}-\frac {\log (x)}{2 a^2}+\frac {3 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )}{2 a^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {59, 631, 210, 31} \begin {gather*} -\frac {\log (x)}{2 a^2}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {2 \sqrt [3]{a^3-b^3 x}+a}{\sqrt {3} a}\right )}{a^2}+\frac {3 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )}{2 a^2} \end {gather*}

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 31

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

Rule 59

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

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 \left (a^3-b^3 x\right )^{2/3}} \, dx &=-\frac {\log (x)}{2 a^2}-\frac {3 \text {Subst}\left (\int \frac {1}{a-x} \, dx,x,\sqrt [3]{a^3-b^3 x}\right )}{2 a^2}-\frac {3 \text {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 \text {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 = 0.05, size = 99, normalized size = 1.34 \begin {gather*} -\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {a+2 \sqrt [3]{a^3-b^3 x}}{\sqrt {3} a}\right )-2 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )+\log \left (a^2+a \sqrt [3]{a^3-b^3 x}+\left (a^3-b^3 x\right )^{2/3}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(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)])/a^2

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 91, normalized size = 1.23

method	result	size
derivativedivides	\(\frac {\ln \left (a -\left (-b^{3} x +a^{3}\right )^{\frac {1}{3}}\right )}{a^{2}}+\frac {-\frac {\ln \left (a^{2}+a \left (-b^{3} x +a^{3}\right )^{\frac {1}{3}}+\left (-b^{3} x +a^{3}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (-b^{3} x +a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a^{2}}\)	\(91\)
default	\(\frac {\ln \left (a -\left (-b^{3} x +a^{3}\right )^{\frac {1}{3}}\right )}{a^{2}}+\frac {-\frac {\ln \left (a^{2}+a \left (-b^{3} x +a^{3}\right )^{\frac {1}{3}}+\left (-b^{3} x +a^{3}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (-b^{3} x +a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a^{2}}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-b^3*x+a^3)^(2/3),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 91, normalized size = 1.23 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.50, size = 90, normalized size = 1.22 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.90, size = 136, normalized size = 1.84 \begin {gather*} \frac {\log {\left (1 - \frac {b \sqrt [3]{- \frac {a^{3}}{b^{3}} + x} e^{\frac {i \pi }{3}}}{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^{i \pi }}{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 {5 i \pi }{3}}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} \end {gather*}

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)*exp_polar(I*pi/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(I*pi)/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(5*I*pi/3)/a)*gamma(1/3)/(3*a**2*gamma(4/3))

________________________________________________________________________________________

Giac [A]
time = 1.38, size = 92, normalized size = 1.24 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.11, size = 104, normalized size = 1.41 \begin {gather*} \frac {\ln \left (9\,a-9\,{\left (a^3-b^3\,x\right )}^{1/3}\right )}{a^2}+\frac {\ln \left (9\,{\left (a^3-b^3\,x\right )}^{1/3}-\frac {9\,a\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^2}-\frac {\ln \left (9\,{\left (a^3-b^3\,x\right )}^{1/3}+\frac {9\,a\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a^3 - b^3*x)^(2/3)),x)

[Out]

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

________________________________________________________________________________________