\(\int \frac {1}{x (-a^3-b^3 x)^{2/3}} \, dx\) [427]

Optimal result

Integrand size = 20, antiderivative size = 76 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt {3} a}\right )}{a^2}-\frac {\log (x)}{2 a^2}+\frac {3 \log \left (a+\sqrt [3]{-a^3-b^3 x}\right )}{2 a^2} \]

[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] (verified)

Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {60, 631, 210, 31} \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=-\frac {\log (x)}{2 a^2}-\frac {\sqrt {3} \arctan \left (\frac {a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt {3} a}\right )}{a^2}+\frac {3 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{2 a^2} \]

[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 60

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[-
Log[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]
&& NegQ[(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*} \text {integral}& = -\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] (verified)

Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt {3} a}\right )-2 \log \left (a+\sqrt [3]{-a^3-b^3 x}\right )+\log \left (a^2-a \sqrt [3]{-a^3-b^3 x}+\left (-a^3-b^3 x\right )^{2/3}\right )}{2 a^2} \]

[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] (verified)

Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.26

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (a -2 \left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )+2 \ln \left (a +\left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}\right )-\ln \left (a^{2}-a \left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}+\left (-b^{3} x -a^{3}\right )^{\frac {2}{3}}\right )}{2 a^{2}}\)	\(96\)
derivativedivides	\(\frac {-\frac {\ln \left (a^{2}-a \left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}+\left (-b^{3} x -a^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (-a +2 \left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a^{2}}+\frac {\ln \left (a +\left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}\right )}{a^{2}}\)	\(99\)
default	\(\frac {-\frac {\ln \left (a^{2}-a \left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}+\left (-b^{3} x -a^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (-a +2 \left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a^{2}}+\frac {\ln \left (a +\left (-b^{3} x -a^{3}\right )^{\frac {1}{3}}\right )}{a^{2}}\)	\(99\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} a - 2 \, \sqrt {3} {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}}}{3 \, a}\right ) - \log \left (a^{2} - {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}} a + {\left (-b^{3} x - a^{3}\right )}^{\frac {2}{3}}\right ) + 2 \, \log \left (a + {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}}\right )}{2 \, a^{2}} \]

[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] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.75 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=\frac {e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {b \sqrt [3]{\frac {a^{3}}{b^{3}} + x}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} - \frac {e^{- \frac {i \pi }{3}} \log {\left (1 - \frac {b \sqrt [3]{\frac {a^{3}}{b^{3}} + x} e^{\frac {2 i \pi }{3}}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} + \frac {\log {\left (1 - \frac {b \sqrt [3]{\frac {a^{3}}{b^{3}} + x} e^{\frac {4 i \pi }{3}}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} \]

[In]

integrate(1/x/(-b**3*x-a**3)**(2/3),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac {\log \left (a^{2} - {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}} a + {\left (-b^{3} x - a^{3}\right )}^{\frac {2}{3}}\right )}{2 \, a^{2}} + \frac {\log \left (a + {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}}\right )}{a^{2}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac {\log \left (a^{2} - {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}} a + {\left (-b^{3} x - a^{3}\right )}^{\frac {2}{3}}\right )}{2 \, a^{2}} + \frac {\log \left ({\left | a + {\left (-b^{3} x - a^{3}\right )}^{\frac {1}{3}} \right |}\right )}{a^{2}} \]

[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] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.45 \[ \int \frac {1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx=\frac {\ln \left (9\,a+9\,{\left (-a^3-x\,b^3\right )}^{1/3}\right )}{a^2}+\frac {\ln \left (9\,{\left (-a^3-x\,b^3\right )}^{1/3}+\frac {9\,a\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^2}-\frac {\ln \left (9\,{\left (-a^3-x\,b^3\right )}^{1/3}-\frac {9\,a\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^2} \]

[In]

int(1/(x*(- b^3*x - a^3)^(2/3)),x)

[Out]

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