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 = 0.0709022, 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 \[ -\frac{3 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{2 a}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt{3} a}\right )}{a}+\frac{\log (x)}{2 a} \]

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)

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Rubi in Sympy [A]  time = 5.80699, size = 63, normalized size = 0.83 \[ \frac{\log{\left (x \right )}}{2 a} - \frac{3 \log{\left (a + \sqrt [3]{- a^{3} - b^{3} x} \right )}}{2 a} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{a}{3} - \frac{2 \sqrt [3]{- a^{3} - b^{3} x}}{3}\right )}{a} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-b**3*x-a**3)**(1/3),x)

[Out]

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

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Mathematica [A]  time = 0.0496431, size = 108, normalized size = 1.42 \[ \frac{-2 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{-a^3-b^3 x}-a}{\sqrt{3} a}\right )+\log \left (-a \sqrt [3]{-a^3-b^3 x}+\left (-a^3-b^3 x\right )^{2/3}+a^2\right )}{2 a} \]

Antiderivative was successfully verified.

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

[Out]

(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)])/(2*a)

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Maple [A]  time = 0.01, size = 101, normalized size = 1.3 \[{\frac{1}{2\,a}\ln \left ( \left ( -{b}^{3}x-{a}^{3} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-{b}^{3}x-{a}^{3}}a+{a}^{2} \right ) }+{\frac{\sqrt{3}}{a}\arctan \left ({\frac{\sqrt{3}}{3\,a} \left ( 2\,\sqrt [3]{-{b}^{3}x-{a}^{3}}-a \right ) } \right ) }-{\frac{1}{a}\ln \left ( a+\sqrt [3]{-{b}^{3}x-{a}^{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-b^3*x-a^3)^(1/3),x)

[Out]

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

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Maxima [A]  time = 1.50924, size = 132, normalized size = 1.74 \[ \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \,{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a} + \frac{\log \left (a^{2} -{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x - a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a} - \frac{\log \left (a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b^3*x - a^3)^(1/3)*x),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 = 0.219406, size = 126, normalized size = 1.66 \[ \frac{2 \, \sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \,{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right ) + \log \left (a^{2} -{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x - a^{3}\right )}^{\frac{2}{3}}\right ) - 2 \, \log \left (a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*(-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 [A]  time = 5.37178, size = 139, normalized size = 1.83 \[ \frac{\log{\left (- \frac{a e^{\frac{2 i \pi }{3}}}{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a \Gamma \left (\frac{2}{3}\right )} + \frac{e^{- \frac{2 i \pi }{3}} \log{\left (- \frac{a e^{\frac{4 i \pi }{3}}}{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a \Gamma \left (\frac{2}{3}\right )} - \frac{e^{- \frac{i \pi }{3}} \log{\left (- \frac{a e^{2 i \pi }}{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a \Gamma \left (\frac{2}{3}\right )} \]

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*gamm
a(2/3)) + exp(-2*I*pi/3)*log(-a*exp_polar(4*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*exp_polar(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/XCAS [A]  time = 0.216275, size = 134, normalized size = 1.76 \[ \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \,{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a} + \frac{{\rm ln}\left (a^{2} -{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x - a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a} - \frac{{\rm ln}\left ({\left | a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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