3.427 \(\int \frac{1}{x \left (-a^3-b^3 x\right )^{2/3}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{\log (x)}{2 a^2}+\frac{3 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{2 a^2}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt{3} a}\right )}{a^2} \]

[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)

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Rubi [A]  time = 0.0626767, 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 \[ -\frac{\log (x)}{2 a^2}+\frac{3 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{2 a^2}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt{3} a}\right )}{a^2} \]

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)

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Rubi in Sympy [A]  time = 6.0539, size = 68, normalized size = 0.89 \[ - \frac{\log{\left (x \right )}}{2 a^{2}} + \frac{3 \log{\left (a + \sqrt [3]{- a^{3} - b^{3} x} \right )}}{2 a^{2}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{a}{3} - \frac{2 \sqrt [3]{- a^{3} - b^{3} x}}{3}\right )}{a} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-b**3*x-a**3)**(2/3),x)

[Out]

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

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Mathematica [A]  time = 0.0167553, size = 112, normalized size = 1.47 \[ \frac{\log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{a^2}-\frac{\log \left (-a \sqrt [3]{-a^3-b^3 x}+\left (-a^3-b^3 x\right )^{2/3}+a^2\right )}{2 a^2}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{-a^3-b^3 x}-a}{\sqrt{3} a}\right )}{a^2} \]

Antiderivative was successfully verified.

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

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Maple [A]  time = 0.005, size = 100, normalized size = 1.3 \[ -{\frac{1}{2\,{a}^{2}}\ln \left ( \left ( -{b}^{3}x-{a}^{3} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-{b}^{3}x-{a}^{3}}a+{a}^{2} \right ) }+{\frac{\sqrt{3}}{{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3\,a} \left ( 2\,\sqrt [3]{-{b}^{3}x-{a}^{3}}-a \right ) } \right ) }+{\frac{1}{{a}^{2}}\ln \left ( a+\sqrt [3]{-{b}^{3}x-{a}^{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-b^3*x-a^3)^(2/3),x)

[Out]

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

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Maxima [A]  time = 1.49841, size = 131, normalized size = 1.72 \[ \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \,{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac{\log \left (a^{2} -{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x - a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a^{2}} + \frac{\log \left (a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [A]  time = 0.220257, size = 128, normalized size = 1.68 \[ \frac{2 \, \sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \,{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right ) - \log \left (a^{2} -{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x - a^{3}\right )}^{\frac{2}{3}}\right ) + 2 \, \log \left (a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b^3*x - a^3)^(2/3)*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^2

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Sympy [A]  time = 5.44929, size = 133, normalized size = 1.75 \[ - \frac{e^{\frac{i \pi }{3}} \log{\left (1 - \frac{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}}{a} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{2} \Gamma \left (\frac{4}{3}\right )} - \frac{e^{\frac{5 i \pi }{3}} \log{\left (1 - \frac{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x} e^{\frac{2 i \pi }{3}}}{a} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{2} \Gamma \left (\frac{4}{3}\right )} + \frac{\log{\left (1 - \frac{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x} e^{\frac{4 i \pi }{3}}}{a} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{2} \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-exp(I*pi/3)*log(1 - b*(a**3/b**3 + x)**(1/3)/a)*gamma(1/3)/(3*a**2*gamma(4/3))
- exp(5*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)*exp_polar(4*I*pi/3)/a)
*gamma(1/3)/(3*a**2*gamma(4/3))

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GIAC/XCAS [A]  time = 0.221649, size = 132, normalized size = 1.74 \[ \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \,{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac{{\rm ln}\left (a^{2} -{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x - a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a^{2}} + \frac{{\rm ln}\left ({\left | a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} \right |}\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b^3*x - a^3)^(2/3)*x),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*ln(a^2 - (
-b^3*x - a^3)^(1/3)*a + (-b^3*x - a^3)^(2/3))/a^2 + ln(abs(a + (-b^3*x - a^3)^(1
/3)))/a^2