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

Optimal. Leaf size=123 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3}}-\frac{1}{a x} \]

[Out]

-(1/(a*x)) - (b^(1/3)*ArcTan[(a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3
]*a^(4/3)) - (b^(1/3)*Log[a^(1/3) - b^(1/3)*x])/(3*a^(4/3)) + (b^(1/3)*Log[a^(2/
3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(4/3))

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Rubi [A]  time = 0.136664, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a - b*x^3)),x]

[Out]

-(1/(a*x)) - (b^(1/3)*ArcTan[(a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3
]*a^(4/3)) - (b^(1/3)*Log[a^(1/3) - b^(1/3)*x])/(3*a^(4/3)) + (b^(1/3)*Log[a^(2/
3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(4/3))

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Rubi in Sympy [A]  time = 30.8007, size = 114, normalized size = 0.93 \[ - \frac{1}{a x} - \frac{\sqrt [3]{b} \log{\left (\sqrt [3]{a} - \sqrt [3]{b} x \right )}}{3 a^{\frac{4}{3}}} + \frac{\sqrt [3]{b} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{4}{3}}} - \frac{\sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(a*x) - b**(1/3)*log(a**(1/3) - b**(1/3)*x)/(3*a**(4/3)) + b**(1/3)*log(a**(2
/3) + a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*a**(4/3)) - sqrt(3)*b**(1/3)*atan(
sqrt(3)*(a**(1/3)/3 + 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(4/3))

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Mathematica [A]  time = 0.056062, size = 114, normalized size = 0.93 \[ -\frac{-\sqrt [3]{b} x \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} x \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{b} x \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )+6 \sqrt [3]{a}}{6 a^{4/3} x} \]

Antiderivative was successfully verified.

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

[Out]

-(6*a^(1/3) + 2*Sqrt[3]*b^(1/3)*x*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] +
2*b^(1/3)*x*Log[a^(1/3) - b^(1/3)*x] - b^(1/3)*x*Log[a^(2/3) + a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2])/(6*a^(4/3)*x)

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Maple [A]  time = 0.007, size = 100, normalized size = 0.8 \[ -{\frac{1}{3\,a}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{6\,a}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{ax}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(-b*x^3+a),x)

[Out]

-1/3/a/(a/b)^(1/3)*ln(x-(a/b)^(1/3))+1/6/a/(a/b)^(1/3)*ln(x^2+x*(a/b)^(1/3)+(a/b
)^(2/3))-1/3/a*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x+1))-1/a/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215844, size = 182, normalized size = 1.48 \[ -\frac{\sqrt{3}{\left (\sqrt{3} x \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} + a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} x \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 6 \, x \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x + \sqrt{3} a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 6 \, \sqrt{3}\right )}}{18 \, a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(sqrt(3)*x*(-b/a)^(1/3)*log(b*x^2 + a*x*(-b/a)^(2/3) - a*(-b/a)^(1
/3)) - 2*sqrt(3)*x*(-b/a)^(1/3)*log(b*x - a*(-b/a)^(2/3)) - 6*x*(-b/a)^(1/3)*arc
tan(1/3*(2*sqrt(3)*b*x + sqrt(3)*a*(-b/a)^(2/3))/(a*(-b/a)^(2/3))) + 6*sqrt(3))/
(a*x)

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Sympy [A]  time = 1.36311, size = 31, normalized size = 0.25 \[ - \operatorname{RootSum}{\left (27 t^{3} a^{4} - b, \left ( t \mapsto t \log{\left (- \frac{9 t^{2} a^{3}}{b} + x \right )} \right )\right )} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(-b*x**3+a),x)

[Out]

-RootSum(27*_t**3*a**4 - b, Lambda(_t, _t*log(-9*_t**2*a**3/b + x))) - 1/(a*x)

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GIAC/XCAS [A]  time = 0.255555, size = 153, normalized size = 1.24 \[ -\frac{b \left (\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} - \frac{\sqrt{3} \left (a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2} b} + \frac{\left (a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*b*(a/b)^(2/3)*ln(abs(x - (a/b)^(1/3)))/a^2 - 1/3*sqrt(3)*(a*b^2)^(2/3)*arct
an(1/3*sqrt(3)*(2*x + (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b) + 1/6*(a*b^2)^(2/3)*ln(x
^2 + x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b) - 1/(a*x)