Optimal. Leaf size=134 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x}{3 a \left (a+b x^3\right )} \]
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Rubi [A] time = 0.132016, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x}{3 a \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^3)^(-2),x]
[Out]
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Rubi in Sympy [A] time = 29.9418, size = 124, normalized size = 0.93 \[ \frac{x}{3 a \left (a + b x^{3}\right )} + \frac{2 \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.128712, size = 118, normalized size = 0.88 \[ \frac{-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac{3 a^{2/3} x}{a+b x^3}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}}{9 a^{5/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^3)^(-2),x]
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Maple [A] time = 0.007, size = 115, normalized size = 0.9 \[{\frac{x}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{2}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{9\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^3+a)^2,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((b*x^3 + a)^(-2),x, algorithm="maxima")
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Fricas [A] time = 0.238362, size = 182, normalized size = 1.36 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (b x^{3} + a\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left (b x^{3} + a\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 6 \,{\left (b x^{3} + a\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x\right )}}{27 \,{\left (a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(-2),x, algorithm="fricas")
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Sympy [A] time = 1.62998, size = 39, normalized size = 0.29 \[ \frac{x}{3 a^{2} + 3 a b x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{5} b - 8, \left ( t \mapsto t \log{\left (\frac{9 t a^{2}}{2} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**3+a)**2,x)
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GIAC/XCAS [A] time = 0.217102, size = 171, normalized size = 1.28 \[ -\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2}} + \frac{x}{3 \,{\left (b x^{3} + a\right )} a} + \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{9 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(-2),x, algorithm="giac")
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