Optimal. Leaf size=238 \[ -\frac{a+b x^3}{a x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt [3]{b} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.246815, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{a+b x^3}{a x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt [3]{b} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{\left (a + b x^{3}\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/((b*x**3+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0651402, size = 133, normalized size = 0.56 \[ -\frac{\left (a+b x^3\right ) \left (\sqrt [3]{b} x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt [3]{b} x \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} \sqrt [3]{b} x \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+6 \sqrt [3]{a}\right )}{6 a^{4/3} x \sqrt{\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]
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Maple [A] time = 0.013, size = 111, normalized size = 0.5 \[ -{\frac{b{x}^{3}+a}{6\,ax} \left ( -2\,\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) \sqrt{3}x-2\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) x+\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) x+6\,\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/((b*x^3+a)^2)^(1/2),x)
[Out]
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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/(sqrt((b*x^3 + a)^2)*x^2),x, algorithm="maxima")
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Fricas [A] time = 0.271098, size = 171, normalized size = 0.72 \[ -\frac{\sqrt{3}{\left (\sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 6 \, x \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 6 \, \sqrt{3}\right )}}{18 \, a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt((b*x^3 + a)^2)*x^2),x, algorithm="fricas")
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Sympy [A] time = 1.45133, size = 29, normalized size = 0.12 \[ \operatorname{RootSum}{\left (27 t^{3} a^{4} - b, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a^{3}}{b} + x \right )} \right )\right )} - \frac{1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/((b*x**3+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.279593, size = 177, normalized size = 0.74 \[ \frac{1}{6} \,{\left (\frac{2 \, b \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a^{2}} + \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a^{2} b} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{a^{2} b} - \frac{6}{a x}\right )}{\rm sign}\left (b x^{3} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt((b*x^3 + a)^2)*x^2),x, algorithm="giac")
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