Optimal. Leaf size=132 \[ -\frac{b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{7/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{7/3}}-\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{7/3}}-\frac{b x}{a^2}+\frac{x^4}{4 a} \]
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Rubi [A] time = 0.202148, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{7/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{7/3}}-\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{7/3}}-\frac{b x}{a^2}+\frac{x^4}{4 a} \]
Antiderivative was successfully verified.
[In] Int[x^3/(a + b/x^3),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{x^{4}}{4 a} - \frac{\int b\, dx}{a^{2}} + \frac{b^{\frac{4}{3}} \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{3 a^{\frac{7}{3}}} - \frac{b^{\frac{4}{3}} \log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{6 a^{\frac{7}{3}}} - \frac{\sqrt{3} b^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{3 a^{\frac{7}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(a+b/x**3),x)
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Mathematica [A] time = 0.0400798, size = 120, normalized size = 0.91 \[ \frac{-2 b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+3 a^{4/3} x^4+4 b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )-4 \sqrt{3} b^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )-12 \sqrt [3]{a} b x}{12 a^{7/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(a + b/x^3),x]
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Maple [A] time = 0.004, size = 115, normalized size = 0.9 \[{\frac{{x}^{4}}{4\,a}}-{\frac{bx}{{a}^{2}}}+{\frac{{b}^{2}}{3\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{b}^{2}}{6\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{b}^{2}\sqrt{3}}{3\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(a+b/x^3),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(x^3/(a + b/x^3),x, algorithm="maxima")
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Fricas [A] time = 0.242816, size = 163, normalized size = 1.23 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3} b \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{b}{a}\right )^{\frac{1}{3}} + \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 4 \, \sqrt{3} b \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) + 12 \, b \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{b}{a}\right )^{\frac{1}{3}}}{3 \, \left (\frac{b}{a}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (a x^{4} - 4 \, b x\right )}\right )}}{36 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(a + b/x^3),x, algorithm="fricas")
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Sympy [A] time = 1.35894, size = 37, normalized size = 0.28 \[ \operatorname{RootSum}{\left (27 t^{3} a^{7} - b^{4}, \left ( t \mapsto t \log{\left (\frac{3 t a^{2}}{b} + x \right )} \right )\right )} + \frac{x^{4}}{4 a} - \frac{b x}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(a+b/x**3),x)
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GIAC/XCAS [A] time = 0.229689, size = 174, normalized size = 1.32 \[ -\frac{b \left (-\frac{b}{a}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} + \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} b \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3}} + \frac{\left (-a^{2} b\right )^{\frac{1}{3}} b{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, a^{3}} + \frac{a^{3} x^{4} - 4 \, a^{2} b x}{4 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(a + b/x^3),x, algorithm="giac")
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