Optimal. Leaf size=136 \[ \frac{b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{8/3}}-\frac{b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{8/3}}-\frac{b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{8/3}}-\frac{b x^2}{2 a^2}+\frac{x^5}{5 a} \]
[Out]
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Rubi [A] time = 0.234764, antiderivative size = 136, 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 \[ \frac{b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{8/3}}-\frac{b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{8/3}}-\frac{b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{8/3}}-\frac{b x^2}{2 a^2}+\frac{x^5}{5 a} \]
Antiderivative was successfully verified.
[In] Int[x^4/(a + b/x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{x^{5}}{5 a} - \frac{b \int x\, dx}{a^{2}} - \frac{b^{\frac{5}{3}} \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{3 a^{\frac{8}{3}}} + \frac{b^{\frac{5}{3}} \log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{6 a^{\frac{8}{3}}} - \frac{\sqrt{3} b^{\frac{5}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{3 a^{\frac{8}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(a+b/x**3),x)
[Out]
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Mathematica [A] time = 0.0685256, size = 122, normalized size = 0.9 \[ \frac{5 b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-15 a^{2/3} b x^2+6 a^{5/3} x^5-10 b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )-10 \sqrt{3} b^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{30 a^{8/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(a + b/x^3),x]
[Out]
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Maple [A] time = 0.008, size = 117, normalized size = 0.9 \[{\frac{{x}^{5}}{5\,a}}-{\frac{b{x}^{2}}{2\,{a}^{2}}}-{\frac{{b}^{2}}{3\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{{b}^{2}}{6\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{{b}^{2}\sqrt{3}}{3\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(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^4/(a + b/x^3),x, algorithm="maxima")
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Fricas [A] time = 0.23887, size = 220, normalized size = 1.62 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3} b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} - b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 10 \, \sqrt{3} b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 30 \, b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (2 \, a x^{5} - 5 \, b x^{2}\right )}\right )}}{90 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(a + b/x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.35074, size = 44, normalized size = 0.32 \[ \operatorname{RootSum}{\left (27 t^{3} a^{8} + b^{5}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a^{5}}{b^{3}} + x \right )} \right )\right )} + \frac{x^{5}}{5 a} - \frac{b x^{2}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(a+b/x**3),x)
[Out]
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GIAC/XCAS [A] time = 0.230588, size = 178, normalized size = 1.31 \[ -\frac{b \left (-\frac{b}{a}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} - \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} b \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, a^{4}} + \frac{\left (-a^{2} b\right )^{\frac{2}{3}} b{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, a^{4}} + \frac{2 \, a^{4} x^{5} - 5 \, a^{3} b x^{2}}{10 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(a + b/x^3),x, algorithm="giac")
[Out]