Optimal. Leaf size=173 \[ \frac{a^2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{5/3}}+\frac{a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{5/3}}-\frac{a^2 \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{54 b^{5/3}}+\frac{1}{6} x^5 \sqrt [3]{a+b x^3}+\frac{a x^2 \sqrt [3]{a+b x^3}}{18 b} \]
[Out]
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Rubi [A] time = 0.238675, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ \frac{a^2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{5/3}}+\frac{a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{5/3}}-\frac{a^2 \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{54 b^{5/3}}+\frac{1}{6} x^5 \sqrt [3]{a+b x^3}+\frac{a x^2 \sqrt [3]{a+b x^3}}{18 b} \]
Antiderivative was successfully verified.
[In] Int[x^4*(a + b*x^3)^(1/3),x]
[Out]
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Rubi in Sympy [A] time = 26.3248, size = 158, normalized size = 0.91 \[ \frac{a^{2} \log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{27 b^{\frac{5}{3}}} - \frac{a^{2} \log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{54 b^{\frac{5}{3}}} + \frac{\sqrt{3} a^{2} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{27 b^{\frac{5}{3}}} + \frac{a x^{2} \sqrt [3]{a + b x^{3}}}{18 b} + \frac{x^{5} \sqrt [3]{a + b x^{3}}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(b*x**3+a)**(1/3),x)
[Out]
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Mathematica [C] time = 0.0586426, size = 78, normalized size = 0.45 \[ \frac{x^2 \left (-a^2 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )+a^2+4 a b x^3+3 b^2 x^6\right )}{18 b \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4*(a + b*x^3)^(1/3),x]
[Out]
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Maple [F] time = 0.042, size = 0, normalized size = 0. \[ \int{x}^{4}\sqrt [3]{b{x}^{3}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(b*x^3+a)^(1/3),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((b*x^3 + a)^(1/3)*x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.242345, size = 232, normalized size = 1.34 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3} a^{2} \log \left (-\frac{b x -{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{1}{3}}}{x}\right ) - \sqrt{3} a^{2} \log \left (\frac{b^{2} x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{1}{3}} b x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (b^{2}\right )}^{\frac{2}{3}}}{x^{2}}\right ) - 6 \, a^{2} \arctan \left (\frac{\sqrt{3} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{1}{3}}}{3 \, b x}\right ) + 3 \, \sqrt{3}{\left (3 \, b x^{5} + a x^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{1}{3}}\right )}}{162 \,{\left (b^{2}\right )}^{\frac{1}{3}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(1/3)*x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.09841, size = 39, normalized size = 0.23 \[ \frac{\sqrt [3]{a} x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{8}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(b*x**3+a)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(1/3)*x^4,x, algorithm="giac")
[Out]