Optimal. Leaf size=59 \[ \frac{3 a (c x)^{2/3} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^2}{a}\right )}{2 c \sqrt [3]{\frac{b x^2}{a}+1}} \]
[Out]
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Rubi [A] time = 0.0675049, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{3 a (c x)^{2/3} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^2}{a}\right )}{2 c \sqrt [3]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(4/3)/(c*x)^(1/3),x]
[Out]
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Rubi in Sympy [A] time = 7.56447, size = 51, normalized size = 0.86 \[ \frac{3 a \left (c x\right )^{\frac{2}{3}} \sqrt [3]{a + b x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{2 c \sqrt [3]{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(4/3)/(c*x)**(1/3),x)
[Out]
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Mathematica [A] time = 0.0591085, size = 81, normalized size = 1.37 \[ \frac{3 x \left (2 a^2 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^2}{a}\right )+3 a^2+4 a b x^2+b^2 x^4\right )}{10 \sqrt [3]{c x} \left (a+b x^2\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(4/3)/(c*x)^(1/3),x]
[Out]
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Maple [F] time = 0.032, size = 0, normalized size = 0. \[ \int{1 \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}}{\frac{1}{\sqrt [3]{cx}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(4/3)/(c*x)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{\left (c x\right )^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(4/3)/(c*x)^(1/3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{\left (c x\right )^{\frac{1}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(4/3)/(c*x)^(1/3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 68.4535, size = 46, normalized size = 0.78 \[ \frac{a^{\frac{4}{3}} x^{\frac{2}{3}} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [3]{c} \Gamma \left (\frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(4/3)/(c*x)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{\left (c x\right )^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(4/3)/(c*x)^(1/3),x, algorithm="giac")
[Out]