Optimal. Leaf size=56 \[ -\frac{3 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{2}{3};\frac{5}{6};-\frac{b x^2}{a}\right )}{c \sqrt [3]{c x} \left (a+b x^2\right )^{2/3}} \]
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Rubi [A] time = 0.0193969, antiderivative size = 56, 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, Rules used = {365, 364} \[ -\frac{3 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{2}{3};\frac{5}{6};-\frac{b x^2}{a}\right )}{c \sqrt [3]{c x} \left (a+b x^2\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{4/3} \left (a+b x^2\right )^{2/3}} \, dx &=\frac{\left (1+\frac{b x^2}{a}\right )^{2/3} \int \frac{1}{(c x)^{4/3} \left (1+\frac{b x^2}{a}\right )^{2/3}} \, dx}{\left (a+b x^2\right )^{2/3}}\\ &=-\frac{3 \left (1+\frac{b x^2}{a}\right )^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{2}{3};\frac{5}{6};-\frac{b x^2}{a}\right )}{c \sqrt [3]{c x} \left (a+b x^2\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0109105, size = 54, normalized size = 0.96 \[ -\frac{3 x \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{2}{3};\frac{5}{6};-\frac{b x^2}{a}\right )}{(c x)^{4/3} \left (a+b x^2\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-{\frac{4}{3}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{2}{3}}}{b c^{2} x^{4} + a c^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.38381, size = 48, normalized size = 0.86 \begin{align*} \frac{\Gamma \left (- \frac{1}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{6}, \frac{2}{3} \\ \frac{5}{6} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{2}{3}} c^{\frac{4}{3}} \sqrt [3]{x} \Gamma \left (\frac{5}{6}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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