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 = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 364
Rule 365
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*}
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Mathematica [A] time = 0.01, 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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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {2}{3}}}{b c^{2} x^{4} + a c^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {2}{3}} \left (c x\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c x \right )^{\frac {4}{3}} \left (b \,x^{2}+a \right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {2}{3}} \left (c x\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (c\,x\right )}^{4/3}\,{\left (b\,x^2+a\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.07, size = 48, normalized size = 0.86 \[ \frac {\Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {2}{3} \\ \frac {5}{6} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {2}{3}} c^{\frac {4}{3}} \sqrt [3]{x} \Gamma \left (\frac {5}{6}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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