Optimal. Leaf size=58 \[ \frac {3 (c x)^{2/3} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^2}{a}\right )}{2 c \sqrt [3]{\frac {b x^2}{a}+1}} \]
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Rubi [A] time = 0.02, antiderivative size = 58, 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} \[ \frac {3 (c x)^{2/3} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^2}{a}\right )}{2 c \sqrt [3]{\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x^2}}{\sqrt [3]{c x}} \, dx &=\frac {\sqrt [3]{a+b x^2} \int \frac {\sqrt [3]{1+\frac {b x^2}{a}}}{\sqrt [3]{c x}} \, dx}{\sqrt [3]{1+\frac {b x^2}{a}}}\\ &=\frac {3 (c x)^{2/3} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^2}{a}\right )}{2 c \sqrt [3]{1+\frac {b x^2}{a}}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 56, normalized size = 0.97 \[ \frac {3 x \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^2}{a}\right )}{2 \sqrt [3]{c x} \sqrt [3]{\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {2}{3}}}{c x}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}}}{\left (c x\right )^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {\left (b \,x^{2}+a \right )^{\frac {1}{3}}}{\left (c x \right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}}}{\left (c x\right )^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^{1/3}}{{\left (c\,x\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.09, size = 46, normalized size = 0.79 \[ \frac {\sqrt [3]{a} x^{\frac {2}{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [3]{c} \Gamma \left (\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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