Integrand size = 11, antiderivative size = 33 \[ \int \sqrt [3]{a+b x^3} \, dx=\frac {x \left (a+b x^3\right )^{4/3} \operatorname {Hypergeometric2F1}\left (1,\frac {5}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{a} \]
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Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {252, 251} \[ \int \sqrt [3]{a+b x^3} \, dx=\frac {x \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\sqrt [3]{\frac {b x^3}{a}+1}} \]
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Rule 251
Rule 252
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{a+b x^3} \int \sqrt [3]{1+\frac {b x^3}{a}} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}} \\ & = \frac {x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\sqrt [3]{1+\frac {b x^3}{a}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.18 (sec) , antiderivative size = 203, normalized size of antiderivative = 6.15 \[ \int \sqrt [3]{a+b x^3} \, dx=\frac {3 \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt [3]{a+b x^3} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{3},-\frac {1}{3},\frac {7}{3},-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}},\frac {i+\sqrt {3}-\frac {2 i \sqrt [3]{b} x}{\sqrt [3]{a}}}{3 i+\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt [3]{b} \sqrt [3]{\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt [3]{\frac {i \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}}} \]
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\[\int \left (b \,x^{3}+a \right )^{\frac {1}{3}}d x\]
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\[ \int \sqrt [3]{a+b x^3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {1}{3}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \sqrt [3]{a+b x^3} \, dx=\frac {\sqrt [3]{a} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \]
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\[ \int \sqrt [3]{a+b x^3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {1}{3}} \,d x } \]
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\[ \int \sqrt [3]{a+b x^3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {1}{3}} \,d x } \]
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Time = 5.94 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \sqrt [3]{a+b x^3} \, dx=\frac {x\,{\left (b\,x^3+a\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ -\frac {b\,x^3}{a}\right )}{{\left (\frac {b\,x^3}{a}+1\right )}^{1/3}} \]
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