Optimal. Leaf size=33 \[ \frac {x \left (a+b x^3\right )^{4/3} \, _2F_1\left (1,\frac {5}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {246, 245} \[ \frac {x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\sqrt [3]{\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rubi steps
\begin {align*} \int \sqrt [3]{a+b x^3} \, dx &=\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*}
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Mathematica [C] time = 0.25, size = 196, normalized size = 5.94 \[ \frac {3 \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt [3]{a+b x^3} F_1\left (\frac {4}{3};-\frac {1}{3},-\frac {1}{3};\frac {7}{3};-\frac {i \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt {3} \sqrt [3]{a}},\frac {-\frac {2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt {3}+i}{3 i+\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt [3]{b} \sqrt [3]{\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt [3]{\frac {i \left (\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt {3}+3 i}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac {1}{3}}, x\right ) \]
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 \[ \int {\left (b x^{3} + a\right )}^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{3}+a \right )^{\frac {1}{3}}\, dx \]
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 \[ \int {\left (b x^{3} + a\right )}^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 37, normalized size = 1.12 \[ \frac {x\,{\left (b\,x^3+a\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ -\frac {b\,x^3}{a}\right )}{{\left (\frac {b\,x^3}{a}+1\right )}^{1/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.98, size = 37, normalized size = 1.12 \[ \frac {\sqrt [3]{a} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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