Optimal. Leaf size=118 \[ \frac {1}{9} (a-3 b) \log \left (\sqrt [3]{a+x^3}-x\right )+\frac {1}{9} \left (3 \sqrt {3} b-\sqrt {3} a\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{a+x^3}+x}\right )+\frac {1}{18} (3 b-a) \log \left (x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}+x^2\right )+\frac {1}{3} x \left (a+x^3\right )^{2/3} \]
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Rubi [A] time = 0.02, antiderivative size = 73, normalized size of antiderivative = 0.62, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {388, 239} \begin {gather*} \frac {1}{6} (a-3 b) \log \left (\sqrt [3]{a+x^3}-x\right )-\frac {(a-3 b) \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{a+x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} x \left (a+x^3\right )^{2/3} \end {gather*}
Antiderivative was successfully verified.
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Rule 239
Rule 388
Rubi steps
\begin {align*} \int \frac {b+x^3}{\sqrt [3]{a+x^3}} \, dx &=\frac {1}{3} x \left (a+x^3\right )^{2/3}-\frac {1}{3} (a-3 b) \int \frac {1}{\sqrt [3]{a+x^3}} \, dx\\ &=\frac {1}{3} x \left (a+x^3\right )^{2/3}-\frac {(a-3 b) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{a+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{6} (a-3 b) \log \left (-x+\sqrt [3]{a+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 98, normalized size = 0.83 \begin {gather*} \frac {1}{3} \left (x \left (a+x^3\right )^{2/3}-\frac {1}{6} (a-3 b) \left (-2 \log \left (1-\frac {x}{\sqrt [3]{a+x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{a+x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {x}{\sqrt [3]{a+x^3}}+\frac {x^2}{\left (a+x^3\right )^{2/3}}+1\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 118, normalized size = 1.00 \begin {gather*} \frac {1}{3} x \left (a+x^3\right )^{2/3}+\frac {1}{9} \left (-\sqrt {3} a+3 \sqrt {3} b\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a+x^3}}\right )+\frac {1}{9} (a-3 b) \log \left (-x+\sqrt [3]{a+x^3}\right )+\frac {1}{18} (-a+3 b) \log \left (x^2+x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 101, normalized size = 0.86 \begin {gather*} \frac {1}{9} \, \sqrt {3} {\left (a - 3 \, b\right )} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + a\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{9} \, {\left (a - 3 \, b\right )} \log \left (-\frac {x - {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{18} \, {\left (a - 3 \, b\right )} \log \left (\frac {x^{2} + {\left (x^{3} + a\right )}^{\frac {1}{3}} x + {\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {1}{3} \, {\left (x^{3} + a\right )}^{\frac {2}{3}} x \end {gather*}
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 \begin {gather*} \int \frac {x^{3} + b}{{\left (x^{3} + a\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {x^{3}+b}{\left (x^{3}+a \right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 170, normalized size = 1.44 \begin {gather*} \frac {1}{9} \, \sqrt {3} a \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {1}{6} \, {\left (2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 2 \, \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right )\right )} b - \frac {1}{18} \, a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {1}{9} \, a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}} a}{3 \, x^{2} {\left (\frac {x^{3} + a}{x^{3}} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3+b}{{\left (x^3+a\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.90, size = 73, normalized size = 0.62 \begin {gather*} \frac {b 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 {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {4}{3}\right )} + \frac {x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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