Optimal. Leaf size=212 \[ \frac {1}{162} \left (a+x^3\right )^{2/3} \left (28 a^2 x-108 a b x-21 a x^4+162 b^2 x+81 b x^4+18 x^7\right )+\frac {1}{243} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (\sqrt [3]{a+x^3}-x\right )+\frac {1}{243} \left (-14 \sqrt {3} a^3+54 \sqrt {3} a^2 b-81 \sqrt {3} a b^2+81 \sqrt {3} b^3\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{a+x^3}+x}\right )+\frac {1}{486} \left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \log \left (x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 171, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {416, 528, 388, 239} \begin {gather*} \frac {1}{162} x \left (28 a^2-87 a b+99 b^2\right ) \left (a+x^3\right )^{2/3}+\frac {1}{162} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (\sqrt [3]{a+x^3}-x\right )-\frac {\left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{a+x^3}}+1}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2-\frac {1}{54} x (7 a-15 b) \left (a+x^3\right )^{2/3} \left (b+x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 239
Rule 388
Rule 416
Rule 528
Rubi steps
\begin {align*} \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx &=\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2+\frac {1}{9} \int \frac {\left (b+x^3\right ) \left (-((a-9 b) b)+(-7 a+15 b) x^3\right )}{\sqrt [3]{a+x^3}} \, dx\\ &=-\frac {1}{54} (7 a-15 b) x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2+\frac {1}{54} \int \frac {b \left (7 a^2-21 a b+54 b^2\right )+\left (28 a^2-87 a b+99 b^2\right ) x^3}{\sqrt [3]{a+x^3}} \, dx\\ &=\frac {1}{162} \left (28 a^2-87 a b+99 b^2\right ) x \left (a+x^3\right )^{2/3}-\frac {1}{54} (7 a-15 b) x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2+\frac {1}{81} \left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \int \frac {1}{\sqrt [3]{a+x^3}} \, dx\\ &=\frac {1}{162} \left (28 a^2-87 a b+99 b^2\right ) x \left (a+x^3\right )^{2/3}-\frac {1}{54} (7 a-15 b) x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2-\frac {\left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{a+x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {1}{162} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 5.15, size = 151, normalized size = 0.71 \begin {gather*} \frac {1}{486} \left (3 x \left (a+x^3\right )^{2/3} \left (28 a^2-3 a \left (36 b+7 x^3\right )+9 \left (18 b^2+9 b x^3+2 x^6\right )\right )+\left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \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 = 1.24, size = 212, normalized size = 1.00 \begin {gather*} \frac {1}{162} \left (a+x^3\right )^{2/3} \left (28 a^2 x-108 a b x+162 b^2 x-21 a x^4+81 b x^4+18 x^7\right )+\frac {1}{243} \left (-14 \sqrt {3} a^3+54 \sqrt {3} a^2 b-81 \sqrt {3} a b^2+81 \sqrt {3} b^3\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a+x^3}}\right )+\frac {1}{243} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )+\frac {1}{486} \left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \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 = 0.96, size = 190, normalized size = 0.90 \begin {gather*} \frac {1}{243} \, \sqrt {3} {\left (14 \, a^{3} - 54 \, a^{2} b + 81 \, a b^{2} - 81 \, b^{3}\right )} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + a\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{243} \, {\left (14 \, a^{3} - 54 \, a^{2} b + 81 \, a b^{2} - 81 \, b^{3}\right )} \log \left (-\frac {x - {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{486} \, {\left (14 \, a^{3} - 54 \, a^{2} b + 81 \, a b^{2} - 81 \, b^{3}\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}{162} \, {\left (18 \, x^{7} - 3 \, {\left (7 \, a - 27 \, b\right )} x^{4} + 2 \, {\left (14 \, a^{2} - 54 \, a b + 81 \, b^{2}\right )} x\right )} {\left (x^{3} + a\right )}^{\frac {2}{3}} \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 {{\left (x^{3} + b\right )}^{3}}{{\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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}+b \right )^{3}}{\left (x^{3}+a \right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 480, normalized size = 2.26 \begin {gather*} \frac {14}{243} \, \sqrt {3} a^{3} \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^{3} - \frac {7}{243} \, a^{3} \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 {14}{243} \, a^{3} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {1}{6} \, {\left (2 \, \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 ) - 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 ) + 2 \, a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {6 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a}{x^{2} {\left (\frac {x^{3} + a}{x^{3}} - 1\right )}}\right )} b^{2} - \frac {1}{18} \, {\left (4 \, \sqrt {3} a^{2} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - 2 \, a^{2} \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 ) + 4 \, a^{2} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {3 \, {\left (\frac {7 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a^{2}}{x^{2}} - \frac {4 \, {\left (x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}\right )}}{\frac {2 \, {\left (x^{3} + a\right )}}{x^{3}} - \frac {{\left (x^{3} + a\right )}^{2}}{x^{6}} - 1}\right )} b + \frac {\frac {67 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a^{3}}{x^{2}} - \frac {77 \, {\left (x^{3} + a\right )}^{\frac {5}{3}} a^{3}}{x^{5}} + \frac {28 \, {\left (x^{3} + a\right )}^{\frac {8}{3}} a^{3}}{x^{8}}}{162 \, {\left (\frac {3 \, {\left (x^{3} + a\right )}}{x^{3}} - \frac {3 \, {\left (x^{3} + a\right )}^{2}}{x^{6}} + \frac {{\left (x^{3} + a\right )}^{3}}{x^{9}} - 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 = -0.00 \begin {gather*} \int \frac {{\left (x^3+b\right )}^3}{{\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 = 3.99, size = 151, normalized size = 0.71 \begin {gather*} \frac {b^{3} 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 {b^{2} 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 )}}{\sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} + \frac {b x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{\sqrt [3]{a} \Gamma \left (\frac {10}{3}\right )} + \frac {x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {13}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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