Optimal. Leaf size=264 \[ -\frac {4 a x \left (a+b x^3\right )^{2/3}}{9 b^2 d}-\frac {x^4 \left (a+b x^3\right )^{2/3}}{6 b d}-\frac {14 a^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{7/3} d}+\frac {2^{2/3} a^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{7/3} d}+\frac {a^2 \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} b^{7/3} d}-\frac {a^2 \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^{7/3} d}+\frac {7 a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{9 b^{7/3} d} \]
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Rubi [A]
time = 0.16, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {489, 596, 544,
245, 384} \begin {gather*} -\frac {14 a^2 \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} b^{7/3} d}+\frac {2^{2/3} a^2 \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{7/3} d}+\frac {a^2 \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} b^{7/3} d}-\frac {a^2 \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^{7/3} d}+\frac {7 a^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{9 b^{7/3} d}-\frac {4 a x \left (a+b x^3\right )^{2/3}}{9 b^2 d}-\frac {x^4 \left (a+b x^3\right )^{2/3}}{6 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 384
Rule 489
Rule 544
Rule 596
Rubi steps
\begin {align*} \int \frac {x^6 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx &=\frac {\left (a+b x^3\right )^{2/3} \int \frac {x^6 \left (1+\frac {b x^3}{a}\right )^{2/3}}{a d-b d x^3} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=\frac {x^7 \left (a+b x^3\right )^{2/3} F_1\left (\frac {7}{3};-\frac {2}{3},1;\frac {10}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{7 a d \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.81, size = 325, normalized size = 1.23 \begin {gather*} -\frac {24 a \sqrt [3]{b} x \left (a+b x^3\right )^{2/3}+9 b^{4/3} x^4 \left (a+b x^3\right )^{2/3}+28 \sqrt {3} a^2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-18\ 2^{2/3} \sqrt {3} a^2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}}\right )-28 a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+18\ 2^{2/3} a^2 \log \left (-2 \sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}\right )+14 a^2 \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-9\ 2^{2/3} a^2 \log \left (2 b^{2/3} x^2+2^{2/3} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{54 b^{7/3} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{6} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{-b d \,x^{3}+a d}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 6.43, size = 701, normalized size = 2.66 \begin {gather*} \left [-\frac {18 \cdot 4^{\frac {1}{3}} \sqrt {3} a^{2} b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} x - 4^{\frac {1}{3}} \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {1}{b}\right )^{\frac {1}{3}}}{3 \, x}\right ) - 42 \, \sqrt {\frac {1}{3}} a^{2} b \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (b^{\frac {4}{3}} x^{3} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{\frac {2}{3}} x\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} + 2 \, a\right ) - 18 \cdot 4^{\frac {1}{3}} a^{2} b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \log \left (-\frac {4^{\frac {2}{3}} b x \left (-\frac {1}{b}\right )^{\frac {2}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 9 \cdot 4^{\frac {1}{3}} a^{2} b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \log \left (-\frac {2 \cdot 4^{\frac {1}{3}} b x^{2} \left (-\frac {1}{b}\right )^{\frac {1}{3}} - 4^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x \left (-\frac {1}{b}\right )^{\frac {2}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 28 \, a^{2} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 14 \, a^{2} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (3 \, b^{2} x^{4} + 8 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b^{3} d}, -\frac {18 \cdot 4^{\frac {1}{3}} \sqrt {3} a^{2} b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} x - 4^{\frac {1}{3}} \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {1}{b}\right )^{\frac {1}{3}}}{3 \, x}\right ) - 18 \cdot 4^{\frac {1}{3}} a^{2} b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \log \left (-\frac {4^{\frac {2}{3}} b x \left (-\frac {1}{b}\right )^{\frac {2}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 9 \cdot 4^{\frac {1}{3}} a^{2} b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \log \left (-\frac {2 \cdot 4^{\frac {1}{3}} b x^{2} \left (-\frac {1}{b}\right )^{\frac {1}{3}} - 4^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x \left (-\frac {1}{b}\right )^{\frac {2}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 84 \, \sqrt {\frac {1}{3}} a^{2} b^{\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (b^{\frac {1}{3}} x + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{b^{\frac {1}{3}} x}\right ) - 28 \, a^{2} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 14 \, a^{2} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (3 \, b^{2} x^{4} + 8 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b^{3} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {x^{6} \left (a + b x^{3}\right )^{\frac {2}{3}}}{- a + b x^{3}}\, dx}{d} \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*} \text {could not integrate} \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 {x^6\,{\left (b\,x^3+a\right )}^{2/3}}{a\,d-b\,d\,x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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