Optimal. Leaf size=174 \[ -\frac {a^2 \sqrt [3]{a+b x^3}}{b^3 d}-\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac {\sqrt [3]{2} a^{7/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^3 d}+\frac {a^{7/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^3 d}-\frac {a^{7/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^3 d} \]
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Rubi [A]
time = 0.13, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 90, 52, 59,
631, 210, 31} \begin {gather*} \frac {\sqrt [3]{2} a^{7/3} \text {ArcTan}\left (\frac {2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^3 d}+\frac {a^{7/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^3 d}-\frac {a^{7/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^3 d}-\frac {a^2 \sqrt [3]{a+b x^3}}{b^3 d}-\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 52
Rule 59
Rule 90
Rule 210
Rule 457
Rule 631
Rubi steps
\begin {align*} \int \frac {x^8 \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2 \sqrt [3]{a+b x}}{a d-b d x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {(a+b x)^{4/3}}{b^2 d}+\frac {a^2 \sqrt [3]{a+b x}}{b^2 (a d-b d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac {a^2 \text {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{a d-b d x} \, dx,x,x^3\right )}{3 b^2}\\ &=-\frac {a^2 \sqrt [3]{a+b x^3}}{b^3 d}-\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (a d-b d x)} \, dx,x,x^3\right )}{3 b^2}\\ &=-\frac {a^2 \sqrt [3]{a+b x^3}}{b^3 d}-\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac {a^{7/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^3 d}+\frac {a^{7/3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^3 d}+\frac {a^{8/3} \text {Subst}\left (\int \frac {1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^3 d}\\ &=-\frac {a^2 \sqrt [3]{a+b x^3}}{b^3 d}-\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac {a^{7/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^3 d}-\frac {a^{7/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^3 d}-\frac {\left (\sqrt [3]{2} a^{7/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{b^3 d}\\ &=-\frac {a^2 \sqrt [3]{a+b x^3}}{b^3 d}-\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac {\sqrt [3]{2} a^{7/3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} b^3 d}+\frac {a^{7/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^3 d}-\frac {a^{7/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 210, normalized size = 1.21 \begin {gather*} -\frac {48 a^2 \sqrt [3]{a+b x^3}+12 a b x^3 \sqrt [3]{a+b x^3}+6 b^2 x^6 \sqrt [3]{a+b x^3}-14 \sqrt [3]{2} \sqrt {3} a^{7/3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+14 \sqrt [3]{2} a^{7/3} \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )-7 \sqrt [3]{2} a^{7/3} \log \left (2 a^{2/3}+2^{2/3} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{42 b^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{8} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{-b d \,x^{3}+a d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 155, normalized size = 0.89 \begin {gather*} \frac {\frac {14 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {7}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{d} + \frac {7 \cdot 2^{\frac {1}{3}} a^{\frac {7}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right )}{d} - \frac {14 \cdot 2^{\frac {1}{3}} a^{\frac {7}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}{d} - \frac {6 \, {\left ({\left (b x^{3} + a\right )}^{\frac {7}{3}} + 7 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2}\right )}}{d}}{42 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.98, size = 174, normalized size = 1.00 \begin {gather*} -\frac {14 \, \sqrt {3} 2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} a^{2} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + 7 \cdot 2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} a^{2} \log \left (2^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right ) - 14 \cdot 2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} a^{2} \log \left (2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) + 6 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + 8 \, a^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{42 \, b^{3} d} \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^{8} \sqrt [3]{a + b x^{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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.65, size = 219, normalized size = 1.26 \begin {gather*} \frac {2^{1/3}\,{\left (-a\right )}^{7/3}\,\ln \left (6\,a^3\,{\left (b\,x^3+a\right )}^{1/3}-6\,2^{1/3}\,{\left (-a\right )}^{10/3}\right )}{3\,b^3\,d}-\frac {a^2\,{\left (b\,x^3+a\right )}^{1/3}}{b^3\,d}-\frac {{\left (b\,x^3+a\right )}^{7/3}}{7\,b^3\,d}-\frac {2^{1/3}\,{\left (-a\right )}^{7/3}\,\ln \left (\frac {6\,a^3\,{\left (b\,x^3+a\right )}^{1/3}}{b^3\,d}+\frac {6\,2^{1/3}\,{\left (-a\right )}^{10/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^3\,d}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^3\,d}+\frac {2^{1/3}\,{\left (-a\right )}^{7/3}\,\ln \left (\frac {6\,a^3\,{\left (b\,x^3+a\right )}^{1/3}}{b^3\,d}-\frac {18\,2^{1/3}\,{\left (-a\right )}^{10/3}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^3\,d}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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