Optimal. Leaf size=175 \[ -\frac {a \left (a+b x^3\right )^{2/3}}{2 b^2 d}-\frac {\left (a+b x^3\right )^{5/3}}{5 b^2 d}-\frac {2^{2/3} a^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^2 d}+\frac {a^{5/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^2 d}-\frac {a^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^2 d} \]
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Rubi [A]
time = 0.11, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 81, 52, 57,
631, 210, 31} \begin {gather*} -\frac {2^{2/3} a^{5/3} \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^2 d}+\frac {a^{5/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^2 d}-\frac {a^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^2 d}-\frac {a \left (a+b x^3\right )^{2/3}}{2 b^2 d}-\frac {\left (a+b x^3\right )^{5/3}}{5 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 52
Rule 57
Rule 81
Rule 210
Rule 457
Rule 631
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x (a+b x)^{2/3}}{a d-b d x} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac {a \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{a d-b d x} \, dx,x,x^3\right )}{3 b}\\ &=-\frac {a \left (a+b x^3\right )^{2/3}}{2 b^2 d}-\frac {\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a+b x} (a d-b d x)} \, dx,x,x^3\right )}{3 b}\\ &=-\frac {a \left (a+b x^3\right )^{2/3}}{2 b^2 d}-\frac {\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac {a^{5/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^2 d}+\frac {a^{5/3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^2 d}-\frac {a^2 \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 )}{b^2 d}\\ &=-\frac {a \left (a+b x^3\right )^{2/3}}{2 b^2 d}-\frac {\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac {a^{5/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^2 d}-\frac {a^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^2 d}+\frac {\left (2^{2/3} a^{5/3}\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^2 d}\\ &=-\frac {a \left (a+b x^3\right )^{2/3}}{2 b^2 d}-\frac {\left (a+b x^3\right )^{5/3}}{5 b^2 d}-\frac {2^{2/3} a^{5/3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} b^2 d}+\frac {a^{5/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^2 d}-\frac {a^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 188, normalized size = 1.07 \begin {gather*} -\frac {21 a \left (a+b x^3\right )^{2/3}+6 b x^3 \left (a+b x^3\right )^{2/3}+10\ 2^{2/3} \sqrt {3} a^{5/3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+10\ 2^{2/3} a^{5/3} \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )-5\ 2^{2/3} a^{5/3} \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 )}{30 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{-b d \,x^{3}+a d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 155, normalized size = 0.89 \begin {gather*} -\frac {\frac {10 \, \sqrt {3} 2^{\frac {2}{3}} a^{\frac {5}{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 {5 \cdot 2^{\frac {2}{3}} a^{\frac {5}{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 {10 \cdot 2^{\frac {2}{3}} a^{\frac {5}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}{d} + \frac {3 \, {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} + 5 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a\right )}}{d}}{30 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.66, size = 181, normalized size = 1.03 \begin {gather*} -\frac {10 \cdot 4^{\frac {1}{3}} \sqrt {3} \left (-a^{2}\right )^{\frac {1}{3}} a \arctan \left (\frac {4^{\frac {1}{3}} \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) + 5 \cdot 4^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} a \log \left (4^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a - 2 \cdot 4^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} a\right ) - 10 \cdot 4^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} a \log \left (-4^{\frac {2}{3}} \left (-a^{2}\right )^{\frac {2}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a\right ) + 3 \, {\left (2 \, b x^{3} + 7 \, a\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{30 \, b^{2} 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^{5} \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(-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 = 4.84, size = 221, normalized size = 1.26 \begin {gather*} \frac {4^{1/3}\,{\left (-a\right )}^{5/3}\,\ln \left (4\,a^4\,{\left (b\,x^3+a\right )}^{1/3}+4\,2^{1/3}\,{\left (-a\right )}^{13/3}\right )}{3\,b^2\,d}-\frac {a\,{\left (b\,x^3+a\right )}^{2/3}}{2\,b^2\,d}-\frac {{\left (b\,x^3+a\right )}^{5/3}}{5\,b^2\,d}-\frac {4^{1/3}\,{\left (-a\right )}^{5/3}\,\ln \left (\frac {4\,a^4\,{\left (b\,x^3+a\right )}^{1/3}}{b^4\,d^2}+\frac {2\,4^{2/3}\,{\left (-a\right )}^{13/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^4\,d^2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^2\,d}+\frac {4^{1/3}\,{\left (-a\right )}^{5/3}\,\ln \left (\frac {4\,a^4\,{\left (b\,x^3+a\right )}^{1/3}}{b^4\,d^2}+\frac {18\,4^{2/3}\,{\left (-a\right )}^{13/3}\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2}{b^4\,d^2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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