Optimal. Leaf size=220 \[ \frac {c^2 \sqrt [3]{a+b x^3}}{d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{4/3}}{4 b^2 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^2 d}+\frac {c^2 \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{10/3}}+\frac {c^2 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac {c^2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3}} \]
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Rubi [A]
time = 0.18, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {457, 90, 52, 60,
631, 210, 31} \begin {gather*} \frac {c^2 \sqrt [3]{b c-a d} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{10/3}}-\frac {\left (a+b x^3\right )^{4/3} (a d+b c)}{4 b^2 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^2 d}+\frac {c^2 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac {c^2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3}}+\frac {c^2 \sqrt [3]{a+b x^3}}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 52
Rule 60
Rule 90
Rule 210
Rule 457
Rule 631
Rubi steps
\begin {align*} \int \frac {x^8 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2 \sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {(-b c-a d) \sqrt [3]{a+b x}}{b d^2}+\frac {(a+b x)^{4/3}}{b d}+\frac {c^2 \sqrt [3]{a+b x}}{d^2 (c+d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {(b c+a d) \left (a+b x^3\right )^{4/3}}{4 b^2 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^2 d}+\frac {c^2 \text {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )}{3 d^2}\\ &=\frac {c^2 \sqrt [3]{a+b x^3}}{d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{4/3}}{4 b^2 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^2 d}-\frac {\left (c^2 (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 d^3}\\ &=\frac {c^2 \sqrt [3]{a+b x^3}}{d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{4/3}}{4 b^2 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^2 d}+\frac {c^2 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac {\left (c^2 \sqrt [3]{b c-a d}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 d^{10/3}}-\frac {\left (c^2 (b c-a d)^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{d}}+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 d^{11/3}}\\ &=\frac {c^2 \sqrt [3]{a+b x^3}}{d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{4/3}}{4 b^2 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^2 d}+\frac {c^2 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac {c^2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3}}-\frac {\left (c^2 \sqrt [3]{b c-a d}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}\right )}{d^{10/3}}\\ &=\frac {c^2 \sqrt [3]{a+b x^3}}{d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{4/3}}{4 b^2 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^2 d}+\frac {c^2 \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{10/3}}+\frac {c^2 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac {c^2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3}}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 265, normalized size = 1.20 \begin {gather*} \frac {\frac {3 \sqrt [3]{d} \sqrt [3]{a+b x^3} \left (-3 a^2 d^2+a b d \left (-7 c+d x^3\right )+b^2 \left (28 c^2-7 c d x^3+4 d^2 x^6\right )\right )}{b^2}+28 \sqrt {3} c^2 \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )-28 c^2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+14 c^2 \sqrt [3]{b c-a d} \log \left ((b c-a d)^{2/3}-\sqrt [3]{d} \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{84 d^{10/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{8} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{d \,x^{3}+c}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.61, size = 282, normalized size = 1.28 \begin {gather*} -\frac {28 \, \sqrt {3} b^{2} c^{2} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}} - \sqrt {3} {\left (b c - a d\right )}}{3 \, {\left (b c - a d\right )}}\right ) + 14 \, b^{2} c^{2} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right ) - 28 \, b^{2} c^{2} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right ) - 3 \, {\left (4 \, b^{2} d^{2} x^{6} + 28 \, b^{2} c^{2} - 7 \, a b c d - 3 \, a^{2} d^{2} - {\left (7 \, b^{2} c d - a b d^{2}\right )} x^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{84 \, b^{2} d^{3}} \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*} \int \frac {x^{8} \sqrt [3]{a + b x^{3}}}{c + d x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 320, normalized size = 1.45 \begin {gather*} \frac {{\left (b^{17} c^{3} d^{4} - a b^{16} c^{2} d^{5}\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b^{17} c d^{7} - a b^{16} d^{8}\right )}} - \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} c^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{3 \, d^{4}} - \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} c^{2} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, d^{4}} + \frac {28 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{14} c^{2} d^{4} - 7 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{13} c d^{5} + 4 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} b^{12} d^{6} - 7 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a b^{12} d^{6}}{28 \, b^{14} d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.93, size = 336, normalized size = 1.53 \begin {gather*} \left (\frac {a^2}{b^2\,d}+\frac {\left (\frac {2\,a}{b^2\,d}+\frac {b^3\,c-a\,b^2\,d}{b^4\,d^2}\right )\,\left (b^3\,c-a\,b^2\,d\right )}{b^2\,d}\right )\,{\left (b\,x^3+a\right )}^{1/3}-\left (\frac {a}{2\,b^2\,d}+\frac {b^3\,c-a\,b^2\,d}{4\,b^4\,d^2}\right )\,{\left (b\,x^3+a\right )}^{4/3}+\frac {{\left (b\,x^3+a\right )}^{7/3}}{7\,b^2\,d}+\frac {c^2\,\ln \left ({\left (a\,d-b\,c\right )}^{1/3}-d^{1/3}\,{\left (b\,x^3+a\right )}^{1/3}\right )\,{\left (a\,d-b\,c\right )}^{1/3}}{3\,d^{10/3}}-\frac {c^2\,\ln \left (\frac {3\,{\left (b\,x^3+a\right )}^{1/3}\,\left (b\,c^3-a\,c^2\,d\right )}{d}-\frac {3\,c^2\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{4/3}}{d^{4/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{1/3}}{3\,d^{10/3}}+\frac {c^2\,\ln \left (\frac {3\,{\left (b\,x^3+a\right )}^{1/3}\,\left (b\,c^3-a\,c^2\,d\right )}{d}+\frac {9\,c^2\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^{4/3}}{d^{4/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^{1/3}}{d^{10/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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