Optimal. Leaf size=201 \[ -\frac {\sqrt [3]{a+b x^3} (a d+b c)}{b^2 d^2}+\frac {\left (a+b x^3\right )^{4/3}}{4 b^2 d}-\frac {c^2 \log \left (c+d x^3\right )}{6 d^{7/3} (b c-a d)^{2/3}}+\frac {c^2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{7/3} (b c-a d)^{2/3}}-\frac {c^2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{7/3} (b c-a d)^{2/3}} \]
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Rubi [A] time = 0.21, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 88, 58, 617, 204, 31} \begin {gather*} -\frac {\sqrt [3]{a+b x^3} (a d+b c)}{b^2 d^2}+\frac {\left (a+b x^3\right )^{4/3}}{4 b^2 d}-\frac {c^2 \log \left (c+d x^3\right )}{6 d^{7/3} (b c-a d)^{2/3}}+\frac {c^2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{7/3} (b c-a d)^{2/3}}-\frac {c^2 \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^{7/3} (b c-a d)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 58
Rule 88
Rule 204
Rule 446
Rule 617
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {-b c-a d}{b d^2 (a+b x)^{2/3}}+\frac {\sqrt [3]{a+b x}}{b d}+\frac {c^2}{d^2 (a+b x)^{2/3} (c+d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {(b c+a d) \sqrt [3]{a+b x^3}}{b^2 d^2}+\frac {\left (a+b x^3\right )^{4/3}}{4 b^2 d}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 d^2}\\ &=-\frac {(b c+a d) \sqrt [3]{a+b x^3}}{b^2 d^2}+\frac {\left (a+b x^3\right )^{4/3}}{4 b^2 d}-\frac {c^2 \log \left (c+d x^3\right )}{6 d^{7/3} (b c-a d)^{2/3}}+\frac {c^2 \operatorname {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^{7/3} (b c-a d)^{2/3}}+\frac {c^2 \operatorname {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^{8/3} \sqrt [3]{b c-a d}}\\ &=-\frac {(b c+a d) \sqrt [3]{a+b x^3}}{b^2 d^2}+\frac {\left (a+b x^3\right )^{4/3}}{4 b^2 d}-\frac {c^2 \log \left (c+d x^3\right )}{6 d^{7/3} (b c-a d)^{2/3}}+\frac {c^2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{7/3} (b c-a d)^{2/3}}+\frac {c^2 \operatorname {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^{7/3} (b c-a d)^{2/3}}\\ &=-\frac {(b c+a d) \sqrt [3]{a+b x^3}}{b^2 d^2}+\frac {\left (a+b x^3\right )^{4/3}}{4 b^2 d}-\frac {c^2 \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^{7/3} (b c-a d)^{2/3}}-\frac {c^2 \log \left (c+d x^3\right )}{6 d^{7/3} (b c-a d)^{2/3}}+\frac {c^2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{7/3} (b c-a d)^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 211, normalized size = 1.05 \begin {gather*} \frac {-\frac {12 \sqrt [3]{a+b x^3} (a d+b c)}{b^2}+\frac {3 d \left (a+b x^3\right )^{4/3}}{b^2}-\frac {2 c^2 \left (\log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )-2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+2 \sqrt {3} \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 )\right )}{\sqrt [3]{d} (b c-a d)^{2/3}}}{12 d^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 246, normalized size = 1.22 \begin {gather*} \frac {\sqrt [3]{a+b x^3} \left (-3 a d-4 b c+b d x^3\right )}{4 b^2 d^2}+\frac {c^2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{3 d^{7/3} (b c-a d)^{2/3}}-\frac {c^2 \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{6 d^{7/3} (b c-a d)^{2/3}}-\frac {c^2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{b c-a d}}\right )}{\sqrt {3} d^{7/3} (b c-a d)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 1156, normalized size = 5.75
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 312, normalized size = 1.55 \begin {gather*} -\frac {b^{10} c^{2} d^{2} \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^{11} c d^{4} - a b^{10} d^{5}\right )}} + \frac {{\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 )}{\sqrt {3} b c d^{3} - \sqrt {3} a 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 \, {\left (b c d^{3} - a d^{4}\right )}} - \frac {4 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{7} c d^{2} - {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{6} d^{3} + 4 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a b^{6} d^{3}}{4 \, b^{8} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (d \,x^{3}+c \right )}\, dx \end {gather*}
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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mupad [B] time = 4.69, size = 292, normalized size = 1.45 \begin {gather*} \frac {{\left (b\,x^3+a\right )}^{4/3}}{4\,b^2\,d}-\left (\frac {2\,a}{b^2\,d}+\frac {b^3\,c-a\,b^2\,d}{b^4\,d^2}\right )\,{\left (b\,x^3+a\right )}^{1/3}-\frac {\ln \left (3\,c^2\,{\left (b\,x^3+a\right )}^{1/3}+\frac {\left (c^2+\sqrt {3}\,c^2\,1{}\mathrm {i}\right )\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{6\,d^{7/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )\,\left (c^2+\sqrt {3}\,c^2\,1{}\mathrm {i}\right )}{6\,d^{7/3}\,{\left (a\,d-b\,c\right )}^{2/3}}+\frac {c^2\,\ln \left (3\,c^2\,{\left (b\,x^3+a\right )}^{1/3}-\frac {c^2\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{3\,d^{7/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )}{3\,d^{7/3}\,{\left (a\,d-b\,c\right )}^{2/3}}+\frac {c^2\,\ln \left (3\,c^2\,{\left (b\,x^3+a\right )}^{1/3}-\frac {c^2\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{d^{7/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{d^{7/3}\,{\left (a\,d-b\,c\right )}^{2/3}} \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}}{\left (a + b x^{3}\right )^{\frac {2}{3}} \left (c + d x^{3}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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