Optimal. Leaf size=206 \[ -\frac {d (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^{8/3}}+\frac {d (b c-a d)^{2/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{8/3}}-\frac {d (b c-a d)^{2/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{8/3}}-\frac {\left (a+b x^3\right )^{2/3} (2 b c-5 a d)}{10 a c^2 x^2}-\frac {\left (a+b x^3\right )^{2/3}}{5 c x^5} \]
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Rubi [C] time = 0.13, antiderivative size = 148, normalized size of antiderivative = 0.72, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {511, 510} \begin {gather*} -\frac {-2 x^3 \left (2 c-3 d x^3\right ) (b c-a d) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+6 x^3 \left (c+d x^3\right ) (b c-a d) \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+c \left (a+b x^3\right ) \left (2 c-3 d x^3\right )}{10 c^3 x^5 \sqrt [3]{a+b x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^6 \left (c+d x^3\right )} \, dx &=\frac {\left (a+b x^3\right )^{2/3} \int \frac {\left (1+\frac {b x^3}{a}\right )^{2/3}}{x^6 \left (c+d x^3\right )} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=-\frac {c \left (a+b x^3\right ) \left (2 c-3 d x^3\right )-2 (b c-a d) x^3 \left (2 c-3 d x^3\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+6 (b c-a d) x^3 \left (c+d x^3\right ) \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{10 c^3 x^5 \sqrt [3]{a+b x^3}}\\ \end {align*}
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Mathematica [C] time = 0.64, size = 148, normalized size = 0.72 \begin {gather*} -\frac {2 x^3 \left (3 d x^3-2 c\right ) (b c-a d) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+6 x^3 \left (c+d x^3\right ) (b c-a d) \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+c \left (a+b x^3\right ) \left (2 c-3 d x^3\right )}{10 c^3 x^5 \sqrt [3]{a+b x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [C] time = 2.50, size = 392, normalized size = 1.90 \begin {gather*} -\frac {i \left (\sqrt {3} d (b c-a d)^{2/3}-i d (b c-a d)^{2/3}\right ) \log \left (2 x \sqrt [3]{b c-a d}+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{6 c^{8/3}}+\frac {\sqrt {\frac {1}{6} \left (-1+i \sqrt {3}\right )} d (b c-a d)^{2/3} \tan ^{-1}\left (\frac {3 x \sqrt [3]{b c-a d}}{\sqrt {3} x \sqrt [3]{b c-a d}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{a+b x^3}-3 i \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{c^{8/3}}+\frac {\left (d (b c-a d)^{2/3}+i \sqrt {3} d (b c-a d)^{2/3}\right ) \log \left (\left (\sqrt {3}+i\right ) c^{2/3} \left (a+b x^3\right )^{2/3}+\sqrt [3]{c} \left (-\sqrt {3} x+i x\right ) \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}-2 i x^2 (b c-a d)^{2/3}\right )}{12 c^{8/3}}+\frac {\left (a+b x^3\right )^{2/3} \left (-2 a c+5 a d x^3-2 b c x^3\right )}{10 a c^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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*} \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{\left (d \,x^{3}+c \right ) x^{6}}\, dx \end {gather*}
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*} \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{6}}\,{d x} \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 {{\left (b\,x^3+a\right )}^{2/3}}{x^6\,\left (d\,x^3+c\right )} \,d x \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 {\left (a + b x^{3}\right )^{\frac {2}{3}}}{x^{6} \left (c + d x^{3}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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