Optimal. Leaf size=260 \[ -\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 b^{4/3} d}+\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3} d}-\frac {c^{4/3} \log \left (c+d x^3\right )}{6 d (b c-a d)^{4/3}}+\frac {c^{4/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d (b c-a d)^{4/3}}-\frac {c^{4/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d (b c-a d)^{4/3}}+\frac {a x}{b \sqrt [3]{a+b x^3} (b c-a d)} \]
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Rubi [C] time = 0.06, antiderivative size = 67, normalized size of antiderivative = 0.26, 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 {x^7 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {7}{3};\frac {4}{3},1;\frac {10}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{7 a c \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 {x^6}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx &=\frac {\sqrt [3]{1+\frac {b x^3}{a}} \int \frac {x^6}{\left (1+\frac {b x^3}{a}\right )^{4/3} \left (c+d x^3\right )} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=\frac {x^7 \sqrt [3]{1+\frac {b x^3}{a}} F_1\left (\frac {7}{3};\frac {4}{3},1;\frac {10}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{7 a c \sqrt [3]{a+b x^3}}\\ \end {align*}
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Mathematica [C] time = 0.45, size = 309, normalized size = 1.19 \begin {gather*} \frac {3 x^4 \sqrt [3]{\frac {b x^3}{a}+1} (b c-a d)^{4/3} F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-2 a c \left (\sqrt [3]{c} \sqrt [3]{a+b x^3} \log \left (\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+\frac {x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+c^{2/3}\right )-2 \sqrt [3]{c} \sqrt [3]{a+b x^3} \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt {3} \sqrt [3]{c} \sqrt [3]{a+b x^3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )-6 x \sqrt [3]{b c-a d}\right )}{12 b c \sqrt [3]{a+b x^3} (b c-a d)^{4/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [C] time = 5.72, size = 500, normalized size = 1.92 \begin {gather*} -\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 b^{4/3} d}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{2 \sqrt [3]{a+b x^3}+\sqrt [3]{b} x}\right )}{\sqrt {3} b^{4/3} d}+\frac {\log \left (\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}+b^{2/3} x^2\right )}{6 b^{4/3} d}-\frac {i \left (\sqrt {3} c^{4/3}-i c^{4/3}\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 d (b c-a d)^{4/3}}+\frac {\sqrt {\frac {1}{6} \left (-1+i \sqrt {3}\right )} c^{4/3} \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 )}{d (b c-a d)^{4/3}}+\frac {\left (c^{4/3}+i \sqrt {3} c^{4/3}\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 d (b c-a d)^{4/3}}+\frac {a x}{b \sqrt [3]{a+b x^3} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 1127, normalized size = 4.33
result too large to display
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 {x^{6}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{\left (b \,x^{3}+a \right )^{\frac {4}{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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}}\,{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 {x^6}{{\left (b\,x^3+a\right )}^{4/3}\,\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 {x^{6}}{\left (a + b x^{3}\right )^{\frac {4}{3}} \left (c + d x^{3}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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