Integrand size = 21, antiderivative size = 62 \[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )^3} \, dx=\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {8}{3},3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 c^3 \left (a+b x^3\right )^{2/3}} \]
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Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {441, 440} \[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )^3} \, dx=\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {8}{3},3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 c^3 \left (a+b x^3\right )^{2/3}} \]
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Rule 440
Rule 441
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{8/3} \left (c+d x^3\right )^3} \, dx}{a^2 \left (a+b x^3\right )^{2/3}} \\ & = \frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} F_1\left (\frac {1}{3};\frac {8}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 c^3 \left (a+b x^3\right )^{2/3}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(515\) vs. \(2(62)=124\).
Time = 11.48 (sec) , antiderivative size = 515, normalized size of antiderivative = 8.31 \[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )^3} \, dx=\frac {x \left (b d \left (36 b^3 c^3-171 a b^2 c^2 d-110 a^2 b c d^2+25 a^3 d^3\right ) x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {4 c \left (\frac {36 b^5 c^3 x^3 \left (c+d x^3\right )^2+9 a b^4 c^2 \left (6 c-19 d x^3\right ) \left (c+d x^3\right )^2+5 a^5 d^4 \left (8 c+5 d x^3\right )+5 a^3 b^2 d^3 x^3 \left (-50 c^2-36 c d x^3+5 d^2 x^6\right )+5 a^4 b d^3 \left (-25 c^2-6 c d x^3+10 d^2 x^6\right )-a^2 b^3 c d \left (189 c^3+378 c^2 d x^3+314 c d^2 x^6+110 d^3 x^9\right )}{a+b x^3}+\frac {4 a c \left (36 b^4 c^4-171 a b^3 c^3 d+540 a^2 b^2 c^2 d^2-235 a^3 b c d^3+50 a^4 d^4\right ) \left (c+d x^3\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-x^3 \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}\right )}{\left (c+d x^3\right )^2}\right )}{360 a^2 c^3 (b c-a d)^4 \left (a+b x^3\right )^{2/3}} \]
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\[\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {8}{3}} \left (d \,x^{3}+c \right )^{3}}d x\]
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {8}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {8}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )^3} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{8/3}\,{\left (d\,x^3+c\right )}^3} \,d x \]
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