Integrand size = 21, antiderivative size = 62 \[ \int \frac {1}{\left (a+b x^3\right )^{5/3} \left (c+d x^3\right )^3} \, dx=\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {5}{3},3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a c^3 \left (a+b x^3\right )^{2/3}} \]
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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} \[ \int \frac {1}{\left (a+b x^3\right )^{5/3} \left (c+d x^3\right )^3} \, dx=\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {5}{3},3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a c^3 \left (a+b x^3\right )^{2/3}} \]
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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 )^{5/3} \left (c+d x^3\right )^3} \, dx}{a \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 {5}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a c^3 \left (a+b x^3\right )^{2/3}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(531\) vs. \(2(62)=124\).
Time = 10.92 (sec) , antiderivative size = 531, normalized size of antiderivative = 8.56 \[ \int \frac {1}{\left (a+b x^3\right )^{5/3} \left (c+d x^3\right )^3} \, dx=\frac {\frac {b d \left (-9 b^2 c^2-16 a b c d+5 a^2 d^2\right ) x^4 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{(-b c+a d)^3}-\frac {4 c \left (4 a c x \left (3 a^3 d^3 \left (6 c+5 d x^3\right )+a b^2 c d \left (54 c^2+35 c d x^3-16 d^2 x^6\right )-9 b^3 c^2 \left (2 c^2+3 c d x^3+d^2 x^6\right )+a^2 b d^2 \left (-54 c^2-43 c d x^3+5 d^2 x^6\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+x^4 \left (9 b^3 c^2 \left (c+d x^3\right )^2-a^3 d^3 \left (8 c+5 d x^3\right )+a b^2 c d^2 x^3 \left (19 c+16 d x^3\right )+a^2 b d^2 \left (19 c^2+8 c d x^3-5 d^2 x^6\right )\right ) \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{(b c-a d)^3 \left (c+d x^3\right )^2 \left (4 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-x^3 \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}}{72 a c^3 \left (a+b x^3\right )^{2/3}} \]
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\[\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {5}{3}} \left (d \,x^{3}+c \right )^{3}}d x\]
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{5/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{5/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a+b x^3\right )^{5/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {5}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x^3\right )^{5/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {5}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{5/3} \left (c+d x^3\right )^3} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{5/3}\,{\left (d\,x^3+c\right )}^3} \,d x \]
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