Integrand size = 21, antiderivative size = 60 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\frac {a x \sqrt [3]{a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {4}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c \sqrt [3]{1+\frac {b x^3}{a}}} \]
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Time = 0.02 (sec) , antiderivative size = 60, 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} \[ \int \frac {\left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\frac {a x \sqrt [3]{a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {4}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c \sqrt [3]{\frac {b x^3}{a}+1}} \]
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Rule 440
Rule 441
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt [3]{a+b x^3}\right ) \int \frac {\left (1+\frac {b x^3}{a}\right )^{4/3}}{c+d x^3} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}} \\ & = \frac {a x \sqrt [3]{a+b x^3} F_1\left (\frac {1}{3};-\frac {4}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c \sqrt [3]{1+\frac {b x^3}{a}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(346\) vs. \(2(60)=120\).
Time = 10.46 (sec) , antiderivative size = 346, normalized size of antiderivative = 5.77 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\frac {x \left (\frac {b (-2 b c+3 a d) x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c}+\frac {4 \left (-4 a c \left (2 a^2 d+a b d x^3+b^2 x^3 \left (c+d x^3\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{\left (c+d x^3\right ) \left (-4 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+x^3 \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}\right )}{8 d \left (a+b x^3\right )^{2/3}} \]
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\[\int \frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{d \,x^{3}+c}d x\]
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Timed out. \[ \int \frac {\left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\int \frac {\left (a + b x^{3}\right )^{\frac {4}{3}}}{c + d x^{3}}\, dx \]
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\[ \int \frac {\left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{d x^{3} + c} \,d x } \]
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\[ \int \frac {\left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{d x^{3} + c} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{4/3}}{d\,x^3+c} \,d x \]
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