Integrand size = 23, antiderivative size = 122 \[ \int \frac {\left (a+b x^3\right )^{3/4}}{\left (c+d x^3\right )^{25/12}} \, dx=\frac {4 x \left (a+b x^3\right )^{3/4}}{13 c \left (c+d x^3\right )^{13/12}}+\frac {9 a x \sqrt [4]{\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{3},\frac {4}{3},-\frac {(b c-a d) x^3}{a \left (c+d x^3\right )}\right )}{13 c^2 \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3}} \]
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Time = 0.04 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {386, 388} \[ \int \frac {\left (a+b x^3\right )^{3/4}}{\left (c+d x^3\right )^{25/12}} \, dx=\frac {9 a x \sqrt [4]{\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{3},\frac {4}{3},-\frac {(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{13 c^2 \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3}}+\frac {4 x \left (a+b x^3\right )^{3/4}}{13 c \left (c+d x^3\right )^{13/12}} \]
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Rule 386
Rule 388
Rubi steps \begin{align*} \text {integral}& = \frac {4 x \left (a+b x^3\right )^{3/4}}{13 c \left (c+d x^3\right )^{13/12}}+\frac {(9 a) \int \frac {1}{\sqrt [4]{a+b x^3} \left (c+d x^3\right )^{13/12}} \, dx}{13 c} \\ & = \frac {4 x \left (a+b x^3\right )^{3/4}}{13 c \left (c+d x^3\right )^{13/12}}+\frac {9 a x \sqrt [4]{\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \, _2F_1\left (\frac {1}{4},\frac {1}{3};\frac {4}{3};-\frac {(b c-a d) x^3}{a \left (c+d x^3\right )}\right )}{13 c^2 \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3}} \\ \end{align*}
Time = 5.70 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^3\right )^{3/4}}{\left (c+d x^3\right )^{25/12}} \, dx=\frac {x \left (a+b x^3\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{3},\frac {4}{3},\frac {(-b c+a d) x^3}{a \left (c+d x^3\right )}\right )}{c^2 \left (1+\frac {b x^3}{a}\right )^{3/4} \sqrt [12]{c+d x^3} \sqrt [4]{1+\frac {d x^3}{c}}} \]
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\[\int \frac {\left (b \,x^{3}+a \right )^{\frac {3}{4}}}{\left (d \,x^{3}+c \right )^{\frac {25}{12}}}d x\]
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\[ \int \frac {\left (a+b x^3\right )^{3/4}}{\left (c+d x^3\right )^{25/12}} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {3}{4}}}{{\left (d x^{3} + c\right )}^{\frac {25}{12}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^3\right )^{3/4}}{\left (c+d x^3\right )^{25/12}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^3\right )^{3/4}}{\left (c+d x^3\right )^{25/12}} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {3}{4}}}{{\left (d x^{3} + c\right )}^{\frac {25}{12}}} \,d x } \]
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\[ \int \frac {\left (a+b x^3\right )^{3/4}}{\left (c+d x^3\right )^{25/12}} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {3}{4}}}{{\left (d x^{3} + c\right )}^{\frac {25}{12}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^3\right )^{3/4}}{\left (c+d x^3\right )^{25/12}} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{3/4}}{{\left (d\,x^3+c\right )}^{25/12}} \,d x \]
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