Integrand size = 21, antiderivative size = 133 \[ \int \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 \, dx=\frac {d (7 b c-2 a d) x \left (a+b x^3\right )^{7/3}}{44 b^2}+\frac {d x \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )}{11 b}+\frac {a \left (44 b^2 c^2-11 a b c d+2 a^2 d^2\right ) x \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{44 b^2 \sqrt [3]{1+\frac {b x^3}{a}}} \]
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Time = 0.06 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {427, 396, 252, 251} \[ \int \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 \, dx=\frac {a x \sqrt [3]{a+b x^3} \left (2 a^2 d^2-11 a b c d+44 b^2 c^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{44 b^2 \sqrt [3]{\frac {b x^3}{a}+1}}+\frac {d x \left (a+b x^3\right )^{7/3} (7 b c-2 a d)}{44 b^2}+\frac {d x \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )}{11 b} \]
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Rule 251
Rule 252
Rule 396
Rule 427
Rubi steps \begin{align*} \text {integral}& = \frac {d x \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )}{11 b}+\frac {\int \left (a+b x^3\right )^{4/3} \left (c (11 b c-a d)+2 d (7 b c-2 a d) x^3\right ) \, dx}{11 b} \\ & = \frac {d (7 b c-2 a d) x \left (a+b x^3\right )^{7/3}}{44 b^2}+\frac {d x \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )}{11 b}-\frac {(2 a d (7 b c-2 a d)-8 b c (11 b c-a d)) \int \left (a+b x^3\right )^{4/3} \, dx}{88 b^2} \\ & = \frac {d (7 b c-2 a d) x \left (a+b x^3\right )^{7/3}}{44 b^2}+\frac {d x \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )}{11 b}-\frac {\left (a (2 a d (7 b c-2 a d)-8 b c (11 b c-a d)) \sqrt [3]{a+b x^3}\right ) \int \left (1+\frac {b x^3}{a}\right )^{4/3} \, dx}{88 b^2 \sqrt [3]{1+\frac {b x^3}{a}}} \\ & = \frac {d (7 b c-2 a d) x \left (a+b x^3\right )^{7/3}}{44 b^2}+\frac {d x \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )}{11 b}+\frac {a \left (44 b^2 c^2-11 a b c d+2 a^2 d^2\right ) x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {4}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{44 b^2 \sqrt [3]{1+\frac {b x^3}{a}}} \\ \end{align*}
Time = 12.00 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.32 \[ \int \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 \, dx=\frac {x \sqrt [3]{a+b x^3} \left (20 a \left (14 c^2+7 c d x^3+2 d^2 x^6\right ) \operatorname {Gamma}\left (-\frac {4}{3}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{3},\frac {10}{3},-\frac {b x^3}{a}\right )-3 b x^3 \left (11 c^2+16 c d x^3+5 d^2 x^6\right ) \operatorname {Gamma}\left (-\frac {1}{3}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {4}{3},\frac {13}{3},-\frac {b x^3}{a}\right )-9 b x^3 \left (c+d x^3\right )^2 \operatorname {Gamma}\left (-\frac {1}{3}\right ) \, _3F_2\left (-\frac {1}{3},\frac {4}{3},2;1,\frac {13}{3};-\frac {b x^3}{a}\right )\right )}{280 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {Gamma}\left (-\frac {4}{3}\right )} \]
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\[\int \left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )^{2}d x\]
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\[ \int \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{2} \,d x } \]
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Result contains complex when optimal does not.
Time = 2.60 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.03 \[ \int \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 \, dx=\frac {a^{\frac {4}{3}} c^{2} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {2 a^{\frac {4}{3}} c d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {a^{\frac {4}{3}} d^{2} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {\sqrt [3]{a} b c^{2} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {2 \sqrt [3]{a} b c d x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {\sqrt [3]{a} b d^{2} x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} \]
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\[ \int \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{2} \,d x } \]
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\[ \int \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{2} \,d x } \]
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Timed out. \[ \int \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 \, dx=\int {\left (b\,x^3+a\right )}^{4/3}\,{\left (d\,x^3+c\right )}^2 \,d x \]
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