Integrand size = 22, antiderivative size = 473 \[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3} a^{7/3} \sqrt [3]{b}}-\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3} a^{7/3} \sqrt [3]{b}}+\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a^2 \left (a+b x^3\right )^{2/3}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^{7/3} \sqrt [3]{b}}-\frac {\log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{6\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\log \left (2 \sqrt [3]{2}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{24\ 2^{2/3} a^{7/3} \sqrt [3]{b}} \]
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Time = 0.28 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {425, 544, 252, 251, 421, 420, 493, 298, 31, 648, 631, 210, 642} \[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3} a^{7/3} \sqrt [3]{b}}-\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3} a^{7/3} \sqrt [3]{b}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{12\ 2^{2/3} a^{7/3} \sqrt [3]{b}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{6\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\log \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{24\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a^2 \left (a+b x^3\right )^{2/3}}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}} \]
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Rule 31
Rule 210
Rule 251
Rule 252
Rule 298
Rule 420
Rule 421
Rule 425
Rule 493
Rule 544
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}-\frac {\int \frac {-3 a b+b^2 x^3}{\left (a-b x^3\right ) \left (a+b x^3\right )^{2/3}} \, dx}{4 a^2 b} \\ & = \frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}+\frac {\int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx}{4 a^2}+\frac {\int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{2/3}} \, dx}{2 a} \\ & = \frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}+\frac {\int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx}{4 a^2}+\frac {\int \frac {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx}{4 a^2}+\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{4 a^2 \left (a+b x^3\right )^{2/3}} \\ & = \frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}+\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{4 a^2 \left (a+b x^3\right )^{2/3}}+\frac {9 \text {Subst}\left (\int \frac {x}{\left (4-a x^3\right ) \left (1+2 a x^3\right )} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt [3]{a+b x^3}}\right )}{4 a^{5/3} \sqrt [3]{b}}+\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{4 a^2 \left (a+b x^3\right )^{2/3}} \\ & = \frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}+\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 a^2 \left (a+b x^3\right )^{2/3}}+\frac {\text {Subst}\left (\int \frac {x}{4-a x^3} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt [3]{a+b x^3}}\right )}{4 a^{5/3} \sqrt [3]{b}}+\frac {\text {Subst}\left (\int \frac {x}{1+2 a x^3} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt [3]{a+b x^3}}\right )}{2 a^{5/3} \sqrt [3]{b}} \\ & = \frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}+\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 a^2 \left (a+b x^3\right )^{2/3}}+\frac {\text {Subst}\left (\int \frac {1}{2^{2/3}-\sqrt [3]{a} x} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^2 \sqrt [3]{b}}-\frac {\text {Subst}\left (\int \frac {2^{2/3}-\sqrt [3]{a} x}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{a} x+a^{2/3} x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^2 \sqrt [3]{b}}-\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} \sqrt [3]{a} x} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt [3]{a+b x^3}}\right )}{6 \sqrt [3]{2} a^2 \sqrt [3]{b}}+\frac {\text {Subst}\left (\int \frac {1+\sqrt [3]{2} \sqrt [3]{a} x}{1-\sqrt [3]{2} \sqrt [3]{a} x+2^{2/3} a^{2/3} x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt [3]{a+b x^3}}\right )}{6 \sqrt [3]{2} a^2 \sqrt [3]{b}} \\ & = \frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}+\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 a^2 \left (a+b x^3\right )^{2/3}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^{7/3} \sqrt [3]{b}}-\frac {\log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{6\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\text {Subst}\left (\int \frac {2^{2/3} \sqrt [3]{a}+2 a^{2/3} x}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{a} x+a^{2/3} x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt [3]{a+b x^3}}\right )}{24\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\text {Subst}\left (\int \frac {-\sqrt [3]{2} \sqrt [3]{a}+2\ 2^{2/3} a^{2/3} x}{1-\sqrt [3]{2} \sqrt [3]{a} x+2^{2/3} a^{2/3} x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^{7/3} \sqrt [3]{b}}-\frac {\text {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{a} x+a^{2/3} x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt [3]{a+b x^3}}\right )}{8 a^2 \sqrt [3]{b}}+\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} \sqrt [3]{a} x+2^{2/3} a^{2/3} x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt [3]{a+b x^3}}\right )}{4 \sqrt [3]{2} a^2 \sqrt [3]{b}} \\ & = \frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}+\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 a^2 \left (a+b x^3\right )^{2/3}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^{7/3} \sqrt [3]{b}}-\frac {\log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{6\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\log \left (2 \sqrt [3]{2}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{24\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{4\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{2\ 2^{2/3} a^{7/3} \sqrt [3]{b}} \\ & = \frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3} a^{7/3} \sqrt [3]{b}}-\frac {\tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3} a^{7/3} \sqrt [3]{b}}+\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 a^2 \left (a+b x^3\right )^{2/3}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^{7/3} \sqrt [3]{b}}-\frac {\log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{6\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\log \left (2 \sqrt [3]{2}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{24\ 2^{2/3} a^{7/3} \sqrt [3]{b}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 10.14 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\frac {x \left (\frac {4}{a^2}-\frac {b x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{a^3}+\frac {48 \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{\left (a-b x^3\right ) \left (4 a \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )+b x^3 \left (3 \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )-2 \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )\right )}\right )}{16 \left (a+b x^3\right )^{2/3}} \]
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\[\int \frac {1}{\left (-b \,x^{3}+a \right ) \left (b \,x^{3}+a \right )^{\frac {5}{3}}}d x\]
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Timed out. \[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=- \int \frac {1}{- a^{2} \left (a + b x^{3}\right )^{\frac {2}{3}} + b^{2} x^{6} \left (a + b x^{3}\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\int { -\frac {1}{{\left (b x^{3} + a\right )}^{\frac {5}{3}} {\left (b x^{3} - a\right )}} \,d x } \]
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\[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\int { -\frac {1}{{\left (b x^{3} + a\right )}^{\frac {5}{3}} {\left (b x^{3} - a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{5/3}\,\left (a-b\,x^3\right )} \,d x \]
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