Integrand size = 19, antiderivative size = 384 \[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx=\frac {\left (1-x^3\right )^{2/3}}{2 b}-\frac {\left (a^3+b^3\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {b^3 x^3}{a^3}\right )}{2 a^2 b^2}+\frac {a^2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}-\frac {\left (a^3+b^3\right )^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a^3+b^3} x}{a \sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}+\frac {\left (a^3+b^3\right )^{2/3} \arctan \left (\frac {1+\frac {2 b \sqrt [3]{1-x^3}}{\sqrt [3]{a^3+b^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}-\frac {\left (a^3+b^3\right )^{2/3} \log \left (a^3+b^3 x^3\right )}{3 b^3}+\frac {\left (a^3+b^3\right )^{2/3} \log \left (-\frac {\sqrt [3]{a^3+b^3} x}{a}-\sqrt [3]{1-x^3}\right )}{2 b^3}-\frac {a^2 \log \left (x+\sqrt [3]{1-x^3}\right )}{2 b^3}+\frac {\left (a^3+b^3\right )^{2/3} \log \left (\sqrt [3]{a^3+b^3}-b \sqrt [3]{1-x^3}\right )}{2 b^3} \]
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Time = 0.32 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {2178, 2177, 245, 2181, 384, 524, 455, 57, 631, 210, 31, 371} \[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx=-\frac {\left (a^3+b^3\right )^{2/3} \arctan \left (\frac {1-\frac {2 x \sqrt [3]{a^3+b^3}}{a \sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}+\frac {\left (a^3+b^3\right )^{2/3} \arctan \left (\frac {\frac {2 b \sqrt [3]{1-x^3}}{\sqrt [3]{a^3+b^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^3}-\frac {\left (a^3+b^3\right )^{2/3} \log \left (a^3+b^3 x^3\right )}{3 b^3}+\frac {\left (a^3+b^3\right )^{2/3} \log \left (-\frac {x \sqrt [3]{a^3+b^3}}{a}-\sqrt [3]{1-x^3}\right )}{2 b^3}+\frac {\left (a^3+b^3\right )^{2/3} \log \left (\sqrt [3]{a^3+b^3}-b \sqrt [3]{1-x^3}\right )}{2 b^3}+\frac {a^2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}-\frac {a^2 \log \left (\sqrt [3]{1-x^3}+x\right )}{2 b^3}-\frac {x^2 \left (a^3+b^3\right ) \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {b^3 x^3}{a^3}\right )}{2 a^2 b^2}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}+\frac {\left (1-x^3\right )^{2/3}}{2 b} \]
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Rule 31
Rule 57
Rule 210
Rule 245
Rule 371
Rule 384
Rule 455
Rule 524
Rule 631
Rule 2177
Rule 2178
Rule 2181
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1-x^3\right )^{2/3}}{2 b}+\frac {\int \frac {b^2-a^2 x}{(a+b x) \sqrt [3]{1-x^3}} \, dx}{b^2}+\frac {a \int \frac {x}{\sqrt [3]{1-x^3}} \, dx}{b^2} \\ & = \frac {\left (1-x^3\right )^{2/3}}{2 b}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}-\frac {a^2 \int \frac {1}{\sqrt [3]{1-x^3}} \, dx}{b^3}+\frac {\left (a^3+b^3\right ) \int \frac {1}{(a+b x) \sqrt [3]{1-x^3}} \, dx}{b^3} \\ & = \frac {\left (1-x^3\right )^{2/3}}{2 b}+\frac {a^2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}-\frac {a^2 \log \left (x+\sqrt [3]{1-x^3}\right )}{2 b^3}+\frac {\left (a^3+b^3\right ) \int \left (\frac {a^2}{\sqrt [3]{1-x^3} \left (a^3+b^3 x^3\right )}-\frac {a b x}{\sqrt [3]{1-x^3} \left (a^3+b^3 x^3\right )}+\frac {b^2 x^2}{\sqrt [3]{1-x^3} \left (a^3+b^3 x^3\right )}\right ) \, dx}{b^3} \\ & = \frac {\left (1-x^3\right )^{2/3}}{2 b}+\frac {a^2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}-\frac {a^2 \log \left (x+\sqrt [3]{1-x^3}\right )}{2 b^3}+\frac {\left (a^2 \left (a^3+b^3\right )\right ) \int \frac {1}{\sqrt [3]{1-x^3} \left (a^3+b^3 x^3\right )} \, dx}{b^3}-\frac {\left (a \left (a^3+b^3\right )\right ) \int \frac {x}{\sqrt [3]{1-x^3} \left (a^3+b^3 x^3\right )} \, dx}{b^2}+\frac {\left (a^3+b^3\right ) \int \frac {x^2}{\sqrt [3]{1-x^3} \left (a^3+b^3 x^3\right )} \, dx}{b} \\ & = \frac {\left (1-x^3\right )^{2/3}}{2 b}-\frac {\left (a^3+b^3\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {b^3 x^3}{a^3}\right )}{2 a^2 b^2}+\frac {a^2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}-\frac {\left (a^3+b^3\right )^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a^3+b^3} x}{a \sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}-\frac {\left (a^3+b^3\right )^{2/3} \log \left (a^3+b^3 x^3\right )}{6 b^3}+\frac {\left (a^3+b^3\right )^{2/3} \log \left (-\frac {\sqrt [3]{a^3+b^3} x}{a}-\sqrt [3]{1-x^3}\right )}{2 b^3}-\frac {a^2 \log \left (x+\sqrt [3]{1-x^3}\right )}{2 b^3}+\frac {\left (a^3+b^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} \left (a^3+b^3 x\right )} \, dx,x,x^3\right )}{3 b} \\ & = \frac {\left (1-x^3\right )^{2/3}}{2 b}-\frac {\left (a^3+b^3\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {b^3 x^3}{a^3}\right )}{2 a^2 b^2}+\frac {a^2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}-\frac {\left (a^3+b^3\right )^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a^3+b^3} x}{a \sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}-\frac {\left (a^3+b^3\right )^{2/3} \log \left (a^3+b^3 x^3\right )}{3 b^3}+\frac {\left (a^3+b^3\right )^{2/3} \log \left (-\frac {\sqrt [3]{a^3+b^3} x}{a}-\sqrt [3]{1-x^3}\right )}{2 b^3}-\frac {a^2 \log \left (x+\sqrt [3]{1-x^3}\right )}{2 b^3}-\frac {\left (a^3+b^3\right )^{2/3} \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a^3+b^3}}{b}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 b^3}+\frac {\left (a^3+b^3\right ) \text {Subst}\left (\int \frac {1}{\frac {\left (a^3+b^3\right )^{2/3}}{b^2}+\frac {\sqrt [3]{a^3+b^3} x}{b}+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 b^4} \\ & = \frac {\left (1-x^3\right )^{2/3}}{2 b}-\frac {\left (a^3+b^3\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {b^3 x^3}{a^3}\right )}{2 a^2 b^2}+\frac {a^2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}-\frac {\left (a^3+b^3\right )^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a^3+b^3} x}{a \sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}-\frac {\left (a^3+b^3\right )^{2/3} \log \left (a^3+b^3 x^3\right )}{3 b^3}+\frac {\left (a^3+b^3\right )^{2/3} \log \left (-\frac {\sqrt [3]{a^3+b^3} x}{a}-\sqrt [3]{1-x^3}\right )}{2 b^3}-\frac {a^2 \log \left (x+\sqrt [3]{1-x^3}\right )}{2 b^3}+\frac {\left (a^3+b^3\right )^{2/3} \log \left (\sqrt [3]{a^3+b^3}-b \sqrt [3]{1-x^3}\right )}{2 b^3}-\frac {\left (a^3+b^3\right )^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 b \sqrt [3]{1-x^3}}{\sqrt [3]{a^3+b^3}}\right )}{b^3} \\ & = \frac {\left (1-x^3\right )^{2/3}}{2 b}-\frac {\left (a^3+b^3\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {b^3 x^3}{a^3}\right )}{2 a^2 b^2}+\frac {a^2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}-\frac {\left (a^3+b^3\right )^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a^3+b^3} x}{a \sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}+\frac {\left (a^3+b^3\right )^{2/3} \arctan \left (\frac {1+\frac {2 b \sqrt [3]{1-x^3}}{\sqrt [3]{a^3+b^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}-\frac {\left (a^3+b^3\right )^{2/3} \log \left (a^3+b^3 x^3\right )}{3 b^3}+\frac {\left (a^3+b^3\right )^{2/3} \log \left (-\frac {\sqrt [3]{a^3+b^3} x}{a}-\sqrt [3]{1-x^3}\right )}{2 b^3}-\frac {a^2 \log \left (x+\sqrt [3]{1-x^3}\right )}{2 b^3}+\frac {\left (a^3+b^3\right )^{2/3} \log \left (\sqrt [3]{a^3+b^3}-b \sqrt [3]{1-x^3}\right )}{2 b^3} \\ \end{align*}
\[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx=\int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx \]
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\[\int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{b x +a}d x\]
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Timed out. \[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{a + b x}\, dx \]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{b x + a} \,d x } \]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{b x + a} \,d x } \]
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Timed out. \[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx=\int \frac {{\left (1-x^3\right )}^{2/3}}{a+b\,x} \,d x \]
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