Integrand size = 22, antiderivative size = 464 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{a-b x^3} \, dx=-\frac {1}{2} x \sqrt [3]{a+b x^3}-\frac {2 \sqrt [3]{2} a^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\sqrt [3]{2} a^{2/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {a x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 \left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} a^{2/3} \log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}+\frac {\sqrt [3]{2} a^{2/3} \log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}-\frac {2 \sqrt [3]{2} a^{2/3} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}+\frac {a^{2/3} \log \left (2 \sqrt [3]{2}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{b}} \]
[Out]
Time = 0.28 (sec) , antiderivative size = 464, 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 = {427, 544, 252, 251, 421, 420, 493, 298, 31, 648, 631, 210, 642} \[ \int \frac {\left (a+b x^3\right )^{4/3}}{a-b x^3} \, dx=-\frac {2 \sqrt [3]{2} a^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\sqrt [3]{2} a^{2/3} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\sqrt [3]{2} a^{2/3} \log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}+\frac {\sqrt [3]{2} a^{2/3} \log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 \sqrt [3]{b}}-\frac {2 \sqrt [3]{2} a^{2/3} \log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 \sqrt [3]{b}}+\frac {a^{2/3} \log \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{3\ 2^{2/3} \sqrt [3]{b}}-\frac {a x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 \left (a+b x^3\right )^{2/3}}-\frac {1}{2} x \sqrt [3]{a+b x^3} \]
[In]
[Out]
Rule 31
Rule 210
Rule 251
Rule 252
Rule 298
Rule 420
Rule 421
Rule 427
Rule 493
Rule 544
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} x \sqrt [3]{a+b x^3}-\frac {\int \frac {-3 a^2 b-5 a b^2 x^3}{\left (a-b x^3\right ) \left (a+b x^3\right )^{2/3}} \, dx}{2 b} \\ & = -\frac {1}{2} x \sqrt [3]{a+b x^3}-\frac {1}{2} (5 a) \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx+\left (4 a^2\right ) \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{2/3}} \, dx \\ & = -\frac {1}{2} x \sqrt [3]{a+b x^3}+(2 a) \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx+(2 a) \int \frac {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx-\frac {\left (5 a \left (1+\frac {b x^3}{a}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{2 \left (a+b x^3\right )^{2/3}} \\ & = -\frac {1}{2} x \sqrt [3]{a+b x^3}-\frac {5 a 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 \left (a+b x^3\right )^{2/3}}+\frac {\left (18 a^{4/3}\right ) \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 )}{\sqrt [3]{b}}+\frac {\left (2 a \left (1+\frac {b x^3}{a}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}} \\ & = -\frac {1}{2} x \sqrt [3]{a+b x^3}-\frac {a 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 \left (a+b x^3\right )^{2/3}}+\frac {\left (2 a^{4/3}\right ) \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 )}{\sqrt [3]{b}}+\frac {\left (4 a^{4/3}\right ) \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 )}{\sqrt [3]{b}} \\ & = -\frac {1}{2} x \sqrt [3]{a+b x^3}-\frac {a 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 \left (a+b x^3\right )^{2/3}}+\frac {\left (\sqrt [3]{2} a\right ) \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 )}{3 \sqrt [3]{b}}-\frac {\left (\sqrt [3]{2} a\right ) \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 )}{3 \sqrt [3]{b}}-\frac {\left (2\ 2^{2/3} a\right ) \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 )}{3 \sqrt [3]{b}}+\frac {\left (2\ 2^{2/3} a\right ) \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 )}{3 \sqrt [3]{b}} \\ & = -\frac {1}{2} x \sqrt [3]{a+b x^3}-\frac {a 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 \left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} a^{2/3} \log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}-\frac {2 \sqrt [3]{2} a^{2/3} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}+\frac {a^{2/3} \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 )}{3\ 2^{2/3} \sqrt [3]{b}}+\frac {\left (\sqrt [3]{2} a^{2/3}\right ) \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 )}{3 \sqrt [3]{b}}-\frac {a \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 )}{\sqrt [3]{b}}+\frac {\left (2^{2/3} a\right ) \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 )}{\sqrt [3]{b}} \\ & = -\frac {1}{2} x \sqrt [3]{a+b x^3}-\frac {a 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 \left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} a^{2/3} \log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}+\frac {\sqrt [3]{2} a^{2/3} \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 )}{3 \sqrt [3]{b}}-\frac {2 \sqrt [3]{2} a^{2/3} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}+\frac {a^{2/3} \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 )}{3\ 2^{2/3} \sqrt [3]{b}}+\frac {\left (\sqrt [3]{2} a^{2/3}\right ) \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 )}{\sqrt [3]{b}}+\frac {\left (2 \sqrt [3]{2} a^{2/3}\right ) \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 )}{\sqrt [3]{b}} \\ & = -\frac {1}{2} x \sqrt [3]{a+b x^3}-\frac {2 \sqrt [3]{2} a^{2/3} \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 )}{\sqrt {3} \sqrt [3]{b}}-\frac {\sqrt [3]{2} a^{2/3} \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 )}{\sqrt {3} \sqrt [3]{b}}-\frac {a 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 \left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} a^{2/3} \log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}+\frac {\sqrt [3]{2} a^{2/3} \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 )}{3 \sqrt [3]{b}}-\frac {2 \sqrt [3]{2} a^{2/3} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}+\frac {a^{2/3} \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 )}{3\ 2^{2/3} \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 = 217, normalized size of antiderivative = 0.47 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{a-b x^3} \, dx=\frac {x \left (-4 \left (a+b x^3\right )+5 b x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )+\frac {48 a^3 \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{\left (a-b x^3\right ) \left (4 a \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )+b x^3 \left (3 \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )-2 \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )\right )}\right )}{8 \left (a+b x^3\right )^{2/3}} \]
[In]
[Out]
\[\int \frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{-b \,x^{3}+a}d x\]
[In]
[Out]
\[ \int \frac {\left (a+b x^3\right )^{4/3}}{a-b x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{b x^{3} - a} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a+b x^3\right )^{4/3}}{a-b x^3} \, dx=- \int \frac {a \sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, dx - \int \frac {b x^{3} \sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (a+b x^3\right )^{4/3}}{a-b x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{b x^{3} - a} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a+b x^3\right )^{4/3}}{a-b x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{b x^{3} - a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{4/3}}{a-b x^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{4/3}}{a-b\,x^3} \,d x \]
[In]
[Out]