Integrand size = 24, antiderivative size = 775 \[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^2} \, dx=\frac {2 x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )}-\frac {11 x}{3 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {\sqrt [3]{2} \sqrt [6]{a} \arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}+\frac {\sqrt [3]{2} \sqrt [6]{a} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 \sqrt {b}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{\sqrt {b}}-\frac {11 \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{2\ 3^{3/4} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {11 \sqrt {2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
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Time = 0.31 (sec) , antiderivative size = 775, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {424, 544, 241, 310, 225, 1893, 402} \[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^2} \, dx=\frac {11 \sqrt {2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {11 \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{2\ 3^{3/4} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{\sqrt {b}}+\frac {\sqrt [3]{2} \sqrt [6]{a} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 \sqrt {b}}+\frac {2 x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )}-\frac {11 x}{3 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )} \]
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Rule 225
Rule 241
Rule 310
Rule 402
Rule 424
Rule 544
Rule 1893
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )}+\frac {\int \frac {-2 a^2 b+\frac {22}{3} a b^2 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{6 a b} \\ & = \frac {2 x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )}+\frac {11}{9} \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx-(4 a) \int \frac {1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx \\ & = \frac {2 x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )}-\frac {\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}+\frac {\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 \sqrt {b}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{\sqrt {b}}-\frac {\left (11 \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{6 b x} \\ & = \frac {2 x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )}-\frac {\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}+\frac {\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 \sqrt {b}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{\sqrt {b}}+\frac {\left (11 \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{6 b x}-\frac {\left (11 \left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{6 b x} \\ & = \frac {2 x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )}-\frac {11 x}{3 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}+\frac {\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 \sqrt {b}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{\sqrt {b}}-\frac {11 \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{2\ 3^{3/4} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {11 \sqrt {2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.16 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.30 \[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^2} \, dx=\frac {x \left (\frac {11 b x^2 \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{a}+\frac {27 \left (2 a-2 b x^2-\frac {9 a^2 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{9 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+2 b x^2 \left (-\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )}\right )}{3 a+b x^2}\right )}{81 \sqrt [3]{a-b x^2}} \]
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\[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {5}{3}}}{\left (b \,x^{2}+3 a \right )^{2}}d x\]
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Timed out. \[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^2} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {5}{3}}}{\left (3 a + b x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^2} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {5}{3}}}{{\left (b x^{2} + 3 \, a\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^2} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {5}{3}}}{{\left (b x^{2} + 3 \, a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^2} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{5/3}}{{\left (b\,x^2+3\,a\right )}^2} \,d x \]
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