Integrand size = 24, antiderivative size = 849 \[ \int \frac {1}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^3} \, dx=\frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {19 x \left (a-b x^2\right )^{2/3}}{1152 a^4 \left (3 a+b x^2\right )}+\frac {19 x}{1152 a^4 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {7 \arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}+\frac {7 \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 )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}-\frac {7 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{864\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {7 \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 )}{288\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {19 \sqrt {2+\sqrt {3}} \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 )}{768\ 3^{3/4} a^{11/3} 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 {19 \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 )}{576 \sqrt {2} \sqrt [4]{3} a^{11/3} 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 = 0.46 (sec) , antiderivative size = 849, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {425, 541, 544, 241, 310, 225, 1893, 402} \[ \int \frac {1}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^3} \, dx=-\frac {19 \left (a-b x^2\right )^{2/3} x}{1152 a^4 \left (b x^2+3 a\right )}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}+\frac {19 x}{1152 a^4 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (b x^2+3 a\right )^2}+\frac {7 \arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}+\frac {7 \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 )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}-\frac {7 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{864\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {7 \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 )}{288\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {19 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\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 )}{768\ 3^{3/4} a^{11/3} b \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}} x}-\frac {19 \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\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 )}{576 \sqrt {2} \sqrt [4]{3} a^{11/3} b \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}} x} \]
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Rule 225
Rule 241
Rule 310
Rule 402
Rule 425
Rule 541
Rule 544
Rule 1893
Rubi steps \begin{align*} \text {integral}& = \frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}-\frac {\int \frac {-15 a b+\frac {11 b^2 x^2}{3}}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^2} \, dx}{48 a^2 b} \\ & = \frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {\int \frac {-6 a^2 b^2-\frac {170}{9} a b^3 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2} \, dx}{128 a^4 b^2} \\ & = \frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {19 x \left (a-b x^2\right )^{2/3}}{1152 a^4 \left (3 a+b x^2\right )}+\frac {\int \frac {\frac {296 a^3 b^3}{3}-\frac {152}{9} a^2 b^4 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{3072 a^6 b^3} \\ & = \frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {19 x \left (a-b x^2\right )^{2/3}}{1152 a^4 \left (3 a+b x^2\right )}-\frac {19 \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx}{3456 a^4}+\frac {7 \int \frac {1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{144 a^3} \\ & = \frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {19 x \left (a-b x^2\right )^{2/3}}{1152 a^4 \left (3 a+b x^2\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}+\frac {7 \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 )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{864\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {7 \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 )}{288\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {\left (19 \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{2304 a^4 b x} \\ & = \frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {19 x \left (a-b x^2\right )^{2/3}}{1152 a^4 \left (3 a+b x^2\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}+\frac {7 \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 )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{864\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {7 \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 )}{288\ 2^{2/3} a^{23/6} \sqrt {b}}-\frac {\left (19 \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 )}{2304 a^4 b x}+\frac {\left (19 \left (1+\sqrt {3}\right ) \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{2304 a^{11/3} b x} \\ & = \frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {19 x \left (a-b x^2\right )^{2/3}}{1152 a^4 \left (3 a+b x^2\right )}+\frac {19 x}{1152 a^4 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}+\frac {7 \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 )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{864\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {7 \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 )}{288\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {19 \sqrt {2+\sqrt {3}} \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 )}{768\ 3^{3/4} a^{11/3} 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 {19 \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 )}{576 \sqrt {2} \sqrt [4]{3} a^{11/3} 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 = 10.18 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^3} \, dx=\frac {x \left (-19 b x^2 \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+\frac {27 a \left (273 a^2+140 a b x^2+19 b^2 x^4+\frac {333 a^2 \left (3 a+b x^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{9 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+2 b x^2 \left (-\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )}\right )}{\left (3 a+b x^2\right )^2}\right )}{31104 a^5 \sqrt [3]{a-b x^2}} \]
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\[\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {4}{3}} \left (b \,x^{2}+3 a \right )^{3}}d x\]
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Timed out. \[ \int \frac {1}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^3} \, dx=\int \frac {1}{\left (a - b x^{2}\right )^{\frac {4}{3}} \left (3 a + b x^{2}\right )^{3}}\, dx \]
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\[ \int \frac {1}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^3} \, dx=\int { \frac {1}{{\left (b x^{2} + 3 \, a\right )}^{3} {\left (-b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {1}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^3} \, dx=\int { \frac {1}{{\left (b x^{2} + 3 \, a\right )}^{3} {\left (-b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^3} \, dx=\int \frac {1}{{\left (a-b\,x^2\right )}^{4/3}\,{\left (b\,x^2+3\,a\right )}^3} \,d x \]
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