Integrand size = 24, antiderivative size = 659 \[ \int \frac {\left (3 a+b x^2\right )^4}{\sqrt [3]{a-b x^2}} \, dx=-\frac {1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac {3794688 a^4 x}{8645 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {1897344 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{8645 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {1264896 \sqrt {2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{8645 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
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Time = 0.36 (sec) , antiderivative size = 659, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {427, 542, 396, 241, 310, 225, 1893} \[ \int \frac {\left (3 a+b x^2\right )^4}{\sqrt [3]{a-b x^2}} \, dx=\frac {1264896 \sqrt {2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{8645 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {1897344 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{8645 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {3794688 a^4 x}{8645 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \]
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Rule 225
Rule 241
Rule 310
Rule 396
Rule 427
Rule 542
Rule 1893
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac {3 \int \frac {\left (3 a+b x^2\right )^2 \left (-78 a^2 b-50 a b^2 x^2\right )}{\sqrt [3]{a-b x^2}} \, dx}{25 b} \\ & = -\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3+\frac {9 \int \frac {\left (3 a+b x^2\right ) \left (1632 a^3 b^2+1344 a^2 b^3 x^2\right )}{\sqrt [3]{a-b x^2}} \, dx}{475 b^2} \\ & = -\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac {27 \int \frac {-25248 a^4 b^3-19168 a^3 b^4 x^2}{\sqrt [3]{a-b x^2}} \, dx}{6175 b^3} \\ & = -\frac {1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3+\frac {\left (1264896 a^4\right ) \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx}{8645} \\ & = -\frac {1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac {\left (1897344 a^4 \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{8645 b x} \\ & = -\frac {1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3+\frac {\left (1897344 a^4 \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 )}{8645 b x}-\frac {\left (1897344 \left (1+\sqrt {3}\right ) a^{13/3} \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{8645 b x} \\ & = -\frac {1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac {3794688 a^4 x}{8645 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {1897344 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{13/3} \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 )}{8645 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 {1264896 \sqrt {2} 3^{3/4} a^{13/3} \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}} 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 )}{8645 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 5 vs. order 4 in optimal.
Time = 15.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.15 \[ \int \frac {\left (3 a+b x^2\right )^4}{\sqrt [3]{a-b x^2}} \, dx=\frac {3 x \left (-941085 a^4+727830 a^3 b x^2+184044 a^2 b^2 x^4+27482 a b^3 x^6+1729 b^4 x^8+2108160 a^4 \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {b x^2}{a}\right )\right )}{43225 \sqrt [3]{a-b x^2}} \]
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\[\int \frac {\left (b \,x^{2}+3 a \right )^{4}}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {\left (3 a+b x^2\right )^4}{\sqrt [3]{a-b x^2}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac {1}{3}}} \,d x } \]
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Time = 2.36 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.25 \[ \int \frac {\left (3 a+b x^2\right )^4}{\sqrt [3]{a-b x^2}} \, dx=81 a^{\frac {11}{3}} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + 36 a^{\frac {8}{3}} b x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + \frac {54 a^{\frac {5}{3}} b^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5} + \frac {12 a^{\frac {2}{3}} b^{3} x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{7} + \frac {b^{4} x^{9} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {9}{2} \\ \frac {11}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{9 \sqrt [3]{a}} \]
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\[ \int \frac {\left (3 a+b x^2\right )^4}{\sqrt [3]{a-b x^2}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {\left (3 a+b x^2\right )^4}{\sqrt [3]{a-b x^2}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\left (3 a+b x^2\right )^4}{\sqrt [3]{a-b x^2}} \, dx=\int \frac {{\left (b\,x^2+3\,a\right )}^4}{{\left (a-b\,x^2\right )}^{1/3}} \,d x \]
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